Re: Could you help me with complex numbers?


nicegirl_...@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.

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?

They aren't vectors, but they can be represented as points
on a plane, called (not surprisingly) the "complex plane".

Furthermore, it is often convenient to define the magnitude
and phase of a complex number, which are defined exactly
the same way as the magnitude and direction of a 2-vector.

They are numbers, a generalization of the real numbers.

Unlike the real numbers, there is no natural "<", ">"
relation. In the real numbers we have an axiom called
trichotomy: Given two numbers a and b, then either
a = b, a < b, or a > b. Complex numbers don't have
that property.

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$ ?

Cost is usually a real number. But that doesn't prevent
you from extending the idea of "number" to things
besides real numbers, any more than restricting
"basketball score" to positive integers means other
things aren't numbers.

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",

Yes, of course they are. But so are real numbers.

Nevertheless, it's an abstraction that is very useful.

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.

Students are encouraged to ask questions.


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

This is the wrong forum to make the claim that liking math is
not normal :-)

However, stereotypes do have some basis in reality. I was
with my wife at a math department picnic once, and when
the softball game started, she was worried about participating
because she's not very athletic. "Honey", I said, "these
are a bunch of MATH majors." We did fine.

- Randy

.

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