Question

Prov that √3+√5 is irational number

Answer 1

We have to prove that (√3 + √5) is an irrational number.

Proof :

Let us consider (√3 + √5) a rational number.

⇒ √3 + √5 = a/b , where both a and b are integers with non-zero b

⇒ √3 = (a/b) - √5

Squaring both sides, we get

3 = a²/b² - (2√5)a/b + 5

⇒ (2√5)a/b = a²/b² + 5 - 3

⇒ (2√5)a/b = a²/b² + 2

⇒ 2√5 = a/b + 2b/a

⇒ √5 = a/(2b) + b/a

Since both a and b are integers, a/(2b) + b/a is a rational number which leads to the contradiction that √5 is an irrational number.

So, our assumption is wrong.

Therefore, (√3 + √5) is an irrational number.

Hence, proved.