Question

Prove the under root 5 is an irrational number​

Answer 1

Step-by-step explanation:

Let we assume that root 5 is rational number and is in the form a/b where a is co-prime and b is not equal to 0

A/b= root 5

A=b×root 5

(squaring both sides)

A2=5b2 (multiple of 5)

A2 is divisible by 5

.: a is also divisible by 5-equation 1

Let a=5c

(squaring both sides)

A2=5c2

A2=25c2

5b2=25c2(from equation 1)

B2=5c2(multiple of 5)

B2 is divisible by 5

.: b is also divisible by 5

Since both A and B are having common number as 5 they are not co-prime

.: root 5 is irrational

Answer 2

Solution-

Let √5 be rational.

Let its simplest form be √5=p/q, where p and q are integers, having no common factor, other than 1 and q not equals to 0.

Then, √5= p/q

=> p^2 = 5q^2 ...(i)

=> p^2 is a multiple of 5

=> p is a multiple of 5.

Let p= 5m for some positive integer m. Then,

p=5m

=> p^2 = 25m^2

=> 5q^2 = 25m^2 (Using (i))

=> q^2= 5m^2

=> q^2 is a multiple of 5

=> q is a multiple of 5.

Thus, p as well as q is a multiple of 5.

This shows that 5 is a common factor of p and q.

This contradicts the hypothesis that p and q have no common factor, other than 1.

Thus, √5 is not a rational number and hence it is irrational.