0 · Mathematical Thinking
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0.1The Big Ideas
This module is about seeing a relationship as a machine that turns inputs into outputs, and an equation as the question "which input lands on this output?"
- 0.1.1.1 A function is a machineDrop a number in the top and exactly one number falls out the bottom, like a coffee machine that pours the same cup every time you press the same button.
- 0.1.1.2 An equation is a balance scaleThe equals sign is the middle of a see-saw, and whatever you do to one side you must do to the other to keep it level while you uncover the hidden weight.
- 0.1.1.3 Two views of one truth"Where does the graph cross zero?" and "what value solves this equation?" are the very same question wearing two different hats.
- 0.1.1.4 The tell-tale signWhenever something changes because something else changes — your bill grows with the groceries — a function is hiding there, waiting to be named.
- 0.1.1.5 Where you'll meet it againFrom linear and quadratic to exponential functions, and every time you set two formulas equal to find where two paths cross.
This module is about turning numbers into pictures and pictures into numbers so each one rescues the other.
- 0.1.2.1 Every number has a home on the lineLay numbers along a long ruler stretching both ways from zero, and "bigger" simply becomes "farther to the right."
- 0.1.2.2 Every equation can become a curvePlot the pairs that fit a rule and dots line up into a line or curve, so you can SEE the answer where the shape touches the floor.
- 0.1.2.3 Read the answer off the pictureTangled algebra can melt away when you just look: two curves crossing is two equations agreeing, with no symbol-pushing required.
- 0.1.2.4 The tell-tale signWhen symbols feel knotted, draw it; when a drawing feels vague, label it with numbers — switching languages unlocks the stuck door.
- 0.1.2.5 Where you'll meet it againReading slope off a line, finding where a parabola crosses zero, proving the Pythagorean theorem with squares on a triangle, and seeing a derivative as a tangent's steepness.
This module is about splitting a messy problem into a few tidy boxes and handling each box on its own.
- 0.1.3.1 Sort before you solveLike sorting laundry into lights and darks before washing, some problems get easy once you split them into "this kind" and "that kind."
- 0.1.3.2 One value can hide several storiesA number can be positive, zero, or negative, and these behave so differently you often must answer each branch separately, like ice, water, and steam.
- 0.1.3.3 Cover everything, overlap nothingGood cases are puzzle pieces: together they fill the whole picture, and none sits on top of another.
- 0.1.3.4 The tell-tale signA hidden "it depends..." — on a sign, a size, an absolute value, or how many — is the problem begging you to make cases.
- 0.1.3.5 Where you'll meet it againRemoving absolute-value bars, the discriminant deciding two roots, one, or none, and splitting a figure by which quadrant it lands in.
This module is about changing a strange new problem into an old friend you already know how to beat.
- 0.1.4.1 Make the new look oldFaced with something unfamiliar, reshape it until it becomes a problem you've cracked a hundred times, like opening a new lock with an old key.
- 0.1.4.2 Legal moves keep the answerDividing by a fraction quietly becomes multiplying by its flip — a hard move swapped for an easy, equal one that changes the look but never the solutions.
- 0.1.4.3 It needn't be simpler, just knownThe transformed problem can even look bigger, as long as it's one you actually know how to finish.
- 0.1.4.4 The tell-tale signWhen you think "if only this were the simpler thing I already know," that wish is your cue to transform it into exactly that.
- 0.1.4.5 Where you'll meet it againSolving by substitution, completing the square to reuse the square-root trick, change of base for logarithms, and clearing fractions to a common denominator.
This module is about peeking at easy examples to guess the big rule, then proving the rule so you can trust every case.
- 0.1.5.1 Try a tiny case firstBefore tackling "any number of people," shake hands with just 3 friends and count; small cases whisper the pattern before it shouts as a formula.
- 0.1.5.2 The general guards the specialOnce you prove a rule for every number, you never check the 4,000th case by hand — the general law covers each one for free.
- 0.1.5.3 Special values are spiesPlug in 0, or 1, or a friendly number to sniff out how a mysterious expression behaves.
- 0.1.5.4 The tell-tale signStuck on 'for every n'? Try n = 1, 2, 3 to catch the pattern; landed a neat formula? Pressure-test it on one fresh case before you trust it.
- 0.1.5.5 Where you'll meet it againFinding the rule of a sequence, the binomial theorem, and guessing a formula then proving it forever by induction.
This module is about taming "forever" by watching where an endless process is heading.
- 0.1.6.1 Forever can have a finish lineAdd 1/2, then 1/4, then 1/8, walking halfway to the wall again and again; you never crash, yet you get as close to exactly 1 as you please.
- 0.1.6.2 A limit is a destination, not a stepA limit is the spot a journey clearly aims for even if it never quite lands, like a ball settling slower and slower to rest.
- 0.1.6.3 Infinity is a direction, not a number"Goes on forever" tells you which way to look, not a place to stand; treat it as a horizon, and describe the endless in finite words.
- 0.1.6.4 The tell-tale signWords like "approaches," "endlessly," "as fine as you like," or "in the long run" mean the infinite has quietly entered the room.
- 0.1.6.5 Where you'll meet it againThe infinite sum of a shrinking geometric series, the area under a curve as endlessly thin slices, and the instant speed behind every derivative.
This module is about separating what MUST happen from what MIGHT happen, and putting a number on the "might."
- 0.1.7.1 Some things are sure, some are maybe"An even plus an even is even" is certain; "this coin lands heads" is only likely — and math handles both kinds of truth honestly.
- 0.1.7.2 Probability is a maybe with a numberLikelihood gets a score from 0 (never) to 1 (always), so "pretty likely" becomes something you can actually compare.
- 0.1.7.3 Random settles downYou can't predict one coin flip, yet flip it ten thousand times and "about half heads" shows up like clockwork — chaos up close, order from afar.
- 0.1.7.4 The tell-tale signWords like "at random," "expected," "on average," or "what are the odds" mean you've stepped from certainty into chance.
- 0.1.7.5 Where you'll meet it againCounting outcomes, expected value, and the bell-shaped curve that explains why averages are so dependable.
This module is about treating a whole chunk as one thing, and noticing what stays the same while everything else moves.
- 0.1.8.1 Hold a clump as a single blockInstead of prying apart (x + y), treat the whole thing as one tile you can slide around; sometimes you never need the pieces at all.
- 0.1.8.2 Symmetry is a fold that matchesFold a butterfly, a face, or a parabola down the middle and the halves land on each other, so understanding one side hands you the other for free.
- 0.1.8.3 Find what refuses to changePour water between differently shaped glasses and the amount stays put; the quantity that won't budge is often the key that unlocks the puzzle.
- 0.1.8.4 The tell-tale signIf the parts look ugly but a combination looks clean, go holistic; if something stubbornly stays equal no matter what you do, you've found an invariant.
- 0.1.8.5 Where you'll meet it againSubstituting a repeated group, Vieta's sum-and-product of roots, a parabola's mirror line, and conserved quantities in geometry.
0.2Methods & Strategies
This module is about reshaping a stubborn quadratic into a perfect square plus a leftover so its secrets fall out.
- 0.2.1.1 Build the missing cornerAn L-shaped patch of tiles is almost a square; add exactly the right corner piece to complete it, then subtract it back so the value never changes.
- 0.2.1.2 A perfect square hides the answerOnce it reads (something)² plus a number, you solve by simply taking a square root.
- 0.2.1.3 The leftover reveals the vertexCompleting the square on a quadratic hands you the parabola's lowest or highest point with no guessing.
- 0.2.1.4 The tell-tale signAn x² and an x term loitering together with no easy factoring is your cue: there's a square waiting to be completed.
- 0.2.1.5 Where you'll meet it againDeriving the quadratic formula, finding a parabola's vertex, the equation of a circle, and the proof behind the basic inequality.
This module is about giving a clumsy chunk a short nickname so the whole problem suddenly looks simple.
- 0.2.2.1 Nickname the messy partLet one letter stand for the ugly repeated lump, and a four-line problem shrinks to a one-liner because your eyes stop tripping over it.
- 0.2.2.2 A good name reveals structureThe right substitution makes a hidden shape — often a plain quadratic — pop right out of the chaos, proof that good notation is a tool for thought.
- 0.2.2.3 Don't forget to change backCrack the simple version first, then swap the real thing back in, or you've answered a different question than the one asked.
- 0.2.2.4 The tell-tale signWhen the same complicated bundle appears two or three times, that repetition is begging for a nickname.
- 0.2.2.5 Where you'll meet it againTurning disguised quadratics like x⁴ − 5x² + 4 into easy ones, integration tricks, and simplifying nested expressions.
This module is about guessing the shape of an answer with blanks in it, then filling the blanks by matching what you know.
- 0.2.3.1 Guess the form, leave blanksYou're sure the answer looks like (blank)x + (blank); now you just need to pin down the two blanks.
- 0.2.3.2 Matching forces the valuesIf two expressions are truly equal, their matching pieces must agree term by term, which hands you a tidy little system to solve.
- 0.2.3.3 The tell-tale signWhen you know the TYPE of answer — a line, a parabola, partial fractions — but not the exact numbers, set up blanks and solve for them.
- 0.2.3.4 Where you'll meet it againFitting a line through points, finding a quadratic from three given points, and splitting a fraction into simpler pieces.
This module is about proving something true by showing the opposite leads to nonsense.
- 0.2.4.1 Assume the opposite, watch it crashSuppose the floor is dry, then notice every umbrella is dripping; the contradiction means it must have rained after all.
- 0.2.4.2 A crash means you were rightIf pretending a claim is false forces an impossibility like 1 = 2, then the claim simply had to be true all along.
- 0.2.4.3 Great for "there is no such thing"It shines at proving something can never exist or happen, where checking every case directly would take forever.
- 0.2.4.4 The tell-tale signClaims with "no," "never," "impossible," "irrational," or "infinitely many" are tailor-made for assuming the reverse and hunting the crash.
- 0.2.4.5 Where you'll meet it againProving √2 isn't a fraction by assuming it is, and showing the primes never run out.
This module is about toppling an endless row of dominoes by knocking the first and guaranteeing each knocks the next.
- 0.2.5.1 Knock over the first dominoCheck the very first case by hand; that's the starting push, the one tile you tip over yourself.
- 0.2.5.2 Each one topples the nextProve that IF any case holds THEN the next one must too, and the toppling chains down the whole infinite row.
- 0.2.5.3 Two steps cover foreverThe first push plus the "each tips the next" promise prove the claim for every counting number at once, with only two checks.
- 0.2.5.4 The tell-tale signA statement claimed "for every positive integer n," like a sum that should work for any number of terms, is induction's home turf.
- 0.2.5.5 Where you'll meet it againProving summation formulas, the laws of exponents, and nailing down patterns in sequences once and for all.
This module is about building the exact example or helper object you need instead of hunting for it.
- 0.2.6.1 Build the thing you needRather than searching, craft a specific number, shape, or function that does exactly the job, like sketching the very triangle that proves your point.
- 0.2.6.2 A helper line works magicDrawing one clever auxiliary line can split a baffling figure into shapes you already understand.
- 0.2.6.3 One counterexample is enoughTo kill a "this always works" claim, you only need to construct a single case where it fails.
- 0.2.6.4 The tell-tale sign"Show there exists..." or "give an example," or a geometry figure that just won't open up, all hint that you should build something.
- 0.2.6.5 Where you'll meet it againAuxiliary lines in triangle proofs, slick counterexamples that sink tempting guesses, and clever setups in inequalities.
This module is about removing unknowns one at a time, and reasoning back from the goal until your path meets what you already know.
- 0.2.7.1 Knock out one unknown at a timeIn a system of equations, add, subtract, or substitute so one variable vanishes, leaving fewer letters to chase, then back-fill the rest.
- 0.2.7.2 Start from the finish lineAsk "what would I need just before the answer?" and step backward, like solving a maze from the exit toward the start.
- 0.2.7.3 Meet in the middlePush forward from what you know and backward from what you want; where the two paths shake hands is your solution.
- 0.2.7.4 The tell-tale signSeveral unknowns tangled across several equations call for elimination; a far-off goal with a foggy start calls for working backwards.
- 0.2.7.5 Where you'll meet it againSolving systems of linear equations, finding curve intersections, and planning multi-step proofs and word problems.
This module is about four quick tactics: list every case, cut-and-paste shapes, plug in clever numbers, and squeeze a value between bounds.
- 0.2.8.1 List them all, miss noneWhen only a handful of cases exist, calmly write out every one; honest exhaustion is a perfectly good proof, like trying every key until a door opens.
- 0.2.8.2 Cut here, paste thereSlice a tricky shape and rearrange the pieces into a friendly rectangle or triangle whose area you can read at a glance.
- 0.2.8.3 Plug in a clever valueSet a variable to 0 or 1 in an identity and watch most terms politely collapse, exposing a hidden coefficient almost for free.
- 0.2.8.4 Trap the answer between two fencesWhen you can't compute a value exactly, show it's bigger than one number and smaller than another, squeezing it into a tight pen.
- 0.2.8.5 Where you'll meet it againCounting outcomes by listing, finding areas by dissection, special values in the binomial theorem, and bounding a sum to prove it stays finite.
0.3Habits of Mind — How to Think Like a Mathematician
This module is about truly grasping what's being asked before you rush to compute anything.
- 0.3.1.1 Name the knowns and the wantedBefore moving, ask plainly: what am I trying to find, and what am I actually handed to work with?
- 0.3.1.2 Say it in your own wordsIf you can retell the problem simply to a friend, you understand it; if you can't, you'll end up solving the wrong thing.
- 0.3.1.3 Draw it or test a tiny caseA quick sketch, a chart, or a small example turns a wall of words into something you can grab hold of.
- 0.3.1.4 Check every conditionNotice each fact you're given; an unused condition is often the exact clue you were missing.
- 0.3.1.5 Plant the look-back seedDecide now what a correct answer should look like, so you'll recognize it when it finally arrives.
This module is about choosing a route before you start walking, by mining what you already know.
- 0.3.2.1 Have you seen its cousin?Search your memory for a similar problem; the method that cracked it may crack this one with small changes.
- 0.3.2.2 Work backward from the goalAsk "what would I need just before the answer?" and trace the steps back to where you stand right now.
- 0.3.2.3 Simplify or specializeCan't see the path? Solve an easier version or a tiny case first to light up the route.
- 0.3.2.4 Pick a tool on purposeSubstitution, a picture, splitting into cases — name your strategy before diving in, not halfway through.
- 0.3.2.5 Break a giant into stepsOne huge leap is scary, but a staircase of small, sure steps gets you to the top without panic.
This module is about executing the plan carefully, neatly, and one justified step at a time.
- 0.3.3.1 One step, then justifyTake a single step and ask "is this actually true?" before the next; speed built on a shaky step collapses later.
- 0.3.3.2 Keep your work readableNeat, lined-up steps leave clear footprints, so you can retrace the path and spot exactly where any slip happened.
- 0.3.3.3 Guard the sneaky errorsA dropped minus sign, a flipped inequality, or a forgotten case is where good plans quietly die.
- 0.3.3.4 Persist, but switch the tool, not the goalIf a good plan stalls, push a little harder before giving up; only when a step truly can't be justified do you back up and try another strategy.
- 0.3.3.5 It's okay to be stuckBeing stuck isn't failure, it's the normal middle of every real problem; productive struggle is where the learning actually happens.
This module is about the step most solvers skip — and the one that builds real skill.
- 0.3.4.1 Does the answer make sense?A person can't be 200 years old, a length can't be negative, and a probability can't top 1; let red flags make you flinch before you celebrate.
- 0.3.4.2 Estimate to sanity-checkA quick "about how much?" guess, made first, warns you instantly when an exact answer comes out wildly off.
- 0.3.4.3 Find a second routeSolving it a different way and landing on the same answer is the most satisfying proof you got it right.
- 0.3.4.4 Extract the reusable takeawayAsk what trick made this work and where else it might apply; that's how one solved problem teaches you ten.
- 0.3.4.5 The habit that compoundsLooking back is the quiet step that slowly turns a problem-doer into a problem-solver.
This module is about the three engines that generate ideas when you're stuck.
- 0.3.5.1 Specialize to get tractionToo abstract? Pin down concrete numbers, solve that, and let the small win light the way.
- 0.3.5.2 Generalize to see the patternOnce a few cases agree, ask "what's the rule behind all of them?" and reach for the broader truth.
- 0.3.5.3 Analogize across topicsA problem about money and one about distance can share the same skeleton, so borrow the solution that fits.
- 0.3.5.4 The tell-tale rhythmFeeling overwhelmed? Specialize. Feeling repetitive? Generalize. Feeling déjà vu? Find the analogy.
- 0.3.5.5 Where you'll meet it againYou'll specialize with a few sequence terms, generalize to a formula, then analogize that very move to series and limits.
This module is about stripping away clutter to find the bare idea, and dressing math back up to capture the real world.
- 0.3.6.1 Keep the structure, drop the clutterForget whether it's apples or dollars; keep only "3 + 2," and one idea ends up solving a thousand situations.
- 0.3.6.2 From five sheep to the number fiveThe leap from "five sheep" to "five" itself is the first and deepest abstraction in all of mathematics.
- 0.3.6.3 Name the patternGiving a recurring idea a symbol — a variable, a function, a set — lets you reason about it cleanly without redrawing it each time.
- 0.3.6.4 Model the real world, then checkTranslate a situation into math, solve it, and translate the answer back, asking whether it honestly fits reality.
- 0.3.6.5 Where you'll meet it againVariables standing for any number, functions for any rule, and modeling that runs through every word problem and optimization.
This module is about daring to guess a pattern boldly, then earning the right to believe it.
- 0.3.7.1 Guess boldly from the evidenceAfter a few examples line up, make a brave, explicit conjecture; a clear guess gives you a target, while vague hoping gives you nothing to aim at.
- 0.3.7.2 Hunt hard for a counterexampleBefore trusting a guess, try your best to break it; a single failing case saves you from proving something false.
- 0.3.7.3 A pattern isn't a proofHolding for a hundred cases is encouraging but not enough; the next case could betray you, so you still owe a real argument.
- 0.3.7.4 Then prove for certaintyProof is the difference between "it seems to always work" and "it must always work, here's exactly why."
- 0.3.7.5 The tell-tale sign"It looks like it's always..." is the precise moment to either find a counterexample or build a proof.
This module is about airtight logic, a steady feel for size, and the wonder that pulls a mathematician forward.
- 0.3.8.1 A chain with no weak linkA proof connects known truths step by unbreakable step; deduction makes the conclusion inescapable, while spotting a pattern (induction of the everyday kind) only suggests it.
- 0.3.8.2 Say exactly what you meanDefine your terms sharply and know your starting rules — the givens, definitions, and axioms everything else is built on — because fuzzy words breed fuzzy thinking.
- 0.3.8.3 Beware the sneaky "obviously""Clearly" often hides the very step that's wrong; slow down and check the part you wanted to skip.
- 0.3.8.4 Carry a number senseKnowing a classroom holds dozens, not millions, lets you smell a wrong answer before you even check it; round first, refine later.
- 0.3.8.5 The mathematician's stanceCuriosity to explore and ask "what if?", courage to conjecture, and honesty to demand proof — carry these into every stage that follows.
Ⅰ · Numbers & Operations
1Numbers & Counting
Build the most basic number sense: counting means pairing each thing with a number, and the last number you say tells you how many there are in all
- 1.1.1 One-to-one matchingGive every cup its own spoon, and if they pair up perfectly, you've got 'the same amount' — that's one-to-one matching
- 1.1.2 Counting in orderTouch one, say one — never skip and never double up — and out comes 1, 2, 3, 4...
- 1.1.3 The last number is the total (the count)When you stop at 'five,' that 'five' stands for the whole pile — it's how many there are
- 1.1.4 More and fewerPair up two piles of candy until one side runs out first; the pile with leftovers has 'more,' and the one that's empty has 'fewer'
- 1.1.5 How many vs. which one (count vs. position)'There are 3 people' tells you how many, while 'you're 3rd in line' tells you where — same 3, different meaning
Understand that 'nothing' is also a number, zero, and discover the magic of bundling a new unit every time you reach ten
- 1.2.1 What zero meansThe plate is empty, everything's been taken away — even this 'nothing at all' is worth writing down, and that's 0
- 1.2.2 The birth of 10: ten makes a bundleTie ten little sticks into one bundle, and count that bundle as a single new 'ten'
- 1.2.3 The teens: a bundle plus a fewOne bundle and three loose sticks is thirteen, written 13 — the 1 in front is the bundle, the 3 behind is the loose ones
- 1.2.4 The tens: several bundlesThree bundles of sticks make thirty, written 30 — the 0 at the end means there's not a single loose stick
- 1.2.5 Counting to one hundredTie ten bundles into one big bundle and you get one hundred, written 100 — ten again, and up one more level
Master the place-value rules for ones, tens, hundreds, thousands... and learn to read and write multi-digit numbers
- 1.3.1 The ones place and the tens placeThe same 5 means 5 sticks in the ones place but 5 bundles in the tens place — its spot decides how big it is
- 1.3.2 Hundreds, thousands, and counting unitsReading right to left — ones, tens, hundreds, thousands — each step left is worth ten of the place before it
- 1.3.3 Zero as a placeholderThree hundred five has no 'tens,' so a 0 holds that spot to write 305 — you can't write it as 35
- 1.3.4 Reading multi-digit numbersRead from the highest place down — 'so many thousands, hundreds, tens, ones' — and read a single 'zero' for a 0 in the middle
- 1.3.5 Writing multi-digit numbersHear 'two thousand and eight,' put 2 in the thousands, fill the empty hundreds and tens with 0, and 8 in the ones — that's 2008
- 1.3.6 Ten thousand and bigger numbersTen thousand is ten one-thousands, and as numbers grow you group every four digits into sections: ones, ten-thousands, hundred-millions
Learn to use place value to quickly compare and order numbers, and bring in symbols to express how they relate in size
- 1.4.1 Finding a number's spot on the number lineLay numbers out like marks on a ruler — the farther right, the bigger — so whoever sits to the right is the larger one
- 1.4.2 How to compare sizesFirst see who has more digits; if they match, compare from the highest place down, and the first place to differ settles it
- 1.4.3 The greater-than, less-than, and equals signsThe pointy end faces the smaller number and the open end faces the bigger one: 5 > 3, 3 < 5, 4 = 4 — the symbol's mouth opens toward the larger number
- 1.4.4 Ordering from small to large and large to smallLine them up by height: shortest in front is ascending order, tallest in front is descending order
- 1.4.5 Finding the largest and smallestOnce they're lined up, the front and the back are the largest and smallest of the bunch
Get to know odd and even numbers by pairing things two by two, and feel the patterns hidden inside numbers
- 1.5.1 Splitting into twos: odd and evenPair the game pieces two by two — if they pair up exactly it's even, and if one is left alone it's odd
- 1.5.2 Telling odd from even by the last digitA ones digit of 0, 2, 4, 6, 8 means even, and 1, 3, 5, 7, 9 means odd
- 1.5.3 The back-and-forth rhythm of odd and evenOdd, even, odd, even take turns — like hopscotch landing on one foot then two — falling into a steady, rhythmic line
- 1.5.4 Finding the pattern in a sequence2, 4, 6, 8... goes up by 2 each step; once you spot that 'step,' you can keep counting right along
Build an intuition for how big numbers are, learn to round to an approximate value, and pave the way for estimating in operations
- 1.6.1 Numbers in everyday lifeAges, house numbers, phone numbers, prices... numbers are all around you — start by noticing what each one stands for
- 1.6.2 Estimating how many there areInstead of counting one by one, glance at the candy in the jar and guess 'about thirty or so' — that's an estimate
- 1.6.3 The nearest ten or hundredOn the number line, 38 sits closer to 40 than to 30, so you say it's 'about 40'
- 1.6.4 RoundingLook at the digit you're cutting off: less than 5, drop it; 5 or more, round up — and tidy the number into a clean ten or hundred
- 1.6.5 The approximately-equal sign and approximate numbers297 ≈ 300 — the wavy equals sign '≈' means 'just about equal,' which makes mental math easier
- 1.6.6 Getting ready to estimateRound numbers to clean tens and hundreds first, so the next step of adding, subtracting, multiplying, or dividing lets you ballpark the answer in your head
2The Four Operations & Estimation
Move from counting to operating, building the real-world meaning of addition and subtraction, their inverse relationship, and column arithmetic for multi-digit numbers.
- 2.1.1 What Addition MeansPour two piles of candy into one bag and count the total — that's putting them together, written a + b.
- 2.1.2 What Subtraction MeansA basket has 8 apples, you take 3 away, and you count what's left — that taking-away is written a − b.
- 2.1.3 Addition and Subtraction Are a Matched PairAdd some and then take it back and you land right where you started, so 5 + 3 = 8 and 8 − 3 = 5 undo each other.
- 2.1.4 Multi-Digit Addition and CarryingAdd the beads in the same place side by side, and whenever they reach ten, carry one over to the left.
- 2.1.5 Multi-Digit Subtraction and BorrowingWhen a place doesn't have enough to subtract, borrow a ten from the neighbor on the left to pay the debt — that's borrowing.
- 2.1.6 Checking Addition and SubtractionAdd your answer back and see if it rebuilds the original — if it matches, you got it right.
Compress repeated addition of equal amounts into multiplication, memorize the times facts, and master multi-digit column multiplication.
- 2.2.1 Multiplication Is Shorthand for Repeated AdditionFive rows of chairs with 4 in each — 4 + 4 + 4 + 4 + 4 is too long, so write it as 5 × 4.
- 2.2.2 The Times FactsLay the dots of a rectangle array into a table and learn it cold, so 7 × 3 = 21 comes out without thinking.
- 2.2.3 Seeing Multiplication as AreaA floor of tiles 3 across and 4 down, counted all the way through, is the product 3 × 4.
- 2.2.4 Multiplying a Multi-Digit Number by One DigitBreak the big number into its places, multiply each one, then stack them up, carrying whenever you hit ten.
- 2.2.5 Multiplying Two Two-Digit NumbersMultiply by the ones first, then the tens, shift one column over, and add — like two floors stacked on top of each other.
- 2.2.6 How 0 and 1 Behave in MultiplicationTimes 1 leaves everything untouched, times 0 wipes it all to zero — an empty pile counts as nothing no matter what.
Understand the two real-world meanings of division, handle remainders, master long division, and see how multiplication and division are inverses.
- 2.3.1 What Division Means: Sharing EquallySpread 12 candies evenly among 3 people to see how many each gets — that even split is 12 ÷ 3.
- 2.3.2 The Other Side of Division: How Many Fit InsidePack 12 candies into bags of 4 and ask how many bags — asking how many 4s fit inside is also 12 ÷ 4.
- 2.3.3 Multiplication and Division Are InversesSplit apart and then combine back and you're whole again, so 3 × 4 = 12 pairs with 12 ÷ 4 = 3.
- 2.3.4 Division with a RemainderShare 7 candies among 3 people — 2 each with 1 left over, and that leftover bit is the remainder r.
- 2.3.5 Long DivisionShare out starting from the highest place, bring down the next digit and keep sharing, writing it neatly as long division.
- 2.3.6 Checking Division with MultiplicationThe quotient times the divisor plus the remainder should equal the number you started with — if it matches, you're safe.
Discover the commutative, associative, and distributive laws, and use them to rearrange expressions and speed up.
- 2.4.1 The Commutative LawSwap two piles and the total stays the same, so a + b = b + a and a × b = b × a.
- 2.4.2 The Associative LawIt doesn't matter which two piles you combine first, so doing (2 + 8) first and then adding 5 feels smoother.
- 2.4.3 The Distributive LawSlice one floor into two strips by width, compute each, then join them: a × (b + c) = a × b + a × c.
- 2.4.4 Computing Cleverly with These LawsRound to nice numbers and split things apart to lighten the work — for 25 × 4, make 100 first.
- 2.4.5 The Fake "Laws" of Subtraction and DivisionSwapping crashes here — 8 − 3 is not 3 − 8 — a reminder that these two can't be reordered freely.
Establish the order of operations with parentheses, and build practical intuition for mental math and estimation.
- 2.5.1 Multiply and Divide Before You Add and SubtractBuying 3 pens at $2 plus a $1 notebook, you work out the pen cost first — multiplication and division go first.
- 2.5.2 Parentheses Come FirstParentheses are like a sticky note saying "do this first," so whatever's inside ( ) gets done before anything else.
- 2.5.3 Three-Step Mixed ExpressionsGo one gate at a time — parentheses, then multiply/divide, then add/subtract — and even tangled expressions stay tidy.
- 2.5.4 Mental Math TricksBreak numbers into easy pieces: for 48 + 27, add 50 then step back 2, and you can do it all in your head.
