Question

prove that root 2 is an irrational number chirkootmaths

Answer 1

Hi friend
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Your answer
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To prove : - √2 is an irrational number.

Proof : -

Let us assume √2 to be rational.

Then, there exists co-primes a and b, such that

√2 = a/b

=> (√2 )² = a²/b² [squaring both sides.]

=> 2 = a²/b²

=> a² = 2b² ......(I)

Here , 2 divides a².

So, 2 divides a.

Again,
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Let, a = 2c for some integer c.

Putting this value of a in eq.(I), we get,

(2c)² = 2b²

=> 4c² = 2b²

=> 2c² = b²

Now,
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2 divides b².

So, 2 divides b.

Thus 2 is a common factor of a and b.

But this contradicts the fact that a and b are co-primes.

This contradiction has arisen due to the incorrect assumption of √2 as rational.

∴ √2 is rational.

Hence, proved.

HOPE IT HELPS