1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 189

DEFINITIONS Polynomial and Root of a Polynomial

A polynomial of degree n in one variable x is a function of the form

where n is a positive integer; an, an_ 1 , ... , a 1 , a 0 are complex numbers; and

an=/; 0.

A root of a polynomial is any complex number r such that p(r) = 0.

As we shall see shortly, the history of so lving the polynomial equation

p(x) = 0- that is, of finding the roots of the polynomial p(x)- is divided

into two distinct searches. One search consisted of trying to find an explicit

formula for the roots, while the other search consisted of developing techniques

for approximating roots.

In order to truly appreciate the algebra of old, we need to realize that

the symbolism we currently use is approximately four hundred years old-

some symbolism being much more recent. The first printed occurrence of

the + and - sign, for example, appeared in an arithmetic book published by

Johann Widman in 1489. However, the symbols were not used to represent

the operations of addition and subtraction but merely to indicate excess and

deficiency. The signs + and - were used to represent operations by the Dutch

mathematician Giel Vander Hoecke in 1514 and probably by others somewhat

earlier. The equal sign, = , appeared in 1557 in The Whetstone of Witte by

Robert Recorde. Our inequality symbols of < and > are due to Thomas

Harriot in 1631. In 1637 the French mathematician Rene Descartes introduced
our custom of using letters early in our alphabet to denote constants and
letters late in our alphabet to represent unknowns. Descartes also introduced
our system of exponents- x^2 for x · x, x^3 for x · x · x, etc.
In 1842, G. H.F. Nesselmann divided the development of algebra into three
stages. The first stage was rhetorical algebra. In rhetorical algebra, problems
were posed and solved using pure prose- no abbreviations or symbolism was
employed. This stage lasted from antiquity until the time of Diophantus
of Alexandria or approximately 250 A.D. The second stage was syncopated
algebra. In this stage, abbreviations were used for some of the quantities and
operations which occurred most frequently. The final stage is called symboli c
algebra. In this stage, the solution of a problem is obtained by purely algebraic
manipulation of mathematical shorthand.
In addition to the changing aspect of algebra, we should also be aware that
different types of numerals and number representation systems were used and