Sunday, February 18, 2007

Maths part 3- the complex numbers

A lot of people don't get complex numbers. It's probably because they think about it too much. The important thing to note about a lot of mathematics is we say "hey, suppose this thing is true" and then see what follows from that. Because mathematicians are generally quite sly, the thing they suppose is true often is true, so their results hold.

So when I say that the root of -1 is i, (some engineers call it j. This is because they are fools), this is a definition. There is no absolute truth behind this, it's simply that whenever I say i, I mean the root of -1. So i^2 is equal to -1. That's all I mean by that. There is absolutely no value that exists that squares to -1, at least not one on our real number line, but that doesn't really matter. We don't care about how useful this thing is just now- we just care that we have defined it thusly. It's a weird concept, I'll admit, but no weirder than the idea that you can't really express the root of 2- as mentioned it's simply impossible to write it in terms of other numbers (other than as the solution to some equation, eg x^2=2).

So, we have our i (I'm going to stop using italics each time I write it, because it's annoying me- you're a smart bunch). So lets do some stuff with it. Well first of all we can write, for example, the root of -2 in terms of i- it will be i multiplied by the root of 2. This way we don't need lots of crazy imaginary numbers to express all of our roots- i will suffice to describe them all.

We can then express all our numbers so far in this form a+ib, where a and b are real numbers. So each number has an imaginary and a real part- a is the real bit, and b is the imaginary bit. In most cases that you'll encounter, b will be zero, and you'll be left with a nice real number, which behaves quite normally.

There are some slightly odd things about the complex numbers- which are numbers of the form a+ib, which are worth noting. An important one is a lack of a sense of order- we know 2<3, but is 2i<3i? Not really- we don't actually know anything about the size of i, so we can't really assign order to the numbers. There is a way of talking about the size of the number, and that's by taking the modulus- when we take the modulus of a real number, we make it positive- so the modulus of 4 is equal to the modulus of -4... it's 4. For the modulus of a complex number, it is defined to be equal to the square root of a^2+b^2.

There is good reason for this- basically you can imagine any number as a line stretching from 0, and the modulus is an attempt to find the length- in the case of -4, it clearly is 4 away from 0. For the complex numbers, we imagine a 2-dimensional graph, with a on the x-axis, and b on the y-axis. To find it's distance from 0 we use pythagoras- c^2=a^2+b^2. This notion of distance, can be, incidentally, extended to a universe with any number of dimensions, even an infinite amount of dimensions!

So we have some kind of frame where our complex numbers live in. But what use are they? Well the thing is is they crop up in this equation-


I have complicated things here by introducing 3 new concepts- cos, sin and exp. Now cos and sin you may well have encountered- they have clearly defined geometric meaning which is why you'll see them when finding angles in a triangle. The exponential (exp) function is quite interesting- basically it's this magical number e^xi. What's e, you might well ask... well thats a story for another day, but for now it's enough to know it's one of the transcendental numbers like pi, and crops up at least as many times as it for various reasons.

Whats neat about this equation is if you plug in x=pi. Cospi=-1, sinpi=0, so we end up with
exp(pi*i)=1. Which is pretty crazy- we put in some imaginary number into our equation, and get a nice ordinary number as a result! It's mad, and the proof is far to involved to get into here, but it's a very important result.

Thanks to this kind of set up being the solution to these things called differential equations, i actually crops up all over mathematics, and in fact in solutions to real life problems. Which is admittedly quite weird, but just goes to show that life does not necessarily work the way you expect it to.

Anyway, that's it for now, next time I will finally be making my way towards infinity, but first describing functions and maps for your enjoyment. I might mention differential equations later on.... if I feel like it.


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