Re: Could you help me with complex numbers?

In article
<1168540300.588468.128060@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
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nicegirl_130@xxxxxxxxx wrote:

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.

What piqued interest was that the polynomial x^2+1 does
not have a real root. By inventing one called i and
exploring the ramifications people discovered that the
complex numbers have many neat properties.

Now having invented negative numbers so we can always
solve x+a = 0, and inventing rational numbers so we can
solve ax + b = 0, and irrational numbers so we can
solve x^2 - 2 = 0; we start to wonder if this will ever
stop? Will we have to invent sqrt(i)? The big surprise
is no! Just multiply ((1+i)/sqrt(2))^2. I'll wait.




(1+i)^2 = 1 + 2 * i -1 = 2 * i
so ((1+i)/sqrt(2))^2 = 2 * i / 2 = i.

Any polynomial has a root in the complex numbers.
Furthermore, a degree n polynomial has exactly n roots
in the complex numbers.

--
Michael Press
.