Chapters 5 and 6: Complex Numbers

Mathematician Adrien Douady explains complex numbers. The square root of negative numbers is explained in simple terms. Transforming the plane, deforming pictures, creating fractal images.

1. The presenter

Complex numbers make up one of the most beautiful chapters in mathematics and have become an essential tool in the sciences. The path to their discovery was not an easy one, and terminology is part of the reason for this; they have been called "impossible" and "imaginary" numbers, and the word "complex'' gives the impression that they are not easy to understand. Happily this is not the case today: we can now present them in a relatively elementary way.

Adrien Douady presents these chapters. An exceptional mathematician, he made a variety of contributions to the field, and he liked to say that all of his research centered on complex numbers. In particular, he was one of a group of mathematicians who revived the theory of complex dynamical systems, about which we will say more later.

One of the characteristics of this theory is that it produces many beautiful fractal pictures, which can now be drawn, thanks to computers. Adrien Douady strongly encouraged these types of pictures, both to help mathematicians in their research and to popularize mathematics.

We also owe to Douady a mathematical animation titled The dynamics of the rabbit (he liked to give mathematical objects surprising names: rabbit, airplane, the cartoon creature shadok, etc.) His recent passing profoundly saddened the mathematical community. For some illustrations of his personality, see this site or this one (in French).

It is clear that even Adrien Douady cannot explain the entire theory of complex numbers in 2 chapters of 13 minutes ... These chapters are not meant to be a substitute for a college level course, a book, or a detailed exposition (see for example this site or this one). One should consider these chapters as supplementary, as illustrations which encourage further learning or maybe as reminders of lessons forgotten long ago. Certainly the film tries above all to show clearly the geometric side of complex numbers.

2. Numbers and transformations

We have seen that the line is one-dimensional, since we can place numbers on the line -- positive numbers to the right and negative to the left of the origin. Points are geometric objects and numbers are algebraic ones. The idea of thinking of numbers as points and points as numbers, that is to say, of mixing algebra and geometry, is one of the most fertile ideas in mathematics. As always, it is not easy to attribute this to one person, but it is generally Descartes to whom we attribute this powerful method of studying geometry by using algebra: this was the birth of algebraic geometry. If the points on a line are numbers, then we can understand geometrically the significance of the elementary operations between numbers: addition and multiplication. The key to this understanding is in the idea of a transformation.

For example, subtracting 1 from a number x, that is to say the transformation x-1, can be seen geometrically as a translation: all the points are translated by 1 to the left. In the same way, multiplication by 2 can be thought of as a dilatation.

Multiplication by -1, which sends each point x to -x can be thought of as a symmetry -- each point is transformed to its symmetric point about the origin. Multiplication by -2 is itself the composition of the two preceding operations. Multiplying two numbers comes down to composing the two associated transformations. For example, the transformation associated to multiplication by -1 is a symmetry, and when we perform this operation twice in a row, we get back the original point, just as the product of -1 with itself is +1. The square of -1 is +1.

The square of -2 is +4 for the same reason. It follows from all of this that the square of any number is always positive. There is no number whose square is -1.

In other words, -1 does not have a square root.

3. The square root of -1

For a long time, the impossibility of finding the square root of -1 was a dogma that could not be discussed. During the Renaissance, certain inventive spirits dared to break the taboo! If we dare to write √-1, then we can also write numbers like, for example, 2+ 3√-1, and we can also play with these numbers in the same formal manner without really trying to understand their meaning. These pioneers boldly went about making computations with these impossible numbers, in an almost experimental fashion. Since their calculations didn't seem to lead to any contradictions, these numbers were gradually accepted by mathematicians, without any real justification.

The story of these numbers is quite long, and it is not our intention to describe the steps that led to their solid foundation. One can consult, for example, this page for a little history. It suffices to say, to simplify in the extreme, that at the turn of the nineteenth century several mathematicians, among them Gauss, Wessel and Argand, became aware of the geometric character of these imaginary numbers. The film shows a simplified presentation of a very simple idea of Argand.