- 2.5.5 Estimation: Get the Rough Picture FirstBefore checking out, round each price and add them up to see if you have enough, using the ≈ sign.
- 2.5.6 Using Estimation to Check AnswersIf your written result doesn't match the ballpark in your head, go back and check — estimation is the sentry on watch.
Learn to round, express precision and significant figures, and use scientific notation to handle large numbers, paving the way for decimals.
- 2.6.1 RoundingLean toward whichever ten the last digit is closer to — 4 and under round down, 5 and up round up.
- 2.6.2 Precision: To Which PlaceSaying your height is "about 1 meter 7" versus "1.72 meters" depends on how fine you need to be.
- 2.6.3 Significant FiguresCounting from the first nonzero digit, the digits that truly carry weight set how solid an approximate number is.
- 2.6.4 Getting a Feel for Very Large NumbersJump up step by step through thousands, millions, billions — each jump tacks another zero onto the end.
- 2.6.5 Scientific NotationFold a long string of zeros into a power of 10, so 300000 is written as 3 × 10⁵.
- 2.6.6 Comparing and Estimating with Scientific NotationFor giants like the speed of light or world population, just compare the exponents to see which is bigger, and estimating gets far easier.
3Fractions & Decimals
Let fractions grow out of the picture of "sharing fairly," understand that a fraction names "part of a whole," and learn to read, write, and place them on the number line.
- 3.1.1 Equal Sharing and the Birth of the FractionCut one pie into equal slices and take some — how many out of how many is exactly what the fraction a/b records.
- 3.1.2 What the Top and Bottom Are Telling YouThe bottom number says how many equal parts you cut into, the top says how many you took, and the bar between them is the knife that does the equal sharing.
- 3.1.3 Where a Fraction Lives on the Number LineSplit 0 to 1 into equal steps, and whichever step you land on is that fraction — fractions live on the number line too.
- 3.1.4 Fractions Built from Pieces: Unit Fractions and Stacking Up1/4 is one piece, so three pieces is three of those 1/4s, which makes 3/4 — every fraction is built up from unit fractions.
- 3.1.5 Proper Fractions, Improper Fractions, and Mixed NumbersLess than a full cup is a proper fraction, full with some spilling over calls for a mixed number, and an improper fraction like 5/4 is just 1¼.
Understand that "cutting finer makes more parts but the same amount," and master equivalent fractions, reducing to lowest terms, and finding common denominators for comparing and calculating.
- 3.2.1 Equivalent Fractions: The Cut Changes, the Amount Doesn'tFold half a sheet again into 2/4 — it looks like more pieces but it's the same amount, so 1/2 = 2/4 = 3/6.
- 3.2.2 Reducing: Saying a Fraction in Its Simplest FormDivide top and bottom by a common factor, like tidying 6/8 down to 3/4, and keep going until nothing more divides them both.
- 3.2.3 Common Multiples and the Common DenominatorTo line up two kinds of grid you need a number of fine cells they can share — that shared count is the common denominator (the least common multiple).
- 3.2.4 Finding a Common Denominator: Turning Them into the Same Size PartCut both 1/2 and 1/3 into 6 parts so they become 3/6 and 2/6 — once the pieces are the same size they're easy to compare and easy to add.
- 3.2.5 Comparing the Size of FractionsSame bottom, the bigger top wins; same top, the smaller bottom wins; otherwise give them a common denominator first, then compare.
Add and subtract fractions once the parts match, handling unlike denominators with a common denominator, mixed-number addition and subtraction, and reducing the result.
- 3.3.1 Same Denominator: Just Add the PartsThey're all 1/8 pieces, so 3 pieces plus 2 pieces is 5 pieces — leave the bottom alone and only add or subtract the tops.
- 3.3.2 Different Denominators: Make Them Match, Then Add1/2 + 1/3 are different sizes and can't just be joined, so turn them into 3/6 + 2/6 = 5/6 first.
- 3.3.3 Mixed Numbers: Carrying and BorrowingHandle the whole numbers together and the fractions together, and when the fraction can't be subtracted, borrow a 1 from the whole number.
- 3.3.4 The Habit of Simplifying the ResultWhen you're done, reduce to lowest terms and turn an improper fraction back into a mixed number — only then is the answer fully said.
Use area and the picture of "a part of a part" to understand fraction multiplication, use "equal sharing" for division, and master canceling and reciprocals.
- 3.4.1 A Fraction Times a Whole Number: That Many of Them3 of 2/5 just means adding 2/5 three times, so keep the bottom and multiply the top by 3 to get 6/5.
- 3.4.2 A Fraction Times a Fraction: Half of a HalfTake a third of a half, draw the grid and count the little cells — multiply top by top and bottom by bottom.
- 3.4.3 Cancel First, Then MultiplyBefore multiplying, cancel common factors across the top and bottom so the numbers shrink — easier work and fewer mistakes.
- 3.4.4 Reciprocals: Flip It Upside DownFlip 3/4 into 4/3, and the two multiply to exactly 1 — that pair are reciprocals of each other.
- 3.4.5 Dividing Fractions: Dividing Means Multiplying by the Reciprocal"Half a pie shared among 1/4-sized portions" is really asking how many 1/4s fit inside, so division turns into multiplying by the reciprocal.
Extend decimals from splitting into ten equal parts, understand that a decimal is really a base-ten fraction, and master place value, reading and writing, comparing, and converting across places.
- 3.5.1 A Decimal Is a Base-Ten FractionSplit 1 into ten equal parts and each part, 0.1, is 1/10 — decimals are really just fractions with denominators of 10, 100, and so on.
- 3.5.2 Decimal Places and Counting UnitsTo the right of the point come tenths, then hundredths, then thousandths — each step right cuts things ten times finer.
- 3.5.3 Reading, Writing, and the Meaning of Decimals0.25 reads "zero point two five" and means 2 of 0.1 plus 5 of 0.01, which is 25/100.
- 3.5.4 A Property of Decimals: Trailing Zeros Don't Change the Value0.3 and 0.30 are the same amount — adding a zero on the end just cuts finer without changing the size.
- 3.5.5 Comparing the Size of DecimalsCompare place by place from the highest down — tenths first, then hundredths — and the bigger digit makes the bigger number.
Master the methods for adding, subtracting, multiplying, and dividing decimals, understand why the point moves where it does, and learn to round and handle repeating cases.
- 3.6.1 Adding and Subtracting Decimals: Line Up the PointLining up the decimal point lines up the matching places, and then you add or subtract in columns just like whole numbers.
- 3.6.2 Multiplying Decimals: Compute as Whole Numbers, Then Place the PointIgnore the point and find the product first, then count the total decimal places in both factors and place the point that many digits from the right.
- 3.6.3 Dividing Decimals: Turn the Divisor into a Whole NumberShift the divisor's point right until it's a whole number, slide the dividend's point the same amount, and the hard problem becomes ordinary whole-number division.
- 3.6.4 Approximate Quotients and RoundingWhen it doesn't divide evenly, keep as many places as you need, look at the next digit to round, and write the result with ≈.
- 3.6.5 Terminating and Repeating DecimalsSome divisions end cleanly as terminating decimals, others have digits that repeat forever, marked with a dot like 0.3̇.
Connect the three representations, introduce percentages as "how many per hundred," and pave the way directly into the next stage on ratios and percentages.
- 3.7.1 Fraction to Decimal: Divide the Top by the BottomThat bar in a fraction was always a division sign, so 3/4 is just 3 ÷ 4 = 0.75.
- 3.7.2 Decimal to Fraction: Write the Denominator from the Place Value0.45 reads "forty-five hundredths," so write it as 45/100 and reduce to 9/20.
- 3.7.3 Percentages: How Many Out of Every HundredTreat the whole as 100 parts, and taking 75 of them is 75% — the symbol % means "per hundred."
- 3.7.4 Converting Among Percentages, Decimals, and FractionsDrop the % and shift the point two places left, so 0.6 = 60% = 3/5 — three ways of writing one number, used interchangeably.
- 3.7.5 Solving Real Problems with All Three FormsFor discounts, hit rates, and shares, reach for whichever form is handiest, and being able to switch builds the bridge straight into "ratios and proportion."
4Ratios, Proportion & Percentages
Building on the fractions you just met, we move from 'how much one quantity is of the whole' to 'which quantity is bigger and by how many times,' giving you the language of ratios and rates.
- 4.1.1 What a Ratio MeansWhen you mix sugar water with 2 scoops of sugar to 3 of water, that 'so-many to so-many' comparison is written 2:3.
- 4.1.2 How Ratios Relate to Fractions and Divisiona to b is really just a divided by b, so a:b = a/b, and the value of the ratio is simply the quotient.
- 4.1.3 Rates and Unit PriceIf 3 kilograms cost 12 dollars, working out 'dollars per kilogram' squeezes the ratio down to a single unit amount, 4 dollars/kg.
- 4.1.4 Simplifying Ratios to Lowest Whole-Number TermsJust like reducing a fraction, divide both parts of 8:12 by 4 to get the cleanest possible 2:3.
- 4.1.5 Continued RatiosWhen you blend three colors at once, comparing red, yellow, and blue together is written a:b:c.
We step from 'one ratio' to 'two ratios that are equal,' learning how to test for a proportion and its basic property, laying the foundation for direct and inverse proportion and for sharing by a ratio.
- 4.2.1 What a Proportion MeansEnlarge a photo and its 'length to width' stays the same, so linking two equal ratios gives a:b=c:d.
- 4.2.2 The Basic Property of ProportionLike a balanced scale, the cross-products must match, so a:b=c:d means ad=bc.
- 4.2.3 Solving a Proportion for an UnknownWith three terms known, the missing one is recovered by cross-multiplying back to find x.
- 4.2.4 Scale on a MapWhen 1 cm on the map stands for 100 m in real life, the ratio of map distance to real distance is the scale 1:10000.
We turn proportion from a still equation into a living relationship between two changing quantities, meeting the two classic kinds of link and getting our first taste of describing change with a graph.
- 4.3.1 Quantities in Direct ProportionBuy more and you spend more while the price holds steady, so y grows in step with x and the ratio y/x=k stays constant.
- 4.3.2 The Graph of Direct ProportionPlot the paired numbers and join them, and direct proportion draws a straight line through the origin.
- 4.3.3 Quantities in Inverse ProportionFor a fixed distance the faster you go the less time it takes, so as one rises the other falls and the product xy=k stays constant.
- 4.3.4 The Graph of Inverse ProportionPlot and join the points of inverse proportion and you get a curving line that never touches the axes.
- 4.3.5 Telling Them Apart and Using ThemFirst ask 'do they rise together or trade off,' then decide by checking whether the ratio k or the product k stays fixed.
We pull out the special ratio that always counts 'out of 100,' understand that a percent is just a kind of ratio, and move freely among fractions, decimals, and percents.
- 4.4.1 What a Percent MeansYour battery is at eight-tenths, and 'how many parts out of every 100' is the percent 80%.
- 4.4.2 Converting Among Percents, Fractions, and Decimals75% is both 75/100 and 0.75, so dropping the percent sign just shifts the decimal point two places right.
- 4.4.3 Finding What Percent One Number Is of AnotherWith 45 passing out of 50, divide the part by the whole and turn it into a percent, 90%.
- 4.4.4 Finding a Given Percent of a NumberTo see what 15% of 200 dollars is, multiply the whole by the percent, 200×15%.
- 4.4.5 Recovering the Whole from a PercentKnowing that the 12% you finished is exactly 6 problems, work backward by dividing by the percent to find the total.
We place percentages into real money and real change, increases and decreases, discounts and interest, and learn to handle every rise or fall with one tidy form, 'original amount times (1 plus or minus rate).'
- 4.5.1 Percent Increase and DecreaseA 20% rise makes the new price 1.2 times the old, all captured by original amount times (1 plus or minus rate).
- 4.5.2 Discounts and MarkdownsTwenty percent off means paying 80% of the original, so sale price = list price times discount rate.
- 4.5.3 Profit, Profit Margin, and Tax RateThe extra you make between buying and selling, measured against cost, is the profit margin = profit divided by cost.
- 4.5.4 Principal, Interest Rate, and Simple InterestThe little extra you earn for keeping money in for a year is the interest, interest = principal times rate times time.
- 4.5.5 A First Look at Compound InterestWhen the interest itself earns interest, money snowballs and is multiplied year after year by (1 plus the rate).
We use ratios and rates to handle the two everyday problems of sharing and converting, building a sense that 'a quantity = number times unit,' rounding off before the next stage brings in signed rational numbers.
- 4.6.1 Sharing by a RatioSplit a bonus by contribution 3:2 by treating the total as 5 parts, then handing each person their share of parts.
- 4.6.2 Continued Sharing and Mixing ProblemsTo mix cement, sand, and gravel at 2:3:5, find the value of one part first, then make up enough of each.
- 4.6.3 Ratios Inside Unit Conversion1 meter = 100 cm is really a conversion ratio, and multiplying by the conversion factor completes the switch.
- 4.6.4 Quantity, Value, and Unit (Dimensional Sense)A quantity equals a number times a unit, and the units on both sides of an equation must line up too, 3m×2=6m.
- 4.6.5 Compound Units and RatesSpeed is 'so many kilometers per hour,' and the units join the arithmetic too, km÷h=km/h.
5Negative Numbers & Rational Numbers
Help learners feel where positive numbers alone get stuck, draw out negative numbers naturally from quantities with opposite meaning, and place every number on the number line.
- 5.1.1 Quantities with Opposite MeaningWalking 3 steps east and 3 steps west are two different things, so opposite directions need opposite signs to tell them apart.
- 5.1.2 Positive and Negative NumbersFive degrees above zero we write as +5, five below as −5, and that little dash out front just means the other direction.
- 5.1.3 Zero Is Neither Positive nor NegativeSea level counts as neither high nor low; 0 is the dividing line between positive and negative and carries no direction of its own.
- 5.1.4 The Three Ingredients of a Number LineThink of a marked thermometer: pick an origin, pick a direction, pick a unit length, and every number gets a home.
- 5.1.5 Showing Numbers on the Number LineGo right from the origin for positives and left for negatives, and however many steps you take is the point where that number sits.
Build two tools, the opposite and the absolute value, around symmetry and distance on the number line, setting up rational-number arithmetic.
- 5.2.1 Opposite NumbersA pair of numbers sitting the same distance on either side of the origin: flip the sign and a becomes −a.
- 5.2.2 Simplifying Stacked SignsFlip direction twice and you're back where you started, so −(−3) is really just +3 — count the dashes.
- 5.2.3 What Absolute Value MeansWhether you go left or right, all we ask is how far you are from the origin, and that distance is the absolute value |a|.
- 5.2.4 Properties of Absolute ValueDistance is never negative, so |a| ≥ 0, and a number and its opposite are always the same distance out.
- 5.2.5 Using Absolute Value for DistanceHow far apart two points sit on the line is just the absolute value of their difference, |a−b|.
Welcome the integers, fractions, and decimals you already know into the big family of "rational numbers," and learn to line them up and compare on the number line.
- 5.3.1 What a Rational Number IsAny number you can write as a ratio of two integers, p/q, is rational — decimals and fractions are all in there.
- 5.3.2 Sorting Rational NumbersLike tidying a drawer: split them by integer versus fraction, or by positive, negative, and zero, and drop each into its own slot.
- 5.3.3 Comparing Size on the Number LineThe farther right a number sits the bigger it is, and a number on the right is always greater than one on the left.
- 5.3.4 How Positives, Negatives, and Zero RankEvery positive is bigger than 0 and every negative is smaller than 0, so any positive beats any negative.
- 5.3.5 Comparing Negatives by Absolute ValueWith two negatives, the one farther from the origin is actually the smaller, so a bigger absolute value means a smaller number.
Understand addition and subtraction through the picture of "keep walking" on the number line, master the sign rules, and turn subtraction into addition.
- 5.4.1 The Picture of AdditionOn the number line, take your first step and then keep going with the second, and where you land is the sum of the two numbers.
- 5.4.2 The Rule for AdditionSame signs: add the absolute values and keep the sign; different signs: subtract them and follow the sign of the bigger one.
- 5.4.3 The Laws of AdditionIt doesn't matter which you add first, so the commutative and associative laws let you group positives and negatives into round numbers first.
- 5.4.4 Turning Subtraction into AdditionOwing money is the same as adding a negative amount, so rewrite a−b as a+(−b) and you're left with only addition.
- 5.4.5 Mixed Addition and SubtractionTreat the whole string as addition, keep each sign glued to its number, and shuffle the order freely to make things round.
Understand the sign rule for multiplication through "repeating in the opposite direction," master the laws and reciprocals, and pave the way to powers.
- 5.5.1 The Picture and Rule for MultiplicationMultiplying by a negative is like flipping direction: same signs give positive, different signs give negative, then multiply the absolute values.
- 5.5.2 Signs with Several FactorsCount the negative signs: an even number gives positive, an odd number gives negative, and a single 0 makes the whole thing 0.
- 5.5.3 The Laws of Multiplication and the Distributive LawReorder and regroup as you like, and the distributive law a(b+c)=ab+ac is handy whether you're opening brackets or pulling out a common factor.
- 5.5.4 ReciprocalsA pair of numbers that fill each other up to 1: a times 1/a comes out to exactly 1.
- 5.5.5 The Rule for DivisionDividing by a number is the same as multiplying by its reciprocal, so division turns into multiplication on the spot, with the same same-positive different-negative sign rule.
Write "multiplying over and over" as a power, sort out the order of operations, use the laws for shortcuts, and slide naturally into roots and real numbers in the next stage.
- 5.6.1 Powers of Rational NumbersMultiply several copies of the same factor and we record it as a power, aⁿ, where n is just how many copies there are.
- 5.6.2 The Sign Pattern of PowersA power of a positive is always positive, while a negative gives positive for even powers and negative for odd ones — watch whether the exponent is even or odd.
- 5.6.3 The Order of Mixed OperationsPowers first, then multiply and divide, then add and subtract, and anything in brackets goes first — like everyone waiting their turn in line.
- 5.6.4 The Laws and Smart ShortcutsRound things off, pull out common factors, split terms apart, and let the laws find a shortcut through a messy expression.
- 5.6.5 A Hint Toward Real NumbersAsk what number raised to some power gives a value and you have to take a root, and some answers slip outside the rationals, pointing toward the real numbers.
6Powers, Roots & Real Numbers
Compress the repeated multiplication of rational numbers from the last stage into power notation, and build the language of base, exponent, and power along with the basic operations
- 6.1.1 Multiplying the Same Number Over and OverA cell doubles every hour, and writing out 2×2×2 gets tiresome, so we shorten it to 2³
- 6.1.2 Base, Exponent, and PowerThink of aⁿ as a multiplying itself n times: the a underneath is the base, the n on top is the exponent, and the result is the power
- 6.1.3 Powers of Negatives and FractionsMultiplying (-2) by itself acts like a flip-flopping sign switch, with the sign set by whether the exponent is odd or even, while a fraction simply raises its top and bottom separately
- 6.1.4 The Sign Pattern of PowersA negative base comes out positive for even exponents and negative for odd ones, like a rotating duty roster for (-a)ⁿ
- 6.1.5 Order of Operations with PowersHandle the powers upstairs first, then multiply and divide, then add and subtract, with parentheses always cutting the line, so every operation knows when to take the stage
Turn the question "given a side length, find the area" around into "given the area, find the side length," leading to square roots and the principal square root
- 6.2.1 Squaring and the Area of a SquareA square with side a has area a×a, which we write as a² and call the square of a
- 6.2.2 Taking a Square Root: From Area Back to SideGiven a square of area 9, asking for its side length is the reverse move: finding x in x²=9
- 6.2.3 Square Roots and the Principal Square RootA square of area 9 could have a side of either +3 or -3, so both are square roots, but the positive one alone, 3, is written √9
- 6.2.4 When √a Exists and Why It's Never NegativeAn area can't be negative, so the a inside √a must be ≥0, and the √a you get out is always ≥0 too
- 6.2.5 Perfect Squares and Estimating RootsNumbers like 1, 4, 9, 16 come out "perfectly square," so for anything caught between them you pin √a down with a lower and upper bound
Lift the square idea into three dimensions to build cubes and cube roots, and contrast the difference in how each handles signs
- 6.3.1 Cubing and the Volume of a CubeA cube with edge a has volume a×a×a, which we write as a³ and call the cube of a
- 6.3.2 Taking a Cube Root: From Volume Back to EdgeGiven a cube of volume 27, asking for its edge length means finding x in x³=27, written ∛27
- 6.3.3 How Cube Roots Handle SignsVolume can be positive or negative, so negative numbers have cube roots too, and ∛(-8)=-2 lets the sign pass right through the root
- 6.3.4 Square Roots versus Cube RootsOne fears negatives and one doesn't: √ demands a non-negative input, while ∛ takes either sign, and in writing one hides a quiet 2 while the other shows a 3
Use roots that never come out evenly to reveal infinite non-repeating decimals, building the ideas of irrational numbers and real numbers and their match with the number line
- 6.4.1 A Root That Won't Come Out: The Mystery of √2The diagonal has length √2, but measure it however you like and it never divides evenly, its decimals running on forever without repeating, too big for the rationals to hold
- 6.4.2 Irrational Numbers: Infinite, Non-Repeating DecimalsNumbers like √2 and π, whose decimals never end and never repeat, are called irrational and can't be written as a fraction p/q
- 6.4.3 What Real Numbers Are and How They SplitPut the rationals and irrationals together and you get the real numbers, sorted into two great families by whether they can be written as a fraction
- 6.4.4 Real Numbers Match the Number Line One-to-OneEvery point on the number line has a real number and every real number has a point, so even √2 can be nailed down exactly on the line
- 6.4.5 Opposites and Absolute Value of Real NumbersPoints mirrored across the origin are opposites and the distance to the origin is the absolute value, and this works for irrational numbers just the same
Treat √a as something you can actually compute with, mastering its properties, simplification, four operations, and rationalizing the denominator
- 6.5.1 What a Square-Root Expression Means and When It's ValidTreat √a as a quantity waiting to be evaluated, which only makes sense when the number under the root, a, is ≥0, since otherwise there's nothing under the root to take
- 6.5.2 Simplest Form and Simplifying RootsShow any perfect-square factor under the root to the door, so √12 becomes 2√3, until nothing inside can be reduced any further
- 6.5.3 Multiplying and Dividing RootsRoots can split apart or join together: √a·√b=√(ab) and √a÷√b=√(a/b), as long as everything stays non-negative
- 6.5.4 Adding and Subtracting Like RootsJust like combining like terms, you simplify first and then add √3 as if it were a unit, so 2√3+√3=3√3
- 6.5.5 Rationalizing the DenominatorA root sitting in the denominator feels awkward, so you multiply top and bottom to invite it up into the numerator, turning 1/√2 into √2/2
- 6.5.6 Mixed Operations with Square-Root ExpressionsTreat the roots as numbers you can compute with and work through a full problem in the usual order: powers, then multiply and divide, then add and subtract
Treat the real numbers as one unified number system for computing, comparing, and estimating, laying the groundwork for algebraic expressions as a universal kind of object
- 6.6.1 The Four Operations on Real NumbersThe rules for adding, subtracting, multiplying, and dividing rationals carry over to the reals unchanged, and irrational numbers follow the very same playbook
- 6.6.2 Comparing the Size of Real NumbersFarther right on the number line means larger, so to compare √5 and 2.3 you compare their squares or trap them in a narrowing range
- 6.6.3 Estimating Irrationals and Choosing PrecisionSqueeze with perfect squares to see that √7 sits between 2.6 and 2.7, and squeeze one layer tighter whenever you want more accuracy
- 6.6.4 Letting a Letter Stand for Any Real NumberSince the reals all obey the same operations, we let a letter like a or x stand for "any real number at all," clearing the way for algebraic expressions
Ⅱ · Expressions & Equations
7Algebraic Expressions & Polynomials
Take the concrete numbers you learned to compute last stage and level them up into letters that can stand for any number, learning to read and write algebraic expressions.
- 7.1.1 Why We Use Letters Instead of NumbersOne apple costs 5 dollars, so n apples cost 5n dollars; letters like a and x are empty blanks waiting to hold any number you like.
- 7.1.2 Writing Expressions: Translating Words into SymbolsTake phrases like '3 more than x' or 'half of x' and translate them, word by word, into x+3 and x/2.
- 7.1.3 How to Write Expressions ProperlyNumbers go before letters, the times sign disappears, and division is written as a fraction, so a·b becomes ab, a tidy convention everyone agrees on.
- 7.1.4 Evaluating an ExpressionDrop a specific number into the letter's blank and follow the order of operations, so 3x+1 becomes 7 when x=2.
- 7.1.5 Describing Patterns with ExpressionsWatch how many matchsticks the nth shape needs, find the expression that changes with n, and capture the whole pattern in a single line of symbols.
Sort and name algebraic expressions, get to know monomials, polynomials, and the broader family, and master the basic terms: coefficient, degree, and like terms.
- 7.2.1 Monomials: Held Together Only by MultiplicationSomething like 3x² or -2ab, just numbers and letters multiplied together, is one single building block called a monomial.
- 7.2.2 The Coefficient and Degree of a MonomialThe number out front is the coefficient (how big the block is), and the letters' exponents added up give the degree (how many layers the block has).
- 7.2.3 Polynomials: Several Blocks Joined TogetherLink a few monomials with plus and minus signs into a string like x²-2x+1, and each piece is called a term.
- 7.2.4 A Polynomial's Terms, Degree, and Constant TermFind the term with the highest power to set the polynomial's degree, and the plain number with no letters is the constant term.
- 7.2.5 The Polynomial Family and Like TermsMonomials and polynomials together make up this family; terms with the same letters and the same exponents are matching blocks, called like terms.
Learn to combine like terms and clear brackets by the rules, so you can add and subtract expressions and arrive at the simplest result.
- 7.3.1 Combining Like Terms3 apples plus 2 apples are still apples; add like terms by adding only the coefficients and leaving the letters untouched.
- 7.3.2 The Rule for Removing BracketsA minus sign before the brackets acts like a mirror, flipping the sign of every term inside, while a plus sign lets them all come out unchanged.
- 7.3.3 The Rule for Adding BracketsRemoving brackets in reverse: tuck several terms inside brackets, and if a minus sign sits in front, every term you tuck in must flip its sign.
- 7.3.4 Adding and Subtracting ExpressionsFirst clear the brackets, then combine like terms, tidying a long string into the simplest few terms like a+b.
- 7.3.5 Putting Addition and Subtraction to WorkWhen you find a perimeter or a price difference, add and subtract the parts' expressions and simplify them into one clean expression.
Extend last stage's exponents into the rules for working with powers, the tools you'll need for multiplying expressions, clearing this gateway first.
- 7.4.1 Multiplying Powers with the Same Basea²·a³ is just 2 copies of a followed by 3 more, so the base stays and the exponents add to give a⁵.
- 7.4.2 Raising a Power to a Power(a²)³ repeats the group 'two a's' three times, so the base stays and the exponents multiply to give a⁶.
- 7.4.3 Raising a Product to a Power(ab)³ means three a's and three b's counted separately, so a power of a product is each factor raised to that power.
- 7.4.4 Combining the Power RulesAll three rules often show up in one problem, so spot whether things are multiplied or stacked and choose rightly between adding and multiplying exponents.
Starting from the power rules, master multiplying monomials and then polynomials, using an area model to understand every step.
- 7.5.1 Monomial Times MonomialMultiply coefficient by coefficient and combine matching letters as same-base powers, like merging two blocks' sizes and layers separately.
- 7.5.2 Monomial Times PolynomialUse the distributive law to hand the outside block to every term inside the brackets, missing none.
- 7.5.3 Polynomial Times PolynomialLike slicing a big rectangle into small cells, multiply every term by every term and add them all up, as in (a+b)(c+d).
- 7.5.4 Understanding Multiplication with an Area ModelDraw the expression as rectangles fitted together, where each small area is one term, so the expansion is plain to see.
Distill the difference of squares and perfect square formulas from polynomial multiplication, use them fluently forwards, and ready your tools for the next stage's factoring.
- 7.6.1 The Difference of SquaresIn (a+b)(a-b) the middle terms cancel, one added and one subtracted, leaving the clean difference a²-b².
- 7.6.2 The Perfect Square FormulaExpanding (a+b)² gives a big square holding two small squares plus two strips, which is a²+2ab+b².
- 7.6.3 Seeing the Formulas GeometricallyFit squares and rectangles together and watch with your own eyes where the two strips that make up 2ab come from.
- 7.6.4 Using the Formulas FlexiblySpot which part is a and which is b, and even a mental sum like 102×98 cracks instantly with the difference of squares.
