Re: history of imaginary numbers

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

Original poster: "Steve Cook" <steve-at-g8cyerichmond.freeserve.co.uk> 

Now I know why my head hurts when I read math rather than build coils. LOL
----- Original Message -----
From: "Tesla list" <tesla-at-pupman-dot-com>
To: <tesla-at-pupman-dot-com>
Sent: Monday, March 22, 2004 11:30 PM
Subject: RE: history of imaginary numbers


 > 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
 >
 >


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