{"version":1,"lectureId":"01M14TZ3CG0A8CY7NEXHKZWWX2","attempt":2,"publication":{"slug":"complex-multiplication-as-rotation","title":"Multiplication as Turning","subject":"mathematics","summary":"Complex numbers taught as turning rather than as impossible square roots. The lecture starts on the number line, where multiplying by a positive number stretches and multiplying by minus one is a half turn, and asks what a turn that is half of a half turn must be. That puts i one step above zero and makes i squared equals minus one a fact about geometry. From there, multiplying by any complex number multiplies lengths and adds angles, and Euler's formula falls out of that picture without a single power series: on the unit circle, turning converts adding into multiplying, which is the law of exponents, and a tiny turn is a step of i, which fixes the base at e. It closes with the three cube roots of one, standing evenly around a circle because three equal turns have to add up to a whole number of full turns.","metaDescription":"Complex numbers as rotation: i as a quarter turn, lengths multiplying while angles add, Euler's formula, and the cube roots of one.","transcript":"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.","watch":{"version":1,"scenes":[{"title":"A Number That Turns","start":0,"end":164.33575,"objects":{"ang":"a VariableNumber","card":"a Title that says \"Complex Numbers — Multiplication as Turning\"","eq_i":"a Math [text] that says \"$i^2 = -1$\"","flip":"a Math [text] that says \"$(-1) dot.op 6 = -6$\"","head_line":"a Heading that says \"What Multiplying Does\"","head_turn":"a Heading that says \"Half of a Half Turn\"","i_mark":"a Point [magenta] labelled \"i\" drawn in plane (location=(0.0, 1.0))","i_note":"a Panel that says \"The quarter turn about zero. 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That is a miserable way to meet it. So let us start again from multiplication, and find out what i actually is.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[15.627,"card is hidden from the screen — left the board."]]},{"start":16.827,"say":"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?","live":null,"does":[[16.827,"head_line is shown on the screen, written out."],[16.827,"line is shown on the screen, written out."],[19.381,"walker is shown on the screen, written out."]]},{"start":28.4215,"say":"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.","live":["line","head_line","walker"],"does":[[28.828,"line moves to a new place on the board."],[28.828,"stretch is shown on the screen, written out."],[30.755,"walker is redrawn as the numbers it depends on change."],[30.755,"pos ticks to 6.0."]]},{"start":40.399,"say":"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.","live":["stretch","line","head_line","walker"],"does":[[41.025999999999996,"flip is shown on the screen, written out."],[45.577,"walker is redrawn as the numbers it depends on change."],[45.577,"pos ticks to -6.0."]]},{"start":54.4905,"say":"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.","live":["stretch","flip","line","head_line","walker"],"does":[[57.602,"twice is shown on the screen, written out."],[63.82399999999999,"walker is redrawn as the numbers it depends on change."],[63.82399999999999,"pos ticks to 6.0."]]},{"start":66.5025,"say":"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.","live":["stretch","flip","twice","line","head_line","walker"],"does":[]},{"start":80.826,"say":"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?","live":null,"does":[[85.85299999999998,"puzzle is shown on the screen, written out."]]},{"start":95.21849999999999,"say":"Put like that, it is not impossible. It is easy. Half of a half turn. A quarter turn.","live":["stretch","flip","twice","puzzle","line","head_line","walker"],"does":[[98.17899999999999,"A box is drawn around puzzle."],[102.48599999999999,"flip is hidden from the screen — left the board."],[102.48599999999999,"head_line is hidden from the screen — left the board."],[102.48599999999999,"line is hidden from the screen — left the board."],[102.48599999999999,"walker is hidden from the screen — line left the board."],[102.48599999999999,"puzzle is hidden from the screen — left the board."],[102.48599999999999,"stretch is hidden from the screen — left the board."],[102.48599999999999,"twice is hidden from the screen — left the board."]]},{"start":103.68599999999999,"say":"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.","live":[],"does":[[103.68599999999999,"head_turn is shown on the screen, written out."],[103.68599999999999,"plane is shown on the screen, written out."],[109.10799999999998,"one_mark is shown on the screen, written out."]]