The rotation problem and Hamilton's discovery of quaternions I | Famous Math Problems 13a

Transcript

Hello everyone, I'm Norman Wildberger. In today's  famous math problem, we're going to look at the   famous Irish mathematician William Rowan Hamilton  and his discovery of quaternions in the context   of what I call the rotation problem, which is  the problem of how do we describe, understand,   and manipulate rotations in three-dimensional 

space algebraically. The motivation for this   problem comes from the intimate connection  between complex numbers and rotations in the   plane. Complex numbers are numbers of the form  a + bi, where i is a kind of imaginary number   satisfying i squared equals minus 1.

Around  Hamilton's time, it was well appreciated that   these complex numbers somehow had an intimate  connection with the geometry of the plane and,   in particular, allowed an efficient  computational calculus for rotations.   So the problem that Hamilton posed to himself was:  what algebraic structure plays an analogous role   for rotations in space? We have rotations in 

the plane, and we know we can rely on complex   numbers to help us with the algebra, but what  about rotations in three-dimensional space,   which are considerably more subtle  and difficult to manipulate? Well, for a long time, Hamilton thought that  the answer lay in extending the idea of complex   numbers from a two-dimensional algebra to a  three-dimensional algebra.

In other words,   he was thinking about vectors of the form t  + ai + bj (forget the ck for a moment) and   trying to find a way of introducing an algebraic  structure on these three-dimensional vectors. He   sought a way of multiplying them that allowed  one to sort of capture the rotational structure   of three-dimensional space.

Then, on some  fateful day in 1843, while walking with his   wife along a canal in Dublin, he was crossing  a bridge when this inspiration came to him. He   realized that the solution was to consider  not a three-dimensional space of vectors,   but rather a four-dimensional space of vectors, 

and that the crucial properties satisfied by   these new numbers i, j, and k could be captured by  these equations: i squared was equal to minus 1,   as was j squared and k squared (that's much  like the complex number situation). In addition,   there's a relationship between i, j, and k which  we can write as i times j times k equals minus 1.   He realized this in a flash of insight and, 

in a small piece of mathematical vandalism,   carved this equation into the stonework of  the bridge. That's a famous mathematical   anecdote—a bit of high drama in  the world of pure mathematics. William Hamilton was no doubt Ireland's most 

famous mathematician, a very brilliant fellow   who made contributions to physics, optics, and  other areas of algebra as well. In addition to   his discovery of quaternions, he also built up  the theory of quaternions into a very powerful and   broad tool for doing physics. However, this came 

into conflict with another approach to vectors,   and ultimately, quaternions lost out to  our current approach using dot products   and cross products. But as we will see, they  are actually closely connected to quaternions   as well. Hamilton was also famous for an 

important insight into classical mechanics.   He took the framework of Lagrange established  for understanding Newton's laws and twisted   it in an important new direction to give  the so-called Hamiltonian formulation of   classical mechanics. This not only brought into  being a subject called symplectic geometry but   also was a very important contributor to the 

20th-century development of quantum mechanics. We're going to describe this problem, and  it's going to take us probably three lectures   to do that. So in today's lecture, I want to  start by setting the framework, so to speak,   by making very clear this connection 

between complex numbers and rotations   in the plane. It turns out that if you look at  that subject in the right way—in other words,   in a rational way—then that makes it much easier  to understand what happens with the story of   quaternions and rotations in three-dimensional  space.

That's what we're going to do today,   mostly, and then in the next lecture, we're  going to talk about rotations and how we think   about rotations in three-dimensional space, and  then we'll get to the actual quaternion algebra   in four dimensions that Hamilton introduced.  So we'd have to go up to four dimensions and   understand a little bit of the geometry of four  dimensions to get at these quaternions.

Now,   this is a subject that is quite important  these days because working with rotations   is something that we do a lot in industrial  work, in graphics, in computer programming,   for example in video game construction. So there  are lots of situations where we want to manipulate   rotations effectively, and it turns out that 

quaternions are probably still the best and most   efficient way of doing that. It's a lovely subject  that undergraduates can learn about, so I'm going   to give you an introduction to Hamilton's  quaternions in these next three lectures. I'm going to start off with this much simpler 

situation: complex numbers and rotations of   the plane. There will be a lot of material in  this lecture, none of it too sophisticated,   but some of it a bit novel because I have  a rational point of view towards things,   and it turns out this rational point of view is  a very good way of understanding things and makes   a lot of higher geometry much simpler.

