Why √2 Cannot Be a Fraction

We will use proof by contradiction: assume √2 can be written as a ratio of integers, then show that assumption forces an impossibility.

Important: the irrational number here is √2, not 2. The number 2 is rational because 2 = 2/1.
Goal
√2 ≠ a / b

for every pair of integers a, b with b ≠ 0.

1

Assume the opposite

Suppose √2 = a/b, and choose the fraction in lowest terms. So a and b have no common factor.

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2

Square both sides

2 = a² / b²
a² = 2b²

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3

Therefore a is even

Since a² = 2b², the number a² is even. A square can be even only if the original integer is even. So write a = 2k.

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4

Substitute a = 2k

a² = 2b²
(2k)² = 2b²
4k² = 2b²
b² = 2k²

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5

Therefore b is even too

Because b² = 2k², b² is even, so b must also be even.

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!

Contradiction

Both a and b are divisible by 2. But we started by saying a/b was already in lowest terms.