Question

proof √2 is an irrational no.​

Answer 1

A proof that the square root of 2 is irrational. Let's suppose √2 is a rational number. Then we can write it √2 = a/b where a, b are whole numbers, b not zero. We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.

Answer 2

SOLUTION :

Let assume that √2 is rational number.

√2 = a/b [ a and b are co-primes ]

b√2 = b

Squaring on both sides

(b√2)² = a²

2b² = a²

b² = a²/2

a² is divisible by 2

If 2 divides a² then it also divides a

Let us assume a = 2c

=> 2b² = (2c)²

=> 2b² = 4c²

=> b² = 2c²

This means that 2 divides b² and 2 divides b.

Therefore, both a and b have 2 as a common factor.

But this contradicts the assumption that a and b are co-prime and have no common factors.

So, our assumption is wrong.

We can conclude that √2 is irrational.