Swing the diagonal of a 1 × 1 square down onto the line: it lands at √2 ≈ 1.414, an exact point that is no fraction at all.
Draw a square one unit on a side and stretch a string across its diagonal. How long is that string? By the rule you already know, its length squared is 12 + 12 = 2, so the diagonal is the number whose square is 2 — that is √2. Reach for a fraction to name it and you will reach forever: √2 is not 12 of anything. In this lesson a brand-new kind of number is born — the irrational number — and once we let it join the family, the number line finally has no holes left in it.
The steady color habit for this lesson: a rational number (one you can write as a fraction) is teal, an irrational number is magenta, the number line and its ticks are blue, and a distance / absolute value is amber.
6.4.1 A root that won't come out: the mystery of √2
You already know how to undo a square. The side of a square of area 9 is √9 = 3; the side of area 16 is 4. These come out clean because 9 and 16 are perfect squares. But what about the diagonal of a unit square? Its length is √2 — the number whose square is exactly 2 — and 2 is not a perfect square. So what is this number?
Try to pin it down by squeezing. Since 12 = 1 and 22 = 4, the number √2 sits between 1 and 2. Test 1.4: 1.42 = 1.96, a touch too small. Test 1.5: 1.52 = 2.25, a touch too big. So √2 is between 1.4 and 1.5. Squeeze again with two decimals: 1.412 = 1.9881 and 1.422 = 2.0164, so it lies between 1.41 and 1.42. You can keep doing this all day — 1.414, 1.4142, 1.41421… — and the squeeze never closes onto a tick. The digits run on forever.
The diagonal of a unit square is √2. Squeezing it between numbers whose squares straddle 2 traps it ever more tightly, but the trap never snaps shut on a clean decimal.
Compare this with an old friend, 13. Its decimal also runs forever: 0.3333…. But look — it repeats. The same block "3" comes back over and over, locked in a pattern, because the long division 1 ÷ 3 keeps giving the same remainder. A repeating decimal is just a fraction in disguise. The decimal of √2 is different in kind: 1.41421356237… shows no repeating block, ever. That is the clue that √2 is no fraction at all.
Watch out
"It goes on forever" is not what makes a number irrational. 13 = 0.333… goes on forever too, yet it is perfectly rational. The dividing line is whether the digits eventually repeat a block (rational) or never settle into any pattern (irrational).
Why √2 can't be a fraction — gently
Suppose √2were a fraction pq in lowest terms. Squaring gives p² = 2q², so p² is even, which forces p itself to be even, say p = 2k. Then 4k² = 2q², so q² = 2k² — meaning q is even too. But p and q can't both be even if the fraction was in lowest terms! The assumption collapses. So no fraction squares to 2: √2 truly is a new kind of number.
🎮 Try itReveal the digits — does it repeat?
Add digits to each number and watch what happens. 1/3 and 1/4 settle down; √2 keeps surprising you.
Now we can name what we have found. An irrational number is a number whose decimal expansion never ends and never repeats — and, equivalently, a number that cannot be written as a fractionpq of two integers. The two descriptions are two faces of the same coin: every fraction produces a decimal that terminates or repeats, so a decimal that does neither cannot have come from a fraction.
The most famous irrational of all is π = 3.14159265358979…, the ratio of a circle's circumference to its diameter. Its digits march on with no pattern anyone will ever find, because there is none to find. The roots of most whole numbers join the club too: √2, √3, √5, √6 are all irrational. Only the roots of perfect squares come out clean and stay rational, like √9 = 3 or √25 = 5.
Every rational decimal either stops or repeats a block. An irrational decimal does neither — its digits never lock into a pattern.
Watch out
Not every square root is irrational, and not every "ugly" number is either. √9 = 3 is rational; 2.5 and 0.375 are rational because they stop. A calculator showing √2 = 1.4142136 has simply rounded — it is not the whole number, only a snapshot of an endless, pattern-free tail.
