history of imaginary numbers

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Original poster: "Robert Jones" <alwynj48-at-earthlink-dot-net> 

Hi all and Terry in particular

The following is a very informative complex number history lesson taken 
from the discussion I posted a link to previously. I assume it was ok to copy.

When I was first introduced to them I figured they were a perversion 
invented by mad professors to confound their students.

It may not be straight text perhaps Terry can try put it up on his site and 
post a link to it.

.

Regards Bob


As you know, people came to natural numbers because of necessity to count 
things. Very soon they have recognized that there exist parts of a whole - 
the fractions. Debt was associated with negative numbers and soon the 
Arithmetic appeared - the Calculus of rational numbers.  Mathematics was 
born. Very soon (after a couple millenniums) Algebra was created - the 
Arithmetic of symbols of numbers. People immediately have recognized that 
there is something wrong with rational numbers: some of them are squares of 
another one, some not. So irrational numbers were discovered. (Please, pay 
attention: "discovered", not "defined"  or "introduced") The completion of 
set of Real numbers had begun.
Soon (after some centuries) the Great Gauss Theorem was proven:

The polynomial algebraic equation of power n

x^n + a1*x^(n-1) + a2*x^(n-2) +... +a(n-1)*x + an = 0

with real coefficients {a1, a2, ... , an} has exactly n roots"

And people immediately noticed that there is something wrong with Reals 
too. Equation like this one

x^2 + 1 = 0

has no solution among all known Reals, at all! So people come to a natural 
conclusion that there should exist in Nature some new type of numbers that 
are not Reals, but can deliver the roots of polynomial algebraic equations. 
So the complex numbers  were discovered. Than was proven that any complex 
number can be represented through a unique one - named i, in the following 
form:

z = x + i*y

where i is one of roots of equation x^2 + 1 = 0 (second root is 1/i): i = 
(-1)^0.5

It was the great Euler, who had discovered the most famous mathematical 
formula - so-called the trigonometric representation of uni-module complex 
number:

exp(i*R) = cos(R) + i*sin(R)

where R is any real number.
Due to this formula any complex number x + i*y can be represented as following

x + i*y = r*exp(i*f)

where r is equal to [x^2 + y^2]^0.5 is called as "the module of complex 
number x + i*y" and f is equal to tangent of ratio y/x and is called  "the 
argument of complex number x + i*y"

As you can see, any real number can be represented in form x + i*y (simply 
at y = 0), but now we have a lot more new type of numbers (with y not equal 
to 0).

After these discoveries people have recognized that now we can say that:

The polynomial algebraic equation of power n

x^n + a1*x^(n-1) + a2*x^(n-2) +... +a(n-1)*x + an = 0

with any complex coefficients {a1, a2, ... , an} has exactly n roots (real 
or complex ones)"

And now there are no exclusions from this theorem. This is a reason, why we 
are saying that the complex numbers are the algebraic closure of set of 
real numbers . (Besides of many of important purely mathematical problems, 
it literally closes the sense of the Great Gauss Theorem.)

Let us notice, that reverse statement to Great Gauss Theorem: "Any number 
can be a root of some algebraic polynomial equation of some finite power 
n"  is wrong. The transcendental numbers are not such numbers: they can not 
be the roots of any algebraic equations.

Also let me notice that we can not place the all complex numbers on the one 
flat plate: many of them require so-called Reimannian surfaces that can be 
imagined as a set of plates that are glued with each other along some 
lines. The theory of complex numbers and functions is much, much more 
complicated and fascinating that the theory of real numbers...