Grand Mathemagician: Welcome back, Apprentice. Are you ready to continue our journey? Can you refresh my memory on what we discussed before?
Apprentice: Well, we talked about some sets of numbers, the whole, or natural, numbers, the integers, the rational numbers and the irrational numbers.
G.M.: Ah yes, so we have discussed the the sets that combine to form the real numbers. What do you know about these Real numbers?
Apprentice: I’ve learned about them from some of the oracles. They usually show me a number line when talking about them.
G.M.: Yes, a number line is useful for imagining the continuum of real numbers. The thing is, these so called Real Numbers have some fairly unrealistic properties. They have no bound, going from negative infinity to positive infinity, and they are uncountable, in the sense that integers are countable. Between any two real numbers there are an infinite number of other real numbers. With real numbers, I can take one, divide it in half, divide the half in half, divide the halved half in half, and so on forever. Then I can take those pieces, and add them up to get one again. Why don’t you grab a saw and a board, and see if you can do the same to it.
Apprentice: I could try, but I’d have to stop at some point.
G.M.: Correct. From a practical standpoint, you will reach a point where you can’t cut anymore. So we’re not actually using real numbers when we measure and cut the board, we’re using approximations, limited by the precision of our tools. In fact, the set of real numbers is a purely mathematical construct. The properties that we define for the real numbers provide a logical foundation for studying more interesting problems with important applications. Speak the the Oracle of Change about calculus sometime and he will tell you more.
Apprentice: So wait. If the real numbers form a foundation for higher maths, and the real numbers are, well, imaginary, then isn’t all of math imaginary? What’s the point?
G.M.: I see you have been spending time with the Oracle of Logic. But you are correct. I prefer word abstraction though. Mathematics is the practice of using abstraction to solve problems. Numbers, variables, functions, shapes, these are all abstract ideas that live in our brains. We have various symbols we use to represent these ideas, but the symbols can change, and do not fully encapsulate the concepts. Abstraction is the point: it allows you to take complicated real life problems, strip away the extraneous information , and break them down into digestible pieces. It allows you to find connections between seemingly disparate problems, and to make new discoveries that were not obvious with all the real life fluff in the way. In fact, for many decisions you make in your life you consider the variables of the situation and play out different scenarios in your head. You can’t predict every variable, so you have to make some assumptions.
Apprentice: So abstraction is a key part of solving problems and making decisions. And mathematics provides a framework for abstraction.
G.M.: Exactly. And by continually building up this framework, we can increase our ability to solve problems and discover new ideas and tools. So now, you tell me, what is the point of imaginary numbers?
Apprentice: Well they’re certainly an abstract idea. I guess they must have some usefulness in solving problems if we’re still talking about them.
G.M.: And they do. However, the idea of imaginary numbers is incomplete: we should instead be talking about complex numbers.
Apprentice: Ok, what’s a complex number?
G.M.: A complex number is a number with two parts: a real component and an imaginary component. When you first see them, they will be represented as x + yi, where x is the real part, y is the imaginary part, and
i =. And this is where it gets a bit weird, because we can’t find . A square with an area of -1 makes no sense. But, remember, a square with an area of exactly 2 is no more realistic. Keep in mind, this is an abstraction. It doesn’t have to have an immediate application to real life, as long as it has some usefulness in solving a problem that does have an application to real life. As it turns out, complex numbers can be very useful when studying many things; for example, periodic functions.
Apprentice: Periodic functions, you mean like waves and stuff?
G.M.: Indeed, waves are an example of periodic motion. In general periodic just means something that repeats, that has some kind of cycle. And we witness periodic phenomena all the time: the motion of the planets, the cycle of the tides, the oscillation of a pendulum, even the alternating current that provides electricity. Further, periodicity is not confined to time: anything which exhibits symmetry is periodic in space. Even elementary particles, electrons, protons and so forth, exhibit wavelike properties, and can be studied using periodic functions.
G.M.: First thing, you can actually forget about the for now. A more useful way of thinking about complex numbers is to think of them as two-dimensional numbers. Instead of the real number line, we will use the complex number plane. The horizontal axis gives the real part and the vertical axis the imaginary. Don’t forget, though, those words, real and imaginary, are pretty much meaningless. As you can see, we can label the same point using an angle theta and the distance from the origin r. And now, for a bit of magic. There’s a beautiful formula which links complex numbers to trigonometry, and our old friend e shows up to the party. Without further ado, here is the formula:
Apprentice: By the name of Odin, where did that come from?
G.M.: I would explain it to you now, but it would take the better part of the evening, and my nose tells me that dinner is almost ready. For now, take it on faith that this formula is true.
Apprentice: I hate it when you teachers say that!
G.M.: As you should, and I am not discouraging you from looking deeper into it, but as with many things in math, the proof requires a foundation of other non-trivial topics that we must discuss first. Speak to the Oracle of Change about Power Series and you will be on your way. Either way, the formula links trigonometry, sines and cosines which we use to study periodic motion, with the complex exponential e to the i times theta. And it so happens that working with complex exponentials is easier than working with sines and cosines, at least when you’re trying to study complicated wave patterns.
Apprentice: So complex numbers just make other math easier?
G.M.: In many cases they do. The study of complex numbers makes mathematics more complete: it provides another piece of the puzzle, a different way of looking at things, and allows for new discoveries. i may be less immediately applicable than say, three, but it is arguably far more powerful.
Apprentice: I think I’m starting to get it, but I still have more questions
G.M.: And you always will. That’s what makes math so much fun! Consider that we currently are unable to provide a complete mathematical rule that describes everything in the universe, and that we likely never will. But we will continue to try, not only to satisfy our curiosity, but also to advance as a civilization. Given these conditions, why should we not invent whatever the hell kind of number we want to, as long as it advances mathematics and humankind?
Apprentice: Well I guess when you put it that way….But you sound like you’ve been spending too much time with the pipeweed.
G.M.: Now that is no concern of yours. And I think I hear the dinner bell ringing, so we are done for today.
Apprentice: Thank you, oh wise one. Let’s go eat.
G.M.: On that we have an understanding.