},{"start":114.73549999999999,"say":"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.","live":["plane","head_turn","one_mark"],"does":[[115.44399999999997,"i_mark is shown on the screen, written out."]]},{"start":125.34349999999999,"say":"Now watch it earn its reputation. 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Now turn another quarter, the very same turn again, and you land on minus one.","live":["plane","head_turn","one_mark","i_mark","ring","mover","spoke"],"does":[[136.48499999999996,"mover is redrawn as the numbers it depends on change."],[136.48499999999996,"spoke is redrawn as the numbers it depends on change."],[136.48499999999996,"ang ticks to 3.141592653589793."],[139.96799999999996,"minus_mark is shown on the screen, written out."]]},{"start":141.47899999999998,"say":"Two quarter turns make a half turn, and a half turn is multiplication by minus one. So i times i is minus one. 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Between the two of them they open out an entire plane, and every single point of it is a number.","live":[],"does":[[164.33575,"heading is shown on the screen, written out."],[164.33575,"plane is shown on the screen, written out."],[171.52275,"grid is shown on the screen, written out."]]},{"start":175.11775,"say":"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.","live":["plane","heading","grid"],"does":[[178.58974999999998,"line is shown on the screen, drawn."],[180.73775,"line_2 is shown on the screen, drawn."],[182.93775,"line is hidden from the screen."],[182.93775,"line_2 is hidden from the screen."],[184.49875,"z_pt is shown on the screen, written out."],[184.49875,"z_vec is shown on the screen, written out."]]},{"start":186.74775,"say":"Any point at all reads the same way. Three across and two down is three minus two i. 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The arrow has not grown and it has not shrunk. It has swung round by exactly a quarter turn, just as the one did.","live":["general","product","plane","heading","grid","z_pt","z_vec"],"does":[[213.89574999999996,"product becomes \"$i dot.op (2 + i) = -1 + 2 i$\"."],[221.96474999999998,"plane: put this frame's data space back where it started (reset_matrix)."],[225.60424999999998,"z_pt: one name gives way to another over the same drawing (label_becomes)."]]},{"start":226.20425,"say":"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.","live":null,"does":[[227.93975,"grid is indicated — a transient flash."],[231.75975,"rule is shown on the screen, written out."]]},{"start":239.40675,"say":"Check that against the coordinates. Two along and one up has become one to the left and two up. 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And a fourth brings the plane home, which is why i to the fourth power is one.","live":null,"does":[[251.29725,"z_pt: take the name off, leaving the drawing (hide_label)."],[252.89374999999998,"plane: put this frame's data space back where it started (reset_matrix)."],[257.52575,"plane: put this frame's data space back where it started (reset_matrix)."],[259.84775,"plane: put this frame's data space back where it started (reset_matrix)."],[262.68075,"powers is shown on the screen, written out."]]},{"start":264.59225,"say":"Four quarter turns, one full circle, and everything is back where it began. There is no memorising in that. You can watch it happen.","live":["general","product","rule","powers","plane","heading","grid","z_pt","z_vec"],"does":[[270.66475,"A box is drawn around powers."]]},{"start":274.12075,"say":"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.","live":null,"does":[[289.83554166666664,"general is hidden from the screen — left the board."],[289.83554166666664,"heading is hidden from the screen — left the board."],[289.83554166666664,"plane is hidden from the screen — left the board."],[289.83554166666664,"grid is hidden from the screen — plane left the board."],[289.83554166666664,"z_pt is hidden from the screen — plane left the board."],[289.83554166666664,"z_vec is hidden from the screen — plane left the board."],[289.83554166666664,"powers is hidden from the screen — left the board."],[289.83554166666664,"product is hidden from the screen — left the board."],[289.83554166666664,"rule is hidden from the screen — left the board."]]