# Complex numbers

By Marcus Wilson 09/09/2009

Sometimes the divide between physics and mathematics is a thin one. Particularly between theoretical physics (which is what I do most of) and mathematics. The difference is that physicists have to keep one foot loosely planted in reality. It’s true that sometimes it just the tip of a little toe that’s behind the reality-line, but look hard enough and you’ll find realism in all that a physicist does.

One example is a physicist’s love of complex numbers. What’s a complex number? Well, think of your square numbers: One times one is one, two times two is four, three times three is nine, etc. Now turn the problem around – for example, what number when squared is 36? Answer, six. But let’s make it awkward. What number squared is minus one? Well, it’s not one (one squared is one), and neither is it minus one (because minus one squared is also one). Put minus one in a calculator and hit the square-root button and you’ll get (if your calculator is like mine) an error. Why? There is no real number that when squared equals minus one.

But this needn’t stop the mathematician. if there isn’t a real solution, he just makes one up, and calls it an ‘imaginary’ number. He calls it by the letter ‘i’. So i squared equals minus one. Beyond reality, yes, but maths is not limited by reality. The mathematician now proceeds to construct a whole algebra (complex algebra) around this concept. For example, you can add, multiply, square, squareroot, take the exponential of etc complex numbers.

Now this last one is why a physicist gets interested. It turns out that the exponential of an imaginary number is related to the cosine and sine functions. And the cosine and sine come almost everywhere in physics. But problems using exponentials are much easier than using sine and cosine. So the physicist turns his problem from a real-world problem to one that is in the ‘imaginary’ (or ‘complex’) world, does his complex algebra on it, then transforms it back into the real world, to get the result. It’s a neater way of doing things. Note that reality remains, because at the end of the journey he’s back in the real world. (It’s only a toe that needs to be behind the line, the rest of the body can well and truly be beyond it.)

If that has confused you, it’s rather like journeying from Auckland to Wellington. You can travel overland, which will take you a long time and you’ll have to cross rivers and go halfway up mountains on the way, but you’ll get there. Or you can go to the airport, leave the ground, travel there quickly, and land again (don’t forget the landing bit – or you’ll be a mathematician). You end up in the reality of Wellington by either route; one just happens to be quicker and simpler, and the fact that it’s off the ground doesn’t matter.