Could you help me with complex numbers?
- From: nicegirl_130@xxxxxxxxx
- Date: 11 Jan 2007 10:31:40 -0800
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.
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.
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.
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.
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.
(Dont get me wrong, though I like Math I'm a normal 15 yo girl)
Thank you
Sharon
.
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