# Prove that 3 + √ 5 is an Irrational Number. - Algebra

Short Note

Prove that 3 +sqrt 5  is an irrational number.

#### Solution

Let us assume that 3 + sqrt 5 is a rational number.

=> 3 + sqrt 5 = p /q, where p and q are the integers and q ≠0.

=> sqrt 5 = p/q - 3 = (p - 3q)/q

Since p , q and 3 are integers. So, (p - 3q)/qis a rational number.

=> sqrt 5 is also a rational number.

but this contradicts the fact that sqrt 5 is an irrational number.

This contradiction has arisen due to the wrong assumption that 3 + sqrt 5 is a rational number.

Hence, 3 + sqrt 5 is an irrational number.

Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics 1 Algebra 9th Standard Maharashtra State Board
Chapter 2 Real Numbers
Practice Set 2.2 | Q 2 | Page 25