WEBVTT 00:00:00.035 --> 00:00:03.597 Somebody once told you that i is the square root of minus one, 00:00:03.680 --> 00:00:07.545 that no such number really exists, and that you should use it anyway. 00:00:07.628 --> 00:00:09.878 That is a miserable way to meet it. 00:00:09.961 --> 00:00:14.920 So let us start again from multiplication, and find out what i actually is. 00:00:16.316 --> 00:00:19.359 Here is the number line, and here is the number two. 00:00:19.555 --> 00:00:22.877 Everything in this lecture comes out of one small question. 00:00:23.073 --> 00:00:26.952 What does multiplying actually do to a point sitting on this line? 00:00:27.748 --> 00:00:31.383 Multiply by three, and the point slides out to six. 00:00:31.579 --> 00:00:35.469 It is further from zero, and it is still pointing the same way. 00:00:35.782 --> 00:00:38.767 Multiplying by a positive number is a stretch. 00:00:39.563 --> 00:00:41.782 Now multiply by minus one. 00:00:42.094 --> 00:00:44.066 The point does not stretch at all. 00:00:44.149 --> 00:00:47.515 It swings straight through zero and comes to rest on the other side, 00:00:47.598 --> 00:00:49.851 exactly as far out as it was. 00:00:50.047 --> 00:00:51.685 That is not a stretch. 00:00:51.882 --> 00:00:53.241 That is a turn. 00:00:54.037 --> 00:00:56.209 A half turn, about zero. 00:00:56.406 --> 00:00:59.089 And that is where the schoolbook rule comes from. 00:00:59.541 --> 00:01:04.276 Two negatives make a positive because a half turn, followed by another half turn, 00:01:04.359 --> 00:01:06.217 brings you back where you started. 00:01:07.014 --> 00:01:09.662 But a line is a cramped place to turn in. 00:01:09.765 --> 00:01:14.178 Everything on it can only do two things: point right, or point left. 00:01:14.374 --> 00:01:16.488 Nothing, and half a turn. 00:01:16.801 --> 00:01:19.693 There is no room on a line for anything in between. 00:01:20.745 --> 00:01:24.449 Which lets us ask the famous question in a completely different way. 00:01:24.646 --> 00:01:28.954 Is there a number which, multiplied by itself, gives minus one? 00:01:29.267 --> 00:01:34.306 In turning language, is there a turn which, done twice, is a half turn? 00:01:35.219 --> 00:01:37.261 Put like that, it is not impossible. 00:01:37.344 --> 00:01:38.633 It is easy. 00:01:38.946 --> 00:01:41.919 Half of a half turn. A quarter turn. 00:01:43.432 --> 00:01:45.870 Here is the plane that line was living in. 00:01:46.183 --> 00:01:51.397 The real numbers run straight across the middle, and here is one, where we started. 00:01:51.594 --> 00:01:54.776 A quarter turn cannot land anywhere along that row. 00:01:55.259 --> 00:01:57.756 So let it land one step above zero. 00:01:57.952 --> 00:02:01.747 That place is what the letter i names, and there is nothing impossible about it. 00:02:01.830 --> 00:02:04.826 It is a point on a page, one unit up. 00:02:05.530 --> 00:02:07.957 Now watch it earn its reputation. 00:02:08.061 --> 00:02:12.973 Start at one, on the circle of radius one, and turn a quarter of the way round. 00:02:13.769 --> 00:02:15.602 There you are, at i. 00:02:15.685 --> 00:02:20.562 Now turn another quarter, the very same turn again, and you land on minus one. 00:02:21.358 --> 00:02:26.070 Two quarter turns make a half turn, and a half turn is multiplication by minus one. 00:02:26.153 --> 00:02:28.952 So i times i is minus one. 00:02:29.149 --> 00:02:33.584 Not because anybody insisted on it, but because that is what turning twice does. 00:02:34.068 --> 00:02:37.749 The thing you were told did not exist is a rotation. 00:02:37.852 --> 00:02:41.113 And once you have a rotation, you can rotate anything at all, 00:02:41.196 --> 00:02:43.124 which is the rest of this lecture. 