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Practice English Speaking&Listening with: Geometry Dimensions 5

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I'm Adrien Douady

My entire life's work in mathematics was centered on

the complex numbers.

My contributions helped to advance both algebraic geometry

and the theory of dynamical systems.

Complex numbers have a long history.

You see here, on the left, Tartaglia and Cardano,

mathematical pioneers, who lived during the Renaissance.

On the right, Cauchy and Gauss,

who consolidated the theory, during the nineteenth century.

Complex numbers are not really as complicated

as the name might lead one to believe!

At first they were called "impossible numbers"

even today they are still sometimes called "imaginary".

Well it's true, it does take a little imagination ...

yet, today these numbers are everywhere in science

and are not really mysterious anymore.

In particular, thanks to them, one can construct

beautiful fractal sets,

something I worked on a lot.

I even produced a film "The dynamics of the rabbit",

it was one of the first animated films in mathematics.

Let me begin by explaining the complex numbers on the blackboard.

Mathematicians just love writing with chalk...

You'ill see in a minute that my ruler, this set square and protractor

behave rather oddly sometimes...

Let's draw a graduated line on the blackboard.

One of the most beautiful ideas in mathematics

is to link geometry to algebra.

This is the starting point of algebraic geometry.

Just as we can add numbers, we can add points.

Here is a red point on the line and another blue one

Let's add these two points.

We get the green point ! One plus two equals three!

When the red and blue points move,

the green point which is the sum must move too.

More interesting still is multiplication of points.

Let's look at multiplication by -2 for instance.

It transforms the point 1 into the point -2, of course.

And, if you multiply once again by -2,

you have to do the same thing:

change sides with respect to the origin

and double the distance from the origin.

You get 4, of course.

If we multiply twice by -2,

we have multiplied by 4.

Multiplying by -1 is very easy.

Each point is sent to the symmetrical point

with respect to the origin,

in other words we do a half-turn,

a rotation by 180 degrees, if you like.

When we multiply a number by itself,

the result is always positive.

For instance, if we multiply by -1,

we make half a turn;

so that if we do it one more time,

well, we come back to the initial point!

This is why -1 times -1 is equal to +1

simple enough.

You see for instance that the multiplication by -1

sends 2 to -2

and that if you multiply one more time by -1,

you come back to 2.

Obvious, isn't it ?

Therefore, there is no number which,

multiplied by itself, yields -1.

Another way of saying this is that -1 has no square root.

But, of course, we are underestimating

the inventiveness of mathematicians!

At the beginning of the nineteenth century, Robert Argand had a really great idea.

He said to himself: "Since multiplying by minus one

is a 180 degrees rotation,

its square root is a rotation by one half of 180 degrees : 90 degrees.

If I do two quarter turns one after another,

I end up doing a half turn!

The square of a quarter turn is a half turn, hence minus one."

It's easy when you know how !

Argand decided therefore that the square root of minus one

is represented by the point which is the image of 1 by a 90 degree rotation.

But of course, this forces us to leave our horizontal straight line,

since we have just agreed to associate a number

to a point in the plane which is not on the line!

As this construction is a bit strange,

we say that this point, the square root of -1, is an imaginary number

and mathematicians denote it by i.

But, once we have the courage to leave the line,

everything else is easy.

We can represent 2i, 3i, and so on...

Each point in the plane represents a complex number

and conversely, each complex number defines a point in the plane.

Points in the plane become numbers in their own right!

These numbers can be added, just like usual numbers.

Look at the red point, which is the point 1+2i.

Let's add 3+i which is the blue point.

Well, you add them

just as schoolchildren do

that gives us 4+3i.

Geometrically, this is just addition of vectors.

You see that it is no problem to add complex numbers

Much more interestingly,

these complex numbers can also be multiplied

just like real numbers.

Lets see

We know how to multiply a complex number by 2 for instance.

Two times 1+i, gives

2+2i .

Geometrically, multiplying by 2 is easy:

it's just scaling up by a factor 2:

if we double the red point, we get the green point!

Multiplying by i is not difficult either

since we know that i corresponds to a quarter turn.

In order to multiply 3+i by i,

we just have to rotate by a quarter turn.

We get -1+3i.

Not so complicated, these complex numbers!

And finally, we can multiply any two complex numbers

with no problem whatsoever.

Let's try for instance to multiply 2+1.5i and -1+2.4 i.

We proceed as usual,

we first multiply by 2 and then by 1,5 i then we add the results.

Therefore we get:

"Two times etc.."

Hence

-2 + 4.8 i - 1.5 i + 3.6 i times i

But, recall that i squared is -1,

since we invented i for this purpose!

This gives:

-2 -3.6 ..etc,

Let's clean up a bit. We find

-2 -3.6 + 4.8 i - 1.5 i,

that's

-5.6 + 3.3 i.

There you are, we know how to

multiply complex numbers

in other words, we can multiply points in a plane!

That's amazing !

we thought that the plane was dimension 2

since two numbers are necessary

to locate a point

and now, I'm telling you that one number is enough!

Of course, we changed our numbers!

and we are now dealing with complex numbers!

It seems the right time to define two notions.

the modulus and the argument of a complex number.

The modulus of a complex number z

is just the distance from the origin to the point that represents z in the plane.

Let's use the ruler to determine the modulus of the red point

which is 2+1.5 i

Let's see, it measures 2.5.

The modulus of 2+1.5 i is therefore 2.5.

For the blue points, I get 2.6.

And for the green point,

which is the product of the two points;

I have 6.5

As a rule : the modulus of a product of two complex numbers

is just the product of the moduli of the two numbers.

The argument of a complex number

is measured by the angle between the abscissa axis

and the straight line joining the origin to the point.

Here for instance, the argument of the red complex number

is 36.8 degrees.

The argument of the blue point is 112.6 degrees.

And for the product, the green point, we get 149.4 degrees:

this is the sum of the arguments of the two numbers...

When we multiply two complex numbers,

moduli are multiplied and arguments are added.

Let's finish up our first encounter with complex numbers

with the stereographic projection.

Consider a sphere tangent to the board at the origin.

Using stereographic projection,

to each point on the board,

that is, to each complex number,

corresponds a point on the sphere.

Only the north pole of the sphere

I mean, the pole from which I'm projecting

has no complex number associated to it.

we say that it corresponds to infinity.

Therefore, mathematicians say that the sphere

is a complex projective line.

Why line?

Because one needs only one number to describe its points!

Why complex?

Because this number is complex.

Why projective?

Because we added a point at infinity, using the projection.

Aren't mathematicians strange

when they try to tell us that the sphere is a straight line?

The Description of Geometry Dimensions 5