I think the blog will change a little from now. I will post once a week, on friday nights (the night is still to be decided upon, but for now fridays are good).
The content and style won't change. However, I will most likely make a second maths blog. In this blog I will be testing out ideas for how to structure a course for Maths Methods 3/4 (the mid-range year 12 maths subject in Victoria), because I intend to put together a short electronic textbook for the subject, to address the woeful inadequacy of the textbooks currently on the market. Potentially more on that later (certainly I'll post a link to that blog once it's underway), but if anyone's interested or wants to help out or anything, please let me know :)
Now that that's out of the way, a quick thing on primes.
Lets start by defining some stuff. A number F is a factor of N if when it is divided N, the result is a whole, positive number (also known as a natural number). We say, then, that N is divisible by N. So for example, 6 is divisible by 1,2,3 and 6, and 5 is divisible by 1 and 5, but not 3. Obviously F can't be greater than N, because then we would get fractions (and they aren't whole numbers, which breaks the definition of a factor)... for example, 6/12=1/2 (in this case N=6 and F=12... the half is just a half. Get over it).
A prime number is a number which has no factors other than 1 and itself. So 2, 3 and 5 are prime, but 4 and 6 are not (because 2x2=4, and 3x2=6).
What I'm going to do is follow in the steps of Euclid (the Greek father of geometry) and prove that there is an infinite number of primes. This isn't necessarily obvious - why SHOULD there be an infinite number? Why can't all the potential gaps eventually be covered up? Well, here's how the explanation goes.
What I'm going to do is this: I will assume we have a finite number of ALL of the primes, and prove that I can find another prime that's not on the list. Because my technique will work for any finite list of primes, then I will have proven that no matter what you do, your list won't cover all of the primes, so there's an infinite number of primes.
But I'm getting ahead of myself. Let's start by assuming that we have a list of all of the primes: 2,3,5,7,11... all the way up to some last number P, which is our 'last prime'. What I'm going to do is multply them together.
So we have some really big number - 2x3x5x7x11x..xP. Let's call it Bob. So Bob is all of the numbers, all the way up to P, all multiplied together. Let's ask a question: what are the factors of this number? Well, we have 2,3,5,...,P... all of the prime numbers are its factors. So clearly Bob is not prime, because it has a bunch of factors other than 1 and Bob.
But what if I add 1? Then, we get Bob+1. Is 2 a factor? No, because Bob is divisible by 2, so Bob plus 1 can't be (imagine 20 and 21 or whatever... even +1 is odd). Is 3 a factor? No, because Bob is divisible by 3, so Bob +1 isn't (again, imagine 18 and 19). In fact, none of the factors we had before are factors. What does this mean?
Well I'm so glad you asked. What it means is that this new number only has 2 factors - 1, and itself (Bob +1), which means it's a new prime, one that wasn't on the list! I know there can't be any others, because if there were, they would have been on the list in the first place... after all, I did assume it covered ALL of the primes. So no matter which list of primes you give me, I can simply multiply all the terms together and add 1 and hey presto! I have a new prime!
So this means that no matter which list of primes you give me, I can find another prime. Imagine this were a game - you give me a list, I find a new one. We could play that game literally forever - you thinking up an ever growing list of numbers, me always finding one not on your list. It's not your fault that you're losing though - you're losing because the game is rigged! So, since we could play the game forever, this means there's an infinite number of primes to be found! Which is what we wanted in the first place. Yay!
This is widely regarded as one of the simplest, beautiful proofs in mathematics. Certainly, G.H Hardy thought so. It shows a simple truth that is otherwise not necessarily so obvious, but when you look at it more closely, you eventually realise (through a simple explanation) that it has to be true! There is an infinite number of primes!
I love playing with buckyballs and figuring out how many I'm holding by using prime factors - I'll explain that one some other time, but suffice to say that buckyballs (or any other equivalent version of the same thing) are pretty cool to play with, in both a 'normal' way, and of course the much more fun mathsy/physicsy way. Get some and muck around!
So I think we have come to several conclusions today.
1) I will be blogging regularly again! Yay!
2) There are an infinite number of primes
3) I like Buckyballs (not this kind, which is also pretty cool... I mean the toys)
4) Euclid is a boss
More to come on Friday (potentially on primes, but maybe not... we'll see). Bye for now :)