00:02:44.214 --> 00:02:49.230 One is over here on the real line, and i is up there, one step above zero. 00:02:49.543 --> 00:02:52.536 Between the two of them they open out an entire plane, 00:02:52.619 --> 00:02:55.244 and every single point of it is a number. 00:02:56.041 --> 00:02:59.385 Reading a point is exactly like reading a map reference. 00:02:59.489 --> 00:03:03.852 Two steps along the real direction, one step up the imaginary direction, 00:03:03.935 --> 00:03:07.164 and the number you have arrived at is written two plus i. 00:03:07.775 --> 00:03:10.283 Any point at all reads the same way. 00:03:10.480 --> 00:03:13.589 Three across and two down is three minus two i. 00:03:13.672 --> 00:03:16.921 There is nowhere in this plane that is not a complex number, 00:03:17.004 --> 00:03:20.128 and in general we write one as a plus b i. 00:03:20.832 --> 00:03:25.964 Now multiply this number by i, using nothing but the algebra you already have. 00:03:26.161 --> 00:03:28.237 i times two is two i, 00:03:28.320 --> 00:03:33.371 and i times i is i squared, which we have just decided is minus one. 00:03:33.796 --> 00:03:36.885 So the product is minus one plus two i. 00:03:37.082 --> 00:03:41.597 Watch what that means on the picture. The arrow has not grown and it has not shrunk. 00:03:41.680 --> 00:03:45.895 It has swung round by exactly a quarter turn, just as the one did. 00:03:46.320 --> 00:03:49.348 And look at what happened to the grid while you were watching the arrow. 00:03:49.431 --> 00:03:54.004 Every point moved. Multiplying by i does not act on one number at a time. 00:03:54.087 --> 00:03:58.093 It turns the whole plane, rigidly, a quarter turn about zero. 00:03:58.518 --> 00:04:00.293 Check that against the coordinates. 00:04:00.376 --> 00:04:03.802 Two along and one up has become one to the left and two up. 00:04:03.998 --> 00:04:07.050 The pair has swapped over, and one sign has flipped, 00:04:07.133 --> 00:04:10.466 which is exactly what a quarter turn does to a map reference. 00:04:11.262 --> 00:04:12.621 So do it again. 00:04:12.725 --> 00:04:17.347 A second quarter turn is a half turn, which is multiplying by minus one. 00:04:17.450 --> 00:04:19.318 A third takes us round to here. 00:04:19.401 --> 00:04:24.231 And a fourth brings the plane home, which is why i to the fourth power is one. 00:04:25.028 --> 00:04:29.845 Four quarter turns, one full circle, and everything is back where it began. 00:04:29.928 --> 00:04:32.007 There is no memorising in that. 00:04:32.110 --> 00:04:33.783 You can watch it happen. 00:04:34.696 --> 00:04:36.633 So here is the first real payoff. 00:04:36.716 --> 00:04:40.232 A complex number is not a strange kind of quantity. 00:04:40.315 --> 00:04:44.330 It is an instruction: turn by this much, stretch by that much. 00:04:44.413 --> 00:04:48.582 And multiplying is carrying out two instructions one after the other. 00:04:49.643 --> 00:04:53.592 So far, every multiplication has either stretched or turned. 00:04:53.788 --> 00:04:58.015 Now let us do a completely general one, and watch it do both at once. 00:04:59.067 --> 00:05:03.271 Here is a number z, drawn as an arrow out from zero. 00:05:03.467 --> 00:05:05.244 Two things pin it down. 00:05:05.441 --> 00:05:09.328 How long the arrow is, which we write with two vertical bars. 00:05:09.411 --> 00:05:10.931 And which way it points: 00:05:11.014 --> 00:05:14.973 its angle, measured round from the positive real direction. 00:05:16.188 --> 00:05:20.775 Here is a second number w, with a length and an angle of its own. 00:05:20.878 --> 00:05:26.104 At the moment w is sitting at one, so it has length one and angle nothing. 