Run powers and multiplication in reverse to learn dividing powers with the same base and dividing monomials and polynomials, easing naturally toward factoring.
- 7.7.1 Dividing Powers with the Same Basea⁵÷a² cancels 2 of the 5 a's, so the base stays and the exponents subtract to leave a³.
- 7.7.2 Zero and Negative ExponentsSince a÷a=1, working backwards gives a⁰=1, and keeping on subtracting makes a⁻ⁿ mean 1/aⁿ.
- 7.7.3 Monomial Divided by MonomialDivide coefficient by coefficient and subtract same-base exponents, the reverse of multiplying monomials, taking the product back apart.
- 7.7.4 Polynomial Divided by MonomialDivide each term of the polynomial by that monomial separately, then add the quotients together.
- 7.7.5 Building a Bridge to FactoringMultiplication expands an expression, and going the other way rewrites a sum back as a product, which is exactly the next stop: factoring.
8Factoring
Building on the polynomial multiplication from the last stage, help learners see that factoring is multiplication run in reverse, and set up the core idea and what counts as a finished factorization.
- 8.1.1 Reading Multiplication BackwardsMultiplying assembles parts into a machine; factoring takes the machine back apart into parts, so a(b+c)=ab+ac is just read right to left.
- 8.1.2 What Factoring Actually IsIt's writing a polynomial as a few things multiplied together, just like writing the number 12 as 3x4, so a²+ab becomes a(a+b).
- 8.1.3 The Mirror Image of MultiplyingExpanding walks to the right and factoring walks to the left along the same line, and you can always expand again to check that your factoring is right.
- 8.1.4 Keep Going Until It Won't Break Down FurtherJust as you break a number down until only primes are left, you factor a polynomial until no common factor can be pulled out of any piece.
- 8.1.5 Spotting the Shadow of "Divides Evenly"If a is one factor of a product, the whole expression divides evenly by a, and that idea is the seed of the canceling you'll do later.
Master the most basic and first-to-try method: find the part every term shares and pull it outside the parentheses, covering coefficients, letters, and whole-expression common factors.
- 8.2.1 Find the Part Everyone CarriesEvery term is holding the same factor, so invite it outside the parentheses: ma+mb=m(a+b).
- 8.2.2 Deciding the Common FactorTake the greatest common divisor of the coefficients and the lowest power of each letter, like finding the one thing the whole class brought, so 6x²+9x gives 3x.
- 8.2.3 Don't Slip on the SignsWhen you pull out a minus sign, every term inside flips, like turning a pocket completely inside out: -a-b=-(a+b).
- 8.2.4 A Whole Group Can Be the Common FactorTreat (x+y) as one bundle and carry the whole thing out: a(x+y)+b(x+y)=(x+y)(a+b).
- 8.2.5 Multiply Back to Double-CheckUse the distributive law to multiply your answer out again and compare with the original, catching any dropped term or sign mistake.
Use the multiplication formulas memorized last stage in reverse, recognizing the difference-of-squares and perfect-square patterns so you can factor quickly.
- 8.3.1 Difference of Squares: One Square Minus AnotherWhenever you see "a square minus another square," split it into the sum times the difference: a²-b²=(a+b)(a-b).
- 8.3.2 Seeing Through a Disguised Difference of Squares4x²-9 is really (2x)²-3², so first rewrite each term as a square, then drop it into the formula.
- 8.3.3 Perfect Square: Squares at the Ends, Double in the Middlea²±2ab+b² is like a sandwich with squares on the outside and twice the product in the middle, folding up into (a±b)².
- 8.3.4 First Check That It's Truly a Perfect SquareMake sure the middle term equals twice the product of the two square roots; only when it matches can you use the formula.
- 8.3.5 Pull Out the Common Factor First, Then Apply the FormulaFor 2a²-2b², pull out the 2 first and then use difference of squares, building the habit of "factor out, then fit the pattern."
Give a systematic splitting technique for quadratic trinomials, moving factoring from "reciting formulas" to "finding the right pair of numbers."
- 8.4.1 What a Quadratic Trinomial Looks LikeIt has the shape x²+px+q, and the goal is to split it into two linear factors multiplied together.
- 8.4.2 Find Two Numbers: Sum Is the Middle, Product Is the LastLook for two numbers that add to p and multiply to q, like guessing a pair from their sum and product: x²+5x+6 becomes (x+2)(x+3).
- 8.4.3 Draw the Cross to CheckWrite the factors on two rows, multiply across the diagonals and add to rebuild the middle term, so the cross diagram is your checking tool.
- 8.4.4 When the Leading Coefficient Isn't 1For ax²+bx+c you also split the leading term, with each side of the cross handling one diagonal product, so the matching takes a bit more care.
- 8.4.5 Signs Decide the Plus-Minus PairingA positive last term means the two numbers share a sign and a negative one means opposite signs, so settling the sign first saves you half the work.
Handle polynomials of four or more terms with no shared common factor by grouping first, factoring each group, and finally pulling out the common bracket.
- 8.5.1 With Four Terms, Group Them FirstWhen there are many terms and no common factor to pull out, pair them up into groups, like sorting people into small squads by similarity.
- 8.5.2 Factor Each Group on Its OwnPull the common factor out of each group so a matching bracket appears: ax+ay+bx+by splits into two groups.
- 8.5.3 Pull Out the Shared BracketBoth groups are left with the same (x+y), so treat it as one piece and pull it out to get (x+y)(a+b).
- 8.5.4 Be Ready to RegroupIf your first grouping doesn't produce a shared bracket, swap the pairings and try again, like changing formation.
- 8.5.5 Grouping Into a Difference of SquaresSometimes after grouping one part becomes a perfect square that then pairs with another term as a difference of squares, blending several methods at once.
Combine all the methods into a standard factoring routine, and use factoring to serve evaluation and canceling, paving the way straight to the rational-expression work in the next stage.
- 8.6.1 The Standard Order for FactoringFirst pull out the common factor, then fit a formula, then try grouping, and finally check it's fully done, like running a repair through a fixed routine.
- 8.6.2 Factoring First Makes Evaluating EasierFactor before plugging in numbers, since turning things into a product often simplifies the work, trading awkward adding and subtracting for easy multiplying.
- 8.6.3 Using Factoring to Prove Divisibility and SimplifyBreak a number expression into a product with a common factor to show it divides evenly: 99²-1=(99+1)(99-1).
- 8.6.4 Laying the Groundwork for CancelingOnce both the top and bottom are factored, matching factors can cancel, and that's the very first step of simplifying a rational expression.
- 8.6.5 Wrap-Up and Self-Check on FactoringReview the trigger signal for each method until "one look at the structure tells you which move to use," and step toward rational expressions.
9Rational Expressions & Equations
Extend last stage's factoring to expressions with letters in the denominator, building the idea of a rational expression and the condition for it to make sense.
- 9.1.1 What Is a Rational ExpressionShare a pizza evenly, but the number of slices is itself a changing letter — an expression with a letter in the denominator is a rational expression A/B.
- 9.1.2 Rational Expressions vs. PolynomialsIf the denominator is just a number like x/3 it's still a polynomial; only when a letter hides in the denominator B does it count as rational.
- 9.1.3 When a Rational Expression Makes SenseYou can't split something among zero people, so the expression only makes sense when the denominator B≠0.
- 9.1.4 When a Rational Expression Equals ZeroNo cake left means zero: the whole expression equals 0 only when the top A=0 and the bottom B≠0.
- 9.1.5 Describing Quantities with Rational ExpressionsDistance divided by speed gives time — turn a real situation into an expression with a letter in the denominator and write the rational expression.
Master the tools for rewriting rational expressions without changing their value — the fundamental property, reducing, and common denominators — to set up the four operations.
- 9.2.1 The Fundamental PropertyScale the top and bottom up or down by the same amount and the share stays the same: A/B=(A·M)/(B·M), M≠0.
- 9.2.2 Reducing to Lower TermsCross out the parts the top and bottom share to make the expression as simple as possible — factor first to spot the common factor, then cancel it.
- 9.2.3 Lowest TermsWhen the top and bottom share no common factor left to cancel, you've reduced all the way to lowest terms.
- 9.2.4 The Least Common DenominatorTo give several expressions the same denominator, find the smallest shared 'container' that holds each denominator's factors — that's the least common denominator.
- 9.2.5 Building to a Common DenominatorTo combine pieces split into different shares, use the fundamental property to multiply each one up to the common denominator.
Carry the rules for multiplying and dividing fractions over to rational expressions, including powers, always reducing before computing.
- 9.3.1 Multiplying Rational ExpressionsMultiply straight across like fractions: top times top, bottom times bottom, (A/B)·(C/D)=AC/BD.
- 9.3.2 Dividing Rational ExpressionsDividing by an expression is the same as multiplying by its reciprocal: flip the second one over, then multiply.
- 9.3.3 Reducing While Multiplying and DividingDon't rush to multiply out — factor the tops and bottoms first and cancel common factors across them to save effort.
- 9.3.4 Raising a Rational Expression to a PowerMultiply the whole expression by itself a few times and the exponent lands on both top and bottom: (A/B)ⁿ=Aⁿ/Bⁿ.
- 9.3.5 Mixing Powers, Multiplication, and DivisionDo the powers first, then the multiplying and dividing, stringing together flipping, reducing, and adding exponents in order.
Finish the last of the four operations — addition and subtraction — and pull everything together into the skill of full simplification.
- 9.4.1 Adding and Subtracting with the Same DenominatorWhen the shares match, just combine the tops and keep the denominator: (A±B)/C.
- 9.4.2 Adding and Subtracting with Different DenominatorsWhen the shares differ, first build to a common denominator, then combine the tops by adding or subtracting.
- 9.4.3 Adding and Subtracting Rational Expressions with PolynomialsTreat a polynomial as a rational expression over 1, then bring everything to a common denominator and compute together.
- 9.4.4 Mixed OperationsLine up parentheses, powers, multiplication and division, then addition and subtraction by rank, and reduce the expression step by step.
- 9.4.5 Simplify, Then EvaluateSimplify a messy expression all the way down before substituting numbers, so you don't plug in early and grind yourself into the ground.
Treat an equation containing rational expressions as something to solve, mastering the idea of clearing denominators and checking for extraneous roots.
- 9.5.1 Meeting Rational EquationsAn equation with the unknown hiding in a denominator is a rational equation, and you'll need a way to coax it out of the denominator.
- 9.5.2 Solving by Clearing the DenominatorMultiply both sides by the least common denominator to turn the rational equation into a familiar polynomial equation you can solve.
- 9.5.3 Where Extraneous Roots Come From and How to CheckIf the thing you multiplied by equals zero, a fake root sneaks in, so once you've solved you must plug back and check no denominator becomes zero.
- 9.5.4 The Full Solving ProcedureClear denominators, solve the polynomial equation, then check the roots — three steps, and the checking is the final gate you can't skip.
- 9.5.5 Applying Rational EquationsIn travel, work, and concentration problems the quantity sits in the denominator, so write a rational equation, clear it, and solve.
10Linear Equations & Systems
Build the idea of an equation and an unknown, master the two basic properties of equality, and get ready to solve any equation
- 10.1.1 Equations and UnknownsA sentence hides a number we don't yet know, so we let a letter stand in for it, and once there's an equals sign we call it an equation
- 10.1.2 The Solution of an EquationDrop your guessed number back in, and if both sides weigh exactly the same, that number is the solution
- 10.1.3 The First Basic Property of EqualityAdd or take away the same thing from both pans and the scale stays level, which is just adding or subtracting the same amount on both sides
- 10.1.4 The Second Basic Property of EqualityScale both pans up or down by the same factor and they still balance, which is multiplying or dividing both sides by the same number (just not zero)
- 10.1.5 Reshaping an Equation Using the PropertiesLike shifting boxes without tipping the scale, these two properties let you step by step leave the unknown alone on one side
Recognize the standard look of a linear equation in one unknown and master the full routine: clear fractions, clear parentheses, move terms, combine, and make the coefficient one
- 10.2.1 What a Linear Equation in One Unknown IsThere's just one unknown raised only to the first power, and once tidied up it takes the shape ax+b=0
- 10.2.2 Moving Terms AcrossCarry a term from one side of the equals sign to the other and it flips its sign as it crosses, which is really just adding or subtracting the same thing on both sides
- 10.2.3 Combining Like Terms and Making the Coefficient OneFirst gather like terms into one pile, then divide both sides by the number in front of the unknown, and x finally stands alone
- 10.2.4 Solving Equations with ParenthesesParentheses are like a packed-up bag, so open them with the distributive law and then move and combine terms
- 10.2.5 Solving Equations with FractionsBuilding on what you learned with fractional equations, multiply both sides by the least common multiple of the denominators and the fractions vanish at once
- 10.2.6 Checking Solutions and Special CasesSubstitute back into the original to see if both sides match; and if x cancels out entirely, the equation may have no solution or hold for every value
Learn to translate a word situation into an equation and use one consistent set-up-and-solve process for all kinds of real problems
- 10.3.1 Steps for Solving Word Problems with EquationsFind the 'equal-quantity relationship' buried in the problem to use as a skeleton, then walk through five steps: name the unknown, write the equation, solve, and answer
- 10.3.2 Sum, Difference, Multiple, and Age ProblemsSo many more, a few years younger, these are all relationships between quantities, translated into two sides of an equals sign
- 10.3.3 Distance, Rate, and Time ProblemsThe yardstick distance = speed × time handles meeting and catching-up problems by giving you the equal-quantity relationship
- 10.3.4 Work and Efficiency ProblemsTreat the whole job as one unit, and the shares each person finishes per day add up to one complete job
- 10.3.5 Profit, Interest, and Concentration ProblemsCost plus profit equals selling price, principal times rate gives interest, and solute over total amount is the concentration
Move from one unknown to two, and understand that a linear equation in two unknowns has infinitely many solutions while a system has a shared solution
- 10.4.1 Linear Equations in Two UnknownsTwo unknowns each to the first power, like x+y=10, where fixing one value pins down the other
- 10.4.2 Solutions of a Linear Equation in Two UnknownsA pair of numbers counts as one solution only if it makes the equation true together, and there are infinitely many such pairs
- 10.4.3 Systems of Linear Equations in Two UnknownsTwo conditions hold at once, and tying the two equations together with a brace gives you a system
- 10.4.4 The Solution of a SystemThe one pair of numbers that satisfies both equations at the same time is the pair the two solution sets share
Master the two core methods, substitution and addition-subtraction, and see that elimination is the single idea of turning two unknowns into one
- 10.5.1 The Idea of EliminationTwo unknowns are too crowded, so find a way to shed one and the problem drops back to the familiar single-unknown equation
- 10.5.2 Elimination by SubstitutionFrom one equation, write one letter in terms of the other, then slot that whole expression into the other equation
- 10.5.3 Elimination by Adding or SubtractingMake one unknown's coefficients equal or opposite, then subtract or add the equations and it cancels right out
- 10.5.4 Adjusting Coefficients Before EliminatingWhen the coefficients don't line up, first multiply a whole equation by some factor to match them, then add or subtract to eliminate
- 10.5.5 Choosing a Method and Checking the SolutionPick whichever method makes a letter easy to express or easy to line up, and once you have a pair of numbers, substitute back into both equations to verify
Use systems to solve real problems with two or three unknown quantities, and carry the elimination idea over to three-unknown systems
- 10.6.1 Solving Word Problems with SystemsWhen a problem has two unknown quantities and two equal-quantity relationships, naming two letters and writing two equations is the smoother path
- 10.6.2 Sum, Difference, Multiple, and Number ProblemsThe sum of two numbers, or how a two-digit number's digits relate, can each be written as two equations solved together
- 10.6.3 Travel, Matching, and Distribution ProblemsMeeting head-on, matching parts in sets, splitting by ratio, when the unknowns come in pairs you write a system
- 10.6.4 An Introduction to Linear Systems in Three UnknownsThree unknowns and three conditions, with three equations bundled inside one brace, make a three-unknown system
- 10.6.5 Solving Three-Unknown Systems by EliminationPair the equations to knock out one letter first, reducing to a two-unknown system, then back-substitute layer by layer to find all three numbers
Tie together the family of linear equations, highlight that they stay 'first power, linear process,' and plant the seed for quadratic equations ahead
- 10.7.1 A Recap of the Linear-Equation FamilyOne, two, or three unknowns all rely on the properties of equality and on elimination, and at heart every unknown reaches only the first power
- 10.7.2 Why 'First Power' Behaves So NicelyA linear equation has at most one solution, clean and tidy; but the moment an x² shows up, the story changes
- 10.7.3 A Preview: When the Unknown Picks Up a SquareWrite x·x as x², and suddenly an equation can have two solutions, which is exactly the doorway to the quadratic equation
11Quadratic Equations
Picking up from the last stage's linear equations, learn to spot equations with a squared term and understand both their standard look and what a "solution" means.
- 11.1.1 Why a Squared Term Shows UpA linear equation tracks "how far you walk," but the area of a square plot changes with its side length — double the side and the area quadruples, and that's where x² is born.
- 11.1.2 What a Quadratic Equation IsIt's a polynomial equation with one unknown whose highest power is 2, and once tidied up it looks like ax²+bx+c=0 (with a≠0).
- 11.1.3 Spotting the Three Coefficients a, b, cLine the equation up into standard form, and the number in front of the squared term is a, the one in front of the single x is b, and the lonely constant left over is c.
- 11.1.4 What "a Root" MeansThe root is the x value that makes both sides of the equation balance exactly — plug it back in and you can check on the spot.
- 11.1.5 Solving by Taking Square RootsWhen it looks like x²=k or (x+m)²=k, it's like knowing the area and working back to the side length — take the square root of both sides to get x=±√k.
Learn to rewrite any quadratic equation as a perfect square so you can take roots, laying the foundation for the quadratic formula.
- 11.2.1 Revisiting the Perfect Square Formula(x+m)²=x²+2mx+m² — read it backwards, and that missing m² is exactly the corner piece of the puzzle you need to fill in.
- 11.2.2 Completing the Square When the Leading Coefficient Is 1Take half of the coefficient of x, square it, and add it in, and x²+bx snaps into the perfect square (x+b/2)².
- 11.2.3 Completing the Square When the Leading Coefficient Isn't 1First factor out a (divide everything by a) to leave the squared term bare, then fill in the corner and complete the square as usual.
- 11.2.4 Solving an Equation Fully by Completing the SquareOnce you've shaped it into (x+m)²=k, take the square root directly, and the plus and minus branches give your two roots, step by step.
- 11.2.5 The Geometric Picture of Completing the SquarePicture x²+bx as a big square missing a small corner — add (b/2)² to make it whole, and that's literally where "completing the square" gets its name.
Push the completing-the-square process through once and for all into a universal formula, and use the expression under the root to judge the roots in advance.
- 11.3.1 Where the Quadratic Formula Comes FromComplete the square on ax²+bx+c=0 in general, and out drops a master key: x=(−b±√(b²−4ac))/(2a).
- 11.3.2 Solving by Applying the FormulaMatch a, b, c into their slots in the formula and crank out both roots mechanically, no need to complete the square from scratch each time.
- 11.3.3 The Discriminant Δ=b²−4acThat piece under the root, b²−4ac, is called the discriminant Δ, and it's a weather forecast that tells you ahead of time what the roots will look like.
- 11.3.4 Using Δ to Judge the RootsΔ>0 gives two different roots, Δ=0 means the two roots merge into one, and Δ<0 means there's no root among the real numbers.
- 11.3.5 Working Backwards from the DiscriminantGiven that there are "two equal roots" or "no real roots," make Δ satisfy the matching condition and solve back for the unknown parameter.
Master the quickest method — factoring — and learn to pick the least-effort approach among the three.
- 11.4.1 The Core Idea: "a Product That Equals Zero"If two numbers multiply to 0, one of them must be 0 — so A·B=0 splits into A=0 or B=0.
- 11.4.2 Solving by Factoring Out and the Difference of SquaresFirst move everything to one side to make it 0, then factor out a common factor or use the difference of squares to split the other side into two brackets multiplied together.
- 11.4.3 Factoring by the Cross MethodSplit x²+bx+c into (x+p)(x+q) by finding two numbers that multiply to c and add to b, pairing them up with a little cross diagram.
- 11.4.4 Comparing and Choosing Among the Three MethodsIf it factors, factoring is fastest; if it won't budge, reach for the formula; and completing the square is the slow, patient way to really see the structure.
Read off the sum and product of the two roots without solving, then work backwards to build an equation from its roots and evaluate symmetric expressions.
- 11.5.1 The Pattern in the Sum and Product of the RootsAdd the two branches of the quadratic formula, then multiply them, and you'll be delighted to find the sum of the roots is −b/a and their product is c/a.
- 11.5.2 Applying Vieta's Formulas DirectlyGiven one root, find the other — or use x₁+x₂ and x₁·x₂ as secret codes to answer without ever solving the equation.
- 11.5.3 Evaluating Symmetric Expressions in the RootsExpressions like x₁²+x₂² or 1/x₁+1/x₂ can all be built from the two building blocks of sum and product.
- 11.5.4 Building an Equation from Its RootsGiven two roots, use Vieta's in reverse: take the sum and product as coefficients to write x²−(sum)x+(product)=0.
Use quadratic equations to solve real problems about area, growth rates, and more, then carry the "equals" mindset toward the next stage's "not equals."
- 11.6.1 The Steps for Modeling a Word ProblemSet the unknown quantity in the situation as x, write an equation from the relationship that must hold, and once you've solved it, don't forget to check whether the answer makes sense.
- 11.6.2 Area and Geometry ProblemsFencing a yard, paving a path, folding a box — the area relationship lines up into a quadratic equation, and a negative root usually has to be thrown out.
- 11.6.3 Average Growth Rate ProblemsWhen a quantity rises or falls by the same rate twice in a row, the total carries a (1+x)² — the "interest on interest" picture naturally brings out a square.
- 11.6.4 Number, Travel, and Mixed ProblemsOnce a product shows up in two-digit-number or meeting-and-catching-up problems, the equation climbs to second degree — set the unknown and solve just the same.
- 11.6.5 From "Equal" Toward "Not Equal"An equation asks "what does it exactly equal," but life often asks "at least" or "no more than" — swap the equals sign for an inequality sign and you step into the next stage.
12Inequalities
Carry over your comfort with the equals sign from the last stage into comparing sizes, and build up the language and symbols of inequalities
- 12.1.1 Unequal Relationships in Everyday LifeBeing taller than your deskmate, a weight limit of no more than five tons - these 'bigger/smaller than' and 'no more than' ideas get written down with > < ≥ ≤
- 12.1.2 Writing Conditions as InequalitiesTurn 'at least 18 years old' and 'speed must not exceed 120' into x≥18 and v≤120, so words become sentences made of symbols
- 12.1.3 Seeing Size on the Number LineInvite the numbers to line up along a horizontal line, bigger ones to the right, so a<b just means a stands to the left of b
- 12.1.4 A Way to Compare Two NumbersTo find out which is bigger, subtract one from the other and check the sign: if a−b>0 then a is bigger than b, and that's the difference method
Get clear on how the size relationship changes when you add, subtract, multiply, or divide on both sides, laying the rules you'll need to solve inequalities
- 12.2.1 Adding or Subtracting the Same on Both SidesPut equal weights on both pans of a balance and the heavier side stays heavier: if a>b then a±c>b±c, and the direction doesn't change
- 12.2.2 Multiplying or Dividing Both Sides by a PositiveDouble everyone's score and the ranking holds: multiply a>b by a positive c and you still get ac>bc, sign unchanged
- 12.2.3 Flip the Sign When You Multiply or Divide by a NegativeLike a mirror swapping left and right: multiply a>b by a negative and the order flips to ac<bc - this is the easiest step to slip on
- 12.2.4 Chaining and Adding InequalitiesIf you're taller than him and he's taller than her, then you're taller than her too: a>b and b>c gives a>c, and same-direction ones can be added
Carry over the steps for solving a linear equation, and learn to solve a one-variable linear inequality and show it on the number line
- 12.3.1 Meeting the Linear InequalitySwap the '=' in an equation for a '<' or '>', like 2x+1>5, where the unknown is to the first power at most - that's one of these
- 12.3.2 Steps for Solving a Linear InequalityJust like solving an equation you clear fractions, move terms, and combine, only flipping the sign at the one step where you divide both sides by a negative
- 12.3.3 The Solution Set and the Number LineThe answer isn't a single number but a stretch: x>2 is the whole length to the right of 2, with an open dot showing 2 itself isn't included
- 12.3.4 Inequalities with Letter CoefficientsFor a coefficient that's a letter, like ax>1, first ask whether a is positive, negative, or zero, then handle each case before drawing a conclusion
Handle several inequality conditions that must all hold at once, learn to find the shared solution set, and tackle simple real-world constraints
- 12.4.1 Several Conditions Holding at OnceYou must both know how to swim and be 12 to enter the deep end - several inequalities tied together with a big brace form a system
- 12.4.2 Finding the Shared Part on the Number LineDraw a stretch for each inequality, and only the overlap that every one of them covers is the solution set of the system
- 12.4.3 Four Quick Rules for the Solution Set'Both bigger take the bigger, both smaller take the smaller, big-and-small take the middle, small-and-big no solution' - check it against the number line and it clicks
- 12.4.4 Pinning Down a Parameter's RangeKnowing an equation's solution has to meet certain conditions, work backwards by writing a system of inequalities to find the range the parameter can take
Lean on the quadratic equations and parabolas from the last stage to read off the solution of a quadratic inequality from a graph
- 12.5.1 Reading Sign from the ParabolaSketch the upward-opening curve y=x²−x−6, and whether it sits above or below the x-axis is exactly whether it's >0 or <0
- 12.5.2 Find the Roots First, Then the IntervalsUse the root-finding skill from last stage to locate the two crossing points, which slice the number line into pieces, and judge the sign piece by piece
- 12.5.3 The Three Discriminant CasesΔ>0 gives two roots splitting inside from outside, Δ=0 just grazes one point, Δ<0 keeps the whole curve on one side, and the solution set shifts to match
- 12.5.4 Fractional and Factorable InequalitiesTreat a fractional or product inequality like (x−1)(x+2)>0 as a few signs multiplied together, and walk it through with the sign-tracking method
Meet the AM-GM inequality as a sharp tool for finding extreme values, understand its geometric picture, and master the condition for equality
- 12.6.1 The Tug-of-War Between Two AveragesThe arithmetic mean (a+b)/2 is never smaller than the geometric mean √(ab), which is why a square fences off the most area with the least material
- 12.6.2 The Geometric Meaning of the Basic InequalityIn a semicircle the radius is no shorter than half a chord, and drawing the height onto the hypotenuse of a right triangle makes √(ab)≤(a+b)/2 plain to see
- 12.6.3 The Condition for EqualityThe two averages are equal only when a=b, and this 'positive, fixed, then equal' is the secret to using it to find extreme values
- 12.6.4 Finding Extremes with the Basic InequalityFixed sum gives the biggest product, fixed product gives the smallest sum: the minimum of x+1/x falls out in one move
Put every tool from before to work on real problems, wrapping up this stage and setting the ground for geometric measurement next
- 12.7.1 Solving Word Problems by Writing InequalitiesSet up 'profit no lower than' or 'enough material' with an unknown and write an inequality, turning words into expressions just as you would for an equation
- 12.7.2 Choosing a Plan and Its Tipping PointTo see which of two packages is the better deal, find the breakeven point where they're equal, then check which one wins on each side
- 12.7.3 Optimization: Spend Less, Produce MoreUse the basic or quadratic inequality to find 'lowest cost' or 'largest area', quietly laying the groundwork for studying measurement of shapes ahead
- 12.7.4 Estimating Error and RangeMeasuring a length allows give or take, so |x−a|<d circles off the reasonable range, pointing toward the geometric quantities you're about to learn
Ⅲ · Plane Geometry
13First Steps in Geometry
Right after finishing algebra and inequalities, we shift our gaze from the number line and symbols to the shapes all around us, and meet what geometry studies: solid figures and flat figures.
- 13.1.1 What Geometry Is AboutAlgebra measures how much, geometry looks at the shape; a Rubik's cube, a soccer ball, a tabletop are all things we study, and we call them geometric figures.
- 13.1.2 Solid FiguresAnything that takes up a chunk of space, like a box, a can, or an ice cream cone, becomes a cuboid, a cylinder, or a cone, the solid figures.
- 13.1.3 Sorting Common SolidsSort the blocks into piles: those built from flat faces are polyhedra (prisms and pyramids), and those with a curved face are solids of revolution (cylinders, cones, spheres).
- 13.1.4 Flat FiguresA shadow or a paper cutout, lying flat on the table with no thickness, like a triangle, a circle, or a rectangle, is a flat figure.
