Question

Prove that√2 is irrational

Answer 1

Let underoot 2 is a rational number.
underoot 2=a/b
2=a2/b2
a2 is dividible by 2 (equation1)
Let a=2c
a2=4c2 (Squaring on both sides)
2b2=4c2
b2=2c2
b is too divisible by 2( equation2)
Both 1st and 2nd equation is divisible by2

So this states a/b is not rational and is irrational.
Hope it helps.
Thanq..

Answer 2

To Prove :-

• √2 is an irrational number.

SoluTion :-

Let's assume on the contrary that √2 is a rational number.

Then, there exists two rational numbers a and b

such that √2 = a/b where, a and b are co primes.

(√2)² = (a/b)²

→ 2 = a²/b²

→ 2b² = a²

2 divides a²

So, 2 divides a.

a = 2k , (for some integer)

a² = 4k²

2b² = 4k²

b² = 2k²

2 divides b²

2 divides b

Now, 2 divides both a and b but this contradicts that a and b are co primes.

It happened due to our wrong assumption.

Hence, √2 is irrational.