(Click on the image to the right to see Argand's original article. )

The number -1 is associated to the symmetry of reflection about the origin, that is to say a rotation by a half-turn. To find a square root of -1 is to find a transformation that, when performed twice in a row, gives a rotation by a half-turn. Argand declared therefore that the square root of -1 must be associated to the rotation by a quarter-turn, quite simply. Doing two rotations by a quarter turn gives a rotation by a half turn, that is to say, multiplying by -1.

If we follow this idea, we want to say that the square root of -1 is obtained by starting at 1 and making a quarter turn. Of course the image of 1 by a quarter turn is not on the line, and we have just decided that the square root of -1 is not a point on the line but on the plane!

The idea is simple and beautiful: to think of the points of the plane as numbers. Then, of course, these are not the same numbers that we are used to. For this reason we say that the "traditional'' numbers are the real numbers, and the numbers that we have just defined, associated to points in the plane, are the complex numbers.

If we express a number in the plane by its two coordinates (x,y), which are real numbers, the line that we have just left is the line whose equation is y = 0, and the point that is the image of (1,0) by a quarter-turn is (0,1). This is therefore the point that Argand considered to be the square root of -1. Mathematicians, still astonished by this "sleight of hand", call this number i, as in "imaginary''. Since we would like to be able to add these numbers, we can consider the number x + i y : this corresponds to the number with coordinates (x,y).

To summarize, Argand encouraged us to consider the points (x,y) in the plane not as a pair of (real) numbers, but rather as a a single (complex) number. This might seem very surprising, and perhaps artificial, but we will see that this idea is very powerful.

4. Complex arithmetic

The following is not difficult. After all these speculations, we have defined a complex number as being given by two real numbers, that is to say a point on the plane, and we denote it by z = x + i y. We are now about to show how to add two complex numbers, how to multiply them, and that all the properties of arithmetic that we are used to are still valid. For example, we have to check that the sum of complex numbers is the same no matter in which order they are added. All this can be done rigorously, but that's certainly not the point of this film ... Here is a presentation of the theory of complex numbers.

For addition this is easy: we have the formula (x+i y) + (x'+i y') = (x+x')+ i (y +y'), so that addition of complex numbers comes down to adding the corresponding vectors.

For multiplication, it is a little more difficult :

but here, it's a small miracle that this formula holds. For example, it is not at all obvious from this formula that we can multiply three complex numbers in any order and get the same result, or even that we can always divide by a nonzero complex number. This small miracle is not explained in the film ... this would have taken us too long!

Two concepts are useful for what follows:

The modulus of a complex number z= x +i y is simply the distance from the corresponding point (x,y) to the origin. We denote this by |z|, and it is equal by Pythagoras's Theorem to √(x²+y²) . For example, the modulus of i is equal to 1 and the modulus of 1+i is √ 2.

The argument represents the direction of z. We denote it by Arg(z), and it is nothing other than the angle between the x-axis and the line joining the origin to (x,y). This argument is only defined when z is not zero. For example, the argument of i is 90 degrees, the argument of 1 is 0, of -1 is 180 degrees, and of 1+i is 45 degrees.

Mathematicians for a long time tried to do the same thing in dimension 3: how to multiply points in space? It took them a long time to understand that this is not possible. In 4-dimensional space, they discovered that this was partially possible, as long as one gives up the idea of a multiplication that satisfies ab=ba ! And they got so far as discovering that in dimension 8 it is still possible, as long as one abandons the idea that (ab)c=a(bc), before understanding around the middle of the twentieth century that, other than in dimensions 1,2,4 and 8, there is absolutely no way to multiply points ! To understand something about the preceding mysterious sentences, see this, this or this.

In summary, each point in the plane is defined by a single number - a complex one. The plane which we have said is 2-dimensional is now 1-dimensional! There is absolutely no contradiction here: the plane has 2 real dimensions but it is a 1-dimensional complex line. Real plane, complex line ... 2 real dimensions, 1 complex dimension. Word games?

