Multiplication as Turning

Somebody once told you that i is the square root of minus one, that no such number really exists, and that you should use it anyway. That is a miserable way to meet it. So let us start again from multiplication, and find out what i actually is. Here is the number line, and here is the number two. Everything in this lecture comes out of one small question. What does multiplying actually do to a point sitting on this line? Multiply by three, and the point slides out to six. It is further from zero, and it is still pointing the same way. Multiplying by a positive number is a stretch. Now multiply by minus one. The point does not stretch at all. It swings straight through zero and comes to rest on the other side, exactly as far out as it was. That is not a stretch. That is a turn. A half turn, about zero. And that is where the schoolbook rule comes from. Two negatives make a positive because a half turn, followed by another half turn, brings you back where you started. But a line is a cramped place to turn in. Everything on it can only do two things: point right, or point left. Nothing, and half a turn. There is no room on a line for anything in between. Which lets us ask the famous question in a completely different way. Is there a number which, multiplied by itself, gives minus one? In turning language, is there a turn which, done twice, is a half turn? Put like that, it is not impossible. It is easy. Half of a half turn. A quarter turn. Here is the plane that line was living in. The real numbers run straight across the middle, and here is one, where we started. A quarter turn cannot land anywhere along that row. So let it land one step above zero. That place is what the letter i names, and there is nothing impossible about it. It is a point on a page, one unit up. Now watch it earn its reputation. Start at one, on the circle of radius one, and turn a quarter of the way round. There you are, at i. Now turn another quarter, the very same turn again, and you land on minus one. Two quarter turns make a half turn, and a half turn is multiplication by minus one. So i times i is minus one. Not because anybody insisted on it, but because that is what turning twice does. The thing you were told did not exist is a rotation. And once you have a rotation, you can rotate anything at all, which is the rest of this lecture.

One is over here on the real line, and i is up there, one step above zero. Between the two of them they open out an entire plane, and every single point of it is a number. Reading a point is exactly like reading a map reference. Two steps along the real direction, one step up the imaginary direction, and the number you have arrived at is written two plus i. Any point at all reads the same way. Three across and two down is three minus two i. There is nowhere in this plane that is not a complex number, and in general we write one as a plus b i. Now multiply this number by i, using nothing but the algebra you already have. i times two is two i, and i times i is i squared, which we have just decided is minus one. So the product is minus one plus two i. Watch what that means on the picture. The arrow has not grown and it has not shrunk. It has swung round by exactly a quarter turn, just as the one did. And look at what happened to the grid while you were watching the arrow. Every point moved. Multiplying by i does not act on one number at a time. It turns the whole plane, rigidly, a quarter turn about zero. Check that against the coordinates. Two along and one up has become one to the left and two up. The pair has swapped over, and one sign has flipped, which is exactly what a quarter turn does to a map reference. So do it again. A second quarter turn is a half turn, which is multiplying by minus one. A third takes us round to here. And a fourth brings the plane home, which is why i to the fourth power is one. Four quarter turns, one full circle, and everything is back where it began. There is no memorising in that. You can watch it happen. So here is the first real payoff. A complex number is not a strange kind of quantity. It is an instruction: turn by this much, stretch by that much. And multiplying is carrying out two instructions one after the other.

So far, every multiplication has either stretched or turned. Now let us do a completely general one, and watch it do both at once. Here is a number z, drawn as an arrow out from zero. Two things pin it down. How long the arrow is, which we write with two vertical bars. And which way it points: its angle, measured round from the positive real direction. Here is a second number w, with a length and an angle of its own. At the moment w is sitting at one, so it has length one and angle nothing. Multiplying by one does nothing at all, which is why the product z w is lying exactly on top of z. Now turn w, and keep your eye on the red arrow. As w swings up away from the real axis, the product swings up by exactly the same amount. Those two marked angles are the same angle. Whatever w's angle is, the product sits that much further round than z. The angles add. And it stays true wherever w goes. There is nothing special about the place we happened to stop. Now the lengths. w still has length one, so z w is exactly as long as z. Watch what happens when I make w longer. The product grows in the same proportion. Make w twice as long, and z w becomes twice as long. Lengths multiply, exactly as angles add. Both at once, then. To multiply two complex numbers, multiply the lengths and add the angles. Notice what has vanished. There is no formula to remember with four products in it. Let us check that against arithmetic you can do by hand. Take one plus i, and square it. By hand it is four terms. One times one is one, then i, then i again, then i times i, which is minus one. The one and the minus one cancel, and what is left is two i. Now the same thing by turning. One plus i is one across and one up, so by Pythagoras its length is the square root of two, and its arrow points at forty-five degrees. Multiply it by itself. The lengths multiply, and root two times root two is two. The angles add, and forty-five plus forty-five is ninety. So the answer has length two, and it points straight up. Length two, straight up, is two i. Which is what the algebra said, and now you can see why it said it. Nothing new was needed there. It is the same multiplication you have always done, looked at from the side where it makes sense.