So even  if you're very familiar with complex numbers,   you are going to learn some important things  in today's lecture. If you haven't seen complex   numbers before, well, this is probably something  of an introduction.

You can also have a look at   the WildTrig 15 video I made quite a few years ago  now, but it also has some information on complex   numbers. You might want to watch that before  you have a look at this, so don't hesitate   to stop the video, go back, and make sure  you understand everything I'm saying here.

Alright, we're starting with complex numbers,  but I adopt a rational point of view because   I don't believe in irrational numbers. It's not a  religious position; it's just that I haven't seen   any, and no one has actually shown me a proper  irrational number, so why should I believe in   them? It's a very good position that I encourage 

you to adopt as well. You might think that it   diminishes one's mathematics, but it turns out  that it does exactly the opposite—it strengthens   one's mathematics because one can then look  carefully and clearly and logically at many   things which previously required waffling.  So, a complex number for us is a pair (a,   b) of rational numbers, and this is what I 

use to denote the type of rational numbers.   I prefer not to think in terms of infinite sets.  Pictorially, we have an XY plane, and the pair (a,   b) can be thought of as this point here  with x-coordinate a and y-coordinate b,   or can also be thought of as the vector from  the origin to this point. Sometimes it's best   to think in terms of a point, and sometimes it's 

better to think in terms of a vector, but in fact,   logically speaking, the complex number is neither  of those—it's just the pair of rational numbers. Complex numbers support operations—actually,  they support all four operations of addition,   multiplication, subtraction, and division—but  here are the two main ones: addition   and multiplication.

Addition is pointwise,  corresponding to the usual vector sum of vectors.   The multiplication is where all the interest lies.  So the formula is that (a, b) times (c, d) is, by   definition, a times c minus b times d—that's the  first entry—and then the second entry is a times   d plus b times c. So we make the definition—that's 

just how we're going to define multiplication of   these ordered pairs of rational numbers—and it  turns out that's a very good choice because it   has lovely properties. Namely, these operations  satisfy the following familiar laws of arithmetic:   first of all, the two operations are commutative:  z plus w equals w plus z, and z times w equals w   times z.

Please check that. We have associativity:  that if you have three complex numbers, it doesn't   matter in what order we pair them. So z plus  (w plus u) is the same as (z plus w) plus u,   and more importantly for multiplication as well: z 

times (w times u) is the same as (z times w) times   u. Then there's a distributive law: z times (w  plus u) equals zw plus zu. And here I'm allowing   myself the shortcut zw means z times w. These laws 

are relatively straightforward to verify, except   perhaps for this one—this is the most interesting  one: the associativity of the multiplication.   That turns out to be somewhat non-trivial; you  actually have to make a calculation for that,   and I please urge you, if you haven't  done this in your mathematical career yet,   make this calculation. Check that this is true. 

Then you might like to think about what happens   if we modify this definition a little bit. Suppose  we decide to make a new multiplication by changing   that minus sign to a plus sign, or maybe that  plus sign into a minus sign, or maybe sticking   a factor of 2 in front of the ad, or some other  variant like that.

You will find that then this   multiplication property here tends to not be  satisfied, so you will get an appreciation for   the beauty of this particular definition. It just  happens to work for that associative property. We'll also define a few special complex 

numbers. So, zero is just the pair (0,   0), and one is the pair (1, 0). You  can check that this number, this one,   has the property that one times z is z times 1,  which is z.

So it's the multiplicative identity,   and this zero has the property that  zero plus z equals (0, 0) equals z. Okay, so we could also define subtraction and  division, but I'm not going to do that. This is   the essence of the algebraic structure of the 

complex numbers, and it's an example of what   is called an algebra in mathematics—a little bit  of an unfortunate name because algebra has these   different meanings, but anyway, that's the complex  numbers. I've now repeated the multiplication here   so we can refer to it, and I want to point out  that there are some other sort of special complex   numbers that play distinguished roles.

First of  all, the numbers of the form (a, 0), where the   second coefficient is 0—those numbers correspond  to points on the x-axis, and their arithmetic   is particularly simple. You can check that,  additively, (a, 0) plus (b, 0) is just (a + b,   0), and (a, 0) times (b, 0), multiplicatively, we 

get the product (ab, 0). So that if we restrict   our attention to this real axis, then the complex  numbers there act just like the usual rational   numbers. So we can think of the ordinary rational  numbers as being embedded in our picture in terms   of the points on the x-axis.

