SECTION
A1
A Brief Review of Algebra
There are many techniques from elementary algebra that are needed in calculus. This
appendix contains a review of such topics, and we begin by examining numbering
systems.
An
integer
is a “whole number,” either positive or negative. For example, 1, 2,
875,
±
15,
±
83, and 0 are integers, while
, 8.71, and
are not.
A
rational number
is a number that can be expressed as the quotient
of two
integers, where
b
±
0. For example,
and
are rational numbers, as are
Every integer is a rational number since it can be expressed as itself divided by 1.
When expressed in decimal form, rational numbers are either terminating or in±nitely
repeating decimals. For example,
A number that cannot be expressed as the quotient of two integers is called an
irrational number.
For example,
are irrational numbers.
The rational numbers and irrational numbers form the
real numbers
and can be
visualized geometrically as points on a
number line
as illustrated in Figure A.1.
FIGURE A.1
The number line.
If
a
and
b
are real numbers and
a
is to the right of
b
on the number line, we say that
a
is greater than
b
and write
a
.
b.
If
a
is to the left of
b
, we say that
a
is less
than
b
and write
a
,
b
(Figure A.2). For example,
5
²
2
±
12
³
0
and
±
8.2
³±
2.4
FIGURE A.2
Inequalities.
a
b
a
>
b
b
a
a
<
b
Inequalities
–5
–4
–3
–2
–1
0
5
4
3
2
1
π
–2.5
–
√
3
2
2
±
2
²
1.41421356
and
´
²
3.14159265
5
8
µ
0.625
1
3
µ
0.33 . . .
and
13
11
µ
1.181818 . . .
±
6
1
2
µ
±
13
2
and
0.25
µ
25
100
µ
1
4
±
4
7
8
5
,
2
3
,
a
b
±
2
2
3
The Real Numbers
642
❘
APPENDIX A
❘
Algebra Review
❘
A-2
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