Interesting maths? Surely an oxymoron (part 1)

To most people, probably. Still, I thought I'd actually blog about some maths on my maths blog (heaven forbid!) and starting with the maths I find interesting would be a good start.


So, what am I going to talk about?


Number systems and how they can be closed or open. I'm not going to rigorously prove things. Partly because I can't and partly because it's not actually that interesting to build up axioms from the bottom up unless you're a proper maths geek.


So, where do we begin? Probably by defining some terms. Firstly a SET is a collection of objects. Sets are denoted by the curly brackets { } for some reason. Stuff inside the brackets is in the set. So the set of all publishers on this blog would be {me}. The set of all even numbers less than 7 would be {2,4,6} and so forth. The symbol ∞ denotes infinity. The => symbol denotes "implies" as in "My umbrella is wet => It is raining".


We can then use this definition of a set to define our first number system: These are the numbers you use every day when you count stuff. They're the most obvious numbers to us, since they clearly exist in the natural world. Thus they get their name; the Natural Numbers. Also called the positive whole numbers. 
They can be (partially) defined as they are the set {0...∞}. Usually this set is labelled by N.
You can add up the natural numbers as much as you like and never reach the end, and you can add any natural number to any other and get a defined result. As multiplication is basically repeated addition, (2x3 is 2+2+2 and so on) you can also multiply any natural number by any other and have a defined result. You can also subtract any number of equal or smaller size from any number. You can't take a larger number from a smaller one though, there is no result in N for this operation. You may be thinking that this sounds a bit complicated and confusing. It looks that way when written down, examples make it simple. For example you can take any number less than or equal to 9 from 9, but can't take 10 from 9, or slightly more formally;
Where a and b are any natural number;
a + b exists in N
a x b exists in N
a - b exists in N if and only if b ≤ a, => a - b ≥ 0

So, the Natural Numbers are not a rich enough number system to contain every operation we might need. This brings us on to our next number system. Somewhere, someone asked "What happens when I take 2 from 1?" It's a fine question. Clearly the answer has to be either zero or some sort of strange less than zero value. If it is a less than zero value what is it? Some historical clever chap thought "why not just extend the natural numbers backwards, and stick a subtraction sign in front of any that are less than zero? This lets us do more stuff and doesn't break anything we know about the natural numbers. Clearly I am a genius. Shower me with riches beyond imagining!" I like to think he (or she) was showered with riches, too.
I'll expand upon this in the second part of this post.