Now, multiplication  by such a real complex number—so often these ones   are called the real complex numbers, which is  possibly a source of confusion because I don't   believe in real numbers in the usual sense—but  that shouldn't prevent me from being able to   use the adjective "real," and so here we're just  going to say that any complex number whose second   coordinate is 0, we're going to call that a real 

complex number, even though in fact the entry a   is actually a rational number. So if we multiply  by such a real complex number (a, 0), then we see   that it acts by scaling by a. So let's check:  if we multiply (a, 0) by (c, d), the rule is we   take a times c minus 0 times d, so that's just a 

times c, and then we get a times d plus 0 times c,   so that's just a times d. So what has happened is  that each of these entries has gotten multiplied   by a, and a is just an ordinary rational number.  Let's agree that we use a sort of vector space   terminology here and notation and agree that  we can pull that a out front if it's common to   both terms, and so we can rewrite this as a times 

(c, d). So here the a is just a rational number;   we're just talking about scaling the vector  by multiplying by a. So that's what happens   when we multiply any complex number by (a, 0). For 

example, if we multiply that complex number by (a,   0), a here is somewhere between 1 and 2, say, then  it would mean that this vector would enlarge by a   factor of a, so it would become roughly twice as  long. So the product would be up there somewhere. On the other hand, it's also interesting 

to consider complex numbers that lie on the   y-axis—they also have a distinguished  role, and let's put them in. So these   are sometimes called—well, they're  called imaginary complex numbers. Imaginary complex numbers.

Alright, the ones that are lying on the y-axis  and have the form (0, a). So what happens if we   multiply by (0, a)? Have a look: here's our basic  law for multiplication. So we get 0 times c minus   a times d for a total of minus a times d, and the 

second entry is 0 times d plus a times c, or ac.   If we pull out the common a, as we did before,  then this is a times (-d, c). So what's the   relationship between (-d, c) and (c, d)? Well, if  here is (c, d), then here is (-d, c). It's really   the same vector except that it's been rotated by 

a quarter turn. So, a quarter of the way all the   way around—that's a right angle there; these two  vectors are perpendicular; that's 90 degrees if   we're measuring an angle spread of 1, if we're  measuring with rational trigonometry. So the   effect of (a, 0) is just dilation, but multiplying 

by (0, a) dilates and rotates by a quarter turn. So we're already seeing some geometry, some  geometrical transformation associated with   the algebraic structure. So the complex number  which is on the y-axis and has coordinate (0,   1) has a special role in the subject, and it's 

usually given a special name. So that's the   complex number i. It's interesting because when  we square it, let's see what happens. i squared   is equal to (0, 1) times (0, 1).

We're going to  get 0 times 0 minus 1 times 1, so that's minus 1,   and we're going to get 0 times 1 plus 1 times 0,  that's 0. So we're getting this number (-1, 0),   which is really like the number, the rational  number, minus 1, because we've said that the   numbers ending in 0 are really acting like 

the rational numbers. So we can agree that   we're going to call this number just minus 1.  In that case, i squared is equal to minus 1.   So we have this algebraic system now where  we have this new number called i, which has   this remarkable property that no rational  number does, namely its square is minus 1.

Now, we can see that geometrically from what  we were just saying in terms of rotation. If   we multiply by i, well, then we just rotate  by 90 degrees or a quarter turn. And so if   we do that twice—if we multiply by i 

squared—we're taking every vector and   changing it to its negative; we're rotating  essentially by 180 degrees or a half turn,   which is essentially to negate any vector. It's  a geometrical interpretation of this equation. Now we're going to use this particular complex 

number to simplify or give an alternate form for   complex numbers. So instead of writing the pair  (a, b), we'll think of this as being a times (1,   0) plus b times (0, 1). And this (1, 0) is really  the number 1, which we don't need to write,   and this (0, 1) is this new number i, so we 

can write this expression as a + bi. That's an   alternate form for complex numbers, which is the  one that we're going to probably use in practice   most. With this notation, this particular equation  is really the only thing that we have to remember   when we're multiplying.

