Re: Could you help me with complex numbers?

Thank you very much!
Sharon


David C. Ullrich wrote:
On 11 Jan 2007 10:31:40 -0800, 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.

Doesn't seem strange to the people here. Of course we seem
strange to others...

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).

Precisely.

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.

It's that simple.

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.

Yup.

Of course there are various other ways to define the complex
numbers. But the way you sketch above is the simplest and
most common, and if we do it that way then everything you've
said is exactly right.

So, whast's the difference between, for example (2,3) and 2 + 3i?

None whatever.

(With the same proviso as before - no difference given this
particular way of constructing the complex numbers.)

The difference is just in your point of view.

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's not a "real" question - it's just a question about
what the word "number" should mean. Words mean what they
_say_ they mean! Most mathematicians would say that yes,
complex numbers are numbers.

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$ ?

No.

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.

There's no real difference. The integers are _constructed_ starting
with the natural numbers, the rationals are _constructed_ starting
from the integers, the reals (rational and irrationals) are
_constructed_ from the rationals, and the complexes are constructed
from the reals.

The only reason it seems different to you right now is that
you've _seen_ the construction of the complexes, while nobody
ever showed you those other constructions I mention.

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,

Who told you that? Don't believe him.

but I'm really
curious.

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

Thank you
Sharon



David C. Ullrich

.

Relevant Pages

  • Re: An uncountable countable set
    ... You are arguing that the naturals are not a subset of the rationals, which are not a subset of the reals. ... That there is an isomorphic image of the system of naturals in some other systems does not mean that the original naturals are in those systems. ... Viewed as a set of points, the real line has as proper subsets both the set of rationals and the set of naturals. ... There are two ways of constructing the rationals from the naturals ...
    (sci.math)
  • Re: Could you help me with complex numbers?
    ... math and try to study a bit more thn required by my school. ... particular way of constructing the complex numbers.) ... the rationals are _constructed_ starting ... from the reals. ...
    (sci.math)
  • Re: Question about a minor recasting of Dedekind cuts.
    ... Now I'm constructing the reals in two ways. ... Your old exercise has been perpetuated onto the human nearly ... is already true for the rationals. ...
    (sci.math)
  • Re: Question about a minor recasting of Dedekind cuts.
    ... Now I'm constructing the reals in two ways. ... that if A and B intersect then the intersection ... is already true for the rationals. ...
    (sci.math)
  • Re: Question about a minor recasting of Dedekind cuts.
    ... Now I'm constructing the reals in two ways. ... A D-cut of the rationals is a pair of sets A,B ... Your old exercise has been perpetuated onto the human nearly ...
    (sci.math)