Re: Could you help me with complex numbers?

So, whast's the difference between, for example (2,3) and 2 + 3i? 2 +
3i suggests a vector notation, as though we could see the complex
numbers as vectors on the plane, like those we study in Physics, like
forces, velocities, etc. Are the complex numbers actual numbers or are
they vectors? That is, does it make sense to measure something in
complex numbers. Sorry for my stupid question, but does it make any
sense to say you bought something for (30 + 50 i) US$ ? If instead of
money it was say, distance, I could understand, since the real axis may
be seen as the horizontal axis and the imaginary axis stands for the
vertical axis.

There are subtle differences between R2 and C. For example, you can
have a function in R2 that is differentiable, and its derivative is
differentiable, but that (the second derivative) is not differentiable.
In C, it's all or nothing, i.e., once you meet the conditions to be
differentiable once, you're differentiable forever.

It seems to me that the complex numbers are somwhat "artificial",
because from what I could read they were say constructed. I mean, fisrt
you have the natural numbers {1,2,3....} and you always could add but
subtraction was impossible if n < m. Then the negative integers were
created and solved this problem, the set Z of all the integers became
available. But then division was not always possible and the set of the
rationals Q was created. But then you couldn't compute things like
sqrt(2) and so came the irrationalls. That is, so far you solved the
problems, but with the complexes, didn't matematician just construct
something rather artificial? They got out of the real line.

Once you have the reals, there are still problems you can't solve. For
example: x^2 + 1 = 0. That's where the complex numbers came from. And
once you have the complex numbers, you can solve all these problems
(and any others that come up).

If you have access to it in your library, check out the Feynman
Lectures on Physics. In the first volume, lecture 20-something, he
discusses the natural numbers, integers, reals, and complexes. It's an
amazing piece, the kind that leaves you thinking "wow, that's so deep."
And it doesn't depend on the other pieces around it, so it's very
easily approachable.

Michael

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