}]},{"title":"Lengths Multiply, Angles Add","start":290.87720833333333,"end":452.7313333333333,"objects":{"ang_rule":"a Math [text] that says \"$op(\"arg\")(z w) = op(\"arg\") z + op(\"arg\") w$\"","angle_p":"an Angle [red] labelled \"phi\" drawn in plane (sides=((0.7324906109962805, 0.5229316444835745), (((1.4 * rw) * cos((…, radius=0.95)","angle_w":"an Angle [green] labelled \"phi\" drawn in plane (sides=((2.6, 0.0), ((rw * cos(beta)), (rw * sin(beta)))), radius=0.55)","beta":"a VariableNumber","ex0":"a Math [text] that says \"$(1 + i)(1 + i) = 1 + i + i + i^2$\"","ex_ang":"a Math [text] that says \"$op(\"arg\")(1 + i) = 45 degree$\"","ex_chk":"a Math [text] that says \"$sqrt(2) dot.op sqrt(2) = 2$\"","ex_len":"a Math [text] that says \"$|1 + i| = sqrt(2)$\"","head_check":"a Heading that says \"Checking It by Hand\"","head_rule":"a Heading that says \"What a General Multiplication Does\"","len_rule":"a Math [text] that says \"$|z w| = |z| dot.op |w|$\"","one_i":"a Vector [blue] labelled \"1 + i\" drawn in plane (end=(1.0, 1.0))","p_arrow":"a Vector [red] labelled \"z w\" drawn in plane (end=(((1.4 * rw) * cos((0.62 + beta))), ((1.4 * rw) * sin((0.62 + b…)","plane":"an Axes (x_range=(-3.2, 3.2), y_range=(-3.2, 3.2), aspect=(1, 1))","ring":"a Circle [gray] drawn in plane","rw":"a VariableNumber (initial_value=1.0)","two_i":"a Vector [red] labelled \"2 i\" drawn in plane (end=(0.0, 2.0))","w_arrow":"a Vector [green] labelled \"w\" drawn in plane (end=((rw * cos(beta)), (rw * sin(beta))))","words":"a Text [text] that says \"Lengths multiply. Angles add.\"","z_arrow":"a Vector [blue] labelled \"z\" drawn in plane (end=(1.1394298393275473, 0.813449224752227))"},"beats":[{"start":290.87720833333333,"say":"So far, every multiplication has either stretched or turned. Now let us do a completely general one, and watch it do both at once.","live":[],"does":[[290.87720833333333,"head_rule is shown on the screen, written out."],[290.87720833333333,"plane is shown on the screen, written out."],[296.35720833333335,"ring is shown on the screen, written out."]]},{"start":299.92920833333335,"say":"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.","live":["plane","head_rule","ring"],"does":[[300.33520833333336,"z_arrow is shown on the screen, written out."]]},{"start":317.08520833333336,"say":"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.","live":["plane","head_rule","ring","z_arrow"],"does":[[317.88620833333334,"w_arrow is shown on the screen, written out."],[329.93720833333333,"p_arrow is shown on the screen, written out."]]},{"start":334.2407083333333,"say":"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.","live":["plane","head_rule","ring","z_arrow","w_arrow","p_arrow"],"does":[[335.0532083333333,"w_arrow is redrawn as the numbers it depends on change."],[335.0532083333333,"p_arrow is redrawn as the numbers it depends on change."],[335.0532083333333,"angle_w is redrawn as the numbers it depends on change."],[335.0532083333333,"angle_p is redrawn as the numbers it depends on change."],[335.0532083333333,"beta ticks to 0.95."]]},{"start":344.98770833333333,"say":"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.","live":null,"does":[[345.78820833333333,"angle_w is shown on the screen, written out."],[346.03820833333333,"angle_p is shown on the screen, written out."],[352.4642083333333,"plane moves to a new place on the board."],[352.4642083333333,"ang_rule is shown on the screen, written out."]]},{"start":353.60470833333335,"say":"And it stays true wherever w goes. There is nothing special about the place we happened to stop.","live":["ang_rule","plane","head_rule","ring","z_arrow","w_arrow","p_arrow","angle_w","angle_p"],"does":[[355.60720833333335,"w_arrow is redrawn as the numbers it depends on change."],[355.60720833333335,"p_arrow is redrawn as the numbers it depends on change."],[355.60720833333335,"angle_w is redrawn as the numbers it depends on change."],[355.60720833333335,"angle_p is redrawn as the numbers it depends on change."],[355.60720833333335,"beta ticks to 2.1."]]},{"start":360.2822083333333,"say":"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.","live":null,"does":[[360.93220833333334,"len_rule is shown on the screen, written out."]]},{"start":369.8447083333333,"say":"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.","live":["ang_rule","len_rule","plane","head_rule","ring","z_arrow","w_arrow","p_arrow","angle_w","angle_p"],"does":[[370.57620833333334,"w_arrow is redrawn as the numbers it depends on change."],[370.57620833333334,"p_arrow is redrawn as the numbers it depends on change."],[370.57620833333334,"angle_w is redrawn as the numbers it depends on change."],[370.57620833333334,"angle_p is redrawn as the numbers it depends on change."],[370.57620833333334,"rw ticks to 1.9."],[376.43920833333334,"words is shown on the screen, written out."]]