Zoom in between 2 and 3: the irrational √5 ≈ 2.236 lands just to the left of the rational 2.3. Same line, same rules — we just have to find out who is bigger.
Picture the real number line as a single, unbroken ruler. The fractions you already trust — 2, 34, 2.3 — sit on it as tidy, nameable points. But so do the "won't-come-out" numbers from Lesson 6.4, like √5 and π. The wonderful news of this lesson is that nothing changes when irrationals join the party: the four operations, the comparison rules, even the idea of a variable all work exactly as before. By the end you will add, subtract, multiply and divide any reals, decide which of two is bigger, pin an irrational between two decimals as tightly as you like, and let a single letter stand for any real number at all.
The color habit for this lesson: a rational number (a fraction or terminating/repeating decimal) is teal; an irrational number is magenta; an estimate, or a pair of bounds that traps a number, is amber; and the number line for comparing is blue.
6.6.1 The four operations on real numbers
Here is the central promise of the real numbers, stated plainly: the rules you learned for adding, subtracting, multiplying and dividing rationals work, without a single change, for every real number. When √2 and π step onto the line, they do not get their own private arithmetic. They obey the very same playbook — the commutative law, the associative law, the distributive law — that 3 and 12 obey.
That means you may treat an irrational symbol like √3 as a single, unbreakable "number-object." To compute 2 + √3 you simply… leave it. There is no tidier way to write it, because √3 is irrational and 2 is rational, and a rational plus an irrational is always irrational. So the exact answer is the expression 2 + √3 itself. If a context needs a decimal, you may then approximate: 2 + √3 ≈ 3.732. The exact form and the rounded form are two different things, and keeping them apart is a habit worth building now.
Multiplication behaves the same way. Three copies of √5 added together is 3·√5 = 3√5 — the distributive law collecting like terms, exactly as 3 apples plus 3 apples is 6 apples. And from Lesson 6.5 you can fold roots together: √2·√8 = √16 = 4, a rational! Multiplying two irrationals can even land you back on a tame teal point.
The same three laws govern both worlds. Swap a teal rational for a magenta irrational and the law does not flinch — that is what "one number system" means.
(b)3·√5 means √5 + √5 + √5 = 3√5 ≈ 6.708. The 3 stays out front; it does not dive under the root.
(c)√2·√8 = √2·8 = √16 = 4. Two irrationals multiplied to a perfect square give a rational.
(d)2(1 + √3) = 2 + 2√3, by the distributive law — multiply both terms inside.
Watch out
A decimal like 1.732 is not√3 — it is a rounded stand-in. Writing "2 + √3 = 3.732" with an equals sign is wrong; use ≈. The exact value 2 + √3 never terminates and never repeats.
Also: √2 + √3 is not√5. You can only merge roots under multiplication (√a·√b = √(ab)), never under addition. Check with a calculator: 1.414 + 1.732 = 3.146, while √5 ≈ 2.236.
Key idea
Irrationals obey the same arithmetic as rationals. Keep an irrational answer in exact form (like 2 + √3 or 3√5); switch to a rounded decimal only when a real-world situation calls for it — and then write ≈, not =.
🎮 Try itBuild a real number, exact vs. rounded
Choose a whole number, an operation, and a root. Watch the exact form on the left and a rounded decimal on the right — and notice they are never quite the same.
Whole number k2
Operation
Radicand n3
6.6.2 Comparing the size of real numbers
On the number line, the rule for "bigger" could not be simpler: the number farther to the right is the larger one. This is true for every kind of real number — fractions, decimals, negatives, irrationals — because they all share one line. So comparing √5 with 2.3 is really just asking: which point sits farther right?
The trouble is that √5 doesn't announce its decimal. We need a way to decide without trusting a calculator's rounding. Here is the cleanest trick. For two non-negative numbers, squaring preserves their order: if a and b are both ≥ 0, then a < b exactly when a² < b². Squaring is "order-friendly" on the non-negative side of the line. So compare the squares instead — and the square of a square root is wonderfully easy.
Square both: (√5)² = 5, and (2.3)² = 5.29. Since 5 < 5.29, we conclude √5 < 2.3. The square root sits just to the left of 2.3 — exactly what the hero picture showed.
Squaring turns a hard comparison into an easy one: compare the areas5 and 5.29. The bigger area belongs to the bigger side, so 2.3 > √5.