We can almost forget  about that original rule. As illustrated here,   suppose we want to multiply (3 + 5i) with (-2  + i). Then if we're just going to do this using   the distributive law, we get 3 times -2 is 

-6, 3 times i is +3i, 5i times -2 is -10i,   and 5i times i is 5i squared. Well, i  squared is equal to minus 1, so we should   replace this thing with a minus 1 to give us  -6 - 5, and then the i's combine as 3 - 10i,   giving us altogether -11 - 7i. You can check 

that this is the same thing that you would   get if you did the original multiplication in  terms of this times this minus this times this,   and this times this plus this times this.  It's just that the advantage is that we   don't actually have to remember the formula for  multiplication—we only have to remember this very   simple law and use natural distributivity.  Alright, and then numbers like this (0, a)   can be just written as a times i, so these are the 

imaginary complex numbers that lie on the y-axis. Alright, now let's introduce a little bit  more terminology and then the main theorem,   the most important fact about complex  numbers. So if z is equal to (a, b),   or in our new notation a + bi, then let's give 

a name to these numbers a and b. Let's call a   the real part of z and denote it by this R(z),  and let's denote the rational number b as the   imaginary part of z and denote it by Im(z).  Notice these are both rational numbers. Let's   define the complex conjugate.

So if we take the  complex number z and we put a little bar over it,   that means complex conjugate, and what you're  going to do is negate the second coefficient.   In terms of a + bi, that changes to a - bi.  So that's the complex conjugate z. That's an   important idea in complex numbers, and we're 

going to use that to define the quadrance of   the complex number. So the quadrance of the  complex number z is Q(z)—it's, by definition,   z times z bar. It's what you get when you multiply  this times this.

Now, that's a difference of   squares. If you multiply a + bi times a - bi, you  get a squared - (bi) squared, but i squared being   minus 1 amounts to a squared + b squared. So what  we're getting is the sum of the squares of the   coefficients of the complex numbers.

If you think  in terms of a diagram and Pythagoras's theorem,   this represents the area of the square that Euclid  would have built on the segment from zero to the   point z. It's the quantity that appears as  the hypotenuse area in Pythagoras's theorem.

Now, an important point: in ordinary  texts, one then uses this—in fact,   one doesn't give this thing a name—one goes  directly to what's considered more fundamental,   namely the length of the complex number z by  taking a square root of this. We do not want   to do that, okay? We do not want to use square 

roots if we can avoid it. Square roots properly   take us outside of pure mathematics because  the square root is actually a very subtle,   problematic construction, as evidenced by the fact  that if I ask you what the square root of 13 is,   you can't give me a precise answer. Your 

calculator will spit out a certain number of   digits, but you don't actually have a number whose  square is 13. In fact, the existence of square   root of 13, as I argue in my Math Foundations  series, is highly suspect. So we're going to   avoid that.

We're not going to mention the length  of vectors or moduli of complex numbers. This is a   big step up conceptually, actually, okay? So not  using that means that we have to make everything   algebraic—we're forced essentially in the right  direction in terms of our thinking.

So I know this   is a little bit novel to many of you, but believe  me, there's much to be said for it. In particular,   we have this main theorem, which looms  as an absolute pillar of the subject,   which is that if you have two complex numbers z  and w, then the quadrance of the product z times w   is the product of quadrances: Q(z times w) = Q(z) 

times Q(w). So the quadrance of a product is the   product of the quadrances. Ha! So let's prove  it. Let's say that z is equal to a + bi and w   equals c + di.

Then the left-hand side—what does  it look like? Well, we have to multiply z and w,   and then there'll be two coefficients, and we  have to take the first coefficient squared plus   the second coefficient squared—that's what the  quadrance of z times w is. So here, I hope you   recognize this as the real part of the product: 

ac - bd, and the imaginary part of the product   is ad + bc. So we're going to take this squared  plus this squared—that's the left-hand side. The   right-hand side is the quadrance of z—that's a  squared + b squared—times the quadrance of w,   which is c squared + d squared.

So the  assertion is that this equals this for   any rational numbers a, b, c, and d, and this  is an identity of Fibonacci, Leonardo of Pisa,   and in fact possibly goes back to Diophantus, 300  years after Christ. Let's check why it's true.

We   square this—we're going to get ac squared,  which is this term here, plus bd squared,   which is this term, minus twice the product. Let's  forget about minus twice the product temporarily.   When we square this thing, we get ad squared,  which is this times this, plus bc squared,   this times this, plus twice the product.

Alright,  so minus twice the product over here is minus 2   times a times c times b times d, and plus twice  the product here is 2 times a times d times b   times c. Those two terms are conveniently exactly  the same with opposite signs, so they cancel,   and equality is obvious.

So this drives this basic  result—very important, crucial identity that's   somehow at the heart of the beauty and usefulness  of complex numbers in terms of rotations. So while quadrance is a rational analogue of  length or distance, what is a rational analogue   of angle? Traditional treatments of complex 

numbers rely heavily on the notion of length and   angle for polar coordinates for complex numbers.  Maybe it's inconceivable to you that you could   study the subject without those two concepts  if you're very familiar with complex numbers,   but in fact, you can. There are rational  analogues of angle, and there are a number   of different possibilities.

