Question

Prove that √5 is irrational

Answer 1

Hey user here is your answer......


--->To prove √5 is irrational


---->let √5 is rational

◆√5=a/b (where a& b are co-prime)

◆√5b=a

Square both side

◆5b^2=a^2

◆It means if 5 divides a^2 then it also divides a

So a is factor of 5

◆let b=5c

◆5b^2=(5c)^2

◆5b^2=25c^2

◆b^2=5c^2

If 5 divides b^2 then 5 divide b

so B is also factor of 5

◆It is not possible that there are two factor of 5 I.e. a& b.

◆This contradiction arises due to our wrong assumption.

◆So,
√5 is irrational


Thanks☺️

Answer 2

HI

Proof:

Let us assume to the contrary that √5 is rational

Therefore, √5 = a/b , where a and b are coprime integers and b ≠ 0

Squaring on both sides,

5 = a²/b²

5b² = a² ......(1)

Since, a² is divisible by 5,

Therefore, a is divisible by 5

Let a = 5c and substitute in eq(1)

5b² = (5c)²

5b² = 25c²

b² = 5c²

Since, b² is divisible by 5,

Therefore, b is divisible by 5

➡️ a and b have at least one common factor i.e. 5

This contradicts the fact that a and b are coprime.

➡️ This contradiction has arisen due to our incorrect assumption that √5 is rational.

Therefore, we conclude that √5 is irrational.

Hence Proved !

Hope it proved to be beneficial....