|Reasons to be cheerful...
||[Dec. 7th, 2006|01:14 pm]
I've got a big smile on my face.|
I've just done a bit of exhibition mathematics (quoting a proof that there's no highest prime number, there's always a bigger one) in a thread following someone's LJ post about proofs, and got a response of "that's beautiful".
It's mathematics, of course it's beautiful. Hurrah for mathematics!
2006-12-07 01:41 pm (UTC)
Thousands of years old and still stunning.
I'm with you on that. Mathematics is beautiful, and the infinite-primes proof particularly so.
Tell it to us, tell it to us
The original thread was asking about proofs-of-nonexistence, particularly of intangible things, and my comment read as follows (slightly edited):A prime number is one that doesn't have any numbers divide into it other than itself and 1. 17, for example, is prime, because if you get any number other than 17 and 1, and try to divide it into 17, there's always something left over.
Numbers which divide into other numbers are called "divisors" or "factors", and I tend to use the terms interchangeably. They're the same thing.
Now the following proof is going to assume something called "existence and uniqueness of prime factorisation", which this comment is too small to contain, but it's on Wikipedia.
Assume that there is a highest prime -- a prime number which doesn't have any other numbers above it which are prime.
Now get all the numbers up to and including this supposed "highest prime" and multiply them all together, and then add one to the result. You'll get quite a big number.
Now think about the numbers which divide into this new big number. Definitely none of the numbers up to or including the alleged "highest prime" we were thinking about earlier won't work -- they'll all leave a remainder of 1.
Now consider our new big number's prime factors. They must all be larger than the "highest prime" that we first thought of -- so that's a contradiction. Unless our new big number is itself prime -- and that's larger than our former "highest prime", and that's a contradiction too.
So our initial assumption that a highest prime exists must be nonsense.
[applauds] Well put, that man.
Absolutely, Maths rocks :D What rocks even more is that by doing it for A level I have saved myself several hours of lectures and practicals in the past two weeks!
The thing I particularly love about that proof is that even my pupils can follow it. I'm sure I discussed it with one of them last year. It's such a lovely idea and requires no great knowledge, just an understanding of what a prime number really is.
I prefer the proof that there can be different sizes of infinity, in which one maps integers to decimal numbers and so on, but I like that one too. :)