Question

If p is a prime no. Then prove that √p is irrational.

Answer 1

Hey


To prove :- √p is an irrational number .

Proof :-

If possible , let √p be a rational number .

So , √p = a/ b

[ where a and b are co - prime numbers ]

Now , squaring both sides , we get ,


p = a² / b²

=> p * b² = a²

=> b² = a² / p


Here p divides a² , so p will also divide a .


Now ,

let a = pc

So ,

b² = p² x² / p


=> b² = px²

=> b² / p = x²


Here also p divides b² , so it will divide b also .


Now , it is contradicting as we have taken a and b are co prime and don't have any other common factor other than 1 .

But by these process , it is shown that a and b has another common factor which is p .


So , our assumption is wrong .

√p is not in the form of a / b .

√p is an irrational number .


♦ Proved ♦

thanks :)

Answer 2

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