1.1 A Little History

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If Complex Numbers had been invented thirty years ago instead of over

three hundred, they wouldn't have been called 'Complex Numbers' at all.

They'd have been called 'Planar Numbers', or 'Two-dimensional Numbers'

or something similar, and there would have been none of this nonsense about

'imaginary' numbers. The square root of negative one is no more and no less

imaginary than the square root of two. Or two itself, for that matter. All of

them are just bits of language used for various purposes.

'Two' was invented for counting sheep. All the positive integers (whole numbers)

were invented so we could count things, and that's all they were invented

for. The negative integers were introduced so it would be easy to

count money when you owed more than you had.

The rational numbers were invented for measuring lengths. Since we can

transduce things like voltages and times to lengths, we can measure other

things using the rational numbers, too.

The Real numbers were invented for wholly mathematical reasons: it was

found that there were lengths such as the diagonal of the unit square which,

in principle, couldn't be measured by the rational numbers. This is of not

the slightest practical importance, because in real life you can measure only

to some limited precision, but some people like their ideas to be clean and

cool, so they went o_ and invented the real numbers, which included the

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10 CHAPTER 1. FUNDAMENTALS

rationals but also _lled in the holes. So practical people just went on doing

what they'd always done, but Pure Mathematicians felt better about them

doing it. Daft, you might say, but let us be tolerant.

This has been put in the form of a story:

A (male) Mathematician and a (male) Engineer who knew each other, had

both been invited to the same party. They were standing at one corner of the

room and eyeing a particularly attractive girl in the opposite corner. 'Wow,

she looks pretty good,' said the Engineer. 'I think I'll go over there and try

my luck.'

'Impossible, and out of the question!' said the Mathematician, who was

thinking much the same but wasn't as forthright.

'And why is it impossible?' asked the Engineer belligerently.

'Because,' said the Mathematician, thinking quickly, 'In order to get to her,

you will _rst have to get halfway. And then you will have to get half of the

rest of the distance, and then half of that. And so on; in short, you can never

get there in a _nite number of moves.'

The Engineer gave a cheerful grin.

'Maybe so,' he replied, 'But in a _nite number of moves, I can get as close

as I need to be for all practical purposes.'

And he made his moves.

***

The Complex Numbers were invented for purely mathematical reasons, just

like the Reals, and were intended to make things neat and tidy in solving

equations. They were regarded with deep suspicion by the more conservative

folk for a century or so.

It turns out that they are very cool things to have for 'measuring' such things

as periodic waveforms. Also, the functions which arise between them are very

useful for talking about solutions of some Partial Di_erential Equations. So

don't look down on Pure Mathematicians for wanting to have things clean

and cool. It pays o_ in very unexpected ways. The Universe also seems

to like things clean and cool. And most supersmart people, such as Gauss,

like _nding out about Electricity and Magnetism, working out how to handle

1.2. WHY BOTHER WITH COMPLEX NUMBERS AND FUNCTIONS?11

calculations of orbits of asteroids and doing Pure Mathematics.

In these notes, I am going to rewrite history and give you the story about

Complex Numbers and Functions as if they had been developed for the applications

we now know they have. This will short-circuit some of the mystery,

but will be regarded as shocking by the more conservative. The same sort of

person who three hundred years ago wanted to ban them, is now trying to

keep the confusion. It's a funny old world, and no mistake.

Your text books often have an introductory chapter explaining a bit of the

historical development, and you should read this in order to be educated,

but it isn't in the exam.

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