Graphics Reference
In-Depth Information
2
Inequalities
Preliminary reading:
Beckenbach and Bellman.
Concurrent reading:
Korovkin, Kazarinoff.
Further reading:
Thurston ch. F., Ivanov ch. 4.
Positive numbers and their properties
Theaddition, subtraction, multiplication and division of numbrs
will work in this course the way they did for you in school. Their
essential algebraic properties are listed in appendix 1 under the heading
'algebraic properties of a field of numbers'. Look at this list briefly.
Examine which of these algebraic properties hold for the set of integers
Z
0,
1,
2,
3,...
,
and then check that the set of rational numbers
satisfies
all
these algebraic properties. There is no need to remember the
list for this course. The letter
p
q
p
,
q
Z,
q
0
Q
is the first letter of the German word
Zahl
, meaning
number
. The letter Q is the first letter of
quotient
; each
rational number being a quotient of integers. The letter
Z
, thefirst
letter of the word
real
, is used to denote the set of real numbers, the
numbers needed for measuring distances along a line. We will presume
that all the algebraic properties of numbers mentioned in appendix 1
hold for the set of real numbers. The difference between Q and R will
be examined in chapter 4.
It is questions of convergence and the finding of limits which
characterise a course in analysis. It is necessary to use inequalities in
order to define these infinite processes. The purpose of this chapter is to
sharpen your awareness of inequalities so that you know how to argue
R
