1.3 What are Complex Numbers?

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Complex numbers are points in the plane, together with a rule telling you

how tomultiply them. They are two-dimensional, whereas the Real numbers

are one dimensional, they form a line. The fact that complex numbers form

a plane is probably the most important thing to know about them.

Remember from _rst year that 2_2 matrices transform points in the plane.

To be de_nite, take

_ x

y _

for a point, or if you prefer vector in R2 and let

_ a c

b d _

be a 2 _ 2 matrix. Placing the matrix to the left of the vector:

_ a c

b d __ x

y _

1.3. WHAT ARE COMPLEX NUMBERS? 13

and doing matrix multiplication gives a new vector:

_ ax + cy

bx + dy _

This is all old stu_ which you ought to be good at by now1.

Now I am going to look at a subset of the whole collection of 2_2 matrices:

those of the form

_ a 􀀀b

b a _

for any real numbers a; b.

The following remarks should be carefully checked out:

_ These matrices form a linear subspace of the four dimensional space of

all 2_2 matrices. If you add two such matrices, the result still has the

same form, the zero matrix is in the collection, and if you multiply any

matrix by a real number, you get another matrix in the set.

_ These matrices are also closed under multiplication: If you multiply

any two such matrices, say

_ a 􀀀b

b a _ and _ c 􀀀d

d c _

then the resulting matrix is still antisymmetric and has the top left

entry equal to the bottom right entry, which puts it in our set.

_ The identity matrix is in the set.

_ Every such matrix has an inverse except when both a and b are zero,

and the inverse is also in the set.

_ The matrices in the set commute under multiplication. It doesn't matter

which order you multiply them in.

_ All the rotation matrices:

_ cos _ 􀀀sin _

sin _ cos _ _

are in the set.

1If you are not very con_dent about this, (a) admit it to yourself and (b) dig out some

old Linear Algebra books and practise a bit.

14 CHAPTER 1. FUNDAMENTALS

_ The columns of any matrix in the set are orthogonal

_ This subset of all 2 _ 2 matrices is two dimensional.

Exercise 1.3.1 Before going any further, go through every item on this list

and check out that it is correct. This is important, because you are going to

have to know every one of them, and verifying them is or ought to be easy.

This particular collection of matrices IS the set of Complex Numbers. I de_ne

the complex numbers this way:

De_nition 1.3.1 C is the name of the two dimensional subspace of the four

dimensional space of 2 _ 2 matrices having entries of the form

_ a 􀀀b

b a _

for any real numbers a; b. Points of C are called, for historical reasons,

complex numbers.

There is nothing mysterious or mystical about them, they behave in a thoroughly

straightforward manner, and all the properties of any other complex

numbers you might have come across are all properties of my complex numbers,

too.

You might be feeling slightly gobsmacked by this; where are all the imaginary

numbers? Where is p

􀀀1? Have patience. We shall now gradually recover

all the usual hocus-pocus.

First, the fact that the set of matrices is a two dimensional vector space

means that we can treat it as if it were R2 for many purposes. To nail this

idea down, de_ne:

C : R2

􀀀! C

by

_ a

b _ ; _ a 􀀀b

b a _

This sets up a one to one correspondence between the points of the plane

and the matrices in C . It is easy to check out:

1.3. WHAT ARE COMPLEX NUMBERS? 15

Proposition 1.3.1 C is a linear map

It is clearly onto, one-one and an isomorphism. What this means is that there

is no di_erence between the two objects as far as the linear space properties

are concerned. Or to put it in an intuitive and dramatic manner: You can

think of points in the plane _ a

b _ or you can think of matrices _ a 􀀀b

b a _ and

it makes no practical di_erence which you choose- at least as far as adding,

subtracting or scaling them is concerned. To drive this point home, if you

choose the vector representation for a couple of points, and I translate them

into matrix notation, and if you add your vectors and I add my matrices,

then your result translates to mine. Likewise if we take 3 times the _rst and

add it to 34 times the second, it won't make a blind bit of di_erence if you

do it with vectors or I do it with matrices, so long as we stick to the same

translation rules. This is the force of the term isomorphism, which is derived

from a Greek word meaning 'the same shape'. To say that two things are

isomorphic is to say that they are basically the same, only the names have

been changed. If you think of a vector _ a

b _ as being a 'name' of a point

in R2 , and a two by two matrix _ a 􀀀b

b a _ as being just a di_erent name

for the same point, you will have understood the very important idea of an

isomorphism.

You might have an emotional attachment to one of these ways of representing

points in R2 , but that is your problem. It won't actually matter which you

choose.

