Prove that squar root of 5 is irrational and hence prove that (2? squar root of 5) is also irrational.

Answer:

Let us assume to the contrary that √5 is a rational number.

It can be expressed in the form of p/q

where p and q are co-primes and q≠ 0.

⇒ √5 = p/q

⇒ 5 = p2/q2 (Squaring on both the sides)

⇒ 5q2 = p2………………………………..(1)

This means that 5 divides p2. This means that 5 divides p because each factor should appear two times for the square to exist.

So we have p = 5r

where r is some integer.

⇒ p2 = 25r2………………………………..(2)

from equation (1) and (2)

⇒ 5q2 = 25r2

⇒ q2 = 5r2

Where q2 is multiply of 5 and also q is multiple of 5.

Then p, q have a common factor of 5. This runs contrary to their being co-primes. Consequently, p / q is not a rational number. This demonstrates that √5 is an irrational number.

Since √5 is an irrational number, the difference between the rational number and irrational number is always an irrational number.

∴ (2 – √5) is also an irrational number.

Leave a Comment

Your email address will not be published. Required fields are marked *

BOOK

Free Class