- 13.1.5 How Solids and Flat Figures ConnectUnfold a paper box to get its net, or look at a solid from one direction to get a view, and that is how solids and flat figures turn into each other.
Take any figure apart to see how it is built up layer by layer from points, lines, surfaces, and solids, and set up the most basic language of geometry.
- 13.2.1 A Point: The Smallest PositionA single star in the night sky, a pin on a map, marking only a spot with no size at all, is a point.
- 13.2.2 A Line: A Moving Point Traces a LineThe mark a pen tip leaves as it slides, the trail of a shooting star, a point set in motion drags out a line.
- 13.2.3 A Surface: A Moving Line Sweeps a SurfaceA wiper sweeping across a windshield, a line sliding sideways, a line in motion spreads out a surface.
- 13.2.4 A Solid: A Moving Surface Builds a SolidA stack of paper growing thick, a rectangle spun full circle, a surface in motion piles up into a solid.
- 13.2.5 How They RelateA solid is bounded by surfaces, a surface edged by lines, a line joined from points; and the other way round, surfaces sweep solids, lines sweep surfaces, points trace lines, all linked together.
Zoom in on the most basic straight figures, tell apart the three siblings (line, ray, segment), and grasp their basic properties as the foundation for every geometric construction ahead.
- 13.3.1 The LineA railway track that runs dead straight to infinity with no end in either direction is a line, written line AB or line l.
- 13.3.2 The RayThe beam from a flashlight, with a starting point and stretching endlessly one way, is a ray, written ray OA starting from endpoint O.
- 13.3.3 The SegmentA piece of rope pulled taut, with a head at each end, is a segment, written segment AB, and the two ends are called endpoints.
- 13.3.4 The Basic Property of a LineHammer two nails into the wall and stretch a string between them, and there is only one: two points determine exactly one line.
- 13.3.5 The Basic Property of a SegmentFrom home to school, walking straight is shortest: between two points the segment is the shortest path, and its length is the distance between them.
- 13.3.6 Positions of Point-and-Line and Line-and-LineA point either sits on a line or lies off it; two lines either cross at a single point or run parallel and never meet.
Learn to compare segment lengths, add and subtract segments, and find midpoints, carrying the thinking from inequalities and algebra over to geometric length.
- 13.4.1 Comparing Segment LengthsWhich chopstick is longer? Line up one end and look at the other, or measure them, to get AB>CD or AB=CD.
- 13.4.2 Copying a Segment with Compass and StraightedgeNo markings needed: open the compass to span a segment, then draw it at a new spot, and the construction reproduces an equal segment.
- 13.4.3 Adding and Subtracting SegmentsJoin two ropes and the lengths add, snip a piece off and the lengths subtract; segments can be added and subtracted too, as in AC=AB+BC.
- 13.4.4 The Midpoint of a SegmentBalance a stick and fold it right at the center so both sides match: that fold is the midpoint, with AM=MB=½AB.
- 13.4.5 Points That Divide a Segment EquallyTie evenly spaced knots to split a rope into three or four parts, and you get the trisection or quadrisection points, each piece equal.
Recognize an angle as the opening between two rays, learn to describe its size with measurement and units, and broaden the view from length to angle.
- 13.5.1 What an Angle IsThe opening of a pair of scissors, the gap between two clock hands: two rays from the same endpoint form an angle, with vertex O and sides OA and OB.
- 13.5.2 Naming an AngleGive that opening a name: ∠AOB, ∠O, or ∠1, with the vertex letter always written in the middle.
- 13.5.3 Measuring Angles and Their UnitsSlice a full turn into 360 equal parts and each part is one degree (1°); finer still are minutes (′) and seconds (″), read off with a protractor.
- 13.5.4 Special AnglesA door cracked open is an acute angle, a square corner is a right angle (90°), a straightened line is a straight angle (180°), and a full turn is a full angle.
- 13.5.5 Converting Degrees, Minutes, and SecondsLike telling time, every 60 carries over to 1: 60″=1′ and 60′=1°, and that rate is all you need to convert between big and small angles.
Carry the whole comparison and add-subtract approach from segments over to angles, introduce the angle bisector, and set angle operations side by side with segment operations.
- 13.6.1 Comparing the Size of AnglesLay the two vertices and one side on top of each other and look at the other side: the wider the opening, the bigger the angle, giving ∠1>∠2 or ∠1=∠2.
- 13.6.2 Adding and Subtracting AnglesPlace two angles side by side to build a bigger one, or carve one out of a big angle to shrink it; angles add and subtract too, as in ∠AOC=∠AOB+∠BOC.
- 13.6.3 The Angle BisectorDraw a line from the vertex that splits the opening right down the middle into two halves, and that line is the angle bisector, with ∠AOC=∠COB=½∠AOB.
- 13.6.4 Constructing and Bisecting Angles with Compass and StraightedgeWith just a compass and straightedge you can copy an angle and even bisect any angle into two equal halves.
Recognize the special numerical relationships between two angles (complementary and supplementary) and their consequences, and use angles to describe direction, paving the way for the paired angles studied next in intersecting and parallel lines.
- 13.7.1 Complementary AnglesWhen two angles fit together into exactly a right angle, they are complementary, summing to 90°, as in ∠1+∠2=90°.
- 13.7.2 Supplementary AnglesWhen two angles fit together into exactly a straight line (a straight angle), they are supplementary, summing to 180°, as in ∠1+∠2=180°.
- 13.7.3 Properties of Complementary and Supplementary AnglesTwo complements (or two supplements) of the same angle must be equal, a property you will lean on again and again when proving lines parallel.
- 13.7.4 Bearings'N30°E' on a compass, measured as an offset from the north-south line, is a bearing that describes direction.
- 13.7.5 Wrapping Up and Looking AheadFrom points, lines, and surfaces to angles and their relationships, the language of geometry is now in place; next, when two lines cross, the paired angles take the stage.
14Intersecting Lines, Parallel Lines & Translation
Building on the lines and angles you met last stage, we explore the pairs of angles that appear when two lines cross and how their sizes relate, laying the language foundation for the reasoning ahead.
- 14.1.1 Intersecting Lines and the Crossing PointWhen two straight roads meet at a junction they form a cross, and that meeting spot is the point where the two lines intersect.
- 14.1.2 Adjacent Supplementary AnglesLay a ruler flat and stick a slanted line into it: on one side the two angles fit together along a straight edge and add up to exactly 180 degrees, and we call them adjacent supplementary angles.
- 14.1.3 Vertical AnglesWhen scissors open up, the back-to-back pair of angles at the blade tips and at the handle tips are equal in size, and that is what we mean by vertical angles being equal.
- 14.1.4 Simple Reasoning with Angle RelationshipsOnce you know one angle, the rules 'adjacent angles sum to 180 degrees' and 'vertical angles are equal' let you find every angle around the point, written as equations like angle 1 + angle 2 = 180 degrees.
We look at the most special kind of crossing, where two lines meet at a right angle, and use it to define the distance from a point to a line, setting up the height of a triangle later on.
- 14.2.1 Perpendicular Lines and the PerpendicularWhere a wall's upright line meets the floor's level line they fit together nice and square at 90 degrees, and that kind of crossing is called perpendicular, written AB perpendicular to CD.
- 14.2.2 Dropping a Perpendicular Through a PointHold one right-angle edge of a set square against the line, draw along the other edge, and you have made the one and only perpendicular through that point.
- 14.2.3 The Perpendicular Segment Is ShortestTo get from a point on the bank to a straight river, going straight across is the shortest walk, and that perpendicular segment is the shortest path.
- 14.2.4 Distance from a Point to a LineTo measure how far a point sits from a line, you measure the length of that perpendicular segment, and that is what we call the distance from a point to a line.
When one line crosses two others, we get to know three kinds of angle pairs (corresponding, alternate interior, and co-interior angles), preparing the language for testing whether lines are parallel.
- 14.3.1 The Three-Line, Eight-Angle SceneA crossbar laid slanting across two rails sprouts four angles at each crossing, eight angles in all, and we need to sort them out and give them names.
- 14.3.2 Corresponding AnglesTwo angles sitting in the same spot, say the upper-right corner, at both junctions are in matching positions and are called corresponding angles, shaped like the letter F.
- 14.3.3 Alternate Interior AnglesA pair of angles tucked between the two lines but on opposite sides of the crossing line face each other across the gap, shaped like the letter Z, and are called alternate interior angles.
- 14.3.4 Co-Interior AnglesA pair of angles tucked between the two lines and crowded on the same side of the crossing line are shaped like the letter U and are called co-interior angles.
- 14.3.5 Telling the Three Kinds ApartFor any pair of angles, first find which crossing each belongs to, then check whether they're on the same or opposite side of the crossing line and inside or outside the two lines, and you can name them.
Starting from the picture of lines that 'never meet,' we introduce parallels, then use the angle relationships from the last module to give workable ways of testing whether lines are parallel.
- 14.4.1 Getting to Know Parallel LinesTwo rails stretch on forever yet never touch, and lines in the same plane that never meet like this are parallel, written a parallel to b.
- 14.4.2 Parallels Exist and Are UniqueDrop a point anywhere off a line, and through it you can draw exactly one parallel that will never collide with the original, not a single one more.
- 14.4.3 Equal Corresponding Angles Mean ParallelWhen a crossbar cuts two rails, if the two matching-position angles are equal the rails must be parallel: equal corresponding angles mean the lines are parallel.
- 14.4.4 Equal Alternate Interior Angles Mean ParallelIf that Z-shaped pair of angles is equal, the lines are parallel too: equal alternate interior angles mean the lines are parallel.
- 14.4.5 Supplementary Co-Interior Angles Mean ParallelIf that U-shaped pair of angles adds up to exactly 180 degrees, the lines are parallel: supplementary co-interior angles mean the lines are parallel.
Now we run it the other way: starting from 'these lines are parallel' we derive the three angle relationships, and learn to combine the tests and the properties for multi-step reasoning.
- 14.5.1 Parallel Gives Equal Corresponding AnglesOnce the two rails are known to be parallel, the matching-position angles cut by the crossbar must be equal: parallel lines give equal corresponding angles.
- 14.5.2 Parallel Gives Equal Alternate Interior AnglesUnder parallel lines, that Z-shaped pair of angles must be equal as well: parallel lines give equal alternate interior angles.
- 14.5.3 Parallel Gives Supplementary Co-Interior AnglesUnder parallel lines, that U-shaped pair of angles must add up to 180 degrees: parallel lines give supplementary co-interior angles.
- 14.5.4 Going Back and Forth Between Tests and Properties'Equal angles means parallel lines' works like a two-way door: missing a line, reason to a line; missing an angle, reason to an angle; just be clear which side is the given.
- 14.5.5 Parallelism Passes AlongAmong three lines, if a is parallel to b and b is parallel to c, then a is parallel to c too, and the parallel relationship carries on down the chain.
- 14.5.6 A Little Combined-Reasoning PracticeString together perpendicularity, vertical angles, and the parallel properties to deduce an unknown angle one reasoned step at a time, a warm-up for proving things about triangles.
On a grid we describe a whole-figure slide using direction and a count of squares, feel how translation leaves shape and size unchanged, and plant a seed for congruence and transformations later on.
- 14.6.1 The Phenomenon of TranslationA sliding door you push, a box gliding past on a conveyor belt: the whole thing slides one way without turning or flipping, and that motion is a translation.
- 14.6.2 Describing a Translation on a GridOn grid paper, saying clearly 'so many squares right, so many squares up' fully spells out how a translation moves.
- 14.6.3 The Basic Properties of TranslationAfter translating, the figure is exactly the same size as before, every point has traveled the same distance in the same direction, and matching segments are parallel and equal.
- 14.6.4 Drawing the Translated FigureMove each corner to its new spot by 'so many right, so many up,' connect them back up, and you have the translated figure.
- 14.6.5 Parallels and Equal Lengths in TranslationThe segments joining each pair of matching points are parallel and equal in length, echoing the parallel lines you just learned, so translation is those ideas in action.
15Triangles
Start from the most basic triangle structure to build up an understanding of its three sides, three angles, and three special inner segments, laying the foundation for later proofs
- 15.1.1 What a Triangle IsJoin three sticks end to end and you get the sturdiest frame there is, the triangle △ABC
- 15.1.2 How the Three Sides RelateThe two shorter routes have to reach around the far point, so any two sides together are longer than the third
- 15.1.3 The Angles Add Up to a Straight AngleTear off all three corners and lay them together and they pave a perfectly straight line, so the angles sum to 180°
- 15.1.4 Sorting Triangles by Sides and by AnglesCheck whether the sides match in length and whether any angle passes 90°, and you can pick out isosceles, acute, obtuse, and right triangles
- 15.1.5 Exterior Angles and Their SumStretch one side past a corner and the angle that opens up equals the two interior angles it doesn't touch, added together
- 15.1.6 Heights, Medians, and Angle BisectorsThe height dropped straight from a corner, the median running to the opposite midpoint, and the line splitting the corner are a triangle's three special segments
Upgrade the geometric intuition from the last stage into rigorous language: learn to write definitions, judge whether statements are true, write proofs, and construct figures precisely with straightedge and compass
- 15.2.1 Definitions and StatementsFirst you pin down what a word means, that's a definition, then you say something that can be judged true or false, that's a statement
- 15.2.2 Hypothesis, Conclusion, and the ConverseBreak a statement into 'if... then...', and swapping the two halves gives you its converse
- 15.2.3 True Statements, False Statements, and CounterexamplesShow one single case where it fails and even the prettiest-sounding claim collapses, and that case is a counterexample
- 15.2.4 What a Proof Looks LikeMove step by step from what you're given, following agreed-upon truths until you reach the conclusion, writing a reason for each step, and that's a proof
- 15.2.5 The Rules of Straightedge-and-Compass ConstructionUsing only an unmarked ruler and a compass, drawing lines and arcs lets you copy lengths and angles exactly
- 15.2.6 The Basic ConstructionsCopying a segment, bisecting an angle, and dropping a perpendicular are the building blocks you assemble with compass and straightedge
Understand the congruence relation of 'identical in shape and size,' master the four keys SSS/SAS/ASA/AAS, and use them in proofs
- 15.3.1 What Congruence MeansIf you can cut out two triangles and they lie perfectly on top of each other, they're congruent, written △ABC≌△DEF
- 15.3.2 Matching Sides and Matching Angles Are EqualWhen they stack, whatever lands on whatever is the match, and matching sides are equal and so are matching angles
- 15.3.3 Side-Side-Side (SSS)Fix all three side lengths and the triangle can only be built one single way
- 15.3.4 Side-Angle-Side (SAS)Lock two sides with the angle between them and the third side and the whole shape have nowhere left to move
- 15.3.5 Angle-Side-Angle and Angle-Angle-Side (ASA, AAS)Know two angles and one side and the triangle is completely pinned down too
- 15.3.6 Using Congruence to Prove Segments and Angles EqualTo show two pieces are the same length, first find a pair of congruent triangles, then bring out the matching equals
Use axes of symmetry and grids to capture the beauty of 'fold and match,' setting up the properties of isosceles triangles (no coordinates involved)
- 15.4.1 Figures with Reflection SymmetryFold along one crease and the two halves cover each other completely, and that crease is the axis of symmetry
- 15.4.2 Two Figures That Are ReflectionsTwo figures sit like a thing and its mirror image, and flipping across the axis between them makes them coincide
- 15.4.3 The Perpendicular-Bisector Property of the AxisThe line joining a pair of mirror points is cut in half at a right angle by the axis, and every point on the axis is equally far from both sides
- 15.4.4 The Perpendicular Bisector of a SegmentEvery point that's the same distance from both ends of a segment lands right on its perpendicular bisector
- 15.4.5 Drawing Symmetric Figures on a GridCount squares to flip each point to the matching spot the same distance across, and the symmetric figure appears
Bring congruence and reflection together to study the properties and tests for isosceles and equilateral triangles, the headline special cases of the triangle
- 15.5.1 The Symmetry of an Isosceles TriangleFold a triangle with two equal legs along its apex and the left and right sides line up exactly
- 15.5.2 Equal Sides Face Equal AnglesWhen the legs are the same length, the two base angles below are bound to be the same size
- 15.5.3 Three Lines in OneIn an isosceles triangle the apex bisector, the height to the base, and the median all crowd onto the very same line
- 15.5.4 Equal Angles Face Equal Sides (the Test)Turn it around: if two angles in a triangle are the same size, the two sides facing them are the same length too
- 15.5.5 Equilateral TrianglesIn a triangle with all three sides equal, all three angles get locked at 60° as well
Focus on right triangles to establish the Pythagorean theorem, its converse, and the HL test, bridging toward diagonals and distance calculations in quadrilaterals
- 15.6.1 The Angles and Sides of a Right TriangleThere's one 90° angle, the two acute angles add up to 90°, and the longest side facing the right angle is called the hypotenuse
- 15.6.2 The Pythagorean TheoremBuild a square on each leg and their areas add up to exactly the square on the hypotenuse, that is a²+b²=c²
- 15.6.3 Finding Lengths with the Pythagorean TheoremKnowing two sides, lean on a²+b²=c² and take a square root to find the third
- 15.6.4 The Converse of the Pythagorean TheoremIf the three sides satisfy a²+b²=c², the triangle is guaranteed to have a right angle
- 15.6.5 Pythagorean TriplesWhole numbers like 3, 4, 5 that fit a²+b²=c² form a set called a Pythagorean triple
- 15.6.6 Right-Triangle Congruence (HL)Two right triangles are congruent when their hypotenuse and one leg match
16Quadrilaterals & Polygons
Take the triangles you just learned, join and generalize them into polygons, and master the language of polygons along with the two core relationships: interior-angle sum and exterior-angle sum
- 16.1.1 Meeting PolygonsThree sticks fence in a triangle, and a few more sticks fence in a polygon; we name each side, vertex, and diagonal in turn
- 16.1.2 Convex Polygons and Regular PolygonsIf a rubber-band outline never caves inward it's convex, and when every side and angle matches we call it a regular n-gon
- 16.1.3 The Interior-Angle Sum of a PolygonDraw diagonals from one vertex to slice it into triangles, an n-gon splits into (n-2) of them, so the interior angles add up to (n-2)·180°
- 16.1.4 The Exterior-Angle Sum of a PolygonWalk once around the edges back to where you started and you've turned through exactly one full circle, so any polygon's exterior angles add up to 360°
- 16.1.5 Tessellation: Tiling a Floor with PolygonsTiles fit together without gaps because the angles meeting at a single point add up to 360°
Focus on the most important quadrilateral, the parallelogram, starting from a feel for its symmetry and deriving every property of its sides, angles, and diagonals
- 16.2.1 Meeting the ParallelogramPush a sliding door out of square and the opposite edges stay parallel; we write this quadrilateral as ▱ABCD
- 16.2.2 Opposite Sides Equal, Opposite Angles EqualTwo parallel rails are the same length and facing angles are the same size; draw a diagonal and a quick congruence proof shows it
- 16.2.3 The Diagonals Bisect Each OtherThe two diagonals act like scissors pinned at their midpoint, and the crossing point cuts each one neatly in half
- 16.2.4 The Distance Between the Two Parallel Lines in a ParallelogramTwo parallel sides run like train tracks, the same width everywhere, and that width is the distance between them
Introduce point symmetry as a fresh way of seeing symmetry, explain why a parallelogram has it, and systematically build up the various ways to test for one
- 16.3.1 Point Symmetry and the Center of SymmetrySpin a figure 180° about a point and if it lands back on itself, that pin point is the center of symmetry
- 16.3.2 Properties of Point SymmetryEvery line joining matching points passes through the same point and is bisected by it, as if each point were bounced across to the other side
- 16.3.3 The Parallelogram Is Point-SymmetricTurn it a half-circle about where the diagonals cross and it covers itself exactly, so that crossing point is its center of symmetry
- 16.3.4 Testing for a Parallelogram from the SidesBoth pairs of opposite sides parallel, or both pairs equal, or one pair both parallel and equal; meet any one and you have a parallelogram
- 16.3.5 Testing for a Parallelogram from the DiagonalsOnce the two diagonals bisect each other, the four vertices are bound to form a parallelogram
Use the parallelogram toolkit to go back and solve the midpoint-connecting problem inside a triangle, arriving at the midsegment theorem that links the two big families
- 16.4.1 What a Triangle's Midsegment IsJoin the midpoints of two sides of a triangle and the short segment you get is called a midsegment
- 16.4.2 The Triangle Midsegment TheoremA midsegment is like a shrunken base: parallel to the third side and exactly half its length
- 16.4.3 Applying the Midsegment TheoremThe small triangle made by joining midpoints is a scaled-down copy of the original, handy for finding lengths and proving lines parallel
Add conditions to the parallelogram step by step to reach the rectangle, rhombus, and square, sorting out their properties, tests, and how they relate
- 16.5.1 Properties of the RectangleBend all of a parallelogram's angles into right angles and you get a rectangle, whose two diagonals are equal in length
- 16.5.2 Testing for a RectangleA parallelogram with one right angle, or with equal diagonals, has to be a rectangle
- 16.5.3 The Median to the Hypotenuse of a Right TriangleAn echo of half a rectangle's diagonal: the median to the hypotenuse of a right triangle equals half the hypotenuse
- 16.5.4 Properties of the RhombusStretch all four sides of a parallelogram to the same length and you get a rhombus, whose diagonals are perpendicular and bisect its angles
- 16.5.5 Testing for a RhombusA parallelogram with four equal sides, or with perpendicular diagonals, is a rhombus
- 16.5.6 Properties and Tests for the SquareBoth a rectangle and a rhombus: four equal sides, four right angles, and diagonals that are both equal and perpendicular
- 16.5.7 The Family Tree of Special QuadrilateralsParallelogram → rectangle/rhombus → square, like a family tree where each added condition makes the shape more special
Build on all the earlier proofs to distill a new tool of argument, proof by contradiction, and close out this stage with it, laying groundwork for the trickier reasoning in similarity
- 16.6.1 The Idea of Proof by ContradictionTo prove something true, first pretend it's false and corner yourself into a dead end (a contradiction), which forces it to be true
- 16.6.2 The Steps of Proof by ContradictionThree steps: first assume the opposite of the conclusion, then derive a contradiction, and finally reject the assumption and affirm the original claim
- 16.6.3 Proving Geometric Statements by ContradictionFor hard-to-prove-directly claims like "through a point off a line there's only one parallel," assume there are two and you crash into a contradiction
- 16.6.4 Paving the Way for Similarity: Invariants Under ScalingEnlarge or shrink a whole figure and parallels stay parallel, angles stay the same; this is exactly the doorway to our next stop, similarity
17Similarity & Dilation
Picking up the parallel lines and congruence from quadrilaterals, we carry the idea of a 'ratio' between numbers over to the lengths of line segments, building the language of proportional segments that lays the foundation for everything about similarity to come.
- 17.1.1 A Look Back at Ratio and ProportionWhich cup of sugar water is sweeter comes down to the ratio of sugar parts to water parts, a:b.
- 17.1.2 The Ratio of Two SegmentsUse one segment as a ruler to measure another, and the number of times it fits is the ratio of the two, AB:CD.
- 17.1.3 Proportional SegmentsWhen four sticks line up so their lengths pair perfectly inside and out, we write a:b=c:d, which means ad=bc.
- 17.1.4 The Basic Properties of ProportionsThe two sides of the equation balance like a seesaw, so cross-multiplying or combining terms keeps things even and gives ad=bc and its rearrangements.
- 17.1.5 The Golden RatioMake one cut in a segment so the short piece is to the long piece as the long piece is to the whole, and that special point is the golden cut, with ratio (sqrt5-1)/2.
Using the parallel lines you already know well as a tool, we discover that 'parallel lines slice by the same ratio,' which is the direct source of the tests for similar triangles.
- 17.2.1 Parallel Lines Cutting Equal SegmentsWhen a ladder's rungs are evenly spaced, the two side rails get chopped into equal pieces, since parallel and equally spaced means equal division.
- 17.2.2 The Parallel-Line Proportionality TheoremA bundle of parallel lines acts like a bread slicer, and the matching pieces it cuts from two crossing lines all share the same length ratio.
- 17.2.3 A Line Parallel to One Side of a TriangleDraw a line across a triangle parallel to its base, and the two other sides get split by the same ratio, giving AD:DB=AE:EC.
- 17.2.4 Using the Converse to Prove Lines ParallelTurn it around: if the two sides are divided in the same ratio, the line joining those points must be parallel to the base.
We introduce what similarity means, the same shape at possibly different sizes, then use the proportionality of parallel lines to derive a few handy tests, the central hub of this whole stage.
- 17.3.1 What Similar Triangles AreThe same photo enlarged or shrunk keeps its angles and scales its sides together, written as triangle ABC ~ triangle A'B'C'.
- 17.3.2 Test: Two Pairs of Equal AnglesTwo set squares with matching angles are the same shape no matter their size, so AA means similar.
- 17.3.3 Test: Two Sides Proportional with Equal Included AngleWhen two sides open out by the same scale around the same angle, the whole figure is stretched proportionally, the SAS kind of similarity.
- 17.3.4 Test: Three Sides ProportionalLengthen all three sides by the same factor and the shape they frame is just an enlarged copy, the SSS kind of similarity.
- 17.3.5 Similarity in Right TrianglesDrop an altitude to the hypotenuse and it splits the right triangle into two smaller triangles, each the same shape as the original.
- 17.3.6 The Geometric Mean RelationsThe altitude from the right angle and the two pieces of the hypotenuse are geometric means of each other, giving relations like h squared = mn and a squared = mc.
Once we can test for similarity, we ask what it actually buys us: how corresponding segments, perimeter, and area change with the scale factor, so learners can both compute and apply.
- 17.4.1 Sides Proportional, Angles EqualUnder a magnifying glass the angles don't budge while every corresponding side is multiplied by the same scale factor k.
- 17.4.2 The Ratio of Altitudes, Medians, and BisectorsEnlarge the whole figure by k and every matching segment inside it grows by k too.
- 17.4.3 Perimeter Ratio Equals the Scale FactorEvery side grows by k, so the full distance around naturally grows by k as well.
- 17.4.4 Area Ratio Equals the Square of the Scale FactorStretch both length and width by k and the area that fills in grows by k times k = k squared.
We extend similarity from triangles to any polygon, making 'same shape' a general idea, and connect it to the dilation that opens the next module.
- 17.5.1 What Similar Polygons AreTwo maps of the same place at different sizes, with equal corresponding angles and proportional corresponding sides, are similar polygons.
- 17.5.2 Testing Polygon Similarity by Cutting It UpSplit a polygon into triangles, and if all the triangles are similar, the whole figure is too.
- 17.5.3 Perimeter and Area Ratios of Similar PolygonsJust as with triangles, perimeter scales by k and area scales by k squared together.
- 17.5.4 Putting the Scale Factor to WorkOnce you know a scale factor k, you can work backward from a known side to find an unknown one.
On top of similarity we add a shared center, giving the dilation transformation that enlarges or shrinks by pure construction, setting up the transformations in circles next and in later coordinate geometry.
- 17.6.1 The Idea of a DilationA bulb casts a shadow whose every side radiates from the same light source, and that point is the center of dilation.
- 17.6.2 Center and Scale Factor of a DilationThe distances from the center out to corresponding points all grow by the same factor k, and that k is the scale factor.
- 17.6.3 Constructing a Dilation (Enlarging and Shrinking)Starting from the center, push each vertex out or pull it in along its ray by a factor of k, then connect them back up.
- 17.6.4 Dilations on the Same Side and Opposite SidesThe enlarged shadow can land on the same side as the light or flip to the other side, depending on whether the center sits between the figures or outside them.
- 17.6.5 How Dilation Relates to SimilarityA dilation is always similar, but similar figures can't always be arranged as a dilation about one shared center.
We use similar triangles to reach heights and distances we can't touch, bringing this stage down to the real world and training the skill of turning a real scene into a similar-figure model.
- 17.7.1 Measuring Height with ShadowsAt the same moment a person's shadow and a tree's shadow stretch in step, so person:person's-shadow = tree:tree's-shadow lets you scale up to the tree's height.
- 17.7.2 Measuring Height with a Mirror or a PoleSpot the tower's top in a mirror and your line of sight builds two similar triangles, so measuring the near one tells you the far one.
- 17.7.3 Measuring a Distance You Can't ReachYou can't cross to the far bank, so build a smaller triangle on your side that's similar to it and scale back up to the true width.
- 17.7.4 Putting It All Together with Similarity ModelsDraw the scene as a figure, label the corresponding sides, set up a proportion, and solve, the all-purpose recipe for similarity problems.
18Circles
Start from the one core idea — every point the same distance from a fixed center — to understand the circle, then unearth its most powerful weapon, symmetry, laying the foundation for every theorem to come.
