Irrational number

A rational number can be written as one integer over another, like 3/4 or -5/1. An irrational number cannot. Its decimal form goes on forever without a repeating pattern. Examples are √2, √3, √5 and π.

The chapter proves that numbers such as √2 are irrational. It uses proof by contradiction. You assume the number is rational, then show this leads to something impossible.

A small example

Prove that 2 + 3√3 is irrational, given that √3 is irrational.

• Assume 2 + 3√3 = a/b, where a and b are integers and b is not 0.
• Rearrange: √3 = (a - 2b)/3b.
• The right side is built from integers, so it is rational. That would make √3 rational.
• This goes against the given fact. So the assumption was wrong, and 2 + 3√3 is irrational.

Where marks go

This is Question 26 of CBSE's 2026-27 Maths Standard sample paper, worth 3 marks. The board's Maths Standard marking scheme for 2026-27 gives 1 mark for the rearranged form, √3 = (a - 2b)/3b. It gives 1 mark for saying that this form is rational. The last mark is for the conclusion: the assumption was wrong, so 2 + 3√3 is irrational.

So two of the three marks are for reasons, not for algebra. Each sentence of the proof is a step the examiner looks for. That is step marking, applied to a proof.

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