🎮 Try itRational or irrational? Sort them.
Tap each number to drop it in the bin you think it belongs to. Watch the feedback — some are sneaky.
6.4.3 What real numbers are and how they split
Put the two families side by side — the rationals (every fraction, which covers all the integers and all the terminating or repeating decimals) and the brand-new irrationals (√2, π, and their endless kin) — and together they form the real numbers. The real numbers are every number you can mark as a point on a line. From here on, "number" without qualification means real number.
The reals split into exactly two non-overlapping families, sorted by a single yes/no question: can this number be written as a fractionpq? Yes → it's rational. No → it's irrational. Every real number lands in one bin or the other, never both, never neither. Inside the rationals there is more structure: the integers (…, −2, −1, 0, 1, 2, …), and inside those the whole numbers (0, 1, 2, …); but plenty of rationals like 34 are not integers at all.
The family tree of the real numbers: one clean split into rationals and irrationals, with the integers and whole numbers nested inside the rationals.
Key idea
The real numbers = the rationals together with the irrationals. The fence between them is a single question: fraction or not?
🎮 Try itExplore the family tree
Tap a family to see who lives there — and which bigger families they also belong to.
6.4.4 Real numbers match the number line one-to-one
Here is the payoff. Before today, the number line had invisible holes. The point exactly one diagonal-of-a-unit-square out from zero had no name — no fraction landed there. The irrationals are precisely the numbers that fill those holes. Now every point on the line is some real number, and every real number is some point. This perfect pairing has a name: the reals and the line are in one-to-one correspondence. There are no gaps left.
And √2 is not floating vaguely "somewhere near 1.4." It has an exact address, fixed by geometry: build the unit square, swing its diagonal down with a compass, and the arc meets the line at the one true point √2. The decimal 1.41421356… is just our attempt to read off that fixed point digit by digit; the point itself was there all along, pinned by the construction.
The compass arc carries the diagonal down to its exact home. √2 lives between 1 and 2; π lives just past 3. No point is missing.
Key idea
The real number line is gap-free: each point ↔ exactly one real number. Irrationals are not "approximate" — they are exact points we can only write approximately.
🎮 Try itConstruct √2 on the line
Drag the slider to swing the diagonal arc down onto the line. Stop where it lands and read the value.
Swing the arc
6.4.5 Opposites and absolute value of real numbers
Everything you learned about opposites and absolute value back in Stage 5 still works perfectly — now including the irrationals. Two points mirrored across the origin are opposites. The opposite of √2 is −√2; one sits about 1.414 to the right of zero, the other about 1.414 to the left. Adding a number to its opposite always gives 0: √2 + (−√2) = 0.
The absolute value|x| is the distance from 0 to the point x — and a distance is never negative. So |√2| = √2 and |−√2| = √2 as well: a point and its opposite are the same distance from zero, just on opposite sides. This is exactly the rule you know for integers — |3| = 3 and |−3| = 3 — applied without change to an irrational. The number being "endless and pattern-free" makes no difference to how far it sits from the origin.
√2 and −√2 are mirror images across the origin; both sit the same distance √2 from zero, so |√2| = |−√2| = √2.
Watch out
Absolute value strips a number down to its distance, so the answer is never negative — but it does not simply "make the number positive by deleting the minus sign" in a careless way. |−√2| = √2 because √2 is the distance; the bars report how far, not which side.
🎮 Try itOpposite & absolute value on the line
Slide a point — including right onto √2 — and watch its opposite and its distance to zero update together.
Point x
★ The big ideas, in one breath
The diagonal of a unit square is √2, a number whose decimal runs on forever without ever repeating — so it is no fraction, and we call it irrational; π and most square roots are irrational too, while terminating and repeating decimals (and all integers) are rational. Bundle the two families and you get the real numbers, split cleanly by the one question fraction or not? Those reals fill the number line with no gaps — every point ↔ exactly one real number — and the old tools still work: √2 and −√2 are opposites, both a distance √2 from the origin.
