Irrationality ProofDate: 04/26/2001 at 12:05:52 From: Brian Rockafellow Subject: Number Theory I need to show that log 2 is irrational. It is in base 10. Date: 04/26/2001 at 16:02:48 From: Doctor Schwa Subject: Re: Number Theory Hi Brian, Here's a hint that should help get you started. If log base 10 of 2 is rational, say it equals p/q, then 10^(p/q) = 2, and therefore 10^p = 2^q. Why is that impossible if p and q are integers? - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ Date: 04/26/2001 at 16:56:00 From: Doctor Rob Subject: Re: Number Theory Thanks for writing to Ask Dr. Math, Brian. As is usual with irrationality proofs, proceed by contradiction. Assume that log(2) = a/b for some positive integers a and b, and that this fraction is reduced to lowest terms. Then that means that 10^(a/b) = 2 10^a = 2^b Now observe that since a > 0, the left-hand side is divisible by 5, but the right-hand side is not, a contradiction. Here we are using the Unique Factorization Theorem. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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