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Complex Numbers and Beyond
Given that Aubrey is really having fun with this mathematical afternoon, let us move forward toward the so called complex numbers. I will introduce them in a somehow non standard way to show the natural ideas behind them, so we will not start with a proper mathematical definition...

So what do we know about numbers ? Well, we know the integers (including the negative ones). Those are used to count. And then we know about the rational numbers like 3/2, like I want to share 3 cakes in 2 parts (yes, this happen), or like 2/3, where you want to share 2 cakes in 3 parts (happens more often).

And then, a bit more than 2500 years ago, some folks discovered that simple numbers like square 2 (the number when you multiply it by itself gives 2) where not rationals numbers, and they have been really really pissed!! (Full story another time).

Anyway, when you get all of those, you have the so called real numbers, the numbers with as many decimals that you want, and more often than you think an infinite number of such digits...

Note that among the irrational numbers (the numbers that are not rational), you have two kinds, the algebraic, and the transcendants.

Let us have a look of the difference between algebraic numbers and the transcendants numbers. So we know that square 2 is not a rational number (not difficult to prove but you will have to believe me on it), but at least square 2 is not so far away from being an everyday life number. Indeed if you multiply it by itself you get this good old 2 that we all know.

People would say that square 2 is solution of the equation X^{2} - 2 = 0. We have here a nice plain simple second degree polynomial with integers as coefficients. When a number is solution/root of such a number we say that it is algebraic.

Another such nice polynomial is X^{5}+6X^{4}+5X-90. Fifth degre polynomial all nice and paceful (beside that I would not be able to give you its exact roots -- and nobody else across the Universe would).

So someone would be tempted to think that all real numbers, including the non rational ones are solution of polynomials with integer coefficients. Well, nothing could be futher from the truth, most real numbers are not algebraic. The most famous one being our daily pi. When a number is not algebraic it is then said transcendental.

When I was a kid, I thought that real numbers where the only ones, I then found always funny when my math teacher was taking time to precise that the numbers he had in mind was reals. One day I asked him why he always wasted time precising. What else than real could they be ?

He said that there were some others, and that day I knew that I had found the love of my life: mathematics.

So we start with a simple innocent second degree equation X^{2}+1=0. This equation doesn't have a solution among real numbers because any number a such that a^{2}+1=0, would be such that a^{2}=-1, and we know that, among other things, the square of a (real) number cannot be a negative number.

So ladies and gentlemen I will perform the extraordinary in front of you: I will expand the universe of numbers and find a solution to our friend equation.

First I ask to you to close your eyes (well, keep one open so you can still read the webpage), and use the power of your mind to create out of nowhere a new number. It is not from Earth, but from another world, an alien number. Not being a real number there is not pre-made name for it, so we will call it i, like imaginary.

So let be i

I already know some folks in the back telling me, "Hey, wait Pascal, you cannot do that! You cannot just create a new number like that out of nowhere without justification.". Well, you would be surprised. Not only I can,but the simple fact that I want is the justification. Welcome to the wonderful world of mathematical postulates.

So coming back to i, what on Earth am I going to do with it ? Well, first of all it is a number, right ? So maybe I can multiply it by 2. We will write this 2 * i. Looks good, and what about multiplying it by 3 ? Well that's easy, and we write this 3 * i. More difficult now, what about multiply it by minus square 2, well still easy - sqrt(2) * i. Hoo wait, what about adding it to 4 ? Well, 4 + i.

So we have a new number which seems to behave correctly and get along well with our real friends actually. So what does it have so special ? Well, using again the power of mathematical postulate, I decide that when you multiply it by itself you get -1. So the only equation I ask our alien number to verify is i * i = -1.

And another time, yes, I can.

In fact, have I not created out of a magician hat a solution to our equation X^{2}+1=0 ? Yes, actually I did. And now I will show you that actually things are going to be ok.

We will call complex numbers numbers of the form x + y * i where x and y are real numbers. We have now a all new field of numbers where the real numbers happen to be a very small part of. A complex number x + y * i is what we knew as a real number everytime y is equal to 0.

To get used to complex numbers, let us reduce the following number : (2 + 7i) + ( 4 -1 i) * (7 - 2i). So yes, we are manipulating complex numbers, but multiplication still have priority over addition...

(2 + 7i) + ( 4 - i) * (7 - 2i)

= (2 + 7i) + (28 -8i -7i -2)
(we reduced one step using the formula (a+b)(c+d) = ac + ad + bc + bd). Also note how -i * -2i = -2

= (2 + 7i) + (26 -15i)

= 28 - 8i

So it looks like it works actually. Well it does work, and in fact complex numbers have allowed physicians, enginneers, navigators and even opticians (well all fields really) to perform computations they would not be able to do otherwise.

There's a field in mathematics, called the theory of analytical functions, where people study the numerical functions from the set of complex numbers to itself. I have to say that it was one of the most beautiful part of my undergraduate studies. A couple of years later I have understood the mathematics behind the JPEG 2000 picture compression algorithm (at a time I still had not touched any computers...)

Anyway, the title said "complex numbers and beyond". You are thinking that complex numbers are already complex enough and there no way we will carry on, right ? After all, with complex numbers we have a solution for all polynomials, so what is the point ?

Well, the point is that mathematicians always think a couple of hundred years ahead of their time and they were looking at problems that were known only to them but needed solutions. We are not going to add one more dimention to our algebra, but two !!

The quaternions are numbers which can be written as x + yi + zj + uk. The rules those new aliens from three different species i, j and k numbers have to follow are given by some equations which I am not going to write down here...

Something wierd with the quaternions is that the given birth algebra is not commutative, and you cannot make it commutative without having everything falling apart. But they are very good in modeling computer graphics and that kind of things where lots of 3 dimentional rotations are involved.

And then...

Now, you are thinking: "Pascal that's enough, you cannot possibly continue like that. What else, 5, 6 dimentions?" Well, brace yourself. The next step is 8 dimentions, with the Cayley Octaves, which only mathematicians who do not sleep at night could talk to you about. I am no longer one of them, I am married now...

The bad news is that it stops there. We have lost associativity with the Octaves, it would be difficult to lose something else wihout not being an algrebra anymore...

So the end ?... Well, yes, the end in that direction yes, but we were simply building vectorial spaces which are also algebras above the set of real numbers. And what if we decided to lose the real numbers and decided to come up with our own (finite or infinite) set of numbers with all kind of funny algebraic rules. This would open the door to..., an infinite number of more or less alien universes (with as many dimentions as we want) to play with...

... Welcome to your first year in mathematics at the University. From here, there's no coming back (^.^)