One of them is the  spread from rational trigonometry, or the turn,   or the half-turn. The turn is also described in  my book; the half-turn I'm going to talk about   in the Math Foundations or the WildTrig series.  It's something else that's also very interesting.   Today, I'm going to tell you about the turn, which 

is a very natural idea that's closely connected   with the geometry of lines in any xy. So here is a  number z = a + bi, x-coordinate a, y-coordinate b,   and here's a vector joining the origin to z. We're  going to define the turn of z to be the slope of   this line.

The slope is, by definition, the change  in the y-coordinate divided by the change in the   x-coordinate, so the y-coordinate is b, the  x-coordinate is a, and that's how we're going   to define the turn of this complex number. So  geometrically, we're going to relate it to a slope   of a line.

Now more generally, what we really want  to do is to define the turn between two vectors.   So this would be the turn between the vector,  say, (1, 0) and z, but more generally, if we have   z1 = a1 + b1i and z2 = a2 + b2i, then it turns  out that this is the formula for the turn, which   is a generalization of this slope of line. What is 

it? So the turn from z1 to z2 is, by definition,   a1b2 - a2b1 in the numerator, and in the  denominator a1a2 + b1b2. Now these two expressions   look a lot like the coefficients of the products  of z1 and z2, and if you think about it actually   for a little while, you'll see that these are  the complementary expressions that are linear   in the various variables—complementary to the two 

expressions that are used in the product. This you   may also recognize as determinants; essentially,  it's an area of a parallelogram formed by the   two vectors, and this here you recognize  as a dot product between the two vectors. Now this particular expression is very nice—what's 

the definition of the turn from z1 to z2,   and in the special case when the first vector  is, say, (1, 0), then you can check that this   thing reduces to this one here. There's one more  thing to be said: there is a slight problem with   this definition if a is 0. In other words, if the 

complex number is perpendicular to the x-axis or   90 degrees, then the turn is either undefined  or infinite. And that will also happen here in   this more general case—if the two vectors are  perpendicular, say z1 and z2 are perpendicular,   then the turn will be undefined. By the way, 

here is the terminology that I like to use.   So it's given by this little straight line with a  little arrow, and there's the "u." The arrow here   denotes that the object has an orientation—it  depends on which complex number is first and   which one is second. You can check that if you  change the order, then the turn negates.

Alright,   so this is how we're going to express the turn  from z1 to z2. It's a number associated with   the two complex numbers that measures somehow how  far apart they are—it's a replacement for angle,   but it's a rational replacement—no transcendental  functions or definitions are required.

And here's   the main theorem for turns: the theorem that  asserts that if we have three complex numbers z,   w, and v, with v not equal to 0, then the turn  from z to w is the same as the turn from zv   to wv. In other words, when we multiply  both z and w by the same complex number,   we get two new complex numbers zv and wv. 

The turn between them is the same as the   turn between the original z and w. So this is  a rational analog to the statement that when   we multiply by a complex number, the angles  between two complex numbers don't change. Alright, so what's the proof? Let's say that z 

is a + bi, w is c + di, and let's suppose that   v is x + yi. So let's calculate this thing here.  Alright, so we have to calculate z times v and w   times v, and then we have to apply the formula  for the turn on the previous page. Alright,   so z times v is going to be—well, there's going to 

be an ax - by—that's going to be the first term,   and then there's going to be ay + bx. And w  times v is going to have terms cx - dy and   the next one cy + dx. Then you can check that this  numerator here is the determinant formed by those   two vectors that we just mentioned.

So the first  coefficient of zv times the second coefficient of   this one, minus the first coefficient of this  one times the second coefficient of that one.   And here in the bottom is the dot product, or  inner product, of those two vectors—the product   of the two first coefficients plus the products  of the two second coefficients.

And now you have   to expand this, alright? So please do this—it's  a good algebraic exercise. Expand the numerator,   expand the denominator, and stare at them  both individually, and convince yourself   that they both factor. The numerator factors 

as x squared + y squared times (ad - bc),   and the denominator factors as x squared +  y squared times (ac + bd). And since we're   assuming this v is non-zero, the x squared + y  squared has to be non-zero—we're talking about   rational numbers here—and so conveniently, these  two terms cancel, and we're left with the turn   between z and w.