},{"start":380.0697083333333,"say":"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.","live":["ang_rule","len_rule","words","plane","head_rule","ring","z_arrow","w_arrow","p_arrow","angle_w","angle_p"],"does":[[390.63420833333333,"ang_rule is hidden from the screen — left the board."],[390.63420833333333,"head_rule is hidden from the screen — left the board."],[390.63420833333333,"len_rule is hidden from the screen — left the board."],[390.63420833333333,"words is hidden from the screen — left the board."]]},{"start":391.8342083333333,"say":"Let us check that against arithmetic you can do by hand. Take one plus i, and square it.","live":["plane","ring","z_arrow","w_arrow","p_arrow","angle_w","angle_p"],"does":[[391.8342083333333,"head_check is shown on the screen, written out."],[391.8342083333333,"z_arrow is hidden from the screen."],[391.8342083333333,"w_arrow is hidden from the screen."],[391.8342083333333,"p_arrow is hidden from the screen."],[391.8342083333333,"angle_w is hidden from the screen."],[391.8342083333333,"angle_p is hidden from the screen."],[395.3872083333333,"one_i is shown on the screen, written out."]]},{"start":398.2042083333333,"say":"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.","live":["plane","ring","head_check","one_i"],"does":[[399.7022083333333,"ex0 is shown on the screen, written out."],[409.66320833333333,"ex0 becomes \"$(1 + i)(1 + i) = 2 i$\"."]]},{"start":411.87670833333334,"say":"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.","live":["plane","ring","ex0","head_check","one_i"],"does":[[417.7632083333333,"ex_len is shown on the screen, written out."],[421.8612083333333,"ex_ang is shown on the screen, written out."]]},{"start":423.9017083333333,"say":"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.","live":["plane","ring","ex0","ex_len","ex_ang","head_check","one_i"],"does":[[427.6632083333333,"ex_chk is shown on the screen, written out."]]},{"start":438.0272083333333,"say":"Length two, straight up, is two i. Which is what the algebra said, and now you can see why it said it.","live":["plane","ring","ex0","ex_len","ex_ang","ex_chk","head_check","one_i"],"does":[[439.32720833333326,"two_i is shown on the screen, written out."],[441.7532083333333,"A box is drawn around ex0."]]},{"start":444.9892083333333,"say":"Nothing new was needed there. It is the same multiplication you have always done, looked at from the side where it makes sense.","live":["plane","ring","ex0","ex_len","ex_ang","ex_chk","head_check","one_i","two_i"],"does":[[451.68966666666665,"ex0 is hidden from the screen — left the board."],[451.68966666666665,"ex_ang is hidden from the screen — left the board."],[451.68966666666665,"ex_chk is hidden from the screen — left the board."],[451.68966666666665,"ex_len is hidden from the screen — left the board."],[451.68966666666665,"head_check is hidden from the screen — left the board."],[451.68966666666665,"plane is hidden from the screen — left the board."],[451.68966666666665,"ring is hidden from the screen — plane left the board."],[451.68966666666665,"one_i is hidden from the screen — plane left the board."],[451.68966666666665,"two_i is hidden from the screen — plane left the board."]]}]},{"title":"Where e Comes In","start":452.7313333333333,"end":689.5743541666666,"objects":{"arc":"a ParametricCurve [gray] drawn in near (function=<function>, t_range=(-0.05, 0.62))","arc_pt":"a Point [magenta] labelled \"E(h)\" drawn in near (location=(cos(h), sin(h)))","base":"a Math [text] that says \"$b = e^i$\"","circle_axes":"an Axes (x_range=(-1.7, 1.7), y_range=(-1.7, 1.7), aspect=(1, 1))","cos_mark":"a Point [blue] labelled \"cos theta\" drawn in circle_axes (location=(cos(theta), 0.0))","e_prop":"a Math [text] that says \"$e^x approx 1 + x$\"","epi":"a Math [text] that says \"$e^(i pi) = -1$\"","epoint":"a Point [red] labelled \"e^(i theta)\" drawn in circle_axes (location=(cos(theta), sin(theta)))","euler":"a Math [text] that says \"$e^(i theta) = cos theta + i sin theta$\"","euler_pre":"a Math [text] that says \"$E(t) = e^(i t)$\"","exp_law":"a Math [text] that says \"$b^s dot.op b^t = b^(s + t)$\"","h":"a VariableNumber (initial_value=0.55)","head_base":"a Heading that says \"Which Base?