Question

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Prove √2 is irrational (class 10)​

Answer 1

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Prove √2 is irrational (class 10)

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Given- √2 is irrational number.

Let √2 = a / b where a,b are integers b ≠ 0 we also suppose that a / b is written in the simplest form

Now √2 = a / b

⇒ 2 = a² / b²

⇒ 2b² = a²

∴ 2b² is divisible by 2

⇒ a² is divisible by 2

⇒ a is divisible by 2

∴ let a = 2c a² = 4c²

⇒ 2b2 = 4c²

⇒ b2 = 2c²

∴ 2c2 is divisible by 2

∴ b² is divisible by 2

∴ b is divisible by 2 ∴a are b are divisible by 2 .

this contradicts our supposition that a/b is written in the simplest form Hence our supposition is wrong

∴ √2 is irrational number.

Answer 2

Let us assume,to the contrary that √2 is rational. So we can find integer a and b( where b is not equal 0) such that √2 =a/b

Suppose a and b have a common factor other than 1. Then we divide by the common factor to get √2=a/b where a and b are coprime.

So, b√2=a.

Squaring on both sides, we get

2b^2= a^2

Here, 2 divides a^2 and 2 divides a also.

So we can write a=2c for some integer c.

Substituting for a, we get 2b^2 =4c^2, that is , b^2=2c^2

This means that 2 divides b^2 and so 2 divides b

Therefore, a and b have at least two as a common factor.

But this Contradicts the fact that a and b have no common factors other than 1.

This contradiction has arisen because of our incorrect assumption that √2 is rational.

So, we conclude that √2 is irrational