The truth about my life

Thu, 04 Nov 2004

Michael D. Alder'in "An Introduction to Complex Analysis for Engineers" adlı kitabını okumaya başladım, 1 yıldan fazla bir süredir bir türlü bitiremediğim Noam Chomsky'nin "Terörizm Kültürü" adlı kitabının yanında.

İlk birkaç sayfadan eğlencelikler aktarmak istiyorum. Matematik ağırlıklı kitaplar arasında bu kadar keyifli bir başlangıca sahip bir kitap daha görmediğimi itiraf etmeliyim:

Chapter 1: Fundamentals

1.1: A Little History

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 cavlled '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 trnasduce 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 off and invented the real numbers, which included the rationals but also filled 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 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 half of the rest of the distance, and then half of that. Ans so on; in short, you can never get there in a finite number of moves.'

The Engineer gave a cheerful grin.

'Maybe so,' he replied, 'But in a finite 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 Differential Equations. So don't look down on Pure Mathematicians for wanting to have things clean and cool. It pays off in very unexpected ways. The Universe also seems to like things clean and cool. And most supersmart people, such as Gauss, like finding about Electricity and Magnetism, working out how to handle 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 appli- cations 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.

Devamı da hayli keyifli gidiyor, ama özellikle bu bölümü paylaşmak istedim.

İyi eğlenceler!

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About me

I'm Enver ALTIN. I'm from the other side of the river.


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