Watch out
The squaring trick is for non-negative numbers only. With negatives it flips: −3 < −2, yet (−3)² = 9 > (−2)² = 4. Farther left can have a bigger square. When negatives are in play, fall back to the number line: farther right wins.
Worked example — trap it instead
Prefer not to square? Trap √5 in a shrinking range. Since 2.2² = 4.84 < 5 and 2.3² = 5.29 > 5, we know 2.2 < √5 < 2.3. The upper trap is 2.3 itself, so √5 < 2.3. Same conclusion, a different road.
🎮 Try itCompare two reals by squaring
Pick a radicand for √n and a decimal d. The widget squares both, places them on the line, and reports <, > or =.
√n, choose n5
decimal d2.3
6.6.3 Estimating irrationals and choosing precision
An irrational number has an exact home on the line, but no terminating decimal name. To estimate one is to squeeze it between two decimals you do understand, then tighten the squeeze until you have all the accuracy you need. The tool is the family of perfect squares, and the idea is the same order-friendly squaring from the last section, run in reverse.
Take √7. First trap it between consecutive whole numbers using perfect squares: 4 < 7 < 9, and since √4 = 2 and √9 = 3, we get 2 < √7 < 3. Now narrow to tenths by testing squares of decimals: 2.6² = 6.76 (too small) and 2.7² = 7.29 (too big), so 2.6 < √7 < 2.7 — that is, √7 ≈ 2.6…. Want more? Add a layer: 2.64² = 6.9696 and 2.65² = 7.0225, so 2.64 < √7 < 2.65, giving √7 ≈ 2.64….
Each new layer shrinks the trap by a factor of ten and buys one more correct digit. You stop when the precision fits the need: for sketching a point, the nearest whole number is plenty; for cutting a board, the nearest tenth; for engineering, more. Estimating well is partly knowing when to stop.
Squeezing √7: the amber trap narrows from a whole unit, to a tenth, to a hundredth. Each zoom pins down one more digit: 2…, 2.6…, 2.64…
Don't round too early. If you call √7 ≈ 2.6 and then square, you get 6.76, not 7. A rounded value is close, not exact — the more arithmetic you pile on a rounded number, the more the error grows. Keep roots exact through a calculation and round only at the very end.
🎮 Try itThe squeeze estimator
Pick a non-square n, then add layers of zoom. Watch the amber interval shrink as the bounds close in on √n.
n =7
precision
6.6.4 Letting a letter stand for any real number
Step back and admire what we have built. There is one number line, and one set of operation rules that every point on it obeys — rational and irrational alike. That uniformity is exactly what makes algebra possible. Because a rule like "double it and add one" works the same whether the input is 3, 12, √2, or π, we can give that input a name — a letter — and let the letter stand for any real number at all.
Write x for "some real number, we're not saying which." Then the expression 2x + 1 is a machine: feed it a real number and it returns a real number. Feed it 3 and out comes 7. Feed it √2 and out comes 2√2 + 1 ≈ 3.828. Feed it π and out comes 2π + 1 ≈ 7.283. The rule is fixed; only the input varies. A letter that roams over the whole real line is called a variable, and an expression built from variables and operations is an algebraic expression — the subject of Stage 7.
This is why everything in Stage 6 mattered. Powers, roots, the birth of irrationals, the comparison and estimation tools — they all confirmed that the reals form one well-behaved system. Now a single letter can carry that whole system at once, and the laws you proved on numbers (commutative, associative, distributive) become the laws you'll use to rewrite expressions.
The expression 2x + 1 as a machine. Any real x goes in; a real number comes out. The letter x is a placeholder for the entire number line at once — the doorway to algebra.
Key idea
Because every real number obeys the same operations, a letter such as x or a can stand for any real number. An expression like 2x + 1 describes a rule, not a single number — that step from "a number" to "any number" is the start of algebra.
🎮 Try itLet a letter be any real number
Slide x across the real line — through integers, a fraction, √2, and π — and watch 2x + 1 evaluate live. The rule never changes; only the input does.
x
★ The big ideas, in one breath
The real numbers are one system: irrationals add, subtract, multiply and divide by the very same laws as rationals, so an answer like 2 + √3 is best kept exact and only rounded (with ≈) when needed. To compare two non-negative reals, square them — squaring keeps the order — or trap the harder one between decimals. To estimate an irrational, squeeze it with perfect squares, one decimal place at a time, stopping at the precision the job demands. And because every real obeys these same rules, a single letter can stand for any of them — which is exactly where algebra begins.
