文档介绍：Section
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The Algebra plex
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At this point we have considered only real-valued functions of a real variable. That is,
all of our work has centered on functions of the form f : R → R, functions which take a
real number to a real number. In this chapter we will plex numbers and the
calculus of associated functions. In particular, if we let C represent the set of plex
numbers, then we will be interested in functions of the form f : R → C and f : C → C .
We will begin the story in this section with a discussion of plex numbers are and
how we work with them.
Perhaps because of their name, it is sometimes thought plex numbers√ are in
some way more mysterious than real numbers, that a number such as i = −1 is not as
“real” as a number like 2 or − or even π. However, all of these numbers are equally
meaningful, they are all useful mathematical abstractions. plex numbers
are a relatively recent invention of mathematics, dating back just over 200 years in their
current form, it is also the case that negative numbers, which were once called fictitious
numbers to indicate that they were less “real” than positive numbers, have only been
accepted for about the same period of time, and we have only started to understand the
nature of real numbers during the past 150 years or so. In fact, if you think about their
underlying meaning, π is a far more “complex” number than i.
plex numbers originate with attempts to solve certain algebraic equa-
tions, such as
x2 + 1 = 0,
we will give a geometric definition which plex numbers with points in the
plane. This definition not only plex numbers a concrete geometrical meaning,
but also provides us with a powerful algebraic tool for working with points in the plane.
Definition plex number is an ordered pair of real numbers with addition defined
by
(a, b) + (c, d) = (a + c, b + d) ()
and multiplication d