- 18.1.1 A Circle Is Born: Always the Same Distance from One PointPin one end of a string, swing the other end all the way around, and every point you reach sits exactly as far from the pin as every other — that is a circle, written as circle O with radius r.
- 18.1.2 Chord, Arc, Diameter, SemicircleA line joining two points on the circle is a chord, the longest chord runs straight through the center and is the diameter, an arc is a curved slice of the rim, and half a circle is a semicircle.
- 18.1.3 The Circle Is the Most Symmetric Shape of AllSpin a circle any amount around its center and it lands on itself, fold it along any diameter and the halves match perfectly — it has both mirror symmetry and rotational symmetry.
- 18.1.4 The Perpendicular Diameter: One Clean CutDrop a cut straight down from the center across a chord and it splits both the chord and the arc it spans into two equal halves — a diameter perpendicular to a chord bisects that chord.
- 18.1.5 Putting the Perpendicular Diameter to WorkKnowing the chord's length and how far the center sits from it, you join up a right triangle and use the Pythagorean theorem to find the radius r.
Build the equal relationships among a central angle, the arc it faces, and the chord it cuts, so that angle, arc, and chord can all be converted into one another.
- 18.2.1 Enter the Central AngleAn angle with its vertex at the center and two radii for sides is a central angle — the wider it opens, the longer the arc across from it.
- 18.2.2 The Measure of an ArcTreat the whole rim as 360 degrees, and however many degrees the central angle facing an arc measures, that is exactly the measure of the arc.
- 18.2.3 Central Angle, Arc, and Chord Match One to OneIn the same circle, equal central angles ↔ equal arcs ↔ equal chords — the three rise and fall together.
- 18.2.4 Bringing in the Chord's Distance from the CenterThe distance from the center to a chord is its chord-center distance, and the longer the chord the closer it sits to the center — this links angle, arc, chord, and chord-center distance into one chain of cause and effect.
Introduce the inscribed angle, with its vertex on the circle, and establish its half relationship with the central angle plus a run of corollaries — this is the heart of the stage.
- 18.3.1 What an Inscribed Angle IsMove the vertex onto the rim and make the two sides chords, and you have an inscribed angle — it looks out over the very same arc as a central angle does.
- 18.3.2 The Inscribed Angle Theorem: Half the Central AngleFor one and the same arc, the angle you see standing on the circle is always half the angle you see standing at the center — inscribed angle = central angle ÷ 2.
- 18.3.3 Inscribed Angles on the Same Arc Are All EqualWherever you stand along the rim to view a given arc, the angle looks the same size, because each one is half of that single central angle.
- 18.3.4 The Angle Facing a Diameter Is a Right AngleLook across a diameter and the angle is always 90 degrees — an inscribed angle that spans a semicircle is a right angle, and the reverse holds too.
- 18.3.5 Opposite Angles of a Cyclic Quadrilateral Add to a Straight AngleIn a quadrilateral with all four corners on the circle, each pair of opposite angles adds up to exactly 180 degrees.
Use one unifying key — comparing a distance against the radius — to judge how a point, a line, and two circles sit relative to one another.
- 18.4.1 Point and Circle: Inside, On, or OutsideCompare the distance d from the point to the center against the radius r — d < r is inside, d = r is on the circle, d > r is outside.
- 18.4.2 Line and Circle: Apart, Tangent, or CrossingCompare the distance d from the center to the line against r — d > r misses entirely, d = r just grazes at one point, d < r cuts through at two.
- 18.4.3 How Two Circles Sit TogetherDepending on how near or far they are, two circles can lie outside, touch outside, cross, touch inside, or nest within — all decided by comparing the distance between centers against the sum and difference of the two radii.
- 18.4.4 How Many Points Fix a CircleThree points not all on one line can be circled by exactly one circle and no other — this is a triangle's circumscribed circle.
Dig deep into the borderline case of tangency, mastering a tangent's properties, its test, and the tangent-length theorem, then apply them to a triangle's inscribed circle.
- 18.5.1 A Tangent's Property: The Radius Must Be PerpendicularWhen a line touches a circle at just one point, the radius drawn to that point is always perpendicular to the line.
- 18.5.2 The Test for a Tangent: Perpendicular Means TangentTurn it around — draw a line through the outer end of a radius at a right angle to it, and that line is guaranteed to be a tangent.
- 18.5.3 The Tangent-Length TheoremDraw two tangents to a circle from a single outside point and the two tangent segments are equal, while the line from that point to the center also bisects the angle between them.
- 18.5.4 A Triangle's Inscribed CircleTuck the largest possible circle inside a triangle so it hugs all three sides at once — its center, where the three angle bisectors meet, is called the incenter.
- 18.5.5 Incenter Versus Circumcenter, Side by SideThe incenter is equidistant from the three sides and always sits inside the triangle, while the circumcenter is equidistant from the three corners and can wander outside it.
Move from the circumference being pi times the diameter to arc length, sector area, and regular polygons, gathering geometry into quantities you can compute, and paving the way for the next stage's use of numbers and intervals to describe shapes.
- 18.6.1 Regular Polygons Share the Circle's RootsDivide the rim into equal parts and join the points, and you get an equilateral triangle, a square, a regular hexagon — all inscribed in the very same circle.
- 18.6.2 A Regular Polygon's Center and ApothemA regular polygon has its own circumscribed and inscribed circles too — center to corner is the radius, center to side is the apothem, and from these you can work out both perimeter and area.
- 18.6.3 The Arc-Length FormulaWhatever fraction of the full circle an arc takes up, its length is that same fraction of the whole circumference 2 pi r — that is, l = n pi r ÷ 180.
- 18.6.4 Sector AreaA sector is like a slice cut from a pizza, and whatever fraction of the whole circle it takes up, its area is that same fraction of pi r squared.
- 18.6.5 A Cone's Side and Its UnrollingCut the side of a cone open and flatten it out and you get a sector, whose arc length is exactly the circumference of the base circle.
- 18.6.6 From Shape to Number: How Do You Pin Down a Point on a CircleTo state precisely where a point on a circle sits, you need a whole language for describing numbers — and that is just what the next stage on sets and intervals will answer.
Ⅳ · Functions
19Sets & Intervals — the Language of Functions
Build the intuition behind a set — bundling "objects that meet the same standard" into a single whole, and learning to write it down
- 19.1.1 What Is a SetPicture every point inside one circle as a single bunch — that "definite collection of objects" is a set, written A.
- 19.1.2 Elements and DefinitenessEach thing is either inside the circle or outside it, never "maybe"; every member inside is called an element, written a.
- 19.1.3 Distinctness and Order-FreeA roll call lists no one twice and doesn't care who comes first, so {1,2,2} is just {2,1}.
- 19.1.4 Names of Common Number SetsNaturals N, integers Z, rationals Q, reals R — like fixed nicknames given to a few big families.
Master listing and rule-based notation, switching freely between "naming members one by one" and "stating one rule"
- 19.2.1 Listing NotationWhen there aren't many members, just call them out one by one inside curly braces, like {2,4,6,8}.
- 19.2.2 Set-Builder NotationWhen there are too many members, post an entry condition instead, writing {x | x is even}, where the bar reads "such that".
- 19.2.3 Switching Between the TwoUnfold a rule into a list, or compress a list into a rule — both describe the very same circle.
- 19.2.4 The Empty SetWhen no one meets the entry condition, the circle holds nothing at all, and this empty circle is written ∅.
- 19.2.5 Element-and-Set MembershipTo say whether something is in the circle: write a∈A if it's in, a∉A if it's not.
Use the "circle inside a circle" picture to understand subsets, proper subsets, and equality, building comparisons between sets
- 19.3.1 SubsetThe small circle sits entirely inside the big one — every member of A is also a member of B, written A⊆B.
- 19.3.2 Proper SubsetThe small circle is inside the big one but the big one has room to spare, written A⊊B.
- 19.3.3 Set EqualityOnly when two circles contain each other and have exactly the same members do we say A=B.
- 19.3.4 The Empty Set Is a Subset of Every SetThe empty circle never causes trouble and fits inside anyone, so ∅⊆A always holds.
- 19.3.5 Counting SubsetsEach element gives two choices — take it or leave it — so n elements yield 2ⁿ subsets.
On "the big canvas of the universal set," learn intersection, union, and complement, and compute them visually with Venn diagrams
- 19.4.1 Universal Set and Venn DiagramsFirst frame the whole scope of discussion as a big canvas U, then let a Venn diagram make the overlap between circles clear at a glance.
- 19.4.2 IntersectionThe shared overlap of two circles — in A and in B at once — is written A∩B.
- 19.4.3 UnionMerge two circles into one pool — in A or in B both count — written A∪B.
- 19.4.4 ComplementScoop the A region out of the big canvas, and what's left is the complement ∁_U A.
- 19.4.5 Mixing and the Rules of OperationsWhen intersection, union, and complement chain together, shade by the Venn diagram to check laws like distributivity without slipping.
Carry the language of sets onto the number line, using intervals and neighborhoods to capture continuous ranges of numbers neatly — a direct route to the domain of a function
- 19.5.1 A Range on the Number LineA continuous stretch of points on the number line is also a set, like {x | 1≤x≤5}.
- 19.5.2 Open and Closed IntervalsBrackets tell you whether the endpoints count: [ ] includes them, ( ) leaves them out, written [1,5] or (1,5).
- 19.5.3 Half-Open and Infinite IntervalsInclude one end but not the other with (1,5]; for an end you can never reach, use +∞ or −∞.
- 19.5.4 Neighborhood: Around a PointZero in on a small stretch of width δ around a point a, and that's its neighborhood (a−δ, a+δ).
- 19.5.5 Expressing Operation Results as IntervalsOn the number line, intersecting and uniting two stretches is just overlapping and joining intervals — a warm-up for function domains.
20Coordinates & First Functions
Stretch the number line and intervals from last stage out into the plane, and learn how a single pair of numbers can lock down any spot on a flat surface
- 20.1.1 How We Describe Where Something IsFinding your movie seat means naming the row and then the seat number; give a direction and a distance, and a single point is pinned down.
- 20.1.2 From the Number Line to the Coordinate PlaneCross two number lines to make a plus sign: the flat one is the x-axis, the upright one is the y-axis, and they meet at the origin O.
- 20.1.3 A Point's Coordinates (x, y)Drop a straight line from the point to each axis, read off the two numbers, and write them in order as (x, y).
- 20.1.4 The Four Quadrants and Points on the AxesThe two axes slice the plane into four fields, numbered one through four going counterclockwise, and the plus or minus signs instantly tell you which field you're in.
- 20.1.5 Coordinates and DistanceHow far a point sits from the x-axis is just |y|, its distance from the y-axis is |x|, and its distance from the origin you piece together with the Pythagorean theorem.
Turn reflection, point symmetry, and translation into simple add-subtract and sign-flip rules on coordinates, laying the groundwork for transforming function graphs later
- 20.2.1 Reflection Across an AxisHold a mirror along the x-axis and left-right stays put while up-down flips, so y switches sign to give (x, -y); mirror along the y-axis and it's x that switches instead.
- 20.2.2 Point Symmetry About the OriginSpin a point a half-turn around the origin and it lands directly opposite, with both coordinates flipping sign together to (-x, -y).
- 20.2.3 Translation in the Coordinate PlaneSlide the whole picture a units right and b units up, and every point simply gains a on its x and b on its y.
- 20.2.4 Combining Translations and ReflectionsFlip then slide, or slide then flip; just apply the sign changes and the add-subtracts to the coordinates in order.
Use whole-plane scaling to understand enlarging and shrinking a shape, and see clearly what multiplying by a ratio means geometrically
- 20.3.1 A Quick Look Back at Similar FiguresA copier blowing up a photo keeps the shape the same and only changes the size, with matching sides all sharing one ratio.
- 20.3.2 Scaling Centered at the OriginStretch the picture out from the origin like a rubber band by a factor of k, and every point's coordinates get multiplied by k to become (kx, ky).
- 20.3.3 The Scale Factor and Its SignA positive k enlarges in the same direction, while a negative k also flips the picture over to the far side of the origin.
- 20.3.4 Coordinate Transformations in a NutshellSign flips do symmetry, adding and subtracting does translation, and multiplying does scaling, with each kind of operation handling its own one job.
Introduce constants and variables, and distill the way one quantity depends on another into the central idea of a function
- 20.4.1 Constants and VariablesA car cruising at a steady speed has that speed fixed as a constant, while the time and the distance keep changing as variables.
- 20.4.2 One Quantity Riding Along with AnotherDistance follows time, so fix the time and the distance is settled too, and that following relationship is the seed of a function.
- 20.4.3 The Idea of a FunctionA machine that takes in one x and spits out exactly one y captures that unique matching, and we say y is a function of x.
- 20.4.4 Independent and Dependent VariablesThe x you dial in yourself is the independent variable, and the y that passively changes along with it is the dependent variable.
Describe the same function through a formula, a table, and a graph, and use intervals to write down the range of inputs and outputs
- 20.5.1 The Formula FormWrite the matching rule as a one-line formula like y = 2x + 1, and plug in an x to compute its y.
- 20.5.2 The Table FormLike a price list with one column of inputs and one of outputs, you glance across to match each x with its y.
- 20.5.3 The Graph FormPlot every (x, y) pair as a point in the coordinate plane, and connecting them traces out the function's graph.
- 20.5.4 The Domain (Written as an Interval)Gather up every x the machine will accept into one set, and write it as the domain using the intervals from last stage.
- 20.5.5 The Range (Written as an Interval)Gather up every y the machine puts out into one set, and writing it as an interval gives you the range.
- 20.5.6 Switching Between the Three FormsA formula can fill a table, a table can be plotted as points, and a graph can reveal the rule, since all three are telling the same story.
Learn to capture a function's rise and fall, symmetry, and high and low points from both the graph and the symbols, getting ready to analyze linear functions
- 20.6.1 Monotonicity: Going Up and Going DownWalk rightward along x and where the graph climbs it's increasing, where it slides down it's decreasing; bigger x giving bigger y is what we call an increasing function.
- 20.6.2 Intervals of Increase and DecreaseSome stretches of the graph go uphill and others downhill, so circle the run of x values that share one trend as a single monotonic interval.
- 20.6.3 Even and Odd FunctionsA graph mirrored across the y-axis is even with f(-x) = f(x), and one symmetric about the origin is odd with f(-x) = -f(x).
- 20.6.4 Maximum and Minimum ValuesAt the graph's highest point y is largest and at its lowest point y is smallest, and these are the function's maximum and minimum values.
- 20.6.5 Putting It Together: Reading a Whole GraphFrom one picture you read the rises and falls, the symmetry, and the peaks and valleys, telling the monotonicity, evenness or oddness, and extreme values all at once.
21Linear Functions
Building on the coordinate plane and the idea of a function, meet the simplest family of functions — direct-proportion functions — and lock in the core picture: a graph that's a straight line through the origin.
- 21.1.1 A Refresher: Watching One Variable Move with AnotherWalk a point across the grid paper once more and feel again that "fix one quantity and the other is settled" — that's exactly what a function y=f(x) means.
- 21.1.2 Two Quantities That Rise TogetherBuy apples: double the pounds and you double the price, and this "locked-together doubling" is written y=kx.
- 21.1.3 What a Direct-Proportion Function Is, and What k MeansThink of the k (k≠0) in y=kx as the fixed stride — how much y climbs every time x goes up by 1.
- 21.1.4 Drawing the Graph of a Direct-Proportion FunctionMark the origin and one other point, lay a ruler across them, and there it is — a direct-proportion graph is a straight line through the origin (0,0).
- 21.1.5 k Sets the Line's Direction and SteepnessWhen k>0 the line climbs to the upper right and when k<0 it slides to the lower right; the bigger |k| is the steeper it gets, and k is the slope.
Generalize from direct-proportion functions to the general linear function y=kx+b, understand what b means geometrically, and draw the graph in full.
- 21.2.1 Giving Direct Proportion a Starting PointA taxi charges a base fare the moment you climb in and then adds on by distance, which we write as y=kx+b.
- 21.2.2 What a Linear Function IsAnything of the form y=kx+b (k≠0) is a linear function, and when b=0 it folds right back into the direct proportion y=kx.
- 21.2.3 b Is Where the Line Crosses the y-AxisSlide that direct-proportion line straight up or down by b units, and where it lands, (0,b), is the intercept b.
- 21.2.4 Drawing a Linear Graph with Two PointsTwo points fix a line: take (0,b) and the spot where it meets the x-axis, connect them, and you've drawn y=kx+b.
- 21.2.5 Finding Where the Line Meets the AxesSet x=0 to get the y-axis crossing, and set y=0 to solve for the x-axis crossing (-b/k, 0).
Systematically describe how the signs of k and b decide a line's direction, whether it rises or falls, and which quadrants it visits — upgrading you from "able to draw" to "able to read."
- 21.3.1 Rising or Falling Comes Down to the Sign of kWhen k>0, larger x means larger y (the line goes up), and when k<0 it's the other way around — this is whether the function increases or decreases.
- 21.3.2 The Signs of k and b Decide Which Quadrants It PassesNotice which way the line climbs and which side of the y-axis it crosses, and you can count off the quadrants it travels through.
- 21.3.3 When Two Lines Are Parallel and When They CrossTwo lines with the same k are like side-by-side slides that never meet — equal slope k means parallel.
- 21.3.4 Shifting and Reflecting the GraphSliding the whole line up or down only changes b, while flipping it like a mirror image changes the signs of k and b — the rules of shifting and reflection.
- 21.3.5 Comparing Function ValuesOn the same graph, whoever's point sits higher has the larger y, so just look and compare y1 and y2.
Master the general technique for recovering y=kx+b from given conditions, opening the reverse route from "graph or data" back to "formula."
- 21.4.1 Thinking in Reverse: Set Up the Formula First, Then Find the CoefficientsIf you don't know k and b yet, just leave them in as unknowns for now — this "set first, solve later" is the method of undetermined coefficients.
- 21.4.2 One Point Pins Down a Direct ProportionA direct proportion is missing only k, so plugging in a single known point is enough to solve for y=kx.
- 21.4.3 Two Points Pin Down a Linear FunctionA linear function has two unknowns, k and b, so plug in two points, set up the pair of equations, and solve for y=kx+b.
- 21.4.4 Recovering the Formula from a GraphRead two clean whole-number points off the picture, substitute them back, and you can reconstruct the line's formula.
- 21.4.5 Recovering the Formula from Properties or a ShiftClues like "parallel to a certain line" or "passing through a certain point" turn into equations in k and b that you solve.
Use the lens of the straight line to unify linear equations and inequalities, turning algebraic solving into "reading heights off the graph."
- 21.5.1 The Line's Crossing of the x-Axis Is the Equation's SolutionThe instant the line touches the x-axis, y=0, and the x there is exactly the solution of the equation kx+b=0.
- 21.5.2 Above or Below the x-Axis Tells You the InequalityWhere the line sits above the x-axis y>0, and where it sits below y<0, so reading the interval solves kx+b>0 or <0.
- 21.5.3 Where Two Lines Cross and the System of EquationsThe point where two lines cross satisfies both equations at once, so the crossing's coordinates are the system's shared solution.
- 21.5.4 Comparing Heights to Solve k1x+b1 > k2x+b2Use the crossing point as the boundary and see which line is higher — that tells you which stretch of x makes one expression bigger than the other.
- 21.5.5 A Wrap-Up: Translating Between Numbers and ShapesAn equation means finding the crossing and an inequality means comparing heights, so solving turns into "finding the spot on the graph."
Understand functions whose domain is split into blocks, with each block described by a different line, and learn to graph them and read values stretch by stretch.
- 21.6.1 Why Split Into PiecesFree parking for the first 30 minutes, then charged by the hour — different stretches follow different rules, so you have to write it in pieces.
- 21.6.2 How to Write a Piecewise FunctionUse one big brace to line up "which formula goes with which stretch of x," noting the range each piece covers.
- 21.6.3 Drawing the Graph of a Piecewise FunctionFor each piece draw only the segment of line it governs, and at the joints use filled or open dots to show clearly whether the endpoint is included.
- 21.6.4 Plugging In by Interval to Find a ValueFirst see which stretch x falls into, then use that stretch's formula to compute y — don't grab the wrong rule.
- 21.6.5 How Absolute Value Connects to Pieces|x| is really just "keep it as is when positive, flip it when negative" — at heart it's a two-piece function stitched together.
Pull together modeling, recovering formulas, reading graphs, and equations and inequalities to solve real situations — and plant a contrast that sets up the nonlinear change of the next stage.
- 21.7.1 Building a Linear Model from a SituationFind the "fixed starting point" and the "increase per unit," fit them into y=kx+b, and your model is built.
- 21.7.2 Graphs in Distance ProblemsDistance changing steadily over time is a slanted straight line, and the slope k is exactly how fast you're going.
- 21.7.3 Comparing Plans and Analyzing Profit and LossDraw a line for each pricing plan; the crossing is the "equally worthwhile" tipping point, so compare heights to pick the cheaper one.
- 21.7.4 Optimization: Finding the Extreme Within a RangeA line is steady with no bends, so its largest and smallest values can only show up at the two endpoints of the variable's interval.
- 21.7.5 Fitting Real Data with Undetermined CoefficientsWhen several measured data points fall roughly along a line, pinning down k and b lets you predict values you never measured.
- 21.7.6 Looking Back at the Line, Ahead to the CurveA linear function changes at a steady pace, so it's a straight line; but if each step's increase itself keeps changing, the graph bends — and that opens the door to inverse proportion.
22Inverse-Proportion Functions
Building on linear functions, draw out the inverse-proportion function from everyday situations where two quantities multiply to a fixed amount, then set up the concept and its expression.
- 22.1.1 Looking Back: How Variables Pair UpA linear function is like walking at a steady pace; this time we switch to "fixed money, how many can you buy?" and meet a new pairing, y=k/x.
- 22.1.2 When the Product Stays the SameFor a rectangle of fixed area, the longer the length the narrower the width, with length times width always the same, so hold onto xy=k.
- 22.1.3 Defining the Inverse-Proportion FunctionDouble one quantity and the other is halved, which we write as y=k/x (k≠0), and that's an inverse-proportion function.
- 22.1.4 Three Equivalent Ways to Write ItThe same relationship in three different outfits: y=k/x, y=kx⁻¹, and xy=k, all saying exactly the same thing.
- 22.1.5 Finding the Constant kGiven one matched pair of numbers, just multiply them to get k, then plug it back into y=k/x to lock down the whole relationship.
- 22.1.6 The Forbidden Spot for xA denominator can never be zero, so x≠0, and this relationship is born with a slit cut right at the origin.
Plot the hyperbola point by point and get to know its shape: two separate branches that never touch the axes.
- 22.2.1 Make a Table and Plot Some PointsPick a few values for x, work out y, connect the dots, and see what the shape looks like first.
- 22.2.2 What a Hyperbola Looks LikeThe graph is two separate curving lines, called a hyperbola, not joined into one piece the way a straight line is.
- 22.2.3 Drawing Ever Closer to the AxesThe curve hugs the x-axis and y-axis closer and closer but never quite touches, so the axes are its asymptotes.
- 22.2.4 When k>0: First and Third QuadrantsWhen k is positive, the two branches sit upper-right and lower-left, with x and y sharing a sign, like a pair of good partners.
- 22.2.5 When k<0: Second and Fourth QuadrantsWhen k is negative, the two branches run off to upper-left and lower-right, with x and y opposite in sign, one rising as the other falls.
Read off increase/decrease behavior, symmetry, and the geometric meaning of k straight from the graph, matching each property to its picture.
- 22.3.1 Rising and Falling Within a BranchOn a single branch, when k>0 a larger x means a smaller y, so remember you can only compare increase and decrease within the same branch.
- 22.3.2 The Trap of Comparing Across BranchesYou can't directly compare sizes between the two branches, so be careful crossing the break at x=0 and don't just assume.
- 22.3.3 Symmetry About the CenterSpin the graph a half-turn around the origin and the two branches swap places exactly, so the hyperbola has central symmetry about the origin.
- 22.3.4 Two Lines of Mirror SymmetryThe graph is also symmetric about y=x and y=-x, as neat and tidy as a reflection in a mirror.
- 22.3.5 k Is That Rectangle's AreaDrop perpendiculars from a point on the graph to both axes to form a rectangle, whose area is always |k|, so remember xy=k.
- 22.3.6 Bigger |k|, Farther From the OriginAs |k| grows, the whole hyperbola backs away outward from the origin, and as it shrinks the curve draws in closer.
Place a line and a hyperbola in the same coordinate system to study their intersections, enclosed areas, and which graph sits above the other.
- 22.4.1 Where the Two Graphs MeetThe spot where the line crosses the hyperbola is the shared solution of y=k/x and y=ax+b solved together.
- 22.4.2 Solving the System for CoordinatesCombine the two equations into one and solve for x, often landing on a quadratic, to find the coordinates of the intersection points.
- 22.4.3 Which One Is On TopTo compare whether k/x or ax+b is larger, just see which curve is drawn higher up, and read off the interval.
- 22.4.4 Triangle Area ProblemsJoin the intersection points to the origin or the axes to form a triangle, then find its area with one-half base times height.
- 22.4.5 Pinning Down Both LinesGiven the intersection points or similar conditions, work backward to find k and a, b, fixing both graphs at once.
Use the inverse-proportion model to describe real quantities whose product stays fixed, building the model, solving it, and respecting the limits on the domain.
- 22.5.1 Building an Inverse-Proportion ModelWhen the distance is fixed, the faster you go the less time it takes, so you recognize an inverse relationship like t=s/v.
- 22.5.2 Inverse Proportion in PhysicsWith a fixed force, a larger contact area means smaller pressure, and p=F/S is exactly inverse proportion in disguise.
- 22.5.3 Situations in Work and EconomicsWith a fixed total amount of work, more people means each does less, so use y=k/x to plan out the resources.
- 22.5.4 The Realistic Range of ValuesHeadcount and length can't be negative, so you draw only the branch that makes sense and keep the domain grounded in reality.
- 22.5.5 Looking Back: The Rate of Change Is No Longer ConstantA line changes at a constant rate while a hyperbola eases off the farther it goes, quietly setting the stage for the curving parabola coming next.
23Quadratic Functions
Pick up the family of functions you've already met, get to know quadratics as a new member, and build their algebraic definition.
- 23.1.1 Revisiting the Function FamilyYou've already walked three roads — direct-proportion, linear, inverse-proportion — each turning inputs into outputs by a different rule; now meet a new rule, y=f(x).
- 23.1.2 A New Relationship Hidden in AreaAs a square's side grows, its area grows faster and faster, and that 'side squared' relationship S=x² is the first shadow of a quadratic.
- 23.1.3 Defining a Quadratic FunctionWhenever the highest power in the expression is a squared term, you've got a quadratic, y=ax²+bx+c (with a≠0).
- 23.1.4 Spotting a, b, and cTidy the expression into its standard shape and read off who is a, who is b, and who is c — and remember a can never be 0.
- 23.1.5 The Simplest Quadratic, y=x²Make a table, plot the points, and watch y=x² trace a symmetric 'bowl' so you can first see what it looks like.
Start from the simplest form y=ax², understand which way it opens and how wide, and build an intuitive picture of the parabola.
- 23.2.1 The Parabola and Which Way It OpensToss a ball upward and its path traces a 'bowl' curve called a parabola; with a>0 the bowl opens upward, with a<0 it opens downward.
- 23.2.2 a Controls How Narrow or WideThe larger |a| is the narrower and steeper the bowl, the smaller it is the wider and gentler — a works like a dial for the opening.
- 23.2.3 Sliding Up and Down with y=ax²+kThe whole bowl moves straight up or down without changing shape: add k to lift it, subtract k to drop it, and the vertex rides along.
- 23.2.4 Sliding Left and Right with y=a(x−h)²The whole bowl shifts sideways: replacing x with x−h slides it h to the right, and the axis of symmetry moves with it.
- 23.2.5 Vertex Form, y=a(x−h)²+kMove it sideways and up-and-down at once, and (h,k) is the bottom (or top) of the bowl — the vertex is visible at a glance.
Move from vertex form to the general form, and master how to find the three geometric essentials along with the graph's properties.
- 23.3.1 The Axis of Symmetry: The Parabola's MirrorA parabola folds like it's been placed against an upright mirror, perfectly matched left and right, and that mirror is the axis x=h.
- 23.3.2 The Vertex: Highest or Lowest PointThe bottom (or top) of the bowl is where the curve turns, and its coordinates (h,k) fix where the whole curve sits.
- 23.3.3 Completing the Square to Find the VertexReshape y=ax²+bx+c into a perfect square and you can pull the vertex form right out of the general form.
