Question

Prove that $\binom{1}{ \sqrt{2} }$ is irrational.​

Answer 1

Answer:

Let us assume 1/√2 is rational.

So we can write this number as

1/√2 = a/b ---- (1)

Here, a and b are two co-prime numbers and b is not equal to zero.

Simplify the equation (1) multiply by √2 both sides, we get,

1= a√2/b

Now, divide by b, we get,

b=a√2

or, b/a = √2

Here, a and b are integers so, b/a is a rational number,

so √2 should be a rational number.

But √2 is a irrational number, so it is contradictory.

Therefore, 1/√2 is irrational number.

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