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Prove that 2 is not a rational number or there is no rational whose square is 2.

Prove that 2 is not a rational number or there is no rational whose square is 2.

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1 Answers

Latika Leekha
askIITians Faculty 165 Points
7 years ago
Let us suppose that there exists a rational number whose square is 2.
Let m/n be a rational number whose square is 2.
Then there exist integers m and n such that 2 = m2/n2
‘m’ and ‘n’ are assumed to be such that they are in the lowest terms. This means that m and n don’t have any common factor except 1.
Then m2 = 2n2
Now 2n2 is even so m2 is even and hence m is even.
Hence, we can write m = 2l, where l is an integer.
Then, m2 = 4l2 = 2n2
This gives n2 = 2l2
Again n2 is even means n is even.
Hence, we have obtained that both m and n are even.
This gives that m/n is not in lowest terms.
But this is a contradiction as we had assumed that m/n is in lowest terms.
2 is not rational.
Thanks & Regards
Latika Leekha
askIITians Faculty

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