RE: history of imaginary numbers

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Godfrey Loudner" <ggreen-at-gwtc-dot-net> 

Hello Bob

The article is disjointed and contains vague statements.
This is not the place to write a treatise on complex
numbers, but I could not resist a few remarks.

 >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)"

The quadratic equation x^2-2x+1 = 0 has exactly one root x = 1.
By factoring, x^2-2x+1 = (x-1)(x-1). By counting x = 1
from each of the two factors, it is said that the equation
has two roots. This is counting roots according to the
multiplicity of factors. In this sense, the theorem is true.

 >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.

True, if one restricts the coefficients of the polynomial equation
to be rational numbers. Example, Pi cannot be the root of any
polynomial equation with only rational coefficients and degree > 0.

 >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...

All the complex numbers can be represented as points on a plane
(complex plane), which is "flat space". Riemann surfaces were
invented to handle multiple-valued functions such as z^(1/2).
By representing such a function on its Riemann surface, it becomes
single-valued. The description of Riemann surfaces from the forum
is commonly presented in textbooks. However, this description is
very inadequate and misleading. A Riemann surface is a very
abstruse concept, requiring the use of manifold theory for its
proper description. In dealing with functions of a complex
variable, one can get along with a Riemann surface as copies
of the complex plane "glued" along "cuts". But one has to be
very very very very careful, or erroneous formulas will come
of it.

Godfrey Loudner