But what about -3? isn't (-3)² also 9? Don't people often write ± 3 as the square root of 9? Whenever you square a number, positive or negative, you end up with a positive, because a positive times a positive is positive, and a negative times a negative is positive too. So all square numbers must be positive.
So what do we make of this? √-9 ? Well, first we know from the rules about square roots, we can express this as √9 × √-1. The square root of any negative number ends up being the square root of a positive number, which we know how to deal with, times the square root of -1.
But this makes no sense at all. There can't be a square root of -1. It's like a little kid drew this who doesn't understand math. It's like studying married bachelors or doing the geometry of square circles. Anyone who wrote √-1 on the board like this you would want to chase out of the room, and tell them to come back when they had learned some math and knew what the symbols meant. But hold on - it turns out that if you can stomach your distaste at this impossibility, you can actually do some pretty good math with this. This raises interesting questions about the nature of mathematical truth. If we grudgingly end up admitting that we can do interesting math based on not just a lie, but something that simply makes no sense at all, what does that say about what we consider "interesting math"?
So how can we do math with a number that isn't really any number at all? How can we do math with something that simply makes no sense? First, we tame it a little by giving it a symbol, i. Then it just becomes another variable we carry around with us, like x or n. The only difference is that it obeys one simple rule.
It becomes -1, since it is the square root of -1.
So, for instance,
When we combine it with regular, real numbers, we end up with things of the form:
where a and b are reals. Because of that troublesome factor i, we can't combine or simplify them any more than that. Examples of numbers that contain parts that are multiples of i in this way are:
There are two names people give to numbers like this, and I don't really like either of them. One, because it is too harsh, and the other because in a sense it is not harsh enough.
The first name is complex numbers. This is the term that I think is too harsh, in that complex means complicated, and complicated means hard. People don't like things that are complex. And a + bi, while more complicated than just a, for instance, really isn't that complicated. I wouldn't have chosen the term "complex" to describe these things.
They also call numbers that involve this number as a factor imaginary numbers. I don't like this term either. In a way, it gives these numbers too much credit. All regular numbers are imaginary. They don't really exist in the real world of tables, chairs, and stars. Imagine a pink unicorn. Not a cartoon one, but picture it as if it were a real animal, in a field, that you could touch or feed a carrot to. Pink unicorns don't actually exist on this planet, but you can imagine them in any amount of detail you like. I can imagine a pink unicorn, or 3. But I can't, and no sane person can, imagine the square root of -1 any more than we can imagine wooden metal. So in a sense, "imaginary" is exactly the wrong word to use, since these numbers are literally unimaginable!
A great thinker, John von Neumann, once said, "In mathematics, you don't understand things. You just get used to them."
Think of it this way. In math, you teach your mind to deal with pure abstractions, like numbers. Up to now, for the most part, the stuff math deals with lines up nicely with your reality as you experience it. Three apples plus two apples equals five apples. In a sense, though, that is math with training wheels, to get you used to the idea of thinking in pure abstractions, like 3 + 2 = 5. There is a ton of math out there that people have figured out that does not line up with experienced reality. It is time to take off the training wheels. As long as you set up some hard rules, and you follow them, you can still do math even if it defies common sense. In a bit, we will see why we would even want to.
What's more, mathmaticians do stuff like this all the time: they come up with nonsensical things that they play with, building elaborate, elegant, and (within their own domains) powerful systems of proofs and theorems. They like it. They'll spend a lifetime proving everything they can discover about something that any fifth grader could tell you makes no sense at all, and that doesn't bother them. In a strange way, they're proud of it. But Nature has a funny way of turning the tables.
Euclid, an ancient Greek, lived over 2000 years ago. He laid out the basis of geometry, starting with five axioms, or postulates. He proved a lot of truths about geometry using just these five basic truths. But off and on for two thousand years, other mathmaticians had a particular doubt about the fifth postulate. It concerned parallel lines, and seemed a bit more complicated than the other four. Some people wondered if it might be provable using the other four postulates, in which case, it would turn out to be a theorem and not a postulate at all. Then all of geometry would rest on only four postulates, not five.