\"","head_euler":"a Heading that says \"Euler's Formula\"","head_law":"a Heading that says \"Turning Obeys the Law of Exponents\"","law":"a Math [text] that says \"$E(s) dot.op E(t) = E(s + t)$\"","near":"an Axes (x_range=(0.55, 1.25), y_range=(-0.08, 0.62), aspect=(1, 1))","neg_one":"a Point [text] labelled \"-1\" drawn in circle_axes (location=(-1.0, 0.0))","one_pt":"a Point [text] labelled \"1\" drawn in near (location=(1.0, 0.0))","p1":"a Point [blue] labelled \"E(s)\" drawn in circle_axes (location=(0.8525245220595057, 0.5226872289306592))","p2":"a Point [green] labelled \"E(t)\" drawn in circle_axes (location=(0.7316888688738209, 0.6816387600233341))","power":"a Math [text] that says \"$E(t) = b^t$\"","psum":"a Point [red] labelled \"E(s + t)\" drawn in circle_axes (location=(0.26749882862458735, 0.963558185417193))","ring":"a Circle [gray] drawn in circle_axes","rise":"a Line [yellow] drawn in near (start=(1.0, 0.0), end=(1.0, 0.6), dashed=True)","sin_mark":"a Point [green] labelled \"sin theta\" drawn in circle_axes (location=(0.0, sin(theta)))","small":"a Math [text] that says \"$E(h) approx 1 + i h$\"","spoke":"a Line [red] drawn in circle_axes (end=(cos(theta), sin(theta)))","theta":"a VariableNumber (initial_value=0.9)","unit_line":"a Text [text] that says \"Every number on this circle has length one, so multiplying by it can only turn.\"","up_pt":"a Point [yellow] labelled \"1 + i h\" drawn in near (location=(1.0, <VariableNumber h = 0.08>))"},"beats":[{"start":452.7313333333333,"say":"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.","live":[],"does":[[452.7313333333333,"head_law is shown on the screen, written out."],[452.7313333333333,"circle_axes is shown on the screen, written out."]]},{"start":461.3423333333333,"say":"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.","live":["circle_axes","head_law"],"does":[[462.2243333333333,"ring is shown on the screen, written out."],[471.7673333333333,"circle_axes moves to a new place on the board."],[471.7673333333333,"unit_line is shown on the screen, written out."]]},{"start":473.13383333333326,"say":"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.","live":["unit_line","circle_axes","head_law","ring"],"does":[[477.63833333333326,"p1 is shown on the screen, written out."],[479.9373333333333,"p2 is shown on the screen, written out."]]},{"start":482.30183333333326,"say":"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.","live":["unit_line","circle_axes","head_law","ring","p1","p2"],"does":[[491.39333333333326,"law is shown on the screen, written out."],[495.87433333333325,"psum is shown on the screen, written out."]]},{"start":497.3683333333333,"say":"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.","live":["unit_line","law","circle_axes","head_law","ring","p1","p2","psum"],"does":[]},{"start":508.3133333333333,"say":"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.","live":null,"does":[[509.6953333333333,"exp_law is shown on the screen, written out."],[515.3373333333333,"exp_law (the \"b^s dot.op b^t\" part) is emphasized."],[517.9843333333333,"exp_law (the \"b^(s + t)\" part) is emphasized."],[517.9843333333333,"exp_law (the \"b^s dot.op b^t\" part) is no longer emphasized."],[520.3763333333333,"exp_law (the \"b^(s + t)\" part) is no longer emphasized."]]},{"start":522.2998333333333,"say":"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.","live":["unit_line","law","exp_law","circle_axes","head_law","ring","p1","p2","psum"],"does":[[529.6253333333333,"power is shown on the screen, written out."],[536.9393333333333,"circle_axes is hidden from the screen — left the board."],[536.9393333333333,"ring is hidden from the screen — circle_axes left the board."],[536.9393333333333,"p1 is hidden from the screen — circle_axes left the board."],[536.9393333333333,"p2 is hidden from the screen — circle_axes left the board."],[536.9393333333333,"psum is hidden from the screen — circle_axes left the board."],[536.9393333333333,"exp_law is hidden from the screen — left the board."],[536.9393333333333,"head_law is hidden from the screen — left the board."],[536.9393333333333,"law is hidden from the screen — left the board."],[536.9393333333333,"power is hidden from the screen — left the board."],[536.9393333333333,"unit_line is hidden from the screen — left the board."]]},{"start":538.1393333333333,"say":"So look very closely at the circle, right next to one.","live":[],"does":[[538.1393333333333,"head_base is shown on the screen, written out."],[538.1393333333333,"near is shown on the screen, written out."],[539.7883333333333,"arc is shown on the screen, drawn."]]},{"start":542.4778333333333,"say":"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.","live":["near","head_base","arc"],"does":[[542.7793333333333,"one_pt is shown on the screen, written out."],[544.6253333333333,"arc_pt is shown on the screen, written out."]]