Is √N Always Irrational for Non-Square Integers?

[bgwyh_88]

I came across this question. How do you show that √N is irrational when N is a nonsquare integer?

Cheers.

 

[hamster143]

It depends on what you're allowed to use.

But the simplest way would be to use the fundamental theorem of arithmetic (that every integer has a unique prime factorization). For any N, if sqrt(N) is rational, you can write that as

N=A^2/B^2

and therefore

B^2 N = A^2

and it's not hard to get from that + the fundamental theorem to the conclusion that N is a square.

 

[bgwyh_88]

Nice. Thanks hammster143. Appreciate it.

bgwyh_88