00:05:26.300 --> 00:05:28.748 Multiplying by one does nothing at all, 00:05:28.831 --> 00:05:33.232 which is why the product z w is lying exactly on top of z. 00:05:34.145 --> 00:05:37.652 Now turn w, and keep your eye on the red arrow. 00:05:37.965 --> 00:05:40.923 As w swings up away from the real axis, 00:05:41.006 --> 00:05:44.246 the product swings up by exactly the same amount. 00:05:45.043 --> 00:05:47.853 Those two marked angles are the same angle. 00:05:48.050 --> 00:05:52.808 Whatever w's angle is, the product sits that much further round than z. 00:05:52.891 --> 00:05:54.529 The angles add. 00:05:55.233 --> 00:05:57.983 And it stays true wherever w goes. 00:05:58.066 --> 00:06:01.491 There is nothing special about the place we happened to stop. 00:06:02.891 --> 00:06:04.251 Now the lengths. 00:06:04.447 --> 00:06:09.173 w still has length one, so z w is exactly as long as z. 00:06:09.486 --> 00:06:12.378 Watch what happens when I make w longer. 00:06:13.290 --> 00:06:15.799 The product grows in the same proportion. 00:06:15.995 --> 00:06:20.243 Make w twice as long, and z w becomes twice as long. 00:06:20.326 --> 00:06:23.427 Lengths multiply, exactly as angles add. 00:06:24.339 --> 00:06:25.777 Both at once, then. 00:06:25.860 --> 00:06:30.955 To multiply two complex numbers, multiply the lengths and add the angles. 00:06:31.038 --> 00:06:32.931 Notice what has vanished. 00:06:33.035 --> 00:06:36.368 There is no formula to remember with four products in it. 00:06:37.579 --> 00:06:40.421 Let us check that against arithmetic you can do by hand. 00:06:40.504 --> 00:06:43.210 Take one plus i, and square it. 00:06:44.007 --> 00:06:46.376 By hand it is four terms. 00:06:46.573 --> 00:06:48.243 One times one is one, 00:06:48.326 --> 00:06:54.201 then i, then i again, then i times i, which is minus one. 00:06:54.397 --> 00:06:58.787 The one and the minus one cancel, and what is left is two i. 00:06:59.583 --> 00:07:01.639 Now the same thing by turning. 00:07:01.743 --> 00:07:07.443 One plus i is one across and one up, so by Pythagoras its length is the square root 00:07:07.443 --> 00:07:11.264 of two, and its arrow points at forty-five degrees. 00:07:12.060 --> 00:07:14.011 Multiply it by itself. 00:07:14.208 --> 00:07:18.296 The lengths multiply, and root two times root two is two. 00:07:18.492 --> 00:07:22.789 The angles add, and forty-five plus forty-five is ninety. 00:07:22.985 --> 00:07:26.736 So the answer has length two, and it points straight up. 00:07:27.533 --> 00:07:30.598 Length two, straight up, is two i. 00:07:30.911 --> 00:07:34.719 Which is what the algebra said, and now you can see why it said it. 00:07:35.423 --> 00:07:37.198 Nothing new was needed there. 00:07:37.281 --> 00:07:40.857 It is the same multiplication you have always done, looked at from the side 00:07:40.857 --> 00:07:42.367 where it makes sense. 00:07:43.372 --> 00:07:46.101 Everything so far has been geometry. 00:07:46.298 --> 00:07:49.245 Now watch a piece of algebra fall out of the geometry, 00:07:49.328 --> 00:07:51.662 and it is the piece with e in it. 00:07:52.366 --> 00:07:54.747 Stay on the circle of radius one. 00:07:54.850 --> 00:07:57.867 Every number sitting on it has length exactly one, 00:07:57.950 --> 00:08:01.028 so multiplying by it cannot stretch anything. 00:08:01.224 --> 00:08:03.338 All it can do is turn. 00:08:04.088 --> 00:08:07.851 Write E of t for the number at angle t on that circle. 00:08:08.012 --> 00:08:10.219 The blue point is E of s. 00:08:10.381 --> 00:08:12.425 The green point is E of t. 00:08:13.187 --> 00:08:14.871 Multiply them together. 00:08:15.033 --> 00:08:20.757 Lengths multiply, and one times one is one, so the answer is still on the circle. 00:08:20.860 --> 00:08:25.621 Angles add, so it sits at s plus t. There it is, in red. 00:08:25.965 --> 00:08:28.880 So E turns adding into multiplying. 