5. Once again: stereographic projection!

Recall stereographic projection: it transforms the 2-dimensional sphere, with the north pole removed, to the plane tangent to the south pole. As a point approaches the north pole, its projection moves away in the plane, so that we say it tends to infinity.

Now, if we think of the plane tangent to the south pole as a complex line, we understand why the 2-dimensional sphere (2 real dimensions!) is often described as the complex projective line. There's a beautiful example of mathematical acrobatics: calling a sphere a line!

Didn't Henri Poincaré say that mathematics consists in giving the same name to different things?

6. Transformations

( See the film: Chapter 6: Complex numbers, continuation )

This chapter proposes to give a little intuition for complex numbers by following certain transformations of the complex line.

A transformation T is an operation that associates to each complex number z, that is each point in the plane, another point T(z). To illustrate, we place the picture of Adrien Douady in the plane and then show its image by the transformation: each pixel that makes up the portrait is transformed by T.

Adrien chose several examples for the transformation T :

T(z) = z/2
Each number is divided by 2. Of course, the image is reduced by a factor of two: a reverse zoom! We call this a homothety.

T(z) = iz
This acts simply by a rotation through a quarter turn, by definition of i...

T(z) = (1+i)z
Since the modulus of 1+i is √ 2 and its argument is 45 degrees, it acts by composition of a rotation by 45 degrees and a homothety by a factor of √ 2. This is called a similarity. This is a big advantage of complex numbers: they allow us to describe very simply similarities as multiplications.

T(z) = z²
Here is our first nonlinear transformation. By placing the photo in two different spots, we become aware of the effect of applying the square to the complex plane: the moduli are squared and the arguments are doubled.

T(z) = -1/z
This acts as a transformation close to what we ordinarily call an inversion. Of course the origin, which corresponds to the number 0, cannot be transformed, but we adopt the convention that the origin is sent to infinity. The reason is very simple: if a complex number z approaches 0, that is to say if its modulus tends to 0, then the modulus of the transformed number -1/z is the inverse of the modulus of z, which tends to infinity. The transformation thus has the property of "exploding'', that is to say of moving small neighborhoods of the origin very far away, past the boundaries of the screen ... Conversely, points that are very far from the origin are "crushed'' very close to the origin.

For a long time, scholarly texts gave great importance to inversion, since it allows one to prove quite beautiful theorems. The principal property of inversion is that it transforms circles to circles or lines. Artists often use this type of transformation, and they've given it the name anamorphosis.

More generally, if we choose 4 complex numbers a,b,c,d, we can consider the transformation

These transformations have several names in mathematics -- Moebius transformations, homographies, projective transformations -- but their principal property is that they send circles to circles or lines. This is the group of transformations of a magnificent geometry called conformal geometry, close to non-Euclidean geometry, but that's another story!

T(z) = z+k/z
This transformation was studied by Joukowsky in his study of the aerodynamics of airfoils ! But Adrien Douady could have chosen other transformations, in particular those that give him a slimmer figure than this one! The point of this illustration is to show a fundamental property of this type of transformation. Of course they no longer transform circles to circles (only Moebius transformations do that), but this is still true on an infinitesimal level. These transformations are called holomorphic or conformal. The greek and Latin roots "holo'' and "con'' mean "same'', and "morph'' of course means "form'' : in other words, these transformations preserve forms. The study of holomorphic functions is one of the most important chapters in mathematics.

6. Holomorphic dynamics

In the second part of Chapter 6, Adrien Douady gives an introduction to a magnificent subject, in which he was one of the major contributors. It is about the study of Julia sets which, beyond their fundamental mathematical interest, are extraordinarily beautiful (and these two features are of course linked). It is rare that a powerful mathematical theory is illustrated in such a beautiful way, and a number of artists have been inspired by these images.