It's a lovely calculation that  replaces the usual fumbling around with cosines   and sines and formulas for cos(a + b), and so  on. Okay, this is without any transcendental   notions. That's a big step up, in fact, so 

it's something well worth thinking about. So the invariance of turn under  multiplication—mathematics is a very   conservative subject, especially pure mathematics,  and many mathematicians have said to me, "Yes,   Norman, that's all very good—rational approach,  probably very interesting—but what about just   the additivity of angles? With angles, we can add 

them—an angle of three degrees plus an angle of   four degrees gives you an angle of seven degrees."  Well, that's true, but it's a very heavy price to   pay—the machinery and the transcendental business  that you have to put into the subject in order to   try to get at that linearity. It's an attempt to  force linearity on the circular structure, which   doesn't really want to go from a theoretical point 

of view. This exercise here gives you a rational   analog to this additive structure. Suppose  that you have three complex numbers z1, z2, z3,   and we measure the turns between all three pairs.  So let's say u1 is the turn between z2 and z3, u2   is the turn between z3 and z1, and u3 is the turn 

between z1 and z2. Remember, the order matters, so   I'm kind of doing this in a cyclical order so it's  all symmetrical. Okay, so a great exercise: prove   then that these three turns satisfy the following  pleasant relation: u1 + u2 + u3 = u1u2u3.   And please do it without any reference to 

tangents of angles, okay? This is a purely   rational result—it deserves a purely rational  proof. Notice that it means, in particular,   that if you know two of those turns, then you  have a linear equation for the third one. So   while it's not quite as simple as just adding the 

two turns, it's not that much more complicated. Alright, now let's have a look at a very  interesting circle of ideas. I want to connect   the unit circle in the complex plane, which plays  a very important role, to the projective line of   lines through the origin, to the idea of rotations 

of the plane centered at zero. So these three   subjects are all intimately connected. I'm going  to remind you of our main theorem with quadrances,   that quadrance of z times w is quadrance of  z times quadrance of w.

And in particular,   if you have a point gamma which lies on this  unit circle, so the unit circle has equation x   squared + y squared = 1 in xy coordinates—in  terms of complex numbers, well, it's those   complex numbers whose quadrance is equal to 1,  or if you like, z times z bar = 1. You know,   three different ways of writing down the equation 

of the unit circle. So suppose that gamma is on   the unit circle, and we consider what happens  if we multiply, say, w by gamma w. So out here   somewhere, if we multiply by gamma, then we get  the quadrance of gamma times the quadrance of w,   but the quadrance of gamma is 1, and that tells 

us that the quadrance of gamma w is the same as   the quadrance of w. In other words, multiplying by  gamma does not change the quadrance of the vector.   So if you have a w out here and you multiply by  gamma, what's going to happen is that you're going   to stay on the big circle through w—you're going  to rotate along that circle.

So multiplication   by gamma is a rotation of the entire plane. So  there's an intimate connection between rotations   and points on the unit circle because, after  all, if we have any rotation centered at 0,   then this number 1 is going to get sent  to another point on the unit circle,   and the rotation is determined by that point that 

we get. So the moral is that rotations of the   plane centered at 0 and points on the unit circle  are intimately connected. For every rotation,   there's a point on the unit circle, and for every  point on the unit circle, there's a rotation.

Now what's more interesting now is I'm going  to throw in a third aspect—the projective line.   Alright, so in projective geometry, the projective  line in this kind of situation is obtained by   looking at lines through the origin—that one there  as an example. Well, we can take any line through   the origin—this space of lines is what's called 

the projective line. So here's a typical line,   let's say, that goes through the point a + bi,  say that point is z. And I remind you again that   all numbers are rational numbers, so a and b  are rational numbers.

We're only considering   lines that go through the origin and a rational  point. Now this line is determined by z, but it's   also determined by any non-zero multiple of z. So  we sometimes write square brackets of z to denote   the line, meaning that we can multiply z by any 

non-zero rational number and it's the same line. Alright, so there's a very important but   somewhat subtle connection between the  projective line and the unit circle. And what is it? Well, if you've been classically  trained, you probably think that what you do is   you take z and you divide by its modulus.

This  is the standard way of getting a unit vector   or a unit complex number from a general complex  number—you divide by the modulus. So that means   we take this and we have to divide by the square  root of a squared + b squared.

But as I tried to   explain, the square root of a squared + b squared  is a highly problematic concept, and if we're   working rationally, as we are here, it doesn't  figure in our picture, so that's not an option   for us. This is a very familiar construction in  standard complex analysis, but it's not available   to us thinking rationally.