00:08:28.984 --> 00:08:32.398 Feed it a sum of angles and you get a product of numbers. 00:08:32.594 --> 00:08:35.684 Now look at that line and tell me what it reminds you of. 00:08:36.480 --> 00:08:41.531 It is the law of exponents, the one you met years ago with powers of two. 00:08:41.844 --> 00:08:46.640 b to the s, times b to the t, is b to the s plus t. 00:08:46.836 --> 00:08:49.171 The same law, letter for letter. 00:08:49.712 --> 00:08:51.803 And that is not a coincidence. 00:08:52.115 --> 00:08:53.938 Anything at all that turns addition 00:08:53.938 --> 00:08:58.200 into multiplication in that way is a power of some fixed number. 00:08:58.396 --> 00:09:03.621 So E of t is b to the t, and the only question left is which b. 00:09:05.018 --> 00:09:08.583 So look very closely at the circle, right next to one. 00:09:09.380 --> 00:09:13.792 Here is one, and here is the point a small angle h round from it. 00:09:14.407 --> 00:09:19.595 The radius is one, so the arc from one up to that point has length exactly h, 00:09:19.678 --> 00:09:21.850 when h is measured in radians. 00:09:22.553 --> 00:09:26.223 And the short way from one to that point is very nearly straight up. 00:09:26.837 --> 00:09:31.319 Straight up from one, by h, is the number one plus i h. 00:09:31.930 --> 00:09:34.590 Watch the two of them as the angle shrinks. 00:09:34.693 --> 00:09:35.994 The curved way and the straight 00:09:35.994 --> 00:09:41.486 way pull together, and the gap between the true point and one plus i h simply closes. 00:09:42.399 --> 00:09:47.020 So for a small angle, turning by h is the same as adding i h. 00:09:47.217 --> 00:09:48.820 Which base does that? 00:09:49.132 --> 00:09:53.719 There is exactly one number whose powers behave that way, and it is e. 00:09:54.144 --> 00:09:55.910 That is what e is for. 00:09:56.106 --> 00:09:59.808 e is the base whose powers climb at exactly the rate one at the start, 00:09:59.891 --> 00:10:03.967 so that e to the x is one plus x whenever x is small. 00:10:04.671 --> 00:10:07.330 So put i h where x is. 00:10:07.527 --> 00:10:12.866 e to the i h is one plus i h, and that is precisely our tiny turn. 00:10:12.949 --> 00:10:15.933 The base we were hunting for is e to the i. 00:10:16.242 --> 00:10:18.043 Which finishes the job. 00:10:18.239 --> 00:10:23.174 E of t, the number at angle t on the unit circle, is e to the i t. 00:10:24.571 --> 00:10:28.089 Back to the whole circle, and read off the coordinates of that point. 00:10:28.286 --> 00:10:30.504 That is all Euler's formula is. 00:10:31.208 --> 00:10:35.235 The point at angle theta on the unit circle is cos theta across, 00:10:35.318 --> 00:10:37.072 and sin theta up. 00:10:37.268 --> 00:10:42.250 That is what cosine and sine mean, and it is the only thing they have ever meant. 00:10:42.733 --> 00:10:47.401 But we have just shown that the very same point is e to the i theta. 00:10:47.597 --> 00:10:49.781 Two names for one point. 00:10:49.977 --> 00:10:55.215 Set them equal, and you have written down the most famous formula in mathematics. 00:10:55.756 --> 00:10:57.856 It is not a mystical identity. 00:10:57.939 --> 00:11:03.289 It says that the number at angle theta has coordinates cosine theta and sine theta, 00:11:03.372 --> 00:11:08.783 and that turning by theta is exactly what raising e to the i theta does. 00:11:09.580 --> 00:11:11.903 Now the famous special case. 00:11:12.355 --> 00:11:14.294 Put theta equal to pi. 00:11:14.491 --> 00:11:16.103 Half a turn round the circle, 00:11:16.186 --> 00:11:20.366 one hundred and eighty degrees, lands you exactly on minus one. 00:11:20.745 --> 00:11:23.857 So e to the i pi is minus one. 00:11:23.961 --> 00:11:25.726 Which we have known since the first minute 00:11:25.726 --> 00:11:29.790 of this lecture, when we swung a point through a half turn on the number line. 00:11:30.929 --> 00:11:32.195 Let us finish by putting all 00:11:32.195 --> 00:11:35.876 of this to work on a question the old story cannot really answer. 