Question

Prove that 3✓2÷5
is irrational
​

Answer 1

Let us suppose that √2 is rational. Then there exist two positive integers a and B such that

√2 = a/b

Where a and B are co primes

Squaring on both side gives us

2=a^2/b^2

2b^2 = a^2

It means 2 is a factor of a^2 and a as well

2c = a. (as 2 is a factor of a)

Squaring on both sides gives us

4c^2 = a^2

4c^2 = 2b^2. ( As proved above)

b^2 = 2c^2

It means 2 is also a factor of B.

Hence it is a contradiction as a and b were co primes.

Hence our supposition is wrong and √2 is irrational.

For further solution see the pic

Answer 2

$\sf \pmb{Answer :}$

Let us suppose that √2 is rational. Then there exist two positive integers a and B such that

√2 = a/b

Where a and B are co primes

Squaring on both side gives us

2=a^2/b^2

2b^2 = a^2

It means 2 is a factor of a^2 and a as well

2c = a. (as 2 is a factor of a)

Squaring on both sides gives us

4c^2 = a^2

4c^2 = 2b^2. ( As proved above)

b^2 = 2c^2

It means 2 is also a factor of B.

Hence it is a contradiction as a and b were co primes.

Hence our supposition is wrong and √2 is irrational.