Re: Could you help me with complex numbers?

nicegirl_...@xxxxxxxxx schrieb:

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.
Many mathematicians had these thoughts too.
Bombelli was the first, who looked at them just as numbers with a
different sign. At his time the negative numbers were just introduced
into europe, so they learned things like
( - 3 ) * ( - 2) = + 6.
Bombelli looked at four different signs : + , - , sqrt (- ), - sqrt (
- ).

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.
You hit the point and You are right.
It can't be that
simple, of course there's something I don't gather, I don't
understand.
It is that simple! It took me more than twenty years to understand,
that some teachers ( some of whom don't know any better) just add some
kabbalistic to math.

The history went like this:
Caspar Wessel wrote in 1798 a text, where he gave to the "imaginary"
numbers and the complex numbers the geometric representation as
arrow-vectors in the plane.
Still in english countries to read his rather simple, short text, You
have to pay a lot of money or You have to go a library. ( It's worth
it.)
B Banner and J Lützen (eds.), Caspar Wessel, On the analytic
representation of direction (1999)

The next step was the invention of writing arrow-vectors as ordered
pairs like ( 3, -4).
This was done by William R. Hamilton, and he did this in 2D with the
addition and multiplication You mentioned. In 1837 he published
Algebraic Couples, and Algebra as the Science of Pure Time
You can read this online here
http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/PureTime/
( the first part is about time, a bit difficult, the second part
concerns us here)
So now one could represent these strange numbers ( they are pairs of
numbers) in the ordinary plane with an ordinary x-axis y-axis cartesian
coordinate-system.
And some great mathematicians did this, like Gauss and Riemann.

But somehow this was not magic enough, so the y-axis was renamed into
i-axis.So now one has two different planes. But if You ask for a
mathematical definition of the i-axis different from the y-axis, nobody
can tell You.


But thats my perception. If I'm right, and probably I'm
wrong, if we see R^2 and C just like sets

Yes R^2 is a set, and when one adds operations to it, one gets
different structures, one is called C.

and dont take into account
those field operations, then apparently they are exactly the same set.


Yes. Now one important point is: ask for proper definitions.
With these You will see, that Your speculation is right.

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

i = sqrt ( - 1 ) = ( 0 , 1 ), so ( 2, 3 ) = 2 + 3 * i ,
one can even do mixed writing like 3 * i * ( 7, 5 ) =( 3, 0 ) * ( 0, 1
) * ( 7, 5 ) = ( - 15 , 21).

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.
Every arrow-vector, denoted by ( 2, 3 ) or in 3D by ( -8, 3.1, 9 ) is
an abstract vector too,
that is an element of a set with some structure to it.

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.

With complex numbers can denote points/ locations in a plane with a
coordinate system, like real numbers can be points on the number line.
With addition/subtraction real numbers are distances betwen two points,
here complex numbers with addition/ subtraction do the same in the
plane, directed distances.
With multiplication real numbers can grow and shrink ( scale ) each
other. This is done by the complex numbers of the form ( a, 0 ).
And a complex numbers of distance one has a direction on the plane (
with regard to the reference x-axis measured as an angle). So such a
number mulitplied to a complex number will change the direction of this
vector-arrow by the given angle.
In short, what a vector is, is given through how he operates on another
vector.

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.

....to another real line perpendicular to the first, which makes it two
real lines.

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)


Keep on like this!
With friendly greetings
Hero
PS More about i - not so imaginary, You can read on my page
http://1iz.de

.

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