- 23.3.4 The Vertex FormulaThe shortcut: the axis is x=−b/2a and the vertex's height is (4ac−b²)/4a, ready to plug in directly.
- 23.3.5 Rising, Falling, and the Overall TrendThe axis splits the curve into a downhill side and an uphill side, and past the vertex the function flips from falling to rising (or rising to falling).
- 23.3.6 Reading the Graph from the Coefficientsa tells you which way it opens, c tells you where it crosses the y-axis, and b together with a tells you which side the axis falls on.
Master the general, vertex, and factored forms, and find the equation from given conditions by solving for the unknown coefficients.
- 23.4.1 What Each Form Is Good ForThe general form shows c, vertex form shows the vertex, and factored form shows the zeros — three outfits for describing the very same curve.
- 23.4.2 Three Points? Use the General FormIf the curve passes through three points, plug in all three coordinate pairs to set up a system and solve for a, b, and c.
- 23.4.3 Know the Vertex? Use Vertex FormIf you know where the bowl's bottom is plus one more point, plug into y=a(x−h)²+k and solving for a is the easiest path.
- 23.4.4 Know Two Crossings? Use Factored FormIf you know the two spots x₁ and x₂ where the curve meets the x-axis, write y=a(x−x₁)(x−x₂) and then pin down a.
- 23.4.5 Equations and Shifting TransformationsSliding a known parabola left, right, up, or down is the same as rewriting the h and k inside its equation.
Connect the parabola's relationship with the x-axis to the roots of a quadratic equation and the solution set of a quadratic inequality.
- 23.5.1 Where the Parabola Meets the x-axisThe spots where the bowl touches the ground are the x-values when y=0, which are exactly the roots of the quadratic equation.
- 23.5.2 The Discriminant Counts the CrossingsΔ=b²−4ac acts like a probe: more than 0 hits two points, equal to 0 hits one, less than 0 never touches down.
- 23.5.3 Solving a Quadratic Equation from the GraphSolving ax²+bx+c=0 just asks where the parabola crosses the x-axis, so read off the x-coordinates of the crossings.
- 23.5.4 Solving a Quadratic Inequality from the GraphAsking when ax²+bx+c>0 just asks which stretch of the curve runs above the x-axis, so read off that interval.
- 23.5.5 One Unified View of the ThreeFunction, equation, and inequality are one family: the graph is the map, the roots are the landmarks, and the solution set is the stretch of road.
Use the vertex to find the maximum or minimum, and translate everyday optimization problems into a quadratic model.
- 23.6.1 The Vertex Is the Extreme ValueThe bowl's bottom is lowest and its top is highest; opening upward the vertex gives the minimum, opening downward the maximum, and the value is k.
- 23.6.2 Extreme Values on a Limited IntervalIf x can only run over part of the line, check whether the axis falls inside that interval before deciding between the endpoints and the vertex.
- 23.6.3 The Largest-Area ProblemWith a fixed length of fence, write the enclosed area as a quadratic in the side length and find the vertex to get the most area.
- 23.6.4 The Largest-Profit ProblemRaise the price and sales drop, lower it and the margin thins, so write total profit as a quadratic and find the most profitable price.
- 23.6.5 Modeling Projectile MotionThe path of a fountain or a jump shot is a parabola, so use a quadratic to find its highest point and where it lands.
- 23.6.6 The Full Modeling WorkflowSet the variable, write the function, fix the range, find the vertex, and answer the original question — five steps to work through any word problem.
Abstract a family of power functions from y=x², recognize their shared pattern, and pave the way for exponentials, logarithms, and other new functions ahead.
- 23.7.1 From Squaring to a General Powery=x² is just one member; swap the exponent for any constant α and you have the whole power-function family, y=xᵅ.
- 23.7.2 What Common Power Functions Look Likex, x², x³, x^(1/2), and x^(−1) each have their own posture, so plot the points and compare their shapes side by side.
- 23.7.3 Odd, Even, and SymmetryEven powers are mirror-symmetric like a bowl while odd powers are symmetric about the origin like a slide, and the exponent decides which.
- 23.7.4 Monotonicity and DomainDepending on whether the exponent is positive or negative, odd or even, the curve's rise and fall and the x-values it allows shift along with it.
- 23.7.5 Does the Base Change, or the Exponent?In a power function the base x changes while the exponent stays fixed; next you'll meet a new function where the base holds still and the exponent moves instead.
24Exponential & Logarithmic Functions
Building on square roots and integer powers, extend exponents to fractions so every rule of powers becomes one unified system, paving the way for exponential functions
- 24.1.1 Revisiting Integer PowersMultiplying the same number several times is written as a to the n-th power, and a negative exponent just flips it into a reciprocal, written a^n and a^(-n).
- 24.1.2 n-th Roots and RadicalsTaking a square root asks what squared gives this number, and we generalize that to asking for an n-th root, written as the n-th root of a.
- 24.1.3 Turning Radicals into Fractional ExponentsTaking a root is like raising to half or a third of a power, so a square root becomes a^(1/2) and an n-th root becomes a^(1/n).
- 24.1.4 What a Fractional Exponent MeansRooting then powering merge into a single fraction, so a to the m/n power means the n-th root of a to the m-th power.
- 24.1.5 One Unified Set of Power RulesSame base multiplied adds exponents, a power of a power multiplies them, and these same rules hold perfectly for fractional exponents too.
- 24.1.6 Estimating Powers with a CalculatorComputing 2 to the 0.5 power or 2 to the 1.7 power gives real numbers, collecting a string of points ready for drawing an exponential curve.
Establish the idea of an exponential function, master its graph and properties, and understand explosive growth and decay
- 24.2.1 The Birth of the Exponential FunctionA cell doubles each generation, so the generation becomes the exponent and the count becomes the result, written y=a^x (a>0 and a≠1).
- 24.2.2 Base Greater Than 1: Faster and FasterLike savings compounding on themselves, the growth gets fiercer the further you go, so when a>1 the graph soars upward and keeps rising.
- 24.2.3 Base Less Than 1: Flatter and FlatterA drug clears the body bit by bit, with half remaining each stretch of time, so when 0<a<1 the graph keeps falling and edges toward zero.
- 24.2.4 What Exponential Curves ShareNo matter the base, every curve passes through (0,1) and stays above the x-axis, which it hugs without ever touching as its asymptote.
- 24.2.5 Comparing Exponential ValuesOn the same rising curve, a point farther right sits higher, so its increasing nature tells you whether a^m or a^n is larger.
- 24.2.6 Shifting and Reshaping the GraphSlide the whole curve left, right, up, or down, and y=a^(x-h)+k carries the basic curve to a new spot.
Introduce logarithms as the inverse of exponentiation, master the idea and its operation rules, the central hub of this stage
- 24.3.1 The Birth of the Logarithm: Finding the ExponentKnowing how many doublings reach 1000, we ask the reverse question of what the exponent is, written log base a of N.
- 24.3.2 Converting Between Logs and ExponentsSaying a to the x power equals N and saying x equals log_a N describe the very same fact, just asked from a different angle.
- 24.3.3 Common Logs and Natural LogsBase 10 fits our decimal world and is written lg, while base e fits natural growth and is written ln.
- 24.3.4 The Log of a Product and a QuotientExponents of a product already add together, so the log of a product equals the logs added, log(MN)=logM+logN.
- 24.3.5 The Log of a Power and Changing the BaseThe exponent of a power moves out front, log_a(M^n)=n·log_a M, and to switch bases you bridge across with the change-of-base formula using division.
- 24.3.6 Simplifying and Evaluating with Log RulesBreak a tangle of multiplications and divisions into added and subtracted logs, and a few steps crack a number that was hard to compute.
Establish the logarithmic function and its properties, revealing that exponential and logarithmic graphs mirror each other across y=x
- 24.4.1 The Graph of a Logarithmic FunctionTreating the logarithm as a function of its input, the curve y=log_a x rises just to the right of the origin, hugging the y-axis.
- 24.4.2 Properties of Logarithmic FunctionsIt only accepts positive inputs and passes through (1,0), rising when a>1 and falling when 0<a<1, with the y-axis as its asymptote.
- 24.4.3 Exponentials and Logarithms as Mirror ImagesOne asks for the result and the other for the exponent, so the graphs of y=a^x and y=log_a x fold over the line y=x as mirror images.
- 24.4.4 How Inverses Cancel OutTaking an exponent then a log brings you back where you started, so a^(log_a x)=x and log_a(a^x)=x act like a pair of on-off switches.
- 24.4.5 Comparing Logarithmic ValuesOn the same logarithmic curve, a point farther right sits higher, so its increasing or decreasing nature tells you which log is larger.
Use exponentials and logarithms to capture real-world growth, decay, and phenomena spanning many orders of magnitude, feeling the value of modeling
- 24.5.1 Compound Interest and DoublingMoney snowballs at a fixed rate, so after n periods the total is the principal times (1+r)^n, a classic case of exponential growth.
- 24.5.2 Radioactive Decay and Half-LifeWith each half-life that passes, an artifact's carbon halves, and exponential decay lets us read back how old it is.
- 24.5.3 Solving for an Unknown Exponent with LogsTo ask how many years until money doubles or when a population tops a hundred million, take the log of both sides to bring the unknown down from the exponent.
- 24.5.4 Seeing Wide Ranges on a Log ScaleEarthquakes, sound, and acidity span several orders of magnitude, so a logarithmic scale squeezes wildly different numbers onto one chart.
- 24.5.5 The Sheer Power of Exponential GrowthFold a sheet of paper a few dozen times and it could reach the moon, showing how unbelievably faster exponential growth is than linear.
Unify the roots of equations through a function lens, master the existence theorem for zeros and the bisection method, laying groundwork for continuity and approximation
- 24.6.1 A Function's Zero Is the Equation's RootAt the point where a curve crosses the x-axis, the x-coordinate is both a zero of the function and a root of the equation f(x)=0.
- 24.6.2 The Existence Theorem for ZerosWith one end below water and the other above, a continuous path must cross the surface somewhere, so if f(a)·f(b)<0 there is a zero in between.
- 24.6.3 Counting Zeros from the GraphLook at where an exponential curve, a log curve, or a line meet, and the number of crossings is the number of zeros.
- 24.6.4 The Idea Behind BisectionLike a guessing game that takes the midpoint and cuts in half each time, you keep halving the interval holding a root to pin down the zero.
- 24.6.5 Finding Approximate Solutions by BisectionShrink the interval step by step until it is precise enough, giving an approximate root of the equation along with its error range.
- 24.6.6 Setting the Stage for Periodic ChangePowers and logs describe one-way rises and falls, but a turning windmill and the ebb and flow of tides need a function that comes back around, which leads us to trigonometry.
25Trigonometry
Carrying the function idea forward, we first build the link between an angle and the ratios of sides inside the most familiar shape — the right triangle — and learn to use it to measure real heights and distances.
- 25.1.1 Slope and AngleThe steeper a hill, the harder the climb, and the ratio 'how far you rise up divided by how far you go across' captures exactly how steep it is.
- 25.1.2 Defining Sine, Cosine, and TangentFix an angle in a right triangle and the proportions of its sides are locked in, so we give those three ratios names: sinA, cosA, and tanA.
- 25.1.3 Trig Values of Special AnglesThink of 30°, 45°, and 60° as three handy building blocks — memorize their sin, cos, and tan and you can snap answers together on the spot.
- 25.1.4 Basic Relationships Among the Same Angle's RatiosThe three ratios of a single angle aren't strangers to each other: sin²A+cos²A=1 and tanA=sinA/cosA tie them all together.
- 25.1.5 Solving a Right TriangleKnowing just one side and one angle lets you pry out all the rest, like using a single key to unlock the whole triangle.
- 25.1.6 Angles of Elevation, Depression, and BearingLook up at a tower's top or down at a passing boat, and the angle between your line of sight and the horizon turns straight into real distances and heights.
We break free of the 'angles only run from 0° to 90°' limit, letting angles spin any number of turns and carry a sign, and we bring in radians — a unit better suited to functions — to set the stage for the unit-circle definition.
- 25.2.1 Any Angle: The Angle as a RotationPicture a pointer sweeping around a center — it can swing half a turn, several turns, forward or backward — so an angle is no longer trapped inside a right triangle.
- 25.2.2 Quadrant Angles and Coterminal AnglesWherever the pointer comes to rest tells you which quadrant the angle is in, and spinning whole extra turns lands you right back, so those angles differ by whole multiples of 360°.
- 25.2.3 Radian Measure: Sizing an Angle by Arc LengthInstead of degrees, we measure an angle by 'how many radii of arc it sweeps across', so one full turn corresponds to an arc of 2π.
- 25.2.4 Converting Between Degrees and Radians180° and π are just two ways of naming the same angle, and once you have that yardstick you can switch back and forth freely.
- 25.2.5 Arc Length and Area of a SectorCut a slice of pizza: arc length = radius × angle (in radians), area = half of 'radius × arc length', and the formulas come out clean precisely because of radians.
We upgrade trig functions from the right triangle to the coordinates of a point on the unit circle, so they make sense for any angle, and we use the circle's symmetry to read off the reduction formulas all at once.
- 25.3.1 Defining Trig Functions on the Unit CircleOn a circle of radius 1, the y-coordinate where the angle's terminal side meets the circle is sinα, and the x-coordinate is cosα.
- 25.3.2 Signs of Trig Values by QuadrantWhich quadrant the point lands in settles whether its x and y are positive or negative, and the signs of sin, cos, and tan follow right along.
- 25.3.3 Reduction Formulas: Using the Circle's SymmetryAdd π, flip to the opposite, or go a full turn, and the point on the circle just reflects into a symmetric spot, so the value only swaps a sign or swaps a name.
- 25.3.4 Memory Tricks and Simplifying with Reduction Formulas'Odd changes the name, even keeps it; the sign comes from the quadrant' lets you boil any angle down to a familiar acute one before you evaluate.
We 'unroll' the turning of a point on the unit circle into a wavy curve, getting a firm grip on period, even/odd symmetry, increasing/decreasing behavior, and extremes, to lay the groundwork for parameter transformations and the laws of sines and cosines.
- 25.4.1 The Birth of the Sine CurveLet the point on the unit circle turn with the angle and trace its height point by point, and those points join into a rolling sine wave.
- 25.4.2 Graphs and Properties of Sine and CosineThe two waves are just shifted sideways by a quarter period from each other, both repeat every 2π, and both stay between -1 and 1.
- 25.4.3 Periodicity and Even/Odd SymmetryThe wave repeats after a fixed stretch — that stretch is the period — and symmetry about the origin makes it odd, while symmetry about the y-axis makes it even.
- 25.4.4 Intervals of Increase/Decrease and ExtremesBetween a crest and a trough the curve climbs or falls the whole way, with crests being maximums and troughs being minimums.
- 25.4.5 Graph and Properties of the Tangent Functiontan looks like a series of curves shooting upward, repeating every π and breaking off at certain angles where it races toward infinity.
We build the rules for combining known angles into new ones — starting from the sum and difference formulas to reach double-angle and auxiliary-angle forms — so messy trig expressions can be simplified, combined, and evaluated.
- 25.5.1 Cosine of a Sum or DifferenceThe cosine of two angles joined together can't simply be added; you have to unpack it with cos(α±β)=cosαcosβ∓sinαsinβ.
- 25.5.2 Sine and Tangent of a Sum or DifferenceFollowing the trail from the cosine formula, the sum-and-difference expansions for sine and tangent come along too.
- 25.5.3 Double-Angle FormulasReplace β with α and the sum formulas instantly become the 'one angle doubled' forms sin2α, cos2α, and tan2α.
- 25.5.4 Power Reduction and Half-Angle UsesRun the cos2α formula backward and you can bring the squares in sin² and cos² 'down a level', which makes integration and simplification easier.
- 25.5.5 The Auxiliary-Angle Formula: Two into Oneasinx+bcosx looks like two waves, but it can be merged into a single Rsin(x+φ) — one angle and one amplitude that say it all.
We treat a sine wave's amplitude, period, and phase as adjustable dials, understand how each parameter reshapes the graph, and use it to describe real back-and-forth motions like springs and sound.
- 25.6.1 Amplitude A: How Tall the Wave IsA acts like a volume knob, stretching the wave up or squashing it down and setting how far it can stray from the midline.
- 25.6.2 Frequency ω: How Tightly Packed the Wave Isω is like a fast-forward button — the bigger it gets, the tighter the wave and the shorter the period, which equals 2π÷ω.
- 25.6.3 Phase φ: Shifting the Wave Left or Rightφ slides the whole wave sideways, deciding where it begins its rise and fall.
- 25.6.4 Finding the Equation from the GraphRead a known wave in reverse — measure its height, its tightness, and its starting point — and you can work back to A, ω, and φ.
- 25.6.5 Simple Harmonic Motion: Sine Waves in the Real WorldA spring bobbing, a pendulum swinging, sound vibrating — the displacement over time is exactly a wave of the form Asin(ωt+φ).
We extend trig functions from the right triangle to any triangle, using the two great laws to convert freely between sides and angles, wrapping up this stage and handing the next, Sequences, a foundation of measurable relationships.
- 25.7.1 The Law of SinesIn any triangle, each side stays in a fixed ratio to the sine of its opposite angle: a/sinA=b/sinB=c/sinC.
- 25.7.2 The Law of CosinesThis is the Pythagorean theorem leveled up: when the angle isn't a right angle, c²=a²+b²-2ab·cosC adds the correction for being 'slanted'.
- 25.7.3 Given Conditions and Judging the Number of SolutionsDepending on which sides and angles you're handed, a triangle may be unique, have two solutions, or none, so check the conditions before you dive in.
- 25.7.4 Area Formulas for a TriangleWith two sides and the angle between them, the area is ½·ab·sinC, where the sine turns 'how wide the angle opens' into area.
- 25.7.5 Measurement and Real-World ApplicationsMeasuring a river's width, a mountain's height, or a ship's position — sketch the real problem as a triangle and solve it with the two great laws.
26Sequences
Narrow the functions you already know down to whole-number inputs only, and build the basic language of sequences, general terms, and recurrence.
- 26.1.1 What Is a SequenceLike people counting off in line, numbers lined up one after another form a sequence {aₙ}
- 26.1.2 Terms and Their PositionThe n-th person in line is the n-th term, with each position n paired to a value aₙ
- 26.1.3 A Sequence Is a Special FunctionLet a function take only the inputs 1, 2, 3, … and its string of outputs becomes a sequence aₙ=f(n)
- 26.1.4 The General-Term FormulaA recipe that turns any position into its value, written aₙ=(an expression in n)
- 26.1.5 Recurrence RelationsInstead of naming values outright, just say how the next term comes from the one before, as aₙ₊₁=f(aₙ)
- 26.1.6 Rising, Falling, and the GraphPlot each term as a lone dot in a row and watch whether the dots climb or sink
Study sequences that grow by a fixed step each time, and master the general term, the key properties, and the sum of the first n terms.
- 26.2.1 What an Arithmetic Sequence IsLike climbing stairs at a steady pace, every step rises by the same amount, and that fixed gap is the common difference d
- 26.2.2 The General-Term FormulaStart at a₁ and take (n−1) steps of size d, so aₙ=a₁+(n−1)d
- 26.2.3 The Arithmetic MeanA number sitting exactly between two others is just their average: 2b=a+c
- 26.2.4 Properties of Arithmetic SequencesTerms balanced around the middle add up the same, so aₘ+aₙ depends only on the sum of the positions
- 26.2.5 The Sum of the First n TermsPair the ends inward into equal totals, then add the sequence to its reverse to get Sₙ=n(a₁+aₙ)/2
- 26.2.6 How Sₙ Depends Quadratically on nRead Sₙ as a quadratic in n, and finding its largest or smallest value is like locating the vertex of a parabola
Study sequences that change by a fixed multiplier each time, mastering the general term, key properties, summation, and the intuition behind its limit.
- 26.3.1 What a Geometric Sequence IsLike cells doubling each generation, every step multiplies by the same factor, and that factor is the common ratio q
- 26.3.2 The General-Term FormulaStart at a₁ and multiply by q a total of (n−1) times, so aₙ=a₁·qⁿ⁻¹
- 26.3.3 The Geometric MeanThe value squarely between two others, averaged through multiplication: b²=a·c
- 26.3.4 Properties of Geometric SequencesTerms balanced around the middle multiply to the same product, so aₘ·aₙ depends only on the sum of the positions
- 26.3.5 The Sum of the First n TermsMultiply by q, shift, and subtract to cancel the middle, giving Sₙ=a₁(1−qⁿ)/(1−q)
- 26.3.6 Infinite Shrinking Sums and Their LimitWhen |q|<1 the running total creeps ever closer to one value, like never quite arriving yet always nearing the finish a₁/(1−q)
Beyond arithmetic and geometric cases, master the general tricks of grouping, splitting, pairing, and cancelling to make tricky sequences summable.
- 26.4.1 Formulas and Grouped SummationSplit a mixed sequence into separate stacks, sum each with a ready-made formula, then add the totals back together
- 26.4.2 Reverse-and-AddWrite the sum forwards, write it backwards, stack them, and the symmetric terms pair into a row of equal values
- 26.4.3 Shift-and-SubtractFor an arithmetic times geometric sum, multiply the whole thing by q, slide it over one slot, and subtract to collapse it into a geometric series
- 26.4.4 TelescopingSplit each term into a front piece minus a back piece, neighboring pieces cancel, and only the two ends survive
- 26.4.5 Choosing the Right MethodLook at the shape of the general term first, then pick the tool: grouping, shifting, and telescoping each suit a different form
Provide a rigorous proof tool for statements that must hold for every positive integer n, circling back to the general terms and sum formulas found earlier.
- 26.5.1 Why We Need InductionChecking the first few terms doesn't prove them all, so we need a way to settle infinitely many n in one stroke
- 26.5.2 The Domino PrincipleIf the first domino falls and each one can knock over the next, then they all fall
- 26.5.3 The Two Steps of InductionFirst check that n=1 holds (the base case), then prove that if n=k holds, so does n=k+1
- 26.5.4 Proving Sums and DivisibilityHand a sum formula or a divisibility claim to the two-step method and run it through rigorously
- 26.5.5 Common Pitfalls in InductionSkip the base case or fail to actually use the assumption, and the dominoes won't fall
Use sequence models to tackle problems of growth and accumulation, and carry the idea of approaching a value forward toward vectors and limits.
- 26.6.1 Arithmetic Models: Steady GrowthSaving a fixed amount each month or getting a flat raise each year are both accumulations with a constant common difference
- 26.6.2 Geometric Models: Compounding and DecayInterest on interest, or radioactive decay, where each period scales by the same factor, is exactly a common ratio q
- 26.6.3 Installments and Recurrence ModelingThis month's balance is born from last month's, so write the recurrence and then solve for the general term
- 26.6.4 Fibonacci and Other Classic SequencesBreeding rabbits and petal arrangements follow each term as the sum of the previous two: aₙ=aₙ₋₁+aₙ₋₂
- 26.6.5 From Ordered Numbers to Ordered PairsA string of numbers carries size and order, and the next step is to give each number a direction too, turning it into a vector
Ⅴ · Vectors, Space & Complex Numbers
27Plane Vectors
Building on the kind of number from sequences that has only size, introduce vectors, which have both size and direction, and set up the basic ideas and how to draw them.
- 27.1.1 Displacement Leads to the VectorSaying "walk 3 steps" isn't enough; you have to attach a direction too, and that pairing is the vector AB→.
- 27.1.2 Drawing a VectorYou draw it as a little arrow: its length is the magnitude |a|, and where the head points is the direction.
- 27.1.3 Magnitude, the Zero Vector, and Unit VectorsThe length is the magnitude |a|; an arrow that goes nowhere is the zero vector 0; and one whose length is exactly 1 is a unit vector.
- 27.1.4 Equal Vectors and Collinear VectorsIf you can slide one arrow onto another so they coincide perfectly, they're equal; and if their directions are the same or exactly opposite, they're collinear (parallel), a∥b.
- 27.1.5 The Idea of a Free VectorThink of a vector as an arrow you can slide anywhere; all that matters is its length and direction, never where it starts.
Define the three linear operations, addition, subtraction, and scalar multiplication, by drawing arrows, and learn the rules that govern them.
- 27.2.1 The Triangle Rule for AdditionWalk along a first, then continue along b, and the straight arrow from your start to your finish is the sum a+b.
- 27.2.2 The Parallelogram Rule for AdditionWhen two forces pull from the same point, build a parallelogram on them and its diagonal is the combined force a+b.
- 27.2.3 Subtracting Vectorsa−b is the arrow pointing from the tip of b to the tip of a, which is the same as a+(−b).
- 27.2.4 Scaling a Vectorλa stretches a by a factor of λ; if λ is negative the arrow flips around, and its length becomes |λ||a|.
- 27.2.5 The Collinear Vector Theoremb being parallel to a in the same direction is exactly the same as saying there's one and only one λ with b=λa.
- 27.2.6 Rules for the Linear OperationsAddition commutes and groups freely, scaling distributes, so you can rearrange these expressions as comfortably as you would ordinary numbers.
Use two non-collinear vectors as a base to describe any vector, then bring in coordinates so geometric operations turn into plain arithmetic.
- 27.3.1 The Fundamental Theorem of Plane VectorsPick any two arrows that aren't parallel as a skeleton, and every vector in the plane can be built from them in exactly one way.
- 27.3.2 A Basis and Orthogonal DecompositionChoose two perpendicular arrows i and j as your basis, and split any vector along its horizontal and vertical edges.
- 27.3.3 Coordinates of a VectorSlide the vector's tail to the origin, and the coordinates (x,y) of its tip are simply the vector's coordinates, a=(x,y).
- 27.3.4 Linear Operations in CoordinatesTo add, subtract, or scale, you just work component by component: (x₁,y₁)+(x₂,y₂)=(x₁+x₂,y₁+y₂).
- 27.3.5 Testing Collinearity with CoordinatesTwo vectors are parallel exactly when their cross-multiplied coordinates match, x₁y₂−x₂y₁=0.
- 27.3.6 Dividing a Segment in a Ratio and the MidpointTake a point that splits a segment in a given ratio and its coordinates are a weighted average of the endpoints, with the midpoint landing halfway.
Introduce the operation that multiplies two vectors into a number, capturing the angle between them, projection, and when they're perpendicular.
- 27.4.1 The Angle Between Vectors and ProjectionThe angle the two arrows open up from a shared start is the angle between them, and the shadow one casts onto the other is its projection.
- 27.4.2 Defining the Dot ProductMultiply length by length by the cosine of the angle to get a single number, a·b=|a||b|cosθ, which measures how much the two are in sync.
- 27.4.3 What the Dot Product Means Geometricallya·b equals the length of a times the projection of b onto a's direction, and its sign tells you whether the angle is acute or obtuse.
- 27.4.4 The Dot Product in CoordinatesWith coordinates it's even easier, a·b=x₁x₂+y₁y₂, just multiply and add to get the answer.
- 27.4.5 Finding Length and Angle with the Dot ProductDot a vector with itself and take the square root to get its length, |a|=√(a·a); then rearrange the cosθ formula to recover the angle.
- 27.4.6 Testing for PerpendicularityTwo arrows meeting at a right angle is exactly the same as their dot product being zero, a·b=0, which in coordinates reads x₁x₂+y₁y₂=0.
- 27.4.7 The Rules, and a Word of CautionThe dot product commutes and distributes, but never try to "cancel a vector" from both sides, because this isn't ordinary multiplication.
Use vectors as a toolkit to prove geometric theorems, compute lengths and angles, handle forces and velocities, and lay the groundwork for vectors in space.
- 27.5.1 Proving Parallel and Perpendicular with VectorsTo show two lines are parallel, find a λ; to show they're perpendicular, make the dot product 0, and the geometry problem becomes an equation.
- 27.5.2 Computing Lengths and Angles with VectorsUse the magnitude for a length and the cosθ formula for an angle, handing the shape over to coordinates to do the work.
- 27.5.3 Proving the Law of Cosines with VectorsWrite the third side as the difference of the other two, square it by dotting it with itself, and c²=a²+b²−2ab cosC falls right out.
- 27.5.4 Proving the Law of Sines with VectorsGo once around the triangle so the vectors sum to zero, project along the altitude, and you find each side's ratio to the sine is equal.
- 27.5.5 The Vector Model for ForcesWhen several forces act at one point, add them with the parallelogram rule to find the resultant, which is zero when everything is in balance.
- 27.5.6 Splitting Velocity and Displacement into ComponentsSplit a boat's speed across a river into a downstream part and a perpendicular part, compute each component on its own, then recombine.
- 27.5.7 Toward Vectors in SpaceAdd one more coordinate z, and the flat (x,y) grows into the solid (x,y,z), bridging into the solid geometry of the next stage.