Coming up next — Stage 7
You just let a letter stand for any real number. In Stage 7 · Algebraic Expressions we put letters to work: building expressions, evaluating them, and rewriting them with the commutative, associative, and distributive laws you trusted here on plain numbers.
✎ Exercises 6.6
Work each one out first, then open the answer to check your thinking. Keep answers exact unless asked to estimate.
Simplify exactly: 3 + √2 + 4.
Show answer
Combine the rationals: 3 + 4 = 7. The lone √2 stays. Exact answer: 7 + √2.
True or false: √2 + √2 = √4. Explain.
Show answer
False. Roots merge under multiplication, not addition. √2 + √2 = 2√2 ≈ 2.828, while √4 = 2. They are not equal.
Compute exactly: 2·√3 + √3.
Show answer
Two √3's plus one more √3 is three of them: 3√3 ≈ 5.196. (This is the distributive law: (2 + 1)√3.)
Multiply: √3·√12. Is the result rational or irrational?
Show answer
√3·√12 = √36 = 6. The result is rational — two irrationals multiplied to a perfect square.
Which is larger, √10 or 3.1? Show your reasoning.
Show answer
Both are non-negative, so square: (√10)² = 10 and 3.1² = 9.61. Since 10 > 9.61, we have √10 > 3.1.
Place these in order, least to greatest: 2.5, √6, 52.
Show answer
52 = 2.5, so two of them are equal. Square √6: 6 vs 2.5² = 6.25, so √6 < 2.5. Order: √6 < 2.5 = 52.
Between which two consecutive whole numbers does √30 lie? Then estimate to the nearest tenth.
Show answer
25 < 30 < 36, so 5 < √30 < 6. Tenths: 5.4² = 29.16 (small), 5.5² = 30.25 (big), so 5.4 < √30 < 5.5. Nearest tenth: 5.5 (since 30 is closer to 30.25 than to 29.16). Indeed √30 ≈ 5.477.
Estimate √7 to the nearest hundredth using perfect squares, and check by squaring your estimate.
Show answer
Tenths: 2.6² = 6.76, 2.7² = 7.29 ⇒ between 2.6 and 2.7. Hundredths: 2.64² = 6.9696, 2.65² = 7.0225 ⇒ between 2.64 and 2.65. Since 7 is nearer 7.0225… check both: 6.9696 is 0.0304 below 7, 7.0225 is 0.0225 above, so √7 ≈ 2.65. (Exact value ≈ 2.6458.)
Let x = √2. Evaluate 2x + 1 exactly, then as a decimal to three places.
A letter a stands for "any real number." A student claims a² is always larger than a. Give one real value of a that breaks this claim.
Show answer
Many work. Take a = 12: then a² = 14, which is smaller than 12. (Also a = 1 gives a² = a, equal not larger; and a = 0 the same.) The claim fails for 0 ≤ a ≤ 1. This is exactly the kind of "for which reals is it true?" question algebra exists to answer.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson serves 8.NS.A.2 (compare the size of irrational numbers and approximate them by rational numbers — e.g. locate √2 and π on a number line and estimate their value) and 8.EE.A.2 (use square-root symbols and evaluate; the perfect-square cases). It then deliberately bridges to algebra (6.EE/7.EE) by letting a letter stand for any real number and evaluating an expression like 2x + 1.
The #1 misconception: treating a rounded decimal as if it were the irrational — writing "√3 = 1.732" with an equals sign, or squaring a rounded value and expecting the original number back. Antidote: insist on the distinction between exact and approximate — keep roots exact through a calculation, always write ≈ for a rounding, and verify estimates by squaring the bounds (not the rounded value) against the radicand. A close runner-up is over-merging roots (√2 + √3 ≠ √5); a quick decimal check defeats it on the spot.
eastmath.com · Stage 6 · 6.6 Operating with & Estimating Reals · Intuition before notation
eastmath.com · 6.6 Operating with and Estimating Real Numbers · 6.6.3 Estimating irrationals and choosing precision