Now, you might say,  "Well, that's a problem for you, Norman. I mean,   we can do something that you can't." Well, whether  you can actually do it or just talk about it is a   question. But the point is that there is something 

better that we can do to replace this idea,   and it's going to be very important for us to  understand what that is. It's a replacement. Okay, so here's what we're going to do. We are 

going to take this line, and we're going to take   the point 1, and we're going to reflect the point  1 in this line—that's like a reflection. Alright,   so how do we do a reflection? Well, we take  a line through here which is perpendicular,   something like that, that's perpendicular, and  then we go up to here.

It's going to meet the   circle at a second point here, and this point  here is the reflection of 1 in this line. And   notice that it's unique—that's quite different  from the idea of taking this thing and dividing   by its modulus to find a unit vector which is on  this line.

In rational geometry, a line through   the origin does not have to meet the circle,  so the existence of this point and this point,   this intersection between the line and circle, is  problematic—maybe it exists and maybe it doesn't;   it depends on a quadratic equation  determined by a and b. But this point   always exists—it's there no matter what; we 

can always reflect this point in that point. Alright, so, well, okay, well, what is this  point? How can we write it efficiently? Well,   it turns out there's a very nice and beautiful  way of doing that, and we can see that by   drawing this line here and going out a little bit  further—imagine it coming out a little bit.

So the   trick is to consider this complex number z and to  multiply it by z. Where will that get us? Well,   if we multiply this by z, we're going to get z  squared, and z squared is going to be up here   somewhere. And in red, up here somewhere, is going 

to be z squared—I'll put it in red so you can see   it—z squared, and it's going to be up there, where  this turn and this turn are equal. This turn,   say u, and this turn, u, are equal, because  multiplying by z will take 1 to this and will   take this to this, so it preserves the turn  here going to the turn there.

That's the main   theorem about turns: the turn between 1 and  z is the same as the turn between z and z   squared. And so that line is going to meet that  point, and that point is equal to this thing   divided by its quadrance. So this point here 

is actually z squared over the quadrance of z. It's the same kind of thing that we were trying  to do here, but because this is z squared, the   quadrance of this is the square of the quadrance  of z, and that's a property of this thing here—the   quadrance of z squared is the quadrance of z  all squared.

So when we divide z squared by   the quadrance of z, we're on the unit circle. So  what a beautiful thing that is! So that's a way of   associating to a line a point, and if you change  that by multiplying it by a scalar—multiplied by   2, or by 3, or by -1—this point is not going to  change.

If you multiply z by lambda, then there's   going to be a lambda squared appearing up there,  and there's going to be a lambda squared appearing   down there, and they're going to cancel. So it  doesn't matter what z you choose on that line,   you perform z squared divided by Q(z),  you're always going to get that point.

This is a very important idea, and it's somehow  at the heart of a certain two-to-oneness which   appears throughout this subject, and it's often  considered a feature of the quaternion business,   but you should appreciate that it's already  existing here in the complex numbers—that if   we want to associate a rotation to this line, the  way to do it is to associate to it the rotation by   z squared over Q(z).

A very good diagram to spend  a couple of hours staring at and thinking about. So let's try to understand this structure by  looking at a sort of special case. But first,   let's identify that what we've managed to do 

is to associate to every complex number z which   is non-zero a rotation. Let's call it phi sub  z, where this rotation is defined as follows:   it's defined as phi sub z acting on a complex  number w is multiplying w by z squared over   Q(z). Because z squared over Q(z) is a unit 

complex number, it lies on the unit circle,   and so this is necessarily a rotation that depends  on z. I'll just mention that there's another nice   way of writing it: remember that Q(z) was equal  to z times the complex conjugate z bar, and so   if we write it that way, then two of the z's on  top and bottom cancel, and we can write it as z   times w over z bar.

It's another way of writing  it. So that's quite nice, and it has the property   that if you multiply z by a rational number, then  the rotation doesn't change. So the rotation only   depends on the line through the origin through 

z. And you can also check, as a nice exercise,   that if you compose two rotations, phi sub z  phi sub w, that's the same as phi sub z times w. Now I want to connect that circle of ideas with a  very classical subject which is close to my heart,   which I've mentioned in quite a few of 

my videos. It's a very important thing   that all undergraduates should be well aware  of, and that's the rational parameterization   of the circle. And in fact, that really drops  out from looking at what we've just done.

