medium difficulty7 min

Proof of irrationality of sqrt(2)

Assume √2 = p/q in lowest terms. After squaring, 2q² = p², so p is even. That leads to q also being even, which contradicts the fraction being in lowest terms.

In one line: Even numerator leads to even denominator; lowest terms cannot survive.

Practice This Concept

Understand it

This is a proof by contradiction. If both p and q become divisible by 2, then the fraction was not in lowest terms after all. That contradiction shows √2 cannot be rational.

Example

If p is even, write p = 2k. Then 4k² = 2q², so q² = 2k² and q is also even.

Formula and rule

Assume √2 = p/q in lowest terms, then 2q² = p².

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Let √2 = p/q in lowest terms.

1Then p² = 2q², so p is even.
2Write p = 2k.
3Substituting gives 4k² = 2q², so q² = 2k² and q is even.
4Both being even is impossible in lowest terms.

Method you can reuse

1Assume √2 = p/q in lowest terms.
2Square both sides.
3Show p is even.
4Substitute p = 2k.
5Show q is even.
6Contradict lowest terms.

Visual tip

A proof ladder with the steps assumption, square, evenness, contradiction, and conclusion works well.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why does the proof of √2 fail once both p and q are even?

Think first, then show answer

Because a fraction with both numerator and denominator even is not in lowest terms, so the starting assumption must be wrong.

Common wrong answer

Saying p is even, so √2 is rational,.

Why it loses marks: The same argument forces q to be even too.

Don't mix this up

Students stop after proving p is even and forget to show that q is even too.

Exam tip

In a proof by contradiction, always end with the words 'this contradicts lowest terms'.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Prove √2 is irrational, or identify the contradiction in the proof.

1
Enough for 1 mark
Assume √2 = p/q in lowest terms. After squaring, 2q² = p², so p is even. That leads to q also being even, which contradicts the fraction being in lowest terms.
2
Add for 2 marks
Example: If p is even, write p = 2k. Then 4k² = 2q², so q² = 2k² and q is also even.
3
Add for 3 marks
This is a proof by contradiction.

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