},{"start":555.7558333333333,"say":"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.","live":["near","head_base","arc","one_pt","arc_pt"],"does":[[559.0873333333333,"rise is shown on the screen, written out."],[562.7683333333333,"up_pt is shown on the screen, written out."]]},{"start":565.4698333333333,"say":"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.","live":["near","head_base","arc","one_pt","arc_pt","rise","up_pt"],"does":[[565.8183333333333,"arc_pt is redrawn as the numbers it depends on change."],[565.8183333333333,"up_pt is redrawn as the numbers it depends on change."],[565.8183333333333,"h ticks to 0.08."],[574.7463333333333,"near moves to a new place on the board."],[574.7463333333333,"small is shown on the screen, written out."]]},{"start":576.3908333333333,"say":"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.","live":["small","near","head_base","arc","one_pt","arc_pt","rise","up_pt"],"does":[[584.1933333333333,"e_prop is shown on the screen, written out."]]},{"start":588.8568333333333,"say":"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.","live":["small","e_prop","near","head_base","arc","one_pt","arc_pt","rise","up_pt"],"does":[[597.5523333333333,"e_prop (the \"1 + x\" part) is emphasized."],[599.9788333333333,"e_prop (the \"1 + x\" part) is no longer emphasized."]]},{"start":600.5788333333333,"say":"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.","live":null,"does":[[609.1703333333332,"base is shown on the screen, written out."]]},{"start":612.3128333333333,"say":"Which finishes the job. E of t, the number at angle t on the unit circle, is e to the i t.","live":["small","e_prop","base","near","head_base","arc","one_pt","arc_pt","rise","up_pt"],"does":[[612.7193333333333,"euler_pre is shown on the screen, written out."],[619.7198333333333,"base is hidden from the screen — left the board."],[619.7198333333333,"e_prop is hidden from the screen — left the board."],[619.7198333333333,"euler_pre is hidden from the screen — left the board."],[619.7198333333333,"head_base is hidden from the screen — left the board."],[619.7198333333333,"near is hidden from the screen — left the board."],[619.7198333333333,"arc is hidden from the screen — near left the board."],[619.7198333333333,"one_pt is hidden from the screen — near left the board."],[619.7198333333333,"arc_pt is hidden from the screen — near left the board."],[619.7198333333333,"rise is hidden from the screen — near left the board."],[619.7198333333333,"up_pt is hidden from the screen — near left the board."],[619.7198333333333,"small is hidden from the screen — left the board."],[619.7198333333333,"circle_axes is shown on the screen, faded in — cast on this board again."],[619.7198333333333,"ring is shown on the screen, faded in — circle_axes came back to the board."],[619.7198333333333,"p1 is shown on the screen, faded in — circle_axes came back to the board."],[619.7198333333333,"p2 is shown on the screen, faded in — circle_axes came back to the board."],[619.7198333333333,"psum is shown on the screen, faded in — circle_axes came back to the board."]]},{"start":620.9198333333334,"say":"Back to the whole circle, and read off the coordinates of that point. That is all Euler's formula is.","live":["circle_axes","ring","p1","p2","psum"],"does":[[620.9198333333334,"head_euler is shown on the screen, written out."],[620.9198333333334,"p1 is hidden from the screen."],[620.9198333333334,"p2 is hidden from the screen."],[620.9198333333334,"psum is hidden from the screen."],[622.0813333333333,"epoint is shown on the screen, written out."],[622.0813333333333,"spoke is shown on the screen, written out."]]},{"start":628.2188333333334,"say":"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.","live":["circle_axes","ring","head_euler","epoint","spoke"],"does":[[632.2123333333333,"cos_mark is shown on the screen, written out."],[633.6523333333333,"sin_mark is shown on the screen, written out."]]},{"start":641.0673333333333,"say":"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.","live":["circle_axes","ring","head_euler","epoint","spoke","cos_mark","sin_mark"],"does":[[648.5903333333333,"euler is shown on the screen, written out."]]},{"start":653.9153333333334,"say":"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.","live":["circle_axes","ring","euler","head_euler","epoint","spoke","cos_mark","sin_mark"],"does":[[659.4303333333332,"euler (the \"cos theta + i sin theta\" part) is emphasized."],[664.9333333333333,"euler (the \"cos theta + i sin theta\" part) is no longer emphasized."],[664.9333333333333,"euler (the \"e^(i theta)\" part) is emphasized."],[667.3713333333333,"euler (the \"e^(i theta)\" part) is no longer emphasized."]]