A-Level Mathematics · Proof · A6

Irrationality of √2.

Follow the classic contradiction proof that √2 cannot be written as a fraction of integers.

A6 Proof10 MinutesAnimated lesson

WATCH THE IDEA MOVE

Animated
explanation.

Pause, replay and follow every logical step at your own pace.

01 / KEY IDEA

Writing √2 as a fraction in lowest terms leads to both numerator and denominator being even.

02 / LEARNING OUTCOMES
  • Use lowest-term assumptions
  • Reason about even squares
  • Complete a contradiction proof

WORKED EXAMPLE

Follow every
reason.

Prove that √2 is irrational.

01

Assume √2 = a/b in lowest terms. Then a² = 2b², so a is even.

02

Write a = 2k. Substitution gives b² = 2k², so b is also even.

03

Then a and b share a factor 2, contradicting lowest terms. Therefore √2 is irrational.

QUICK CHECK

Test the
logic.

Choose the statement that describes a valid mathematical proof.

What makes a mathematical proof valid?