Question

prove that ✓5 is an irrational number​

Answer 1

Answer:

Let √5 be rational in the form of a/b

Step-by-step explanation:

Now, √5 = a/b

Squaring on both sides,

5 = a²/b²

5b² = a² .......(1)

5 divides a²

5 divides a

Let a = 5c, for some integer c.

Putting a = 5c in eq.(1)

5b² = 25c²

5 divides b²

5 divides b [ Hence, 5 is prime

and 5 divides a²]

Thus, 5 is a common factor of a and b.

But, this contradicts the fact that a and b have no common factor other than 1.

The contradiction arises by assuming that √5 is rational.

Hence, √5 is irrational.