Question

1. Prove that √5 is irrational.​

Answer 1

Answer

To prove:

√5 is irrational.

Proof:

Let's assume that √5 is irrational.

so,

√5 = a ('a' and 'b' , b not equal to 0

1 b

are co-primes).

i.e., no common factors other than 5.

=> √5b = a

squaring on both sides-

(√5b)² = a²

5b² = a². ........ (i)

b² = a²

5

Hence, 5 divides (a)². (by theorem)

Hence, C = a , where C is some integer.

5

So, a = 5c, put the value of a in (i)

5b² = (5c)²

5b² = 25c²

b² = 5c²

c² = b²

5

Hence, 5 is divided by b²

So, 5 divides b

Hence, 5 is factor of a and b.

so, a and b are not co-primes.

Thus our assumption is wrong.

and by contradiction √5 is irrational.

hope this helps you.