


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 onedimensional,
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 x1,
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.


Click
on the image for a film. 


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 halfturn.
To find a square root of 1 is to find a transformation that, when
performed twice in a row, gives a rotation by a halfturn. Argand
declared therefore that the square root of 1 must be associated to the
rotation by a quarterturn, 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 quarterturn 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).


Click
on this image for a film. 

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 :
(x+i y).(x'+i y') =
xx' + i xy' + i yx'
+ i^{2} yy' = (xx'yy') + i (xy'+x'y)
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!


Click on this image for a
film. 

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 √(x^{2}+y^{2})
. 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 xaxis
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
4dimensional 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 2dimensional
is now 1dimensional! There is absolutely no contradiction here: the
plane has 2 real
dimensions but it is a 1dimensional 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
2dimensional 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 2dimensional
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) = z^{2}
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.


Click on this image for a
film. 


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
T(z) = (az+b)/(cz+d).
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 nonEuclidean 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)
= z^{2}
+ 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 z_{1}=
T_{c}(z). From there we
consider the transformed value of the transformed value z_{2}=
T_{c}(z_{1}),
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)=z^{2}.
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 filledin 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.


Click on this image for a
film. 


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.


Click
on this image for a film. 

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_{n2},
B=z_{n}z_{n1}
and C=z_{n1}z_{n2}.
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 selfsimilarity, a
frequent characteristic of fractal sets. To better understand this, see
for example this page.


