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Irrationality Proof

Date: 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   

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   
Associated Topics:
High School Number Theory

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