Mathematics › 📐 Grade 8 › Irrational Numbers & Rational Approximation

Irrational Numbers & Rational Approximation

Some numbers refuse to settle into a repeating pattern of digits — but you can still corner them between two fractions as tightly as you like.

Grade 8Pre-Algebra / Algebra 1

Irrational Numbers & Rational Approximation — illustration

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The big idea: Every fraction, written as a decimal, either stops (0.5) or falls into an endlessly repeating pattern (0.333…). Numbers like √2 and π do neither — their digits run forever with no repeating block, which means they cannot be written as a ratio of two integers at all. These irrational numbers are not vague or unknowable, though: by squaring guesses and narrowing the gap, you can trap one between two fractions as closely as you want, which is exactly how a calculator produces its decimal approximation.

🎯 By the end, you'll be able to

Define rational numbers as terminating or repeating decimals, and irrational numbers as neither
Explain why √2 and π are irrational, using them as the canonical examples
Locate an irrational number on the number line using nested intervals and bracketing by squaring
Distinguish an exact irrational value from a calculator's rounded decimal approximation

📎 You should already know

The Pythagorean theorem
Square roots

What every fraction's decimal looks like

Take any fraction — a ratio of two integers, like ½ or ⅓ — and turn it into a decimal by long division. Something predictable always happens: either the division ends cleanly (½ = 0.5) or it falls into a block of digits that repeats forever (⅓ = 0.333…). There is no third option for a fraction. A number whose decimal does one of these two things is called rational — literally, expressible as a ratio of integers.

🔑 Rational vs. irrational

A number is rational if it can be written as a ratio of two integers a/b (b ≠ 0); its decimal always terminates or repeats. A number is irrational if it cannot be written that way — its decimal goes on forever with no repeating block. The two canonical irrational numbers you'll meet constantly are √2 and π.

Where √2 comes from — and why it can't be a fraction

You've already met √2: it's the length of the hypotenuse of a right triangle with both legs equal to 1, by the Pythagorean theorem (1² + 1² = 2, so the hypotenuse is √2). It is a perfectly real, measurable length — you could draw it. Yet it cannot be written as any fraction a/b in lowest terms. Informally: if a/b squared equalled 2, then a² = 2b², which forces a to be even; but writing a as 2k and simplifying forces b to be even too — so a/b was never actually in lowest terms after all, a contradiction. No fraction, however carefully chosen, hits exactly 2 when squared.

⚠️ A calculator shows an approximation, not the number

Your calculator displays √2 as 1.414213562 and stops — but that is a rounded approximation, not the exact value. The true decimal expansion of √2 continues forever without repeating. Never write √2 = 1.414213562 with an equals sign; write √2 ≈ 1.414213562, or leave it as √2 exactly.

Cornering √2 between two integers

Even without a calculator, you can pin down √2's location on the number line by squaring guesses and comparing to 2. This method — bracketing by squaring — never gives up on precision; it just keeps narrowing the net.

\[ 1^{2}=1 < 2 < 4=2^{2}\ \Longrightarrow\ 1<\sqrt{2}<2 \]

Since 1² is below 2 and 2² is above 2, √2 itself must sit strictly between 1 and 2.

📝 Worked example: Narrow √2 to one decimal place using nested intervals.

Start from 1 < √2 < 2. Test tenths between them: try 1.4 and 1.5.
Square each: 1.4² = 1.96, which is less than 2; 1.5² = 2.25, which is more than 2.
So 1.96 < 2 < 2.25 puts √2 strictly between 1.4 and 1.5.

✓ 1.4 < √2 < 1.5 — the interval has been nested down to one decimal place.

🎮 Square-Root Bracket Zoom LIVE

Zoom in on the number line, squaring each new guess to trap √2 inside an ever-narrower interval.

📝 Worked example: Continue narrowing √2 to two decimal places, starting from 1.4 < √2 < 1.5.

Test hundredths inside that interval: try 1.41 and 1.42.
Square each: 1.41² = 1.9881, which is less than 2; 1.42² = 2.0164, which is more than 2.
So 1.9881 < 2 < 2.0164 places √2 strictly between 1.41 and 1.42, and the process could continue forever without √2's decimal ever settling into a repeating pattern.

✓ 1.41 < √2 < 1.42, matching a calculator's √2 ≈ 1.41421…

π: irrational for a different reason

π, the ratio of a circle's circumference to its diameter, is irrational for reasons far harder to prove than √2's — but the consequence is identical: 3.14 and 22/7 are both useful rational approximations, and neither is exactly π. The same bracketing idea applies: 3 < π < 4, then 3.1 < π < 3.2, then 3.14 < π < 3.15, and so on, closing in without ever arriving.

✨ Bracketing works for any irrational square root

The same nested-interval technique locates √3, √5, or any non-perfect-square root: find the two consecutive integers whose squares straddle the target, then test tenths, then hundredths, tightening the bracket one decimal place at a time. It never “finishes” — it just gets arbitrarily precise, which is exactly what an irrational number demands.

Check your understanding

1. Which of these is irrational?

0.75 terminates and 0.333… and 7/9 are repeating decimals, so all three are rational. √2's decimal never terminates or repeats, so it is irrational.

2. Why can't √2 be written as a fraction a/b?

The classic proof by contradiction: if a/b (lowest terms) squared equals 2, algebra forces both a and b to share a factor of 2, contradicting the assumption that a/b was already fully reduced.

3. A calculator displays √2 as 1.414213562. What is the most accurate way to describe this?

√2's true decimal expansion never ends and never repeats, so any calculator display is necessarily a rounded approximation, not the exact number.

4. Using bracketing by squaring, between which two consecutive integers does √5 lie?

2² = 4 is less than 5, and 3² = 9 is more than 5, so 4 < 5 < 9 places √5 strictly between 2 and 3.

5. Which statement about π is correct?

π is irrational — its decimal never terminates or repeats. 3.14 and 22/7 are convenient rational numbers that approximate it closely but are not equal to it.

✅ Key takeaways

A rational number's decimal always terminates or repeats; an irrational number's decimal does neither, no matter how far you compute.
√2 (the hypotenuse from the Pythagorean theorem with legs 1 and 1) and π are the canonical irrational numbers.
√2 cannot equal any fraction a/b: assuming it does forces a contradiction about a/b being in lowest terms.
Nested intervals bracket an irrational number by squaring guesses — 1 < √2 < 2, then 1.4 < √2 < 1.5, then 1.41 < √2 < 1.42 — narrowing forever without ever landing exactly.
A calculator's decimal display, like 1.414213562 for √2, is always a rounded approximation, never the exact irrational value.