Prove that √2 is an irrational number. Hence show that 3/√2 is al

1.	Prove that √2 is an irrational number. Hence show that 3/√2 is also an irrational number.
Answer» Let √2 be a rational number. ∴ √2 = a/b , (a, b are co-prime integers and b ≠ 0) a = √2 b Squaring, a² = 2b² ⇒ 2 divides a² ⇒ 2 divides a. So we can write a = 2c for some integer c, substitute for a, 2b² = 4c² , b² = 2c² This means 2 divides b² , so 2 divides b. ∴ a and b have ‘2’ as a common factor. But this contradicts that a, b have no common factor other than 1. ∴ Our assumption is wrong. Hence, √2 is irrational. Let 3/√2 be rational 3/√2 = a/b , where a and b are integers, b ≠ 0 3b = √2a √2 = 3b/a 3b/a is rational but √2 is not rational. ∴ Our assumption is wrong. ∴ 3/√2 is irrational.

Prove that √2 is an irrational number. Hence show that 3/√2 is also an irrational number.

Answer»

Let √2 be a rational number.

∴ √2 = a/b , (a, b are co-prime integers and b ≠ 0)

a = √2 b

Squaring, a² = 2b²

⇒ 2 divides a²

⇒ 2 divides a.

So we can write a = 2c for some integer c, substitute for a, 2b² = 4c² , b² = 2c²

This means 2 divides b² , so 2 divides b.

∴ a and b have ‘2’ as a common factor.

But this contradicts that a, b have no common factor other than 1.

∴ Our assumption is wrong.

Hence, √2 is irrational.

Let 3/√2 be rational

3/√2 = a/b , where a and b are integers, b ≠ 0

3b = √2a

√2 = 3b/a

3b/a is rational but √2 is not rational.

∴ Our assumption is wrong.

∴ 3/√2 is irrational.

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