Proving the Square Root of a Prime is IrrationalDate: 07/15/98 at 18:20:51 From: April Smothers Subject: (Proof) square root of p is irrational if p is prime Will you please help me prove that if p is prime then the square root of p is irrational? Is there a way to do this using the Fundamental Theorem of Arithmetic? Thank you! April Date: 07/24/98 at 10:50:05 From: Doctor Anke Subject: Re: (Proof) square root of p is irrational if p is prime Hi April, Well, in my opinion, the best way to prove this is by contradiction. There are definitely other ways as well. I haven't checked to see if there is one using the Fundamental Theorem of Arithmetic. Let's assume that p is prime and the square root of p is rational. This means there are (positive) integers a, b such that sqrt(p) = a/b. Therefore: p = (a/b)*(a/b) = (a^2)/(b^2) This shows that a/b already has to be a (positive) integer, so that (a^2)/(b^2) is also one. (If a/b is not an integer, multiplying it by itself wouldn't create one, since no elements would come in that you could cancel the numerator and the denominator with.) So we have shown that (a^2)/(b^2) = (a/b)^2 = p. But this means that p isn't prime, because it has a/b (an integer) as a divisor, so we have a contradiction of the given fact that p is prime. This makes our assumption that sqrt(p) is rational false, and therefore proves that if p is prime, sqrt(p) is irrational. I hope this helps. If have further questions, please feel free to write again. - Doctor Anke, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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