Coming up next — 6.5
Now that irrationals like √2 are real, honest numbers, we'll learn to operate on square-root expressions — simplifying things like √12 = 2√3 and tidying 1√2 into √22.
✎ Exercises 6.4
Work each one out first, then open the answer to check your thinking.
Write 13 as a decimal. Does it terminate, repeat, or neither? Is it rational or irrational?
Show answer
1/3 = 0.333… The block "3" repeats forever. A repeating decimal is a fraction in disguise, so 1/3 is rational.
Is √9 rational or irrational? What about √2?
Show answer
√9 = 3, a whole number, so it is rational. √2 = 1.41421… never ends and never repeats, so it is irrational. The root of a perfect square is rational; the root of a non-perfect-square is irrational.
True or false: "Every decimal that goes on forever is irrational." Explain.
Show answer
False.0.333… goes on forever yet is rational because it repeats. The test for irrational is "never ends and never repeats."
Between which two consecutive whole numbers does √5 lie?
Show answer
22 = 4 and 32 = 9, and 5 is between 4 and 9, so √5 is between 2 and 3. (In fact √5 ≈ 2.236.)
Sort into "rational" and "irrational": 2.5, π, √16, √3, 78.
What is the opposite of √2? What is √2 + (the opposite of √2)?
Show answer
The opposite of √2 is −√2 (mirror image across zero). A number plus its opposite is always 0: √2 + (−√2) = 0.
Evaluate |−√2| and |√2|. Are they equal? Why?
Show answer
Both equal √2. Absolute value is the distance to the origin, and a number and its opposite sit the same distance from 0 — just on opposite sides. So |−√2| = |√2| = √2 ≈ 1.414.
Use squeezing to locate √2 to two decimal places. Show the two squares that trap it.
Show answer
1.412 = 1.9881 (just under 2) and 1.422 = 2.0164 (just over 2). Since these straddle 2, √2 is between 1.41 and 1.42 — that is, √2 ≈ 1.41 (rounding to two places).
A square has area 2 square units. Explain why its side length is irrational, but a square of area 4 has a rational side.
Show answer
Side of area-2 square is √2, and 2 is not a perfect square, so the side is irrational. Side of area-4 square is √4 = 2, and 4 is a perfect square (2²), so the side is the rational number 2. A clean side appears exactly when the area is a perfect square.
Decide if each statement is true or false: (a) Every integer is a real number. (b) Every real number is rational. (c) √2 is an exact point on the number line, not just an approximation.
Show answer
(a) True — integers are rationals (e.g. 3 = 3/1), and rationals are reals. (b) False — √2 and π are real but not rational. (c) True — the diagonal construction pins √2 to one exact point; the decimal 1.41421… is just how we read that fixed point off, digit by digit.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
This lesson serves the Common Core grade-8 number-system standards: 8.NS.A.1 — know that numbers that are not rational are called irrational, understand informally that every number has a decimal expansion, and that rational numbers are exactly those whose expansions terminate or eventually repeat; 8.NS.A.2 — use rational approximations of irrationals to compare their sizes and locate them approximately on a number line; and 8.EE.A.2 — knowing that √2 is irrational. The #1 misconception is believing that "a decimal that never ends" means "irrational." Antidote: put 1/3 = 0.333… right next to √2 = 1.41421356… and ask only one question — does a block of digits repeat? 1/3 repeats (rational); √2 never settles (irrational). Reinforce that a calculator's 1.4142136 is a rounded snapshot, not the number itself, and that √2 names one exact point fixed by the unit-square diagonal.
eastmath.com · Stage 6 · 6.4 Irrational & Real Numbers · Intuition before notation
eastmath.com · 6.4 The Birth of Irrational and Real Numbers · 6.4.1 A root that won't come out: the mystery of √2