Everything so far has been geometry. Now watch a piece of algebra fall out of the geometry, and it is the piece with e in it. Stay on the circle of radius one. Every number sitting on it has length exactly one, so multiplying by it cannot stretch anything. All it can do is turn. Write E of t for the number at angle t on that circle. The blue point is E of s. The green point is E of t. Multiply them together. Lengths multiply, and one times one is one, so the answer is still on the circle. Angles add, so it sits at s plus t. There it is, in red. So E turns adding into multiplying. Feed it a sum of angles and you get a product of numbers. Now look at that line and tell me what it reminds you of. It is the law of exponents, the one you met years ago with powers of two. b to the s, times b to the t, is b to the s plus t. The same law, letter for letter. And that is not a coincidence. Anything at all that turns addition into multiplication in that way is a power of some fixed number. So E of t is b to the t, and the only question left is which b. So look very closely at the circle, right next to one. Here is one, and here is the point a small angle h round from it. The radius is one, so the arc from one up to that point has length exactly h, when h is measured in radians. And the short way from one to that point is very nearly straight up. Straight up from one, by h, is the number one plus i h. Watch the two of them as the angle shrinks. The curved way and the straight way pull together, and the gap between the true point and one plus i h simply closes. So for a small angle, turning by h is the same as adding i h. Which base does that? There is exactly one number whose powers behave that way, and it is e. That is what e is for. e is the base whose powers climb at exactly the rate one at the start, so that e to the x is one plus x whenever x is small. So put i h where x is. e to the i h is one plus i h, and that is precisely our tiny turn. The base we were hunting for is e to the i. Which finishes the job. E of t, the number at angle t on the unit circle, is e to the i t. Back to the whole circle, and read off the coordinates of that point. That is all Euler's formula is. The point at angle theta on the unit circle is cos theta across, and sin theta up. That is what cosine and sine mean, and it is the only thing they have ever meant. But we have just shown that the very same point is e to the i theta. Two names for one point. Set them equal, and you have written down the most famous formula in mathematics. It is not a mystical identity. It says that the number at angle theta has coordinates cosine theta and sine theta, and that turning by theta is exactly what raising e to the i theta does. Now the famous special case. Put theta equal to pi. Half a turn round the circle, one hundred and eighty degrees, lands you exactly on minus one. So e to the i pi is minus one. Which we have known since the first minute of this lecture, when we swung a point through a half turn on the number line.

Let us finish by putting all of this to work on a question the old story cannot really answer. Which numbers, cubed, give one? The old story says one, and stops there. But cubing means multiplying three times over, and multiplying is turning, so let us ask it as a question about turns. Suppose z cubes to one. Lengths multiply, so the length of z, cubed, must be one. The only positive length whose cube is one is one itself, so z has to sit on the unit circle. That is half the answer, and we have not touched the angle yet. Every cube root of one is somewhere on this circle. Angles add, so cubing an angle triples it. Here is z, in yellow, and here in red is z cubed. Watch the red one as I walk z round. z has gone a third of the way round. The cube has gone the whole way round, a complete turn, and landed back on one. So that angle works. One hundred and twenty degrees. Another third of the way round, and the cube goes round a second full turn and lands on one again. Two hundred and forty degrees works too. And a third third brings z home to one, with the cube having gone round three times. So there are the three angles: nothing, a hundred and twenty, and two hundred and forty. Three cube roots of one, not a single one. And they are evenly spaced around the circle. They have to be, because three equal turns can only add up to a whole number of full turns, and there are exactly three ways to do that. Written in the coordinates you were taught, that second root is minus a half, plus root three over two, times i. Cubing that by hand is a page of algebra and a good chance to make a mistake. Cubing it by turning is one line. Three times a hundred and twenty degrees is three hundred and sixty, which is no turn at all, which is one. And the same argument hands you every root of every number. The n-th roots of one are n points, evenly spaced around the unit circle. Always. So that is the whole lecture. Multiplying is turning and stretching. i is the quarter turn. e to the i theta is the turn by theta. And the roots of one are the corners of a regular polygon. None of it needed a number that does not exist. It needed a number line with enough room to turn.