Question

Prove the following number is Irrational/Not Rational: 5+3?2​

Answer 1

5 + 3 √2 is an irrational number is to be proved.

Let 5 + 3 √2 is a rational number

So p/q = 5 + 3 √2

p = 5q + 3q√2

p – 5q = 3q√2

p – 5q / 3q = √2

p/2q – 5q/3q = √2

p/3q = 5/3 = √2

Here 5/3 is a rational number. Both terms are rational and therefore 5 + 3 √2 is an irrational number

Answer 2

Given:3 + 2√5

To prove:3 + 2√5 is an irrational number.

Proof:

Letus assume that 3 + 2√5 is a rational number.

Soit can be written in the form a/b

3 + 2√5 = a/b

Here a and b are coprime numbers and b ≠ 0

Solving3 + 2√5 = a/b we get,

=>2√5 = a/b – 3

=>2√5 = (a-3b)/b

=>√5 = (a-3b)/2b

This shows (a-3b)/2b is a rational number. But we know that But √5 is an irrational number.

so it contradictsour assumption.

Our assumption of3 + 2√5 is a rational number is incorrect.

3 + 2√5 is an irrational number

Hence proved