Question

(Introductory Real Analysis) I'm trying to show that sqrt(p) is irrational if p is prime, but I have no idea where to start; could anybody give me suggestions without outright telling me what to do? e.g. should I start from a specific type of argument, contrapositive/counterexample/contradiction, etc, or should I mess around with some field axioms?

Answer 1

contradiction would be your best bet i reckon? dan is the master so listen to him

Answer 2

i really dont know what to do yet xD

Answer 3

but im thinking contradiction is the easiest too

Answer 4

That's what I'm thinking too, messing around with it on a whiteboard. Prime numbers cannot be even, so p must be odd; both a and b must be either both odd or both even because of p being odd.

Answer 5

Now I'm thinking of trying to prove a contradiction from this, or something. I'll see in a minute.

Answer 6

A and b can't be even, because they wouldn't be in lowest terms (where a and b are the integers forming a fraction in lowest terms, equaling sqrt(p).

Answer 7

So a and b must both be odd. Now I'll try to contrive a contradiction from this, but IDK if this will work the same way as it does with something like sqrt(3).