Question

Prove that
1/√2
is an irrational​

Answer 1

To prove that 1/√2 is a irrational number.

Let us assume 1/√2 as a rational number.

We know that,

Rational number can be written in the form of a/b.

=> 1/√2 = a/b

• Where, a and b are the positive integers. (b ≠ 0).

=> b/a = √2

Here, b/a is a rational number as it is expressed or written in the form of a/b.

But, We know that √2 is an irrational number.

• Rational number ≠ Irrational number.

Therefore, Our assumption that 1/√2 is a rational number is false or wrong.

∴ 1/√2 is an irrational number.

Hence proved.

Answer 2

★ Concept ★

Here in this question, concept of assumption or contrary method is used. We have to prove the given number as irrational. We can prove it irrational by following steps.

So let's start!!

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Let us assume to the contrary that $\sf{\dfrac{1}{\sqrt 2}}$ is a rational number. So we can write it in the form of $\sf \dfrac{P}{Q}$ where P and Q are integers, Q≠0 and P & Q are co-prime.

So:-

$\sf\dfrac{P}{Q}=\dfrac{1}{\sqrt 2}$

By cross multiply:

$\sf P\sqrt2 =Q$

By arranging it:

$\sf\sqrt2=\dfrac{P}{Q}$

Here $\sf\dfrac{Q}{P}$ is a rational number but $\sf\sqrt2$ is irrational number.

We know that irrational numbers can't be equal to rational numbers.

This contradicts the fact that √2 is irrational.

This contradiction has arisen because of our incorrect assumption that $\sf\dfrac{1}{\sqrt2}$ is a rational number.

Since our assumption is wrong $\sf\dfrac{1}{\sqrt2}$ is an irrational number.

$\large\underline{\boxed{\sf{\red{Hence\; Proved}}}}\star$

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★ More to know ★

Co-prime numbers

Any pair of numbers which doesn't have any common factor except 1 are called co-prime numbers.

For example:-

• 7 & 9 are a pair of co-prime numbers, where they don't have any common factor.

• 5 & 6 are a pair of co-prime numbers, where 5 and 6 don't have any common factor.