When we think of numbers, our intuitive sense is to think of them in terms of
quantity: counting, measuring, or comparing quantities. And that’s a good intuition for real numbers. But when you start working with more advanced math,
you find out that those numbers – the real numbers – are just a part of the picture. There’s more to numbers than just quantity.
As soon as you start doing things like algebra, you start to realize that
there’s more to numbers than just the reals. The reals are limited – they exist
in one dimension. And that just isn’t enough.
In terms of algebra – we know that if you have a polynomial of power n,
then it has n solutions. But if you look at many polynomials, you find that if
you limit yourself to real numbers, you don’t get n solutions. For
a trivial example, the equation “x2+1=0″ should have two roots. But there are no real numbers that are solutions to that polynomial.
The solution to that polynomial is the square root of -1, commonly called i, or the imaginary number: it is a number which is not a member of the reals. It is a number – it is real. But it’s not part of the set of real numbers. Once you actually grasp the idea that i exists, then you can start doing some really cool things.
Instead of the numbers described by algebraic equations being points on a line, suddenly they become points on a plane. Complex numbers are really two dimensional; and just like the integer “1″ is the unit distance on the axis of the “real” numbers, “i” is the unit distance on the axis of the “imaginary” numbers. As a result numbers in general become what we call complex: they have two components, defining their position relative to those two axes. We generally write them as “a + bi” where “a” is the real component, and “b” is the imaginary component.
The complex fix more than just the problem of some polynomials not having enough roots. The real numbers are not closed algebraically under multiplication and addition. With the addition of i, multiplicative algebra becomes closed: every operation, every expression in algebra becomes meaningful: nothing escapes the system of the complex numbers.
Arithmetic and algebra work beautifully on complex numbers – you just treat them
as if they were polynomials, and follow the same procedures you would for doing
addition, subtraction, multiplication, and division on polynomials. For example,
Complex numbers are real; but they’re not part of the set that we call real numbers, which is endlessly frustrating to math geeks like me.
Why do I insist that they are real? Because there are real phenomena in the world that
behave in ways that can only be described using complex numbers. If you try to
avoid the use of the complex numbers, you’ll only wind up re-inventing them under another name. They’re real, they exist, and they describe real phenomena.
What’s interesting about the complex numbers in an abstract mathematical sense is
that they can be treated as a superset of the reals; the reals are the set of abstract
numbers whose imaginary component is 0 – all numbers of the form a+0i. What the complex
numbers do is add a second dimension to the number. A complex number is a
number with two dimensions, which is a fascinating idea – numbers become more than just
a line – they became a plane when you use the complex numbers. We’ve expanded our horizons, and can talk about things that just wouldn’t make sense described using real numbers.
There are numerous real phenomena which are described using real numbers. Alternating current – like the current that’s probably powering your computer as you read this article – requires the use of complex numbers to describe it. To perform
computations describing most phenomena involving waves – sound, light, etc., you
inevitably wind up encountering complex numbers.
However, that second dimension comes at a cost. By adding the second dimension, we
wind up with a set of numbers that is still a field – but which does not have
a total ordering. All of our ordering properties are out the window – they no longer
make sense. (Is 1+0i greater than 0+1i? It’s a meaningless comparison.)
You can take the idea of adding dimensions to numbers, and create 4 dimensional numbers. They’re called quaternions, and they’re quite real too. A quaternion has four components: a + bi + cj + dk. They’re very useful for describing rotation. But with quaternions, by gaining those dimensions, you lose even more – they’re not commutative. That is, given two quaternions X and Y, it’s no longer
true that X×Y=Y×X. So quaternions are no longer a field – and a lot of
algebra gets tossed out the window.
You can keep going. There are also 8 dimensional numbers, called octonions. Octonions lose associativity: (A×(B×C)) is not equal to (A×B)×C.
John Baez (he of the n-category cafe) has described these families of multidimensional numbers with a great metaphor:
The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.