Over the centuries, they tried and failed to prove the fifth postulate using the other four. Eventually, they got the bright idea of doing it by contradiction. If, in fact, the fifth was a logical consequence of some combination of the first four, then its contradiction would contradict the first four. So what if we started doing geometry using a nonsensical system of the first four postulates, plus the opposite of the fifth one? If the fifth postulate could be derived from the first four, then you would quickly run into basic contradictions as you developed your nonsense geometry (called non-Euclidean geometry).
However, they found no contradictions. The fifth postulate, then, is completely independent of the other four postulates, as both it and its contradiction are perfectly consistent with the other four. The geometry that resulted from this attempt made no sense at all, of course, and was therefore completely useless if you are trying to build a bridge or design a steam engine, but the mathmaticians had great fun exploring this new abstract world.
The punch line here is that some decades later, Einstein comes along and says, "You know that non-Euclidean geometry you guys played with a while back? Yeah, well, about that. Turns out that it's true. That's actually the way spacetime looks."
In the original Arthur Conan Doyle Sherlock Holmes stories, the evil genius Moriarty was a professor of non-Euclidean geometries.
In the same way, it turns out that complex numbers are true, in that we need them to do all kinds of real world computations, including some involving quantum mechanics. Are complex numbers, as used by actual physicists, just some kind of calculational aid, or is the square root of negative one somehow built into the physical universe at some deep level? And more fundamentally, what would the difference between these be? I honestly don't know.
The mathmatician and quantum physicist Freeman Dyson once said:
The discoverers of the system of complex numbers thought of it as an artificial construction, a useful and elegant abstraction from real life. It never entered their heads that this artificial number system that they had invented was in fact the ground on which atoms move. They never imagined that nature had got there first.
Remember that there are the whole numbers, and the integers:
Then there are the rationals that fit in between the integers:
Then there are the reals that fit even between the rationals:
By the way, did you know that there are holes between the rationals? There are some numbers that can't be expressed with a fraction, no matter how big you make the denominator and the numerator? You might think you could zero in on any number and express it with a fraction, but the ancient Greeks figured out that there is no way to express √2 as a fraction.
Assume √2 = a/b, and a and b are relatively prime integers, i.e. a/b is in its simplest form. Square both sides: a²/b² = 2. Then a² = 2 b². Therefore a² is even, therefore a is even, therefore a = 2k for some integer k. So we can write a²/b² = 2 as (2k)²/b² = 2, or 4k²/b² = 2. Then 4k² = 2 b², so 2k² = b². Therefore b² is even, therefore b is even. But if a and b are both even, then that contradicts our original hypothesis that a and b are relatively prime.
You might have thought that the reals covered it, there were no more gaps to be filled in, and that was as many numbers as we could ever have. But to draw a picture of a complex number, the way we can draw an integer, rational, or real number on a number line, we need what we call the complex plane.
By convention, the horizontal axis is the real axis, and the vertical axis is the imaginary axis. To draw a number like this: -7 + 3i we put a point here:
Since the real and imaginary parts don't mix, you just have to give them each a dimension on the plane.
In ordinary math up to now, you use the plane to show pictures of functions like:
or
But now we are using the plane in an entirely different way. We are not showing how one number varies as a function of another number; we need the whole plane just to show the numbers at all.
How big is any complex number? It is a little harder to say than with regular numbers, but notice that when we are talking about integers, rationals, or reals, the size of a number is its distance from the origin on the number line. So let's continue with that and just say that the distance from the origin on the complex plane is the magnitude of a complex number. So i, -i, 1 and -1 are all the same "size". They are all 1 away from the origin.
You all know the Pythagorean Theorem?
The distance from the origin of a point on the complex plane is just the hypotenuse of a triangle, where the legs are the real component and the imaginary component. So the "size" of 3 -2i is √(3² + -2²) = √(9 + 4) = √13, or a little less than 4.
OK, let's do a little algebra.