},{"start":667.9713333333333,"say":"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.","live":null,"does":[[667.9713333333333,"cos_mark is hidden from the screen."],[667.9713333333333,"sin_mark is hidden from the screen."],[672.6963333333333,"epoint is redrawn as the numbers it depends on change."],[672.6963333333333,"spoke is redrawn as the numbers it depends on change."],[672.6963333333333,"theta ticks to 3.141592653589793."],[677.2823333333333,"neg_one is shown on the screen, written out."]]},{"start":679.0088333333333,"say":"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.","live":["circle_axes","ring","euler","head_euler","epoint","spoke","neg_one"],"does":[[680.5873333333333,"epi is shown on the screen, written out."],[683.0723333333333,"A box is drawn around epi."],[688.5326875,"circle_axes is hidden from the screen — left the board."],[688.5326875,"ring is hidden from the screen — circle_axes left the board."],[688.5326875,"epoint is hidden from the screen — circle_axes left the board."],[688.5326875,"spoke is hidden from the screen — circle_axes left the board."],[688.5326875,"neg_one is hidden from the screen — circle_axes left the board."],[688.5326875,"epi is hidden from the screen — left the board."],[688.5326875,"euler is hidden from the screen — left the board."],[688.5326875,"head_euler is hidden from the screen — left the board."]]}]},{"title":"The Three Cube Roots of One","start":689.5743541666666,"end":841.0839791666665,"objects":{"ang_arg":"a Math [text] that says \"$3 theta = 0 degree, 360 degree, 720 degree$\"","cube":"a Point [red] labelled \"z^3\" drawn in plane (location=(cos((3.0 * theta)), sin((3.0 * theta))))","eq":"a Math [text] that says \"$z^3 = 1$\"","head_all":"a Heading that says \"Every Root of Every Number\"","len_arg":"a Math [text] that says \"$r^3 = 1 arrow.r r = 1$\"","n_note":"a Panel that says \"The $n$-th roots of one are $n$ points, evenly spaced around the unit circle.\"","omega":"a Math [text] that says \"$omega = -frac(1, 2) + frac(sqrt(3), 2) i$\"","plane":"an Axes (x_range=(-1.6, 1.6), y_range=(-1.6, 1.6), aspect=(1, 1))","question":"a Panel that says \"Which numbers, cubed, give one?\"","r0":"a Point [magenta] labelled \"1\" drawn in plane (location=(1.0, 0.0))","ring":"a Circle [gray] drawn in plane","roots":"a Math [text] that says \"$theta = 0 degree, 120 degree, 240 degree$\"","seed":"a Point [yellow] labelled \"z\" drawn in plane (location=(cos(theta), sin(theta)))","theta":"a VariableNumber","tri":"a Polygon [green] drawn in plane (vertices=((1.0, 0.0), (-0.5, 0.8660254037844386), (-0.5, -0.866025403784…, filled=False)","w1":"a Point [magenta] labelled \"omega\" drawn in plane (location=(-0.5, 0.8660254037844386))","w2":"a Point [magenta] labelled \"omega^2\" drawn in plane (location=(-0.5, -0.8660254037844386))"},"beats":[{"start":689.5743541666666,"say":"Let us finish by putting all of this to work on a question the old story cannot really answer. Which numbers, cubed, give one?","live":[],"does":[[689.5743541666666,"question is shown on the screen, written out."]]},{"start":697.7788541666666,"say":"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.","live":["question"],"does":[[701.2963541666666,"eq is shown on the screen, written out."],[707.1133541666666,"plane is shown on the screen, written out."],[707.1133541666666,"ring is shown on the screen, written out."]]},{"start":708.6413541666666,"say":"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.","live":["eq","plane","question","ring"],"does":[[711.2073541666666,"len_arg is shown on the screen, written out."]]},{"start":722.0703541666666,"say":"That is half the answer, and we have not touched the angle yet. Every cube root of one is somewhere on this circle.","live":["eq","len_arg","plane","question","ring"],"does":[[728.3053541666666,"ring is indicated — a transient flash."]]},{"start":729.7408541666666,"say":"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.","live":null,"does":[[733.5833541666666,"seed is shown on the screen, written out."],[736.3933541666665,"cube is shown on the screen, written out."],[739.8413541666666,"seed is redrawn as the numbers it depends on change."],[739.8413541666666,"cube is redrawn as the numbers it depends on change."],[739.8413541666666,"theta ticks to 2.0943951023931953."]]},{"start":742.1248541666666,"say":"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.","live":["eq","len_arg","plane","question","ring","seed","cube"],"does":[[750.8203541666666,"w1 is shown on the screen, written out."]]