Re: Could you help me with complex numbers?

Hi,

David Ullrich gave great answers to your questions, so I won't repeat them other than to second his suggestion that you roundly ignore whoever told you that 15-year-old girls have no business being interested in complex numbers (or anything else that piques your intellectual curiosity).

On second thought I have given a partial answer to your question about what complex numbers measure, see below.

On Thu, 11 Jan 2007, nicegirl_130@xxxxxxxxx wrote:

Hi, I'm a 15 year old girl, and, though this may seem strange, I love
math and try to study a bit more thn required by my school. The complex
numbers have always made me curious, so I tried to understand them to
the extent it's possible for someone my age.

Formal education will try to distill this natural and healthy impulse to learn out of you. Don't let it!

At first, I was introduced to the misteryous i = sqrt(-1), something
really kabbalistic to me. Then, I got to know what mathematicians did
was extend those algebraic laws of the real field (I've read something
about groups, rings and fields) to the vector space R^2. Not sure if
this is correct, but that's what I think. In R^2 we can add and
multiply by a scalar, but we can't multiply, for example, (2,3) and
(3,7). But then it was defined that, in the complex plane, (Argand
Gauss plane, right?) (a,b) * (c,d) = (ac - bd , ad + bc). So, it seems
to me that, just like R^2, the complex plane C is formed by orderd
pairs of elements extracted from the real line. And the only difference
between them is that R^2 doesn't have all of those algebraic laws and C
does. In other words, C is a field and R^2 is not. It can't be that
simple, of course there's something I don't gather, I don't
understand. But thats my perception. If I'm right, and probably I'm
wrong, if we see R^2 and C just like sets and dont take into account
those field operations, then apparently they are exactly the same set.

As David suggested elsewhere in this thread, the distinction is entirely a matter of point of view. You are entirely correct. Most mathematicians use R^2 to denote a two dimensional real vector space and C to denote the field, but one can set everything up so that they have the same underlying set.

You may be interested to know that there are theorems about R and C that guarantee that even if you have a different construction of C (or R) in mind than I do, we are still talking about fundamentally the same objects: all of the a priori different things that get called C are isomorphic to each other.

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 is one thing that comes to mind, for what it's worth: in the theory of AC electrical circuits, the quantity of _impedance_ is represented as a complex number.

http://en.wikipedia.org/wiki/Electrical_impedance

This may not be very enlightening but at least it is a partial answer to the question "can complex numbers measure things?" Perhaps not in the sense of rulers or voltmeters, but it is very useful to be able to imagine them doing so in a way consistent with the theory.

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.

They did get out of the real line, but in a natural extension of the process you describe. The desire for solutions to

k = m - n (or, if you like, m + n = 0)

for arbitrary positive integer m and n leads to the negative integers in a way similar to the way that the desire for solutions to

z^2 + x = 0

for arbitrary real numbers x leads to the complex numbers.

It is a bit surprising when first meeting these objects when one realizes that the expansion of the system of the real numbers that allows for solutions to x^2 + 1 = 0 actually allows you to solve ALL polynomial equations

a_n x^n + ... + a_1 x + a_0 = 0,

even when the a's are complex! See "Fundamental Theorem of Algebra."

You may be interested in further progress in this direction. Google for "quaternions", "octonions", "normed division algebra," and, if you're feeling particularly enthusiastic, "Clifford algebra". Given what you say in your post, I should not think an understanding of the basics of any of these ideas to be beyond you (with the possible exception of the Clifford algebras, at least without some more reading). At the worst it will be confusing; at best it will introduce you to some of the most beautiful and fascinating mathematics around.

http://en.wikipedia.org/wiki/Quaternion
http://en.wikipedia.org/wiki/Octonion
http://en.wikipedia.org/wiki/Normed_divison_algebra
http://en.wikipedia.org/wiki/Clifford_algebra

Well, thank you all and sorry, I know a 15 yo girl is not supposed to
ask such silly questions to the top dogs of Math, but I'm really
curious.

Ha. Ask whatever you like, we're happy to help. We like math so much that many of us do it for a living and all of us kill time on sci.math. At least in part we are here because we (sadly) talk to relatively few people in real life who express such interest in the things that also interest us.

(Dont get me wrong, though I like Math I'm a normal 15 yo girl)

I venture to suggest the two ought not to be thought mutually exclusive.

Best,
-dave

***********************************************************
"Papa, can you multiply triples?"
"No, I can only add and subtract them."

-an (apocryphal) conversation between W.R. Hamilton and
his young son
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