All we've done is change the x into an i, so the answer should be exactly the same. But the answer we get in the case where we used i can be simplified. Remember that the one and only thing we actually know about i is that i² = -1. So:
which is just another complex number.
What happens when you square a real number? If it is bigger than 1, it gets bigger. If you square the result, it gets bigger still. But look what happens when we square complex numbers:
It gets bigger, but do you see how the i, when squared, kind of drags the whole thing down by turning negative and thus subtracting from the real part? Think of it as a little negativity bomb that doesn't go off until you square it.
Let me present an algorithm to you. Do you all know what an algorithm is? It is a sequence of instructions, like a recipe or directions to get someplace. Consider this one:
Take some complex number, let's call it A. Square it, then add the original number, A, back in. Take that result, square it, then add the original number, A, back in. Keep doing this, over and over again.
What we find if we try this with different starting complex numbers (A) is that for some of them, the result gets pretty big pretty quickly, flying away from the origin. Other starting points, however, just dance around the origin but never get too far from it no matter how many spins through the algorithm we do.
Given this distinction, then, let's make the following rule. If, at any point, the result gets big, like more than, say, 5 away from the origin, throw it out. We consider A to be a bad number. If, on the other hand, we do this procedure 500 times and at the end of it the number we are left with is still within 5 of the origin, we keep it and consider A a good number.
What can we say about the good numbers? Can we draw a picture of them on the complex plane? How would be do this? We would have to create a grid of pixels, dividing the complex plane very finely into many, many points. Then we would have to walk through all those points, row upon row, and perform the algorithm on each point (with up to 500 spins for each point). This sounds pretty tedious. It would take a lifetime. The guy who first invented this algorithm didn't have access to a computer, so he had to do it the hard way. He worked it for points here and there on the plane and found something kind of interesting.
In general, if the point you start with is far from the origin, it is more likely to be a bad number. Our first guess might be that if you start out a certain distance or more from the origin, you are bad, but within that distance you are good. In this case, the set of all the good points is a circle.
But a few test cases shows this not to be true. While there are some large circles of which we can say all points outside the circle are bad, and there are some small circles of which we can say that all points inside are good, any time we try out a good candidate circle for our magic border, we find that there are some points inside the circle that are bad, but other points outside of it that are good. That is, some good points are farther out than some bad points.
To see this, let's try out the points exactly 1 away from the origin in all four directions.
Note that all but one of these quickly fall into an infinitely repeating pattern, and so are good, but the second one, 0i + 1, spins out of control, getting bigger and bigger each run through the loop, and is thus a bad point.
So right off the bat, we have four points, each the same distance from the origin, and three are good but one is bad, disproving the circle idea.
So what is the shape?
The set of all the "good" points is called the Mandelbrot set after Benoit Mandelbrot, who discovered it. In the 1970's he was working on some then-obscure mathematical objects called Julia sets (also made of complex numbers), which some mathmaticians had worked on half a century earlier, and which had been largely forgotten.
Mandelbrot could only do this with, at best, a pocket calculator back then, so he could not see the actual shape at first, only plot a point here and there. Fortunately, we have invented devices that are very good at performing thousands of mathematical calculations with perfect accuracy in the blink of an eye.
I have a very crude, old-school program that I wrote in a computer language called C, that will allow you to type in the real part, then the imaginary part of a complex number, and it will show you its work as it crunches through. It only does ten runs through the loop, but this is often enough to see if a point is going to spin off into the stratosphere.
Finally, in maybe 1978 or 1979, Mandelbrot got his hands on a computer and was able to make it walk through the whole grid, and print out, in very low resolution, the picture of his set. It drew the shape in the form of x's on a printout. I have written a computer program to compute the shape on a more modern computer with a higher resolution display. It runs through the points, and gives up after 500 spins for each point. I want to emphasize that this program is not using canned data in any way. It is actually running through all these calculations as you watch.
What about this shape? Is there anything that would have led you to expect anything like this to come out? Remember, this is all based on i, the square root of -1, and as such, it is nonsensical and silly and we shouldn't even be wasting our time with such foolishness. But here it is. Why this?