},{"start":752.1398541666666,"say":"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.","live":["eq","len_arg","plane","question","ring","seed","cube","w1"],"does":[[754.2763541666666,"seed is redrawn as the numbers it depends on change."],[754.2763541666666,"cube is redrawn as the numbers it depends on change."],[754.2763541666666,"theta ticks to 4.1887902047863905."],[759.1873541666666,"w2 is shown on the screen, written out."]]},{"start":759.7873541666665,"say":"Two hundred and forty degrees works too. And a third third brings z home to one, with the cube having gone round three times.","live":["eq","len_arg","plane","question","ring","seed","cube","w1","w2"],"does":[[763.6533541666665,"seed is redrawn as the numbers it depends on change."],[763.6533541666665,"cube is redrawn as the numbers it depends on change."],[763.6533541666665,"theta ticks to 6.283185307179586."]]},{"start":768.1773541666665,"say":"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.","live":null,"does":[[769.2573541666666,"ang_arg is shown on the screen, written out."],[769.7573541666666,"roots is shown on the screen, written out."],[774.2613541666666,"r0 is shown on the screen, written out."]]},{"start":776.7943541666666,"say":"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.","live":["eq","len_arg","ang_arg","roots","plane","question","ring","seed","cube","w1","w2","r0"],"does":[[777.5203541666666,"tri is shown on the screen, written out."],[787.3188541666666,"plane moves to a new place on the board."],[787.3188541666666,"ang_arg is hidden from the screen — left the board."],[787.3188541666666,"eq is hidden from the screen — left the board."],[787.3188541666666,"len_arg is hidden from the screen — left the board."],[787.3188541666666,"question is hidden from the screen — left the board."],[787.3188541666666,"roots is hidden from the screen — left the board."]]},{"start":788.5188541666666,"say":"Written in the coordinates you were taught, that second root is minus a half, plus root three over two, times i.","live":["plane","ring","seed","cube","w1","w2","r0","tri"],"does":[[788.5188541666666,"head_all is shown on the screen, written out."],[788.5188541666666,"seed is hidden from the screen."],[788.5188541666666,"cube is hidden from the screen."],[789.4123541666665,"omega is shown on the screen, written out."]]},{"start":795.9108541666666,"say":"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.","live":["plane","ring","w1","w2","r0","tri","omega","head_all"],"does":[[801.7153541666665,"w1 is indicated — a transient flash."]]},{"start":809.4673541666666,"say":"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.","live":null,"does":[[813.9483541666666,"n_note is shown on the screen, written out."]]},{"start":819.7958541666666,"say":"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.","live":["plane","ring","w1","w2","r0","tri","omega","n_note","head_all"],"does":[[832.5903541666665,"tri is indicated — a transient flash."]]},{"start":834.1653541666666,"say":"None of it needed a number that does not exist. It needed a number line with enough room to turn.","live":null,"does":[[840.0423124999999,"head_all is hidden from the screen — left the board."],[840.0423124999999,"n_note is hidden from the screen — left the board."],[840.0423124999999,"omega is hidden from the screen — left the board."],[840.0423124999999,"plane is hidden from the screen — left the board."],[840.0423124999999,"ring is hidden from the screen — plane left the board."],[840.0423124999999,"w1 is hidden from the screen — plane left the board."],[840.0423124999999,"w2 is hidden from the screen — plane left the board."],[840.0423124999999,"r0 is hidden from the screen — plane left the board."],[840.0423124999999,"tri is hidden from the screen — plane left the board."]]}]}]},"durationSeconds":841,"chapters":[{"title":"A Number That Turns","startSeconds":0,"narration":"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."},{"title":"A Plane of Numbers","startSeconds":164.33575,"narration":"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."},{"title":"Lengths Multiply, Angles Add","startSeconds":290.87720833333333,"narration":"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."},{"title":"Where e Comes In","startSeconds":452.7313333333333,"narration":"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."},{"title":"The Three Cube Roots of One","startSeconds":689.5743541666666,"narration":"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."}]}}