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# Use proof by contradiction to prove that log2(3) is irrational. watch

1. (Original post by nuodai)
Proof by contradiction is a method of proof whereby you assume the conclusion is false, and then show this assumption leads to something which can't be true (e.g. 1=0 or "2 is odd").

A number is rational if it is in the form , where are integers ().

Piecing this together, we want to show that is irrational; i.e. that it can't be written in the form for any integers . So, we start our proof by assuming that there exist integers p,q (q nonzero) such that .

By the definition of logarithms, this gives , and raising both sides to the power of q gives . This can only happen if , but we can't have so this can't be true, so the assumption can't be true, so it must be false; hence the proposition is true.
Why is it that only q can not equal 0? Shouldn't p and q can not equal 0?
2. (Original post by Punit21)
Why is it that only q can not equal 0? Shouldn't p and q can not equal 0?
q is in the denominator of the fraction, but p isn't, so the restriction applies only to q

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Updated: January 24, 2013
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