Question

prove that √5 is an irrational number?​

Answer 1

Step-by-step explanation:

: √5

√5

:

√5 .

/ , - ≠0

⇒ √5 = /

,

⇒5 = ²/²

⇒5² = ² —————–()

²/5 = ²

5

5

⇒ = 5

⇒ ² = 25² ————-()

() (), ,

5² = 25²

⇒ ² = 5²

⇒ ² 5

⇒ 5

, , 5. -. , /

√5 .

Answer 2

Step-by-step explanation:

If possible , let root 5 is rational number and its simplest form be a , where b is not equal to 0.

b

Then,a and b are integers having no common factor other than 1.

Now ,

root 5=a

b

5=a²

b²

5b²=a²....(i)

Therefore, 5 divides a²

5 divides a

5c= a

putting the value of a in equation 1

5b²=(5c)²

5b²=25c²

b²=5c²

Therefore,5divides b²

5divides b

This, contradicts the fact that a and b are integers having no common factor other than 1.

This, contradiction arises by assuming root 5 be rational.

So,our assumptions are incorrect.

Hence,root 5 is an irrational number.

THANK YOU.

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