00:11:36.189 --> 00:11:38.976 Which numbers, cubed, give one? 00:11:39.889 --> 00:11:42.676 The old story says one, and stops there. 00:11:42.872 --> 00:11:45.541 But cubing means multiplying three times over, 00:11:45.624 --> 00:11:50.373 and multiplying is turning, so let us ask it as a question about turns. 00:11:51.076 --> 00:11:53.388 Suppose z cubes to one. 00:11:53.584 --> 00:11:57.985 Lengths multiply, so the length of z, cubed, must be one. 00:11:58.437 --> 00:12:01.686 The only positive length whose cube is one is one itself, 00:12:01.769 --> 00:12:04.417 so z has to sit on the unit circle. 00:12:05.213 --> 00:12:08.976 That is half the answer, and we have not touched the angle yet. 00:12:09.288 --> 00:12:12.354 Every cube root of one is somewhere on this circle. 00:12:13.151 --> 00:12:16.634 Angles add, so cubing an angle triples it. 00:12:16.738 --> 00:12:21.418 Here is z, in yellow, and here in red is z cubed. 00:12:21.615 --> 00:12:24.599 Watch the red one as I walk z round. 00:12:25.396 --> 00:12:27.623 z has gone a third of the way round. 00:12:27.706 --> 00:12:32.897 The cube has gone the whole way round, a complete turn, and landed back on one. 00:12:33.000 --> 00:12:34.731 So that angle works. 00:12:35.528 --> 00:12:37.479 One hundred and twenty degrees. 00:12:37.675 --> 00:12:39.392 Another third of the way round, 00:12:39.475 --> 00:12:43.121 and the cube goes round a second full turn and lands on one again. 00:12:44.173 --> 00:12:46.670 Two hundred and forty degrees works too. 00:12:46.983 --> 00:12:49.303 And a third third brings z home to one, 00:12:49.386 --> 00:12:52.115 with the cube having gone round three times. 00:12:53.167 --> 00:12:57.055 So there are the three angles: nothing, a hundred and twenty, 00:12:57.138 --> 00:12:58.810 and two hundred and forty. 00:12:59.007 --> 00:13:02.166 Three cube roots of one, not a single one. 00:13:02.869 --> 00:13:05.387 And they are evenly spaced around the circle. 00:13:05.470 --> 00:13:09.382 They have to be, because three equal turns can only add up to a whole number 00:13:09.382 --> 00:13:13.203 of full turns, and there are exactly three ways to do that. 00:13:14.716 --> 00:13:18.812 Written in the coordinates you were taught, that second root is minus a half, 00:13:18.895 --> 00:13:21.961 plus root three over two, times i. 00:13:22.758 --> 00:13:27.867 Cubing that by hand is a page of algebra and a good chance to make a mistake. 00:13:28.064 --> 00:13:30.221 Cubing it by turning is one line. 00:13:30.304 --> 00:13:33.379 Three times a hundred and twenty degrees is three hundred and sixty, 00:13:33.462 --> 00:13:35.913 which is no turn at all, which is one. 00:13:36.709 --> 00:13:40.016 And the same argument hands you every root of every number. 00:13:40.099 --> 00:13:45.557 The n-th roots of one are n points, evenly spaced around the unit circle. 00:13:45.660 --> 00:13:46.764 Always. 00:13:47.677 --> 00:13:49.567 So that is the whole lecture. 00:13:49.650 --> 00:13:51.843 Multiplying is turning and stretching. 00:13:51.926 --> 00:13:53.935 i is the quarter turn. 00:13:54.132 --> 00:13:57.639 e to the i theta is the turn by theta. 00:13:57.743 --> 00:14:01.633 And the roots of one are the corners of a regular polygon. 00:14:02.545 --> 00:14:05.344 None of it needed a number that does not exist. 00:14:05.448 --> 00:14:08.537 It needed a number line with enough room to turn.