An answer, in two stages: stop after first if you think you can see rest: Stage 1:
So as SimonM did above, we do this:
SimonM
log23=qp⇔3q=2p...
What we are doing here is assuming that log23 is a rational number: i.e. is the rato of two integers p and q. This implies that for some integer p, q, 3q=2p (by rearrangement really).
An answer, in two stages: stop after first if you think you can see rest: Stage 1:
So as SimonM did above, we do this:
What we are doing here is assuming that log23 is a rational number: i.e. is the rato of two integers p and q. This implies that for some integer p, q, 3q=2p (by rearrangement really).
Stage 2:
Spoiler
Bingo!
Thanks for that I'm still very baffled on why it came up on the C3 paper though, we've never been through this
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 qp, where p,q are integers (q=0).
Piecing this together, we want to show that log23 is irrational; i.e. that it can't be written in the form qp for any integers p,q. So, we start our proof by assuming that there exist integers p,q (q nonzero) such that log23=qp.
By the definition of logarithms, this gives 3=2p/q, and raising both sides to the power of q gives 3q=2p. This can only happen if p=q=0, but we can't have q=0 so this can't be true, so the assumption can't be true, so it must be false; hence the proposition is true.
nuodai, would you be able to just explain the last stage of this? I don’t quite see why there are no integers p and q for which 2^p=3^q 10^4=100^2 but there the bases have common factors. Is that the point?
but surely that only proves there is only one solution of 3^x=2^x (ie x=0). But it doesn't imply that there cannot be two different integers (p and q) for which 2^p=3^q does it?
Sorry, if I'm being thick
2^p=3^q=n
If p and q are not zero then n has two different prime factorisations. This is not possible. You might want to check the proof of that statement.