28Solid Geometry & Views
Building on flat figures, develop an intuitive sense of solid shapes, and learn to draw them on paper using sketches and projections, then rebuild the solid from the drawing.
- 28.1.1 From the Plane into SpaceThe figure on your page stands up: add one more direction, height, to the flat plane and you've stepped into three dimensions.
- 28.1.2 Meeting the Basic SolidsLike naming shapes in a box of blocks: prism, pyramid, frustum, cylinder, cone, truncated cone, and sphere, one at a time.
- 28.1.3 Prisms and CylindersStack a deck of identical cards and you get a prism; spin a rectangle around and you get a cylinder, each made of a base plus a side surface.
- 28.1.4 Pyramids, Frustums, and Solids of RevolutionA pyramid tapers to a point like the great pyramids; slice the tip flat and it becomes a frustum; spin any figure around an axis and it sweeps out a solid of revolution.
- 28.1.5 Sketching Solids: Oblique ProjectionTo make a flat drawing look solid, keep horizontal lengths, halve the vertical ones, and tilt the depth, the trick known as oblique projection.
Use the idea of projection to understand where the three views come from, master drawing them and rebuilding the solid from them, and use them to compute measurements.
- 28.2.1 What Projection IsA lamp casts an object's shadow onto the wall: when the rays run parallel it's a parallel projection, and when they strike straight on it's an orthographic projection.
- 28.2.2 Where the Three Views Come FromTake one straight-on photo from the front, side, and top, and you get the front view, side view, and top view.
- 28.2.3 Drawing the Three Views of a SolidLength lines up, height stays level, width matches: the three views are like three siblings whose measurements all agree with each other.
- 28.2.4 Rebuilding a Solid from Three ViewsGiven three photos, work backward to the real thing: piece the outlines from three directions together in your mind to reform the solid.
- 28.2.5 Three Views and MeasurementRead the true edge lengths and heights off the three views, then go compute the surface area or volume.
Use nets to flatten a solid's surface into a plane figure, turning surface-area problems into the flat-area calculations you already know.
- 28.3.1 The Net of a SolidCut a box along its edges and lay it flat: the solid's skin spreads out into a plane figure, and that's its net.
- 28.3.2 Unfolding Prisms, Cones, and FrustumsA cylinder's side unrolls into a rectangle and a cone's into a sector, and the arc length of that sector traces back to the base's circumference.
- 28.3.3 Surface Area of Prisms and PyramidsSurface area is just the area of every face added up: the lateral area plus the base area gives the whole thing.
- 28.3.4 Surface Area of Cylinders, Cones, and Truncated ConesThe lateral-area formulas come straight from the net: 2πrl for a cylinder, πrl for a cone, and the larger circle minus the smaller one for a truncated cone.
- 28.3.5 Surface Area of a SphereLike orange peel wrapping the ball, remember S = 4πr², exactly the area of four great circles.
Starting from the principle that equal heights and matching cross-sections mean equal volume, make clear where the volume formulas for prisms, cones, and spheres come from, rather than memorizing them.
- 28.4.1 The Intuition of VolumePour water into a container and the amount of water is the volume; start simple with a box, where V = length × width × height.
- 28.4.2 Cavalieri's PrincipleTwo stacks of books of equal height, with matching cross-sections at every level, are equally thick: same height and same cross-sections mean equal volume.
- 28.4.3 Volume of a Prism or CylinderUse Cavalieri's principle to straighten the solid upright, and base area times height gives its volume, V = Sh.
- 28.4.4 Volume of a Cone or PyramidThree cones with equal bases and heights fit together into one prism, so a cone is a third of it, V = ⅓Sh.
- 28.4.5 Volume of a SphereUse Cavalieri's principle to compare a hemisphere with a cylinder that has a cone carved out, and you arrive at V = 4/3πr³.
Build the language and basic facts of spatial position, sorting out how points, lines, and planes intersect or run parallel, to lay the groundwork for judging parallelism and perpendicularity.
- 28.5.1 Describing Points, Lines, and PlanesA point at the corner of a room, a line where two walls meet, a whole wall as a plane: space is built from these three things.
- 28.5.2 The Basic Facts of a PlaneWhy a tripod stands steady: three non-collinear points fix one plane, and once two points lie in a plane, the whole line through them does too.
- 28.5.3 How Two Lines Sit in SpaceTwo pipes in a room either cross, run parallel, or do neither, that last case being skew lines.
- 28.5.4 The Angle Between Skew LinesSlide two skew lines together so they meet, measure the angle there, and that's the angle between them, in (0, 90°].
- 28.5.5 Line-and-Plane, Plane-and-PlaneA rod and the floor: the line can lie in the plane, cross it, or stay parallel to it, while two planes either meet or run parallel.
Master systematically the tests and properties of parallel lines, line-and-plane, and plane-and-plane, and learn to convert freely among them.
- 28.6.1 Test for a Line Parallel to a PlaneFor a line outside a plane, as long as it's parallel to some line inside the plane, it's parallel to the whole plane.
- 28.6.2 Property of a Line Parallel to a PlaneOnce a line is parallel to a plane, cut through it with a plane and the line where they meet must be parallel to it.
- 28.6.3 Test for Two Parallel PlanesIf one plane holds two intersecting lines each parallel to another plane, then the two planes are parallel.
- 28.6.4 Property of Two Parallel PlanesWhen two parallel planes are sliced by a third, the two lines of intersection are always parallel, like the two faces of a sliced loaf of bread.
- 28.6.5 Converting Between Kinds of ParallelismLine-to-line, line-to-plane, and plane-to-plane parallelism can step up and down: proving parallelism in space often means shuttling among all three.
Master the tests and properties of perpendicularity for line-to-line, line-and-plane, and plane-and-plane along with the key angles, completing the pure-geometry approach and connecting to the vector method ahead.
- 28.7.1 Tests for Perpendicular Lines and a Line Perpendicular to a PlaneIf a flagpole is perpendicular to two intersecting lines on the ground at once, it's perpendicular to the whole ground.
- 28.7.2 Property of a Line Perpendicular to a PlaneTwo poles perpendicular to the same plane must be parallel to each other, both standing equally straight.
- 28.7.3 Dihedral AnglesThe angle an open book makes: draw a perpendicular into each page from the spine, and the angle you measure is the dihedral angle.
- 28.7.4 Tests and Properties of Perpendicular PlanesIf one plane contains a line perpendicular to another plane, the two planes are perpendicular; run the reasoning backward to get a line perpendicular to a plane.
- 28.7.5 Distance and Perpendicularity CombinedThe distance from a point to a plane is the length of that perpendicular segment, and perpendicularity is the key to finding distances and heights.
- 28.7.6 Toward the Vector MethodProving perpendicularity by pure geometry often means hunting for auxiliary lines; next we'll set up coordinates in space and let vectors settle it with a calculation.
29Spatial Vectors & Solid Geometry
Carry the familiar arrow of a plane vector into three dimensions, build up addition, subtraction, scalar multiplication, and tests for collinear and coplanar vectors in space, picking up where the last stage's feel for solid shapes left off.
- 29.1.1 Arrows in SpaceDraw a directed arrow inside the box from the last stage and you have a spatial vector a — it carries both a size and a direction.
- 29.1.2 Adding and Subtracting Spatial VectorsWalk along one edge then another, tip to tail, and the combined journey is a+b — the parallelogram rule still works just fine.
- 29.1.3 Scalar Multiplication of Spatial VectorsStretch the arrow longer or flip it around, and scaling its length by λ gives λa, with the sign of λ deciding the direction.
- 29.1.4 Collinear Vectors and Testing DirectionTwo arrows that point along the same line share one direction, written b=λa, which is exactly what collinear means.
- 29.1.5 The Coplanar Vectors TheoremIf three arrows can all be laid flat on a single sheet of paper, then p=xa+yb holds and we call them coplanar.
- 29.1.6 The Dot Product of Spatial VectorsThe smaller the angle between two arrows the more "in sync" they are, and a·b=|a||b|cosθ measures just how in sync.
Use three non-coplanar vectors as a "skeleton" to describe every vector in space, then set up a 3D coordinate system that turns both vectors and points into three numbers, laying the groundwork for the calculations ahead.
- 29.2.1 The Basis Theorem for Spatial VectorsPick three non-coplanar arrows as a basis, and any vector at all can be assembled as xa+yb+zc.
- 29.2.2 Orthonormal Bases and Unit VectorsChoose three mutually perpendicular unit arrows i, j, k, and assembling vectors from them is as tidy and effortless as it gets.
- 29.2.3 The 3D Coordinate SystemTreat the three edges at a corner as the x, y, and z axes, and every point in space gets its own three labels (x,y,z).
- 29.2.4 Coordinates of a Vector and Coordinate OperationsSplit an arrow along i, j, k and record it as (x,y,z); then adding, subtracting, and scaling are just done component by component.
- 29.2.5 The Coordinate Formula for the Dot ProductMultiply matching components and add them up — a·b=x₁x₂+y₁y₂+z₁z₂ — with no need to look at a picture anymore.
- 29.2.6 Formulas for Length, Distance, and AngleThe square root of the summed squared coordinate differences is the distance between two points, and cosθ falls straight out of the coordinates too.
Translate lines and planes into two handy "keys," the direction vector and the normal vector, and use vectors being collinear or perpendicular to settle parallelism and perpendicularity among lines and planes.
- 29.3.1 The Direction Vector of a LineWhichever way a line points, use an arrow d aimed the same way to stand in for the whole line.
- 29.3.2 The Normal Vector of a PlaneThe arrow n stuck perpendicular into a plane represents the entire surface, much like a flagpole stands for the ground it's planted in.
- 29.3.3 Finding a Plane's Normal VectorMake n perpendicular to two vectors lying in the plane at once, and solving n·a=0 and n·b=0 hands you n.
- 29.3.4 Testing Parallelism with VectorsDirection vectors being multiples means parallel lines, normal vectors being multiples means parallel planes — both are just collinearity speaking.
- 29.3.5 Testing Perpendicularity with VectorsA dot product of 0 between two direction vectors means the lines are perpendicular, while d parallel to n means the line is perpendicular to the plane.
- 29.3.6 Combined Proofs of Parallel and PerpendicularSwap every tricky picture problem for the question of whether dot products of d and n are zero, and the proof becomes plain arithmetic.
Use the angle formulas for direction and normal vectors to turn abstract spatial angles into coordinate calculations, working out angles between lines, between a line and a plane, and dihedral angles in a systematic way.
- 29.4.1 The Angle Between Skew LinesTake the acute angle between the two direction vectors, with cosθ=|d₁·d₂|/(|d₁||d₂|).
- 29.4.2 The Angle Between a Line and a PlaneThe line–plane angle is the "complement" of the angle between direction d and normal n, found with sinθ=|d·n|/(|d||n|).
- 29.4.3 The Dihedral Angle and Its Vector DescriptionTwo half-planes are like an open book, and the angle between them is pinned down by the angle between the two faces' normal vectors.
- 29.4.4 Finding a Dihedral Angle with Normal VectorsCompute cosθ between the two normal vectors, then read the picture to decide whether to keep that angle or take its supplement.
- 29.4.5 Combined Applications of Angle ProblemsSet up axes and write coordinates first, then find d and n, then plug into the formula, and any spatial angle comes out in one smooth pass.
Use the idea of projection to find the distance from a point to a plane, pull together the full routine of setting up axes, assigning coordinates, and finding vectors, and pave the way for the next stage where analytic geometry makes geometry fully algebraic.
- 29.5.1 The Projection of a VectorCast one arrow's shadow onto another, and the length of that shadow is the projection of a in the direction of b.
- 29.5.2 The Distance from a Point to a PlaneRun a segment from the point to the plane and measure its projection along the normal n, giving d=|AP·n|/|n|.
- 29.5.3 Point-to-Line and Line-to-Plane DistancesWhen things are parallel, take the point-to-plane distance of any one point, which reduces line-to-plane and plane-to-plane distances to a point-to-plane distance.
- 29.5.4 The General Routine for Solving by CoordinatesFind a perpendicular corner and set up axes, write coordinates, find the direction and normal vectors, then plug into the formula — five steps that handle it all.
- 29.5.5 Solving Combined Solid Geometry Problems with VectorsParallelism, perpendicularity, angles, and distances all bundle into one problem, solved in a single stroke by coordinates and dot products.
- 29.5.6 Toward Analytic GeometrySince points can be written as coordinates and relationships as equations, the next step is to use equations to draw the curves themselves.
30Analytic Geometry
Carry the vector thinking forward from the last stage, build the big idea of describing shapes with coordinates and equations, and revisit the tools of the plane.
- 30.1.1 Geometry Does the Bookkeeping: The Idea of Analytic GeometryLay a shape on grid paper, give every point a street address (x,y), and suddenly you can compute the figure with an equation.
- 30.1.2 The Distance Between Two PointsStretch a rubber band between two points, walk across then up to form a right triangle, and its length is √((x₂−x₁)²+(y₂−y₁)²).
- 30.1.3 Midpoint and the Dividing Point in a Given RatioThe exact middle of a segment sits like a seesaw's pivot, so you average the coordinates; nudge it off-center and you weight them by a ratio λ.
- 30.1.4 Vectors Back in the Plane: Direction and CoordinatesFlatten last stage's space vectors back onto the page, where a single arrow becomes (a,b) and quietly sets up the direction of a line.
Start from how steep a line is, master slope and the various forms of a line's equation, and learn to write down any line in the coordinate plane.
- 30.2.1 Angle of Inclination: How Steeply a Line TiltsThe angle α a line makes as it tips up off the horizontal axis, running from 0° up to just under 180°, captures its pose.
- 30.2.2 Slope: How Much You Climb for Each Step RightIt's the steepness of the climb: step 1 to the right and you rise k, where k=tanα=(y₂−y₁)/(x₂−x₁).
- 30.2.3 Point-Slope and Slope-Intercept FormsKnow one point and the steepness and you can draw the whole line, y−y₀=k(x−x₀), and where it lands on the vertical axis gives y=kx+b.
- 30.2.4 Two-Point and Intercept FormsPin down two points and the line is fixed; measure the footprints a and b it leaves on the two axes and write x/a+y/b=1.
- 30.2.5 The General Form Ax+By+C=0One tidy pocket that holds every line, even the vertical ones the other forms can't carry.
Study how lines relate to each other, judge their positions, find intersections, and compute distances, finishing the toolbox for lines.
- 30.3.1 Parallel and Coincident: Same SteepnessTwo lines that tilt the same and never meet are parallel, with k₁=k₂ but different intercepts; match the intercepts too and they're the very same line.
- 30.3.2 Perpendicular: Slopes Are Negative ReciprocalsPin two lines into a right angle and as one climbs the other must drop to match, so k₁·k₂=−1.
- 30.3.3 Intersection: Solving the SystemThe point where two lines meet lies on both at once, so solve their two equations together to find it.
- 30.3.4 Distance from a Point to a LineDrop the shortest perpendicular from the point to the line, and its length is d=|Ax₀+By₀+C|/√(A²+B²).
- 30.3.5 Distance Between Two Parallel LinesTwo parallel rails stay the same width forever, so pick any point on one and measure straight across to the other.
Write the set of points equidistant from a fixed point as an equation, master the circle's two forms, and judge tangency and intersection with lines and other circles.
- 30.4.1 What Path Is a Circle?Pin one end of a string and swing the other end full around, and every point at the same distance r from the center traces a circle.
- 30.4.2 The Standard Equation of a CircleJust announce the center (a,b) and radius r and you get (x−a)²+(y−b)²=r².
- 30.4.3 The General Equation of a CircleWhen it's all expanded and jumbled into x²+y²+Dx+Ey+F=0, completing the square recovers the center and radius.
- 30.4.4 Line and Circle: Apart, Tangent, or CrossingCompare the distance d from the center to the line against the radius r, and which is bigger decides whether they touch or cut through.
- 30.4.5 Positional Relationships Between Two CirclesAs two hoops drift together, the distance d between centers against the sum and difference of the radii sorts them into five cases, from fully apart to externally tangent to overlapping.
- 30.4.6 Tangent Lines and Chord LengthFrom a point outside the circle two tangent lines reach out like arms; a chord is bisected by the perpendicular, so use d and r to build a right triangle and find its length.
Introduce the ellipse from the constant sum of distances to two fixed points, build its standard equation, and read off its geometric properties.
- 30.5.1 Where the Ellipse Comes From: Two Pins and a StringLoop a slack string around two pins and trace all the way around, and the path keeps the sum of distances to the two foci fixed at 2a.
- 30.5.2 The Standard Equation of an EllipseSet the ellipse upright and centered in the coordinate plane and it reads x²/a²+y²/b²=1, with the long and short semi-axes plain to see.
- 30.5.3 The Relationship Among a, b, and cThe semi-major axis, semi-minor axis, and semi-focal distance form a right triangle, so a²=b²+c².
- 30.5.4 Eccentricity: How Flat the Ellipse IsThe farther the foci sit from the center the flatter it gets, with e=c/a between 0 and 1, looking more like a rugby ball as it nears 1.
- 30.5.5 Symmetry and Range of the EllipseAn ellipse is symmetric about both axes and the center, and its points stay boxed within ±a and ±b.
Introduce the hyperbola from the constant difference of distances to two fixed points, and master its standard equation, asymptotes, and eccentricity.
- 30.6.1 Where the Hyperbola Comes From: A Fixed DifferenceWhen the absolute difference of distances to the two foci stays fixed at 2a, you get two branches opening away from each other.
- 30.6.2 The Standard Equation of a HyperbolaSet upright it reads x²/a²−y²/b²=1, and that one-positive, one-negative minus sign is its signature.
- 30.6.3 The Relationship Among a, b, and cHere it flips: c is the longest, c²=a²+b², and the foci sit outside the real axis.
- 30.6.4 Asymptotes: Hugging Close but Never TouchingThe farther the curve reaches, the closer it hugs the two slanted lines y=±(b/a)x, like rails it approaches forever without meeting.
- 30.6.5 Eccentricity and OpeningWith e=c/a greater than 1, the larger it grows the wider the branches flare, setting how far open the hyperbola spreads.
Introduce the parabola and, through equal point-line distance and eccentricity, thread all three curves into one system, wrapping up the coordinate method.
- 30.7.1 Where the Parabola Comes From: Equal Distance to Point and LineEvery point as far from one focus as from a fixed directrix joins into an open arc, like the arc of a fountain's water.
- 30.7.2 The Standard Equation of a ParabolaPut the vertex at the origin opening to the right and it reads y²=2px, with a bigger p opening it wider.
- 30.7.3 Focus, Directrix, and Geometric PropertiesThe focus and directrix bracket the curve front and back, and its reflective property funnels every incoming signal to the focus.
- 30.7.4 The Conic Sections Are One FamilySlice a cone at different angles and out come the ellipse, parabola, and hyperbola, all springing from the same source.
- 30.7.5 Eccentricity Unifies the Three CurvesWith the same single e: less than 1 is an ellipse, equal to 1 is a parabola, and greater than 1 is a hyperbola.
- 30.7.6 Looking Back at the Coordinate Method: Geometry Is AlgebraEvery figure has been translated into an equation to compute, and this back-and-forth between shape and symbol quietly seeds the imaginary axis ahead.
31Complex Numbers
Picking up the coordinate geometry from the last stage, we start from a genuine dead end — the real number line just isn't enough — and from there introduce the imaginary unit and the idea of complex numbers
- 31.1.1 The Point You Can't Find on the Number LinePicture this: x²=−1 has no answer anywhere on the number line, like asking "which length, when squared, comes out negative" — and that pushes us to invent a brand-new number i with the rule i²=−1
- 31.1.2 The Birth of the Imaginary Unit iPicture this: we give "the square root of negative one" a name, i; it doesn't live on the number line, but the single rule i²=−1 keeps everything else working as before
- 31.1.3 What a Complex Number Looks Like: a+biPicture this: take a real number and attach a little "imaginary remainder" — every complex number can be written as a+bi, where a is the real part and b is the imaginary part
- 31.1.4 When Complex Numbers Are Equal, and How We Sort ThemPicture this: two ID cards count as the same person only when both real part and imaginary part match; b=0 makes it a real number, a=0 makes it purely imaginary
Borrowing the coordinate plane from the last stage, we give every complex number its own "address," turning it from a symbol into a visible point and arrow
- 31.2.1 The Complex Plane: an Address for Every Complex NumberPicture this: the horizontal axis tracks the real part and the vertical axis tracks the imaginary part, so a+bi lands exactly at the point (a,b)
- 31.2.2 A Complex Number Is Just an ArrowPicture this: the arrow pointing from the origin out to that point looks exactly like a vector from the last stage — that's the geometric identity of a complex number
- 31.2.3 Modulus: How Long the Arrow IsPicture this: measure the length of the arrow and you have the modulus, |a+bi|=√(a²+b²), which can never be negative
- 31.2.4 Argument: How Far the Arrow Has TurnedPicture this: the angle the arrow makes with the positive real axis is its argument, and one full turn brings it back, so arguments differing by 2π point the same way
We build the rules for adding, subtracting, multiplying, and dividing — learning both to grind through them algebraically and to see what addition and subtraction mean geometrically
- 31.3.1 Addition and Subtraction: Joining Arrows Tip to TailPicture this: just like with vectors, add real part to real part and imaginary to imaginary; geometrically you're sliding arrows together, (a+bi)±(c+di)
- 31.3.2 Multiplication: Just Substitute i²=−1Picture this: multiply it out like expanding brackets, and every time you hit i² swap in −1, and naturally you land on (ac−bd)+(ad+bc)i
- 31.3.3 The Cycle of Powers of iPicture this: i, −1, −i, 1 roll around like four seasons, repeating every fourth step, so for iⁿ you only need the remainder of n divided by 4
- 31.3.4 Division: Rationalizing the Denominator, Complex-StylePicture this: to clear the i out of the denominator, multiply top and bottom by the conjugate to "wash" the denominator into a real number, then split into real and imaginary parts
We get a systematic feel for the conjugate and how it relates to the modulus, treating it as a central tool for working with complex numbers
- 31.4.1 The Conjugate: Flip It in the MirrorPicture this: reflect the arrow across the real axis so b becomes −b; the conjugate of a+bi is written a−bi, real part unchanged, imaginary part flipped
- 31.4.2 Conjugate Meets Modulus: z·z̄=|z|²Picture this: multiply a number by its own mirror image and the imaginary parts cancel out perfectly, leaving a real number that equals exactly the modulus squared
- 31.4.3 Modulus and Operations: |z₁z₂|=|z₁||z₂|Picture this: multiply two arrows and their lengths simply multiply too; the modulus of a product is the product of the moduli, and conjugation spreads across operations the same way
- 31.4.4 Solving Problems with Conjugate and ModulusPicture this: faced with an equation mixing z and z̄, either write z as x+yi or cleverly use z·z̄=|z|² to turn a tangle into something simple
We upgrade a complex number from a "coordinate address" to a "length plus direction," revealing that multiplication and division are really rotation and scaling — and laying groundwork for deeper material to come
- 31.5.1 Trigonometric Form: Length Paired with DirectionPicture this: rather than report coordinates, report "how long the arrow is and which way it turns"; a+bi=r(cosθ+i sinθ), where r is the modulus and θ is the argument
- 31.5.2 What Multiplication Really Does: Add the Angles, Multiply the LengthsPicture this: multiply two complex numbers and the arrow lengths multiply while the arguments add — multiplying by a complex number is just one rotation plus one scaling
- 31.5.3 What Division Really Does: Subtract the AnglesPicture this: division runs it backwards, dividing the lengths and subtracting the arguments; a complex number packages "rotate and scale" into a single multiply or divide
- 31.5.4 Multiplying by i Is a 90° Counterclockwise TurnPicture this: i has modulus 1 and argument 90°, so multiplying by it leaves the arrow's length alone and just turns it a right angle — the simplest algebraic switch for rotation
We use complex numbers to complete the roots of real-coefficient equations, reclaiming what a negative discriminant used to cost us, and paving the way for counting and polynomials ahead
- 31.6.1 A Negative Discriminant Is No Longer a Dead EndPicture this: when Δ<0 a negative slips under the square root, which we once called "no solution," but now one stroke of i drops the roots right onto the complex plane
- 31.6.2 Complex Roots of Real-Coefficient EquationsPicture this: rewrite √(negative) as i√(positive) and a quadratic always yields a pair of complex roots, x=(−b±i√(−Δ))/(2a)
- 31.6.3 Complex Roots Come in Pairs: Conjugate RootsPicture this: the complex roots of a real-coefficient equation always arrive as a matched mirror pair; if a+bi is a root, then a−bi must be one too
- 31.6.4 The Full Picture of a Quadratic's RootsPicture this: whether Δ is positive, zero, or negative, a quadratic always has two roots (counting a repeated one); complex numbers finally make "the number of roots" perfectly uniform
- 31.6.5 Onward to Counting and PolynomialsPicture this: now that no roots ever go missing, it's natural to start counting "how many kinds, how many of each" — and the next stop is counting, permutations, and combinations
Ⅵ · Counting, Probability & Statistics
32Counting, Permutations, Combinations & the Binomial Theorem
Build the two basic habits of mind — add when you sort into kinds, multiply when you go step by step — and learn to tell whether a task is a 'sort' or a 'sequence', laying the foundation for permutations and combinations
- 32.1.1 Why Learn to CountComing back from complex numbers to plain 'counting': when you could list everything but there's just too much, you need a way to find the count N without listing it all
- 32.1.2 The Addition Principle (Counting by Cases)To travel out of town you can take a train or a bus — the choices don't overlap and each is counted on its own, so the total is just the cases added up, N = m1 + m2 + ...
- 32.1.3 The Multiplication Principle (Counting by Steps)Pairing a shirt with pants takes two steps, and you only have an outfit once both are picked, so the count is the steps multiplied, N = m1 x m2 x ...
