Question

Prove that root 5 is irrational no.

Answer 1

let root 5 be rational

then it must in the form of p/q [q is not equal to 0][p and q are co-prime]

root 5=p/q

=> root 5 * q = p

squaring on both sides

=> 5*q*q = p*p ------> 1

p*p is divisible by 5

p is divisible by 5

p = 5c [c is a positive integer] [squaring on both sides ]

p*p = 25c*c --------- > 2

sub p*p in 1

5*q*q = 25*c*c

q*q = 5*c*c

=> q is divisble by 5

thus q and p have a common factor 5

there is a contradiction

as our assumsion p &q are co prime but it has a common factor

so √5 is an irrational

Answer 2

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✅Here is your answer ✅

Let us assume,to the contrary,that √5 is rational.

So,we can find integer r and s (≠0) such that √5= r/s.

Suppose r and s have a common factor other than 1.Then we divide by the common factor to get √5= a/b, where a and b are coprime.

So,b√5=a.

On squaring both side and rearranging,

we get

5b²= a². Therefore,5 divides a².

5 divides a.

So,We can write a= 5c for some integer c.

Substituting for a,we get

5b²= 25c²,that is, b²=5c².

This means that 5 divides b² and so 5 divides b .

Therefore, a and b have atleast 5 as a common factor.

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

This contradiction has arisen because of our incorrect assumption that √5 is rational.

So,We conclude that√5 is irrational.

✌️Hope it helps you ✌️

♥️ Thank you♥️

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