But the unexpectedness of the particular shape isn't even what is most remarkable about the Mandelbrot set. I wrote this program so that if you click on a part of it, it will redraw the screen, zooming in on that part you clicked on. If you right click, you zoom back out.
The computer itself is limited. Eventually, if you zoom in enough, the graphics start to get bad looking just because the computer itself can't handle such tiny numbers. But the set itself, in a pure mathematical world, never flattens out. It is infinitely complex. You zoom in enough, you feel you are lost, and indeed, if you zoomed back out again, you could never find your way back to where you were. But it's OK, since any place on the set is just as complex, just as beautiful.
This program is in black and white, but some people have programs to do it in color. Instead of just coloring the points in the set black and the ones that get big before you get to 500 (or whatever your limit is) white, you can still make the points in the set black, but make all the points outside the set a different color depending on how many spins it took them to get a certain distance from the origin: color them based on how "bad" they are. Some people have written such programs and run them on powerful computers that can handle very tiny numbers very precisely, and they compute thousands of zooms, each one just a little more zoomed in that the last, and when they put them together, they are one smooth animated zoom. Then they set it to music and throw it on YouTube. I'm going to play you one. It goes on for a while, so at first I'm going to be quiet and we'll just watch, but after a minute or two I'll start talking again and we'll just let the video run in the background.
If you ever find yourself abducted on an alien spaceship with no clothes and no way to talk to the aliens, and your life depends on convincing them that you are intelligent and come from a reasonably advanced civilization, scratch the Mandelbrot set into the wall of your cell. You simply can't know what this shape looks like without a computer. And the aliens will have discovered it too, they will recognize it.
The Mandelbrot set is an example of a fractal. Benoit Mandelbrot himself coined the term, from the latin adjective "fractus", derived from the verb "frangere", to break (like fracture and fraction). A fractal is something that is self-similar at many different scales. That is, it looks the same from far away as it does close up. Coastlines and ferns and mountain ranges exhibit this kind of self-similarity.
Fractals in general and the Mandelbrot set in particular are often spoken of in the same breath as chaos. Chaos is the term that describes non-linear systems, that is, situations where things don't clump together the way you might expect. With normal mathematical functions, like y = x², x's that are close together will become y's that are close together. Clumped inputs lead to clumped outputs.
But in chaotic systems, neighbors do not map to neighbors. The Mandelbrot set resists generalizations about the kinds of points that turn out to be in the set as opposed to the kinds of points that are outside it. On its edge, two points can be arbitrarily close, or nearly infinitely close, and one could spin off to infinity, and the other stays well behaved. People sometimes speak of the butterfly effect: the idea that a butterfly flapping its wings in Brazil could cause a hurricane in the Atlantic Ocean. The butterfly effect describes situations in which a tiny change in initial conditions leads to huge changes in outcomes: neighbors do not map to neighbors.
This math, as I keep saying, makes no sense. Did Mandelbrot discover some deep truth of the universe, or did he just invent a clever way of manipulating symbols that gives us a computationally simple way of drawing pretty pictures? If imaginary numbers and complex numbers make no sense, what is all this? Is this all fake somehow? What does it mean? It's almost like God played some kind of trick, like He left an Easter egg in reality for us to find.
Is math something we invent, or is it something we discover? Was this all here since the beginning of time, or did we make it up? We didn't exactly make it up the way Shakespeare made up his sonnets. If Benoit Mandelbrot had never been born, this still would have been there, and someone else would eventually have discovered (or created) it. There are rules that our minds seem bound to follow. But still, it seems that we created this. If math exists, for all time, even without mathmaticians, what form does it take? What is it made of? Where is it? Is it real, or is it only in our heads? Then again, a beloved philosopher once said, "Of course it's inside your head, but why on earth should that mean that it is not real?" [Albus Dumbledore, "Harry Potter and the Deathly Hallows", page 723, last sentence of Chapter 35]