Question

prove that 1/√2 is irrational? briefly.

Answer 1

Solution :-

Let us assume 1/√2 is rational

i.e, 1/√2 = a/b Where a, b are co - primes and b ≠ 0

⇒ √2/1 = b/a

⇒ √2 = b/a

⇒ a√2 = b

Squaring on both sides

⇒ (a√2)² = (b)²

⇒ a²(√2)² = b²

⇒ a²(2) = b²

⇒ 2a² = b² ....(1)

⇒ a² = b²/2

Here 2 is a prime number.

If 2 divides b², then 2 divides b according to theorem(Let p be a prime number, if p divides a², then p divides a)

For 2 to divide b, b should be a multiple of 2.

⇒ b = 2c

Substitute b = 5c in (1)

⇒ 2a² = (2c)²

⇒ 2a² = 4c²

Cancelling 2 on both sides

⇒ a² = 2c²

⇒ a²/2 = c²

If 2 divides a², then 2 divides a.

So a must be a multiple of 2

⇒ a = 2d

Therefore b = 2c, a = 2d

Both a and b have 2 as their common factors.

This contradicts that a and b are co primes.

This contradiction has arised because of a wrong assumption that 1/√2 is rational.

So we can conclude that 1/√2 is not a rational number but an irrational number.

Hence proved.

Answer 2

Answer:

i dont know

Step-by-step explanation: