Rational & Irrational
The number line is mostly holes — and √2 falls straight through one
Free trail · Math · ages 13–14 · 6 stations
The Number Families
Every number you have ever written belongs to a family, and the families nest inside one another like boxes. The interesting question is not what is in each box — it is what the box cannot hold.
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Naturals are 1, 2, 3, … Integers add zero and the negatives: …−2, −1, 0, 1, 2… Integers are closed under + − ×, but not under ÷: 3 ÷ 2 escapes the box.
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A number you can write as p/q where p and q are integers and q ≠ 0. It has nothing to do with being sensible — the word comes from ratio.
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A rational number's decimal always terminates (0.375) or recurs (0.428571428571…). It can never wander forever without a pattern.
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Do the long division of p ÷ q. Every remainder is smaller than q, so after at most q steps a remainder must repeat — and once a remainder repeats, the digits repeat forever. That is a genuine proof, not a hand-wave.
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The Number Families
Flip cards
Every number you have ever written belongs to a family, and the families nest inside one another like boxes. The interesting question is not what is in each box — it is what the box cannot hold.
Naturals sit inside integers sit inside rationals. Flip every card to see where each family ends.
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Rational or Irrational?
Sorting baskets
The test is simple to state and easy to get wrong: can this number be written as p/q with p and q integers? A square-root sign does not automatically mean irrational, and a decimal that looks messy is not automatically irrational either.
Do not trust how a number looks — work out what its decimal actually does.
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Proving √2 Is Irrational
This is proof by contradiction, and it is about 2,400 years old. We assume √2 is rational, then follow perfectly ordinary algebra until we reach something impossible. Since the algebra is faultless, the assumption must have been the problem. One fact you need first: if n² is even then n is even, because odd × odd is always odd.
Assume the opposite, follow the logic, and watch it destroy itself. Tap the steps in order.
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Chasing √2 With Fractions
Slider explorer
If √2 is irrational, no fraction can ever equal it. But fractions can get arbitrarily close. Slide the denominator and watch the best possible approximation, and its error, as they close in.
Slide the denominator up and watch the best fraction creep closer to √2 — but never arrive.
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The Classic Traps
Try it
Real numbers are full of statements that feel right. Each round below is a place where intuition and mathematics disagree — and mathematics wins.
Five statements that sound obvious. Only one in each round survives inspection.
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Practice Zone
Practice Zone
Rational and irrational numbers, from every angle.
Ten questions on this topic — a proper practice set, fresh every time.
Free printable worksheet
Ten questions from this topic on one page, with an answer key at the bottom — a PDF you can print for homework, a car journey or a classroom. No account needed and nothing to sign up for. The Practice Zone in the app builds a fresh set every visit; this is a fixed set you can hold.
More Math for ages 13–14
🔺 Exponents & Standard Form · ⬛ Squares, Cubes & Roots · 🔤 Algebraic Expressions · ⚖️ Linear Equations · ✖️ Simultaneous Equations · ↔️ Inequalities · 🧩 Factorising · 🟰 Algebraic Identities
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