Of course, the matrix form uses up twice as much ink and space, so you'd be

a bit weird to prefer the matrix form, but as far as the sums are concerned,

it doesn't make any di_erence.

Except that you can multiply the matrices as well as add and subtract and

scale them.

And what THIS means is that we have a way of multiplying points of R2 .

Given the points _ a

b _ and _ c

d _ in R2 , I decide that I prefer to think of

them as matrices _ a 􀀀b

b a _ and _ c 􀀀d

d c _, then I multiply these together

16 CHAPTER 1. FUNDAMENTALS

to get (check this on a piece of paper)

_ ac 􀀀 bd 􀀀(ad + bc)

ad + bc ac 􀀀 bd _

Now, if you have a preference for the more compressed form, you can't multiply

your vectors, Or can you? Well, all you have to do is to translate your

vectors into my matrices, multiply them and change them back to vectors.

Alternatively, you can work out what the rules are once and store them in a

safe place:

_ a

b _ _ _ c

d _ = _ ac 􀀀 bd

ad + bc _

Exercise 1.3.2 Work through this carefully by translating the vectors into

matrices then multiply the matrices, then translate back to vectors.

Now there are lots of ways of multiplying points of R2 , but this particular

way is very cool and does some nice things. It isn't the most obvious way

for multiplying points of the plane, but it is a zillion times as useful as the

others. The rest of this book after this chapter will try to sell that idea.

First however, for those who are still worried sick that this seems to have

nothing to do with (a + ib), we need to invent a more compressed notation.

I de_ne:

De_nition 1.3.2 For all a; b 2 R; a +ib = _ a 􀀀b

b a _

So you now have three choices.

1. You can write a+ib for a complex number; a is called the real part and

b is called the imaginary part. This is just ancient history and faintly

weird. I shall call this the classical representation of a complex number.

The i is not a number, it is a sort of tag to keep the two components

(a,b) separated.

1.3. WHAT ARE COMPLEX NUMBERS? 17

2. You can write _ a

b _ for a complex number. I shall call this the point

representation of a complex number. It emphasises the fact that the

complex numbers form a plane.

3. You can write

_ a 􀀀b

b a _

for the complex number. I shall call this the matrix representation for

the complex number.

If we go the _rst route, then in order to get the right answer when we multiply

(a + ib) _ (c + id) = ((ac 􀀀 bd) + i(bc + ad))

(which has to be the right answer from doing the sum with matrices) we can

sort of pretend that i is a number but that i2 = 􀀀1. I suggest that you might

feel better about this if you think of the matrix representation as the basic

one, and the other two as shorthand versions of it designed to save ink and

space.

Exercise 1.3.3 Translate the complex numbers (a+ib) and (c+id) into matrix

form, multiply them out and translate the answer back into the classical

form.

Now pretend that i is just an ordinary number with the property that i2 = 􀀀1.

Multiply out (a + ib) _ (c + id) as if everything is an ordinary real number,

put i2 = 􀀀1, and collect up the real and imaginary parts, now using the i as

a tag. Verify that you get the same answer.

This certainly is one way to do things, and indeed it is traditional. But it

requires the student to tell himself or herself that there is something deeply

mysterious going on, and it is better not to ask too many questions. Actually,

all that is going on is muddle and confusion, which is never a good idea unless

you are a politician.

The only thing that can be said about these three notations is that they each

have their own place in the scheme of things.

The _rst, (a + ib), is useful when reading old fashioned books. It has the

advantage of using least ink and taking up least space. Another advantage

18 CHAPTER 1. FUNDAMENTALS

is that it is easy to remember the rule for multiplying the points: you just

carry on as if they were real numbers and remember that i2 = 􀀀1. It has the

disadvantage that it leaves you with a feeling that something inscrutable is

going on, which is not the case.

The second is useful when looking at the geometry of complex numbers,

something we shall do a lot. The way in which some of them are close to

others, and how they move under transformations or maps, is best done by

thinking of points in the plane.

The third is helpful when thinking about the multiplication aspects of complex

numbers. Matrix multiplication is something you should be quite comfortable

with.

Which is the right way to think of complex numbers? The answer is: All

of the above, simultaneously. To focus on the geometry and ignore the

algebra is a blunder, to focus on the algebra and forget the geometry is an

even bigger blunder. To use a compact notation but to forget what it means

is a sure way to disaster.

If you can be able to ip between all three ways of looking at the complex

numbers and choose whichever is easiest and most helpful, then the subject

is complicated but fairly easy. Try to _nd the one true way and cling to it

and you will get marmelised. Which is most uncomfortable.

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