An explanation of Complex Numbers by Prof M. Alder at UWA. It's quite funny.
"There is a certain mysticism about complex numbers, much of which comes
from meeting them relatively late in life. The student has it explained that
there is an imaginary number, i which is the square root of −1 and this is
certainly imaginary because it is obvious to any fool that −1 doesn’t have a
square root. When much younger, you might have been troubled by 4−7 =
−3 which is just as obviously impossible and meaningless because you can’t
take seven from four. Since you met this when you were young and gullible
and the teacher assured you that −3 is a perfectly respectable number, just
not one which is used for counting apples, you went along with it and bought
the integers. Now, when your brains have nearly ossified you are asked to
believe in square roots of negative numbers and those poor little brains resist
the new idea with even more determination that when you first met negative
numbers.
The fact is all numbers are imaginary. The question is, can we find a use for
them? If so, we next work out what the rules are for messing around with
them. And that’s all there is to it. As long as we can devise some consistent
rules we are in business. Naturally, when we do it, all the practical people
scream ‘you can’t do that! It makes no sense!’. After a few years they get
used to it and take them for granted. I bet the first bloke who tried to sell
arabic notation to a banker had the hell of a time. Now try selling them the
Latin numerals as a sensible way to keep bank balances. Up to about 1400
everybody did, now they’d laugh at you.
It’s a silly old world and no mistake.
We got from the natural numbers (used for counting apples and coins and sheep and daughters, and other sorts of negotiable possessions) to the integers
by finding a use for negative numbers. It certainly meant that we could
now subtract any two natural numbers in any order, so made life simpler
in some respects. You could also subtract any two integers, so the problem
of subtraction was solved. The extra numbers could be used for various
things, essentially keeping track of debts and having a sort of direction to
counting. They didn’t seem to do any harm, so after regarding them with
grave suspicion and distrust for a generation or so, people gradually got used
to them.
We got from the integers to the rational numbers by finding that although you
could divide four by two you couldn’t divide two by four. So mathematicians
invented fractions, around five thousand years ago. While schoolies all around
were saying Five into three does not go, some bright spark said to himself But
what if it did? Then you could divide by pretty much anything, except zero.
Dividing by zero wasn’t the sort of thing practical folk felt safe with anyway,
so that was OK. And people found you could use these new-fangled numbers
for measuring things that didn’t, like sheep and daughters, naturally come
in lumps. Then the Greeks found (oh horror!) that the diagonal of the unit
square hasn’t got a length- not in Q anyway. The Pythagorean Society, which
was the first and last religious cult based on Mathematics, threatened to send
out the death squads to deal with anyone spreading this around. The saner
mathematicians just invented the real numbers (decimals) which of course,
to a Pythagorean, were evil and wicked and heretical and the product of the
imagination- and a diseased one at that.
So there is a long history of people being appalled and horrified by new numbers
and feeling that anyone spouting that sort of nonsense should be kicked
out if not stoned to death. Square root of minus one indeed! Whatever rubbish
will they come up with next? Telling us the world isn’t flat, I shouldn’t
wonder.
Meanwhile back in the mathematical world where we can make up whatever
we want, let’s approach it from a different angle.
The real line is just that– a line. We can do arithmetic with the points on
the line, adding and subtracting and multiplying and dividing any pair of
points so long as we don’t try to divide by zero.
Suppose you wanted to do the same on something else. Say a circle. Could
you do that? Could you make up some rules for adding and multiplying
points on a circle? Well you could certainly regard the points as angles from
some zero line and add them up. If you did it would be rather like the clock
arithmetic of the last chapter except you would be working mod 2 instead
of mod 12. You could also do multiplication the same way if you wanted.
The mathematician yawns. Boooooring! Nothing really new here.
Could you take R2 and make up a rule for adding and multiplying the points
in the plane? Well, you can certainly add and subtract them because this
is a well known vector space and we can always add vectors. Confusing the
vectors with little arrows and the points at the end of them with the arrows
themselves is fairly harmless."
If you want me to post the rest, let me know. Otherwise, that's all I will copy and paste since no-one will read very much.