Okay,   so a bit of complex analysis allows us to rethink  this rational parameterization in a very pleasant   way. And the idea is that, well, if, let's say,  we're interested in lines through the origin, the   space of lines through the origin can be described  as essentially a line plus a point at infinity.   So if I take this line here, this green line, 

which has the equation real part of z equals 1,   all the complex numbers on here are of the form  z = 1 + it, where t is some number. Alright,   if we take any z like this, then the point on  it, joined to the origin, we get a line, and   all lines through the origin are of that form for  some t, except for the sort of special case where   we're just looking at this parallel one, which 

is sort of the case when t is infinite. Alright,   but otherwise, all lines meet this green line  exactly at one point, let's call it 1 + it. Alright, now that's a rather interesting  point—let's have a look at what happens if we   do this construction that we talked about, where 

we calculate z squared over Q(z) when z happens   to be this special complex number 1 + it. So if z  is equal to 1 + it, then z squared over Q(z)—what   is it? Well, first of all, what is z squared?  z squared will be 1 - t squared + i times 2t,   1 - t squared + 2t times i. But we have to 

divide by Q(z)—what's Q(z)? It's the sum of   the squares of the coefficients, so it's 1 +  t squared. So we get (1 - t squared)/(1 + t   squared) plus (2t)/(1 + t squared) i, and we  know that that's a point on the unit circle,   and moreover, it's exactly the point  that we get by doubling this angle,   if you like, or taking this turn and applying 

an equal turn to it. There it is right there. So while z squared is up here generally, then  dividing by Q(z) scales it down so that it gets   on the unit circle, and this point is also  the reflection of the point 1 in this line.

Now this is the familiar rational parameterization  of the unit circle that goes back essentially to   Euclid—every rational point on the unit circle  is exactly this form. That's the x-coordinate,   that's the y-coordinate, for some rational  number t, except for the special case when   you have minus one, which sort of 

corresponds to t equals infinity. There's a few other things I would just like  to mention about this which are interesting   too. The other property is that if you take the  line from -1 to this point z squared over Q(z),   that's actually parallel to the line 

that we started with. Why is that? Well,   it's because here you see that's u, that's  a turn of u, and that's a turn of u,   and so this turn subtended by this chord at the  center is related to this turn subtended by that   same chord at the circumference—that's going to  be a turn of u as well—elementary geometry of the   circle.

So this line is actually parallel to this  line. And if we just move everything over by 1,   then the point 1 + it, if we translate it  over by 1, it just becomes the point it. And so the meaning of t, if you like, you can 

think of it here, is it's the point on the y-axis   that you need to choose, and then connect it to  -1, and then that line will meet the unit circle   at this point: (1 - t squared)/(1 + t squared),  (2t)/(1 + t squared) i. And then connecting   things with the rotation phi sub z, once we  have this point on the unit circle associated   with that is the rotation multiplication by 

that point, or what we're calling phi sub z,   and that's the rotation that physically takes  the point 1 and rotates it to that point there,   takes this point and rotates it to there—so it's a  rotation like that. That's the rotation phi sub z. And for future reference, I'd like to also remind 

you that that rotation can also be thought of as   a product of two reflections. Alright, so  there's this line here, and there's this   line here that we're considering. If we take  the product of the reflection in this line,   followed by the reflection in this line, what do 

we get? Take some point here and you reflect it   first in the x-axis, and then you reflect it in  this line here, you're going to get a point here,   which is the same as the rotation of the  original point phi sub z. So this rotation,   determined by this unit circle, is  also associated with this line in   another way—that is, basically the product 

of this reflection times this reflection. Alright, so that's quite a lot of material.  If you haven't seen complex numbers before,   of course it'll probably be overwhelming to you.  If you have seen some complex numbers before,   I hope this provides a somewhat different view  on things, and this two-to-one aspect is going   to be very important for us when we try to 

understand quaternions. So this is a very   good thing to try to understand first—it makes  understanding quaternions so much easier. As   good practice for you, you might like to calculate  the turn associated with the vector, say (1, 0),   and this unit vector that we found here.

So  in other words, the turn u, this turn here,   is t because that's t and that's 1, so  the slope is t. So that's a turn of u,   and that's a turn of u. Then what's the combined 

turn from here to here? In fact, you can think   about what happens if you combine more turns of u,  three turns of u, four turns of u, and so on. What   is the turn of the composite turn? It's a very  nice little formula, and of course we all want to   do that without any transcendental functions—no  circular functions, no mention of angles.

So in our next video, we're now going to go up  to three-dimensional space and start discussing   rotations—the problem of rotations. How do we  describe rotations in three-dimensional space,   and in particular, how do we compose  rotations? I hope you'll join me for   that.

I'm Norman Wildberger. Thanks for listening.