1.2 Why Bother With Complex Numbers and Functions?

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In mastering the material in this book, you are going to have to do a lot of

work. This will consist mainly of chewing a pencil or pen as you struggle to

do some sums. Maths is like that. Hours of your life will pass doing this,

when you could be watching the X-_les or playing basketball, or whatever.

There had better be some point to this, right?

There is, but it isn't altogether easy to tell you exactly what it is, because

you can only really see the advantages in hindsight. You are probably quite

glad now that you learnt to read when you were small, but it might have

seemed a drag at the time. Trust me. It will all be worth it in the end.

If this doesn't altogether convince you, then talk to the Engineering Lecturers

about what happens in their courses. Generally, the more modern

and intricate the material, the more Mathematics it uses. Communication

Engineering and Power Transmission both use Complex Functions; Filtering

Theory in particular needs it. Control Theory uses the subject extensively.

Whatever you think about Mathematicians, your lecturers in Engineering

are practical people who wouldn't have you do this course if they thought

they could use the time for teaching you more important things.

Another reason for doing it is that it is fun. You may _nd this hard to believe,

but solving problems is like doing exercise. It keeps you _t and healthy and

has its own satisfactions. I mean, on the face of it, someone who runs three

12 CHAPTER 1. FUNDAMENTALS

kilometres every morning has to be potty: they could get there faster in a

car, right? But some people do it and feel good about themselves because

they've done it. Well, what works for your heart and lungs also applies to

your brain. Exercising it will make you feel better. And Complex Analysis is

one of the tougher and meatier bits of Mathematics. Tough minded people

usually like it. But like physical exercise, it hurts the _rst time you do it,

and to get the bene_ts you have to keep at it for a while.

I don't expect you to buy the last argument very easily. You're kept busy

with the engineering courses which are much more obviously relevant, and

I'm aware of the pressure you are under. Your main concern is making sure

you pass the examination. So I am deliberately keeping the core material

minimal.

I am going to start o_ by assuming that you have never seen any complex

numbers in your life. In order to explain what they are I am going to do a

bit of very easy linear algebra. The reasons for this will become clear fairly

quickly.

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