The starting idea is very simple: we choose an arbitrary complex number c. Then we consider the transformation T_c(z) = z² + c. It acts first by squaring the number z and then by translating the answer by c. Starting at the initial point z, its transformed value is a point z₁= T_c(z). From there we consider the transformed value of the transformed value z₂= T_c(z₁), and we continue towards infinity, producing a sequence z_n of complex numbers where each number in this sequence is the transformed value of the preceding number. We say that the sequence z_n is in the orbit of the initial point z under the transformation T_c. To study the behavior of the sequence z_n, is to understand the dynamics of T_c . It comes down to a very simple example, but this example is sufficiently rich to give rise to some very beautiful mathematics.

First consider the case where c = 0. This acts in effect by repeating the transformation T_c(z)=z². The modulus of each z_n is therefore the square of the preceding modulus. If the modulus of z is less than or equal to 1, that is if z is inside the disk of radius 1 centered at the origin, then all of the z_n will stay in the disk. If, on the other hand, the modulus of z is strictly greater than 1, then the moduli of the z_n will keep increasing and even tend toward infinity: the orbit of z will end up leaving the screen!

In the first case, we say that the orbit is stable : it stays in a bounded region of the plane. In the second case it is unstable : it tends towards infinity. The set of points z for which the orbit is stable is therefore the disk.

More generally, for each value of c, we can also distinguish between two types of orbits for points z. The orbit of z by T_c is stable if it stays in a bounded region of the plane, or unstable otherwise. The set of z for which the orbit is stable is called the filled-in Julia set of the transformation T_c. Understanding the structure of these Julia sets and the way that they vary as c varies is a major goal of the theory of holomorphic dynamical systems. First, Adrien Douady shows us a few examples of Julia sets, for various values of c. Some of them have exotic names, for example "the rabbit'' (do you see its ears?) for the value c= -0.12+0.77i.

We've known since the beginning of the twentieth century that the Julia set can be one of two types. It can have, as in the examples that we've just shown, a single component -- connected as mathematicians would say -- or else it is totally disconnected, consisting of infinitely many separate pieces, each having empty interior, that is to say we don't see them in a picture! Consequently, there are values of c for which we see the Julia set and others for which we don't see it at all (even though it exists). the set of values of c for which we see the Julia set (for which the Julia set is connected) is called the Mandelbrot set , in honor of Benoît Mandelbrot. Adrien Douady has done a lot of work to understand this set; he has for example contributed to showing that it itself is connected, and he would have loved to show (as would many others) that it is locally connected...

The end of the chapter is devoted to diving into the Mandelbrot set, diving deeply since the factor of dilatation is on the order of two hundred billion! We can look at this scene in two ways. We can simply admire it: it is beautiful enough for that! But we can also ask ourselves some questions ...

For example, what is the significance of the colors? An old theorem shows that the Julia set is not connected (in other words c is not in the Mandelbrot set) if and only if the orbit of 0 under T_c is unstable. For a given value of c we can therefore look at the orbit of z=0 by T_c and observe its behavior for large values of n. If z_n becomes large very rapidly, this means that c is not in the Mandelbrot set, and even that it is pretty far from it. If the sequence z_n tends toward infinity, but more slowly, the point c is still not in the Mandelbrot set, but is in some sense closer to it. The color of the point c depends on the speed at which the sequence z_n, tends to infinity, which also shows its "proximity'' to the Mandelbrot set. If on the other hand z_n stays in a bounded region, then c is in the Mandelbrot set and it is colored black.

The Mandelbrot set in the figure above has been colored in this fashion, but there are dozens of other methods. In the film, we use the method called "the triangle inequality'': when the modulus of z_n becomes bigger than a certain value, we calculate the moduli A=|z_n-z_n-2|, B=|z_n-z_n-1| and C=|z_n-1-z_n-2|.
The quantity A/(B+C) is always a number between 0 and 1, and we use this number to find a position on a color wheel.

Why at certain times do we have the impression of seeing new little black copies of the Mandelbrot set? That is much too difficult to explain, and that is one of the important discoveries of Adrien Douady: the Mandelbrot set has the property of self-similarity, a frequent characteristic of fractal sets. To better understand this, see for example this page.