- 32.1.4 'Cases' or 'Steps'?If one choice finishes the job, add; if it takes a relay of choices to finish, multiply — the whole question is whether the task is an 'or' or an 'and'
- 32.1.5 Tree Diagrams and ListingDraw every choice as a branching tree, count the leaves, then look back and check that your addition and multiplication gave the same answer
- 32.1.6 Putting Both Principles TogetherLicense plates, passwords, route problems: first split into broad cases, then go step by step inside each case, nesting cases and steps together
Understand counting where the selection is ordered, master the permutation formula and factorials, and learn to spot permutation problems that carry restrictions
- 32.2.1 What a Permutation IsPicking a few people and then lining them up in order, where who stands in front and who stands behind makes a different arrangement — ordered selection is called a permutation
- 32.2.2 The Permutation Formula A(n,m)To line up m people chosen from n, the first spot has n choices, the second n-1, and so on; multiply them together and you get the permutation count A(n,m)
- 32.2.3 Factorials and Full PermutationsLine up all n in a row and multiply from n all the way down to 1, written n! — the fullest permutation there is
- 32.2.4 Permutations with Restrictions (I)Someone must stand at the front, someone can't be at the back: settle the 'special spot or special person' first, then arrange everyone else
- 32.2.5 Permutations with Restrictions (II): Together and ApartIf two must be side by side, 'bundle' them into one block; if they mustn't touch, arrange the others first and then 'slot' them into the gaps
- 32.2.6 Permutation Problems CombinedForming numbers from digits, arrangements with a fixed order, and the like: translate each restriction into 'which step comes first', then apply the permutation formula
Tell combinations apart from permutations, master the combination formula and its two key properties, and solve combinations with restrictions plus simple distribution problems
- 32.3.1 What a Combination IsHandshakes, picking a team: once the group is chosen you're done, and who came first makes no difference — selection without order is called a combination
- 32.3.2 The Combination Formula C(n,m)A combination is just a permutation with the internal line-up repeats removed, so divide A(n,m) by m! and you get C(n,m)
- 32.3.3 Two Properties of CombinationsChoosing m is the same as 'rejecting the other n-m', so C(n,m) = C(n,n-m); add to that C(n+1,m) = C(n,m) + C(n,m-1)
- 32.3.4 Telling Permutations and Combinations ApartJust ask 'if I shuffle the order, is it still the same selection?' — if yes it's a combination, if no it's a permutation
- 32.3.5 Combinations with Restrictions'Must include this person', 'at most this many': solve it either by sorting into cases directly or by 'total minus the ones that don't qualify'
- 32.3.6 Grouping and Distribution ProblemsSplit people into piles and then hand out tasks: when the piles are equal in size, divide by the factorial of the number of piles so identical piles aren't counted twice
Use combination numbers to explain the expansion of (a+b)^n, master the general term and the pattern of the coefficients, and connect it to Pascal's triangle, paving the way for the binomial distribution in the next stage on probability
- 32.4.1 Starting from Expanding (a+b)^nMultiply the n copies of (a+b) one by one, and each term is one way of choosing 'a or b' out of every bracket
- 32.4.2 The Binomial TheoremThe coefficient of each term is just the combination number C(n,k) for 'how many of the n brackets you pick b from' — that is the binomial expansion
- 32.4.3 The General Term T(k+1)To pull out one term directly, use the general term T(k+1) = C(n,k)·a^(n-k)·b^k, without expanding the whole string
- 32.4.4 Properties of the Binomial CoefficientsThe coefficients are small at the ends and big in the middle, symmetric left to right, and adding them all up gives exactly 2^n (set a=b=1)
- 32.4.5 Pascal's TriangleA number triangle where 'each entry is the sum of the two above its shoulders', and every row is precisely the line of coefficients for (a+b)^n
- 32.4.6 Finding a Specific Term or CoefficientFor the constant term or the coefficient of a certain power, set the exponent of x in the general term equal to your target, solve for k, and substitute back
- 32.4.7 Toward Probability: Expanding Success and FailureIf a and b stand for the probabilities of 'success' and 'failure', the terms of (p+q)^n foreshadow the binomial distribution coming in the next stage
33Probability
Start from random phenomena, where "you can't be sure of the outcome," and build up probability as a number that measures "how likely" something is
- 33.1.1 Random Phenomena and the Sample SpaceBefore you flip a coin nobody can tell you heads or tails, so toss every possible outcome into one bag, and that bag is the sample space Ω
- 33.1.2 Random Events"Rolling an even number" is just a description of an outcome, and it picks out a chunk of what's in the bag, that chunk being the event A
- 33.1.3 Frequency: Try a Thousand Times and Watch the RatioFlip a coin a thousand times and see what share comes up heads, and that measured share is the frequency f = m/n, which steadies as the count grows
- 33.1.4 From Frequency to ProbabilityThe frequency wobbles around but eventually settles near one number, and that "bullseye" is the probability P(A), the event's built-in likelihood
When every outcome is equally likely, turn finding a probability into the "counting" you just practiced
- 33.2.1 Equally Likely Outcomes and the Classical ModelA fair die gives all six faces an equal shot, and this "everyone's on equal footing" setup is the classical probability model
- 33.2.2 The Classical Probability FormulaDivide the number of favorable outcomes by the number of all outcomes to get the probability, P(A) = (count in A) / (count in Ω)
- 33.2.3 Counting Probabilities with Permutations and CombinationsTake the number of ways to draw two red balls over the number of ways to draw any two, bringing your earlier C and A straight in as numerator and denominator
- 33.2.4 A Glimpse of Geometric ProbabilityToss a pebble onto a field and ask how likely it lands in the pond, swapping a ratio of counts for a ratio of areas, P = (shaded area) / (total area)
Treat events like sets you can add and subtract, building the addition formula and the language of mutually exclusive and complementary events
- 33.3.1 Intersection, Union, and Complement of Events"Rainy and windy" is the intersection A∩B, "rainy or windy" is the union A∪B, and "not rainy" is the complement Ā
- 33.3.2 Mutually Exclusive and Complementary Events"Rolling a 3" and "rolling a 5" can't both happen, so they're mutually exclusive, while "winning" and "not winning" leave no in-between, so they're complementary
- 33.3.3 The Addition Formula for ProbabilityFor either of two events happening, add their probabilities and then take back the overlap, P(A∪B) = P(A) + P(B) − P(A∩B)
- 33.3.4 Finding Probabilities via the ComplementWhen the direct count is a headache, count the opposite instead, so "at least one" becomes 1 minus "none at all," P(A) = 1 − P(Ā)
Tell apart "they don't affect each other" from "knowing one, now look at the other," and master the multiplication, conditional, and total probability rules
- 33.4.1 Independent EventsToday's coin flip remembers nothing of yesterday's, so when events don't affect each other you just multiply, P(A∩B) = P(A)P(B)
- 33.4.2 Conditional ProbabilityYou already know the roll was even and now ask whether it's a 6, shrinking the stage down to just the even outcomes, P(B|A) = P(A∩B)/P(A)
- 33.4.3 The Multiplication FormulaDraw one card without putting it back, then draw another, and chain the two step probabilities together, P(A∩B) = P(A)·P(B|A)
- 33.4.4 The Total Probability FormulaA ball might come from box one, two, or three, so weight each box by its share and add it all up, P(B) = ΣP(Aᵢ)P(B|Aᵢ)
- 33.4.5 A First Look at Bayes' FormulaGiven that you drew a white ball, work backward to which box it most likely came from, tracing the cause from the result P(Aᵢ|B)
When direct calculation is hard, use random numbers and repeated trials to "try out" the probability, echoing the frequency idea
- 33.5.1 Generating Random NumbersA calculator or spinner coughs up a random number, as if flipping a coin or rolling a die on our behalf
- 33.5.2 The Idea Behind Random SimulationUse random numbers to mimic a real trial and run it many times, then count the success rate and let frequency close in on the probability
- 33.5.3 Estimating Geometric Probability by SimulationScatter points at random inside a square and watch the fraction landing in a circle, which can actually estimate π, and gets sharper the more you scatter
Stick number labels on random outcomes, and use a distribution table to fully capture how likely each value is
- 33.6.1 Discrete Random VariablesRecord things like "the sum of two dice" or "how many defects you drew" as a single number X that changes at random
- 33.6.2 The Distribution TableLay out a table giving the probability of X taking each value, like drawing the random variable a "menu of possibilities"
- 33.6.3 Properties of the Distribution TableEvery probability on the menu is non-negative, and they all add up to exactly 1, no more and no less
Sum up a random variable with two numbers: where it lands on average and how much it swings, paving the way for the sample numbers in statistics
- 33.7.1 Mathematical ExpectationOver many rounds, how much you take home per round on average is the probability-weighted mean, the expectation E(X) = Σxᵢpᵢ
- 33.7.2 Properties of ExpectationDouble every outcome and toss in a fixed bonus, and the expectation shifts and scales right along with it, E(aX+b) = aE(X)+b
- 33.7.3 Variance and Standard DeviationWhether outcomes hug the average or swing wildly is measured by the squared deviation from the mean D(X), and its square root is the standard deviation
- 33.7.4 Properties and Uses of VarianceBetween two plans with the same expectation, the one with smaller variance is steadier, handy for weighing risk when you decide
Get to know three classic distribution models, then step from discrete to continuous, laying the groundwork for population distributions in statistics
- 33.8.1 The Two-Point DistributionA single trial ends in only success or failure, with X taking 0 or 1, the simplest building block of every distribution
- 33.8.2 The Binomial DistributionRepeat the same trial independently n times and count the successes, X~B(n,p), with the probability being exactly one term of a binomial expansion
- 33.8.3 Expectation and Variance of the Binomial DistributionOver n repeats each with success rate p, you average np successes, plugging straight into E(X)=np and D(X)=np(1−p)
- 33.8.4 The Hypergeometric DistributionGrab n items in one handful from N that include M defects, and count the defects you got without replacing any, with the distribution computed from combinations
- 33.8.5 Continuous Random Variables and the Density CurveHeight and weight can be any real number, so probability no longer sits at single points but is read as the area under a curve
- 33.8.6 The Normal DistributionCountless tiny factors stack into a bell-shaped curve, high in the middle and low at the sides, with its mean μ and standard deviation σ setting its width and position
- 33.8.7 The 3σ Rule and Applications of the Normal DistributionAlmost all the data falls within three σ on either side of the mean, and anything outside counts as "unusual," quietly setting the stage for statistical judgment
34Statistics
Build the habit of answering questions with data, understand where data comes from and how to sample without bias, laying the raw-material foundation for everything that follows
- 34.1.1 Why We Do StatisticsProbability calculates the unknown tomorrow, statistics reads the yesterday that already happened, turning a pile of numbers x₁,…,xₙ into a conclusion
- 34.1.2 Population, Individual, and SampleTo know if the whole pot of soup is salty, you taste one spoonful; that spoonful is the sample, the whole pot is the population N
- 34.1.3 Census and SamplingCalling the entire roll is a census, asking just a few is sampling; the sample size is written n, and the bigger it is the closer to the truth
- 34.1.4 Simple Random SamplingWrite everyone's name on a slip, crumple them up, and draw lots so each person has an equal chance, that's simple random sampling
- 34.1.5 Stratified and Systematic SamplingSplitting into men and women first and then drawing in proportion is stratified, while taking every k-th one is systematic sampling
- 34.1.6 Representativeness and BiasAsking about height only at a basketball court runs high; if the sample is skewed, the conclusion is skewed too
Learn to pick the right chart for the data type, read it correctly, and stay alert to misleading graphics, making messy data visible at a glance
- 34.2.1 Frequency and Frequency Distribution TablesSort scores into ranges and count heads, the count in each range is the frequency and its share is the relative frequency f = count/n
- 34.2.2 Bar, Pie, and Line ChartsUse bars to compare amounts, pies to show shares, and lines to track trends, each with its own job
- 34.2.3 Frequency Distribution HistogramsGroup continuous data and stack each group into a bar whose area is that group's relative frequency, with the total area equal to 1
- 34.2.4 Choosing and Reading ChartsFirst ask whether the data is categorical or continuous, whether you're comparing or tracking a trend, then pick the chart that speaks
- 34.2.5 Charts That LieWhen the axis starts at 90, a tiny gap looks like a chasm; before trusting a chart, check whether the scale has been tampered with
Represent a pile of numbers with a single number, master the mean, median, and mode along with their trade-offs, and understand when to use which
- 34.3.1 The MeanPour everyone's money together and split it evenly, and each person's share is the mean x̄ = (Σxᵢ)/n
- 34.3.2 The MedianLine everyone up and take the value of the one standing in the middle; with an even count, average the middle two
- 34.3.3 The ModeThe value that shows up most often; the best-selling shoe size is the mode of shoe sizes
- 34.3.4 Comparing and Choosing Among the ThreeWhen a billionaire walks into a small shop, the average salary shoots up, and in such extreme-value cases the median is more reliable
- 34.3.5 The Weighted MeanWhen the final exam counts for more than the homework, you score by weight; x̄ = Σwᵢxᵢ, where each weight wᵢ is how much that piece counts
Describe a distribution from both position and variability, master quartiles, range, variance, and standard deviation, paving the way for box plots and spread comparisons
- 34.4.1 PercentilesWhen a ranking says you "beat 90% of people," that dividing line is the 90th percentile
- 34.4.2 Quartiles and the Interquartile RangeThe three points that split the line into four equal parts are Q₁, Q₂, Q₃, and the span of the middle half is IQR = Q₃ − Q₁
- 34.4.3 Box PlotsDraw a box with two whiskers to show five key numbers, revealing the distribution's shape and spread at a single glance
- 34.4.4 Range and the Feel of DispersionTwo classes can share the same average, one perfectly even and the other swinging high and low; the largest minus the smallest is the range
- 34.4.5 Variance and Standard DeviationHow far each number sits from the mean, squared and then averaged, is the variance s², and its square root is the standard deviation s
- 34.4.6 Comparing Stability of DataWho do you pick? When averages are close, the one with the smaller standard deviation is steadier and more dependable
Understand why a small sample can speak for a large population, build the ideas of estimation and sampling variability, and prepare the ground for correlation and testing
- 34.5.1 Estimating the Population from a SampleTaste one spoonful to judge the whole pot; the sample mean x̄ is used to estimate the population mean μ
- 34.5.2 Estimating Probability from Sample FrequencyThe more times you flip a coin, the closer the frequency of heads sits to the true probability 0.5, and that's the stability of frequency
- 34.5.3 Variability in EstimatesEvery round of sampling comes out a little different; the larger the sample, the steadier the estimate and the closer to the true value
- 34.5.4 The Bell-Shaped Intuition of the Normal DistributionHeights and weights tend to be high in the middle and low on both sides, a bell shape written N(μ,σ²), where μ sets the center and σ sets the width
- 34.5.5 The 3σ RuleNearly all normal data falls within three standard deviations above and below the mean, and anything beyond that is extremely rare
Move from a single variable to paired data, describe correlation, fit a line, use it to predict, and understand that correlation is not causation
- 34.6.1 Paired Data and Scatter PlotsPair each height with a weight and plot them as dots, and the drift of the dot cloud reveals the relationship between the two
- 34.6.2 Positive, Negative, and No CorrelationDots drifting up to the right are positive, sliding down to the right are negative, and scattered every which way means almost no relationship
- 34.6.3 The Correlation Coefficient rUse a single number from −1 to 1 to measure how strong the linear relationship is, and the closer to either end the more it looks like a straight line
- 34.6.4 The Simple Linear Regression EquationThread the line that hugs the dot cloud best, ŷ = b̂x + â, so the total deviation is as small as possible
- 34.6.5 The Idea of Least SquaresMake the sum of the squared vertical distances from each point to the line as small as possible, and only then does the line fairly represent the trend
- 34.6.6 Predicting with Regression and ResidualsPlug in x to get the prediction ŷ, the gap between actual and predicted is the residual, and small residuals mean a good fit
- 34.6.7 Correlation Is Not CausationIce cream sales and drownings both rise together, but the real culprit is summer; a link doesn't have to mean a cause
Handle paired categorical data, use a contingency table and chi-square to judge whether two traits are related, and carry the inference mindset toward logical judgment, leading into the next stage
- 34.7.1 Categorical Variables and Contingency TablesPair "smokes or not" with "sick or not" and fill them into a two-by-two grid, and that grid is called a contingency table
- 34.7.2 Spotting Association from FrequenciesIf the smoking group has a clearly higher illness rate, it makes you suspect the two are not entirely unrelated
- 34.7.3 The Meaning of Independence and Expected CountsIf they were truly unrelated, each cell's count should fall out naturally in proportion, and that is the expected count
- 34.7.4 The Chi-Square Statistic χ²Square the gap between actual and expected and add it all up into one number χ², which grows larger the bigger the gaps are
- 34.7.5 The Test of Independence and the VerdictOnce χ² exceeds the critical value you can confidently say the two are related, and this step of "reaching a verdict" leads into the language of logic
Ⅶ · Logic, Analysis & Beyond
35Logic & Quantifiers
Gather the experience of drawing conclusions from data into something that can be judged true or false — a statement — and set up the basic objects and language of logic.
- 35.1.1 What Is a StatementOnly a sentence you can firmly call true or false counts: a claim with a definite truth value is a statement p.
- 35.1.2 Hypothesis and ConclusionSplit a sentence into "if... then...": the first half is the hypothesis p, the second half is the conclusion q.
- 35.1.3 True Statements and False StatementsOne counterexample is enough to knock it down: p is true only when no counterexample exists and it all holds together.
- 35.1.4 Rewriting a StatementSay the same thing a new way: tidy an everyday sentence into the standard form "if p then q."
Capture the relationship between hypothesis and conclusion with "is it enough to have it, can you do without it" — and learn to identify all three kinds of conditions.
- 35.2.1 Sufficient Condition"Having it is enough": if p holds, q is guaranteed, written p⇒q, so p is a sufficient condition for q.
- 35.2.2 Necessary Condition"Without it, nothing works": for q to hold you must first have p, q⇒p, so p is a necessary condition for q.
- 35.2.3 Necessary and Sufficient Condition"Have it and you're set, lose it and you can't": p and q guarantee each other, p⇔q, fully equivalent.
- 35.2.4 Seeing Conditions as SetsA small circle inside a big one: the p-region sits inside the q-region, so the small one is sufficient and the big one is necessary.
- 35.2.5 Identifying the Type of ConditionCheck each direction once: see whether p⇒q, q⇒p, or both, and call it sufficient, necessary, or both.
Tell apart "everyone does" from "at least one does," and bring in the two quantifiers and their symbolic forms.
- 35.3.1 The Universal Quantifier"All of them, every single one, hold": every x has the property, written with ∀ for a universal statement.
- 35.3.2 The Existential Quantifier"You can find at least one": some x meets the condition, written with ∃ for an existential statement.
- 35.3.3 Judging Quantified Statements True or FalseUniversal must cover every case, existential needs just one: hunt for a gap, or find a single witness.
- 35.3.4 Writing Quantified Statements in SymbolsTranslate the words into symbols: ∀x∈M, p(x) and ∃x∈M, p(x) each capture one kind.
Master the negation of statements, especially the pattern for statements with one quantifier, laying the foundation for rigorous reasoning ahead.
- 35.4.1 The Negation of a StatementFlip the whole sentence on its head: the negation of p is written ¬p, and its truth value is exactly opposite.
- 35.4.2 Negating a Universal Statement"Not everyone does" means "someone doesn't": ∀ becomes ∃, and the conclusion turns negative.
- 35.4.3 Negating an Existential Statement"There isn't a single one" means "every one fails": ∃ becomes ∀, and the conclusion turns negative.
- 35.4.4 Common Traps in NegationDon't just swap the quantifier and forget the conclusion: the negation of "all greater than" is "some not greater than," not "less than."
Link conditions, quantifiers, and negation into reasoning — using counterexamples, the contrapositive, and proof by contradiction — and pave the way for rigorous arguments about sequences and function limits.
- 35.5.1 Converse and ContrapositiveTurn it around, then flip and turn it: from "if p then q," the contrapositive "if ¬q then ¬p" has the same truth value.
- 35.5.2 Putting Equivalent Statements to WorkWhen the direct proof is hard, prove the contrapositive instead: switch to an equivalent phrasing and the path often clears at once.
- 35.5.3 First Steps in Proof by ContradictionPretend the conclusion is false: derive a contradiction, which shows the conclusion had to hold all along.
- 35.5.4 Disproving with a CounterexampleWant to overturn "it always holds"? One example where it fails is enough.
- 35.5.5 The "Getting Infinitely Close" Intuition in Reasoning"As close as you like" is itself a claim: a warm-up for pinning down limits with ε in the next stage.
36Limits, Derivatives & Their Applications
Build the gut feeling of "getting infinitely close," turning the vague "closer and closer" into a precise, workable language of limits that sets the stage for rates of change.
- 36.1.1 From "Necessary and Sufficient" to "Infinitely Close"Last stage you judged whether things were close enough; now let that closeness keep going forever, and that's the arrow in x→a.
- 36.1.2 The Limit of a SequenceA string of numbers crowds in toward a single spot like stepping stones converging on one point, and that landing spot is written lim aₙ.
- 36.1.3 The Limit of a Function at a PointWalk in toward x=a along the curve from both the left and the right, and if the height heads for the same value either way, that value is lim f(x).
- 36.1.4 The Arithmetic of LimitsWhen two quantities each settle toward their own value, their sum, difference, product, and quotient settle right along with them, so you can break a limit into pieces.
- 36.1.5 Continuity: Never Lifting the PenIf your pen never jumps off the paper as you draw, the function's value lands exactly on its limit, and we say f is continuous there.
Use a limit to squeeze "the average pace over a stretch" down into "the pace at this very instant," leading naturally to the heart of the derivative.
- 36.2.1 The Average Rate of ChangeTake how much height you gained over a stretch of time and divide, and you get the average steepness Δy/Δx.
- 36.2.2 The Slope of a Secant LineThe slope of the straight line joining two points on a curve is exactly the average rate of change across that stretch.
- 36.2.3 Letting the Gap Shrink to ZeroAs the two points slide together, the secant tips over into a tangent, and the average rate squeezes into the instant rate as Δx→0.
- 36.2.4 Instant Rate of Change and the Birth of the DerivativeHow fast is it changing right now? It's the limit of Δy/Δx as Δx→0, written f′(x₀).
- 36.2.5 Instant Speed: Our First ApplicationThe number on the speedometer at one instant is just position differentiated with respect to time, s′(t).
Promote the derivative from "a number at one point" to "a brand-new function," and nail down its geometric meaning once and for all: the slope of the tangent.
- 36.3.1 The Definition of the DerivativeWrite the whole approaching process as one formula: f′(x₀)=lim[f(x₀+Δx)−f(x₀)]/Δx.
- 36.3.2 The Slope of the Tangent LineZoom in on the curve until it looks like a straight line, and the steepness of that little piece is f′(x₀).
- 36.3.3 Finding the Equation of a TangentOnce you know the point of contact and the slope f′(x₀), the point-slope form hands you the tangent in one step.
- 36.3.4 The Derivative Function: The Derivative Is a Function TooEvery point has its own steepness, so connect them all up and you get a new function f′(x).
- 36.3.5 How Differentiability and Continuity RelateA curve with a sharp corner has no single tangent, so being differentiable always means being continuous, but being continuous doesn't guarantee differentiable.
Turn point-by-point limit-taking into looking things up and applying rules, so differentiation is fast and accurate across common functions and nested structures.
- 36.4.1 Derivatives of the Basic FunctionsPowers, exponentials, logarithms, and trig functions each have a fixed "steepness template," like (xⁿ)′=nxⁿ⁻¹, ready to use once you know them.
- 36.4.2 The Sum and Difference RuleStack a few curves together and their steepnesses just add up too: (u±v)′=u′±v′.
- 36.4.3 The Product RuleWhen two quantities multiply, let each one change in turn and add the results: (uv)′=u′v+uv′.
- 36.4.4 The Quotient RuleThe steepness of a fraction follows a fixed recipe: (u/v)′=(u′v−uv′)/v².
- 36.4.5 Composite Functions and the Chain RuleA nested function is like gears in a chain where the turning rates multiply: the outer derivative times the inner derivative.
- 36.4.6 Mixed Differentiation PracticeFirst spot whether the structure is a sum, a product, or a nesting, then apply the matching rules in order and you'll land f′ every time.
Read a function's rise and fall from the sign of its derivative, pin down its peaks, valleys, and extremes, and see the shape of a curve at a glance.
- 36.5.1 The Sign of the Derivative and Increasing or DecreasingWhen the steepness points up (f′>0) you're climbing, and when it points down (f′<0) you're descending.
- 36.5.2 Finding Intervals of Increase and DecreaseSolve f′(x)>0 and f′(x)<0, then mark off the uphill stretches and the downhill stretches.
- 36.5.3 Extrema: Peaks and ValleysTurning from uphill to downhill gives a peak, the reverse gives a valley, and at the turn f′=0.
- 36.5.4 How to Identify ExtremaWatch how f′ changes sign as it crosses zero: plus to minus is a maximum, minus to plus is a minimum.
- 36.5.5 Extremes on a Closed IntervalThe overall highest and lowest can only sit at a peak, a valley, or one of the two endpoints, so just compare them one by one.
- 36.5.6 Sketching the Graph of a FunctionCombine the rises, the falls, and the extrema, then connect the dots to trace the curve's true path.
Translate real-world "cheapest" and "largest" problems into finding extremes of a function, use derivatives to deliver the best plan, and look back over the whole journey to open the door ahead.
- 36.6.1 Modeling an Optimization ProblemTurn "least material used, most profit earned" into variables, then write down the objective function and the range it's allowed to take.
- 36.6.2 Geometric Extremes: Area and VolumeHow do you fold a sheet of metal into the roomiest box? Differentiate the volume function and find the extreme point.
- 36.6.3 The Best Choice in Everyday LifeThe point where cost, time, or return works out best is right where the derivative is zero.
- 36.6.4 A Summary of the Optimization MethodModel it, differentiate it, find the extreme, check the range: four steps walk you to the best answer at f′=0.
- 36.6.5 Looking Back and the Door to MoreFrom approaching to optimizing, the first shape of calculus has appeared, and from here the next leg is yours to take in whatever direction draws you in.
37Beyond (Optional)
Starting from the picture of finding an area or adding things up, build the idea of the definite integral and bridge it to the derivative, so the differentiation you just learned flows naturally into integration.
- 37.1.1 From Derivatives to IntegralsIf you know the speed and want the distance, you run the previous stage's differentiation backwards, and that's exactly what integration does.
- 37.1.2 Area Under a Curve and Riemann SumsSlice a curved region into countless thin strips and add them up, the narrower the strips the closer you get, and that limit is written ∫f(x)dx.
- 37.1.3 Defining the Definite Integral and Its Geometric MeaningThe limit of those thin-strip areas is the definite integral, and ∫ₐᵇf(x)dx equals the net area under the curve.
- 37.1.4 Basic Properties of the Definite IntegralArea can be split, stacked, and have constants pulled out, giving rules like ∫ₐᵇ = ∫ₐᶜ + ∫ᶜᵇ.
- 37.1.5 Antiderivatives and the Indefinite IntegralFind a function whose derivative is exactly f and you've found its antiderivative, written ∫f(x)dx = F(x) + C.
- 37.1.6 The Fundamental Theorem of CalculusAstonishingly, finding an area is just taking the difference of an antiderivative, ∫ₐᵇf = F(b) − F(a), so differentiation and integration undo each other.
- 37.1.7 Using Integrals to Find Area and DistanceTreat the fundamental theorem as a tool and the area under a curve or the total distance of variable-speed motion drops right out.
Express direction and transformation with vectors and matrices, understand linear transformations, determinants, and solving systems of equations, and lay the groundwork for higher dimensions and abstract structures.
- 37.2.1 The Intuition Behind Vector SpacesA collection of arrows you can freely add and stretch makes up a vector space, closed under addition and scaling.
- 37.2.2 Linear Combinations and a BasisA few basic arrows build the whole space, and any vector can be written as their linear combination Σcᵢvᵢ.
- 37.2.3 Matrices as Linear TransformationsA matrix is like a stretch-and-rotate machine: feed in a vector x and it comes out as Ax, with the grid neatly reshaped.
- 37.2.4 Matrix Multiplication and OperationsDoing two transformations one after another corresponds to multiplying their matrices AB, and the order can't just be swapped.
- 37.2.5 Determinants and Area ScalingHow many times the transformation enlarges the area of a unit square is exactly the determinant det(A).
- 37.2.6 Solving Linear Systems with MatricesWrite several equations as Ax = b, and if the determinant isn't zero there's a unique solution, found by elimination or the inverse matrix.
Starting from divisibility and remainders among the integers, meet the prime numbers as building blocks and congruence as a clock arithmetic, and feel the beauty of pure structure.
- 37.3.1 Divisibility and FactorsIf a can be split evenly into parts of size b, we say b divides a, written b∣a, with remainder zero.
- 37.3.2 Primes and the Fundamental Theorem of ArithmeticPrimes are the building blocks that can't be broken down further, and every integer splits uniquely into a product of primes, like 12 = 2²·3.
- 37.3.3 Greatest Common Divisor and the Euclidean AlgorithmKeep dividing the larger number by the smaller and taking the remainder until it hits zero, and you've found gcd(a, b).
- 37.3.4 Congruence: Arithmetic on a ClockJust as a clock loops back to 1 after 12, numbers sort into classes by their remainder mod m, and a ≡ b (mod m) means they're congruent.
- 37.3.5 Working with Congruences and Their UsesCongruences can be added and multiplied, and modular arithmetic is handy for checking divisibility, figuring out the day of the week, or verifying a last digit.
Make the why behind a result clear, master the language of mathematical statements and the three main styles of proof, and move from being able to compute to being able to prove.
- 37.4.1 Statements, Conditions, and the Contrapositive"If p then q" is the basic sentence of mathematics, and it's true or false in exactly the same cases as its contrapositive "if not q then not p."
- 37.4.2 Universal and Existential Quantifiers"For all" is written ∀ and "there exists" is written ∃, and negating a statement swaps the two and flips what follows.
- 37.4.3 Mathematical InductionKnock over the first domino and make sure each one topples the next, and they all fall, which is induction on n.
- 37.4.4 Proof by ContradictionAssume the conclusion is false, push until you hit a contradiction, and that forces the original to be true, like proving √2 is irrational.
- 37.4.5 Construction and CounterexamplesTo prove something exists, build one with your own hands, and to knock down a "for all" claim you only need a single counterexample.
Return to the limits you met earlier and use the ε–δ language to say exactly what "getting infinitely close" means, closing out the whole path and opening the door to higher mathematics.
- 37.5.1 Why Make Limits Rigorous"Getting closer and closer" is too vague, so we have to ask how close and how close can we demand, until there's no room for doubt.
- 37.5.2 The ε–N Definition of a Sequence LimitFor any error ε you name, there's some index N beyond which every term lands within that error, and that's the precise meaning of aₙ → A.
- 37.5.3 The ε–δ Definition of a Function LimitTo keep f(x) within ε of L, it's enough to keep x close enough to a (within δ), and that's the hard-edged definition of a limit.
- 37.5.4 The Rigorous Definition of ContinuityWhen the limit at a point equals the function's value there, you can draw straight through without lifting your pen, and the function is continuous at that point.
- 37.5.5 Global Properties of Continuous FunctionsA function continuous on a closed interval must attain a maximum and minimum and must hit every value in between, with the intermediate value and extreme value theorems bringing the journey to a close.