Algebra Review

119 Pages·1978·2.12 MB·English

by  Charles Denlinger and Elaine Jacobson (Auth.)

AGEBR/4
REVIEW
to accompany Anton/Kolman's
APPLIED FINITE MATHEMATICS,
second edition, and APPLIED FINITE
MATHEMATICS with CALCULUS
Charles Denlinger
Elaine Jacobson
ACADEMIC PRESS
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London
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COPYRIGHT © 1978. BY ACADEMIC PRESS, INC.
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PREFACE
This review is designed as a background supplement to Applied
Finite Mathematics and Applied Finite Mathematics with Calculus by
Howard Anton and Bernard Kolman. It is not intended as a textbook
for a first course in algebra. Rather, it is a guide by means of
which a student can brush up on rusty skills. Every effort has
been made to keep this review brief; topics have been chosen for
their usefulness in understanding the main text, with the focus on
how to perform algebraic manipulations and not on the underlying
theory.
The authors wish to thank Dr. Leo W. Lampone for reviewing
this manuscript and solving the problems.
Charles Denlinger
Elaine Jacob son
Unit 1
The Number Systems
of Algebra
1.1 INTRODUCTION
Mathematics begins with the system of natural numbers, the
numbers used in counting:
1, 2, 3, 4, 100, 101, 102, ••• .
This is the number system we all learn first, and it is adequate
for keeping account of "how many".
In accounting procedures there is a need for negative numbers
(deficits) and zero. We thus create the system of integers (or
whole numbers):
3,-2,-1, 0, 1, 2, 3,
consisting of positive integers (natural numbers), negative inte­
gers, and zero. To handle negative quantities we must respect
certain rules:
Table 1
(-m)n = m(-n) = -(mn)
(-m) (-n) = mn
- (m + n) = -m - n
- (m - n) = n - m
In attempting to divide a number of objects equally among a
number of people (say 5 apples among 20 people) we are led to a
new kind of number: a "rational number" or "fraction". When
2 .
dividing m units equally among n people, each person gets ^ units.
The ratio (or fraction) ^ is called a rational number. Recall
that we never allow the denominator n to equal 0. In addition to
obeying all the rules established for the system of integers, the
rational number system follows these rules:
Table 2
Rule Explanation Examples
Both numerator and
denominator may be
ac a 6 3 34 2
multiplied or divi­
be = b 8 4 9 51 3
ded by the same
non-zero number.
Add or subtract
b a + b 4 7
= fractions with h
c c c 5 5
the same denomin­
ator by adding or
a_ b a-b 8 17 -9 3
= subtracting their
c c c 15 "" 15 15 "5
numerators,
To add or subtract 2 9 .8 17
{•
fractions with 3 12 12 12
c ad+ cb
different denomin­
d bd
ators, change them 5 4 5 8 3 1
to fractions with 6 3 6 " 6 " 6" 2
a c ad - cb
the same denomina­
b " d ~ bd
tor and then add -5 7 -15 14 -29
or subtract. 12 "" 18 ~ 36 " 36 36
To multiply frac­ 1 2 2 1
— •
a c ac tions, multiply 2 3 " 6 " 3
their numerators
b * d = bd
and multiply their 2 4 8 2
denominators. " 4 5 "20 5
5 . 1 5 2 .
— as — • — =s S
2 T 2 2 1^
a . c a d To divide one frac­
b " d b * c tion by another, 1 . 2 1 3 _ 3
invert the divisor 2 T 3 2*2 4
ad and multiply.
be
3 . , 31 —3
7 * ~ 7 # -4 " 28
3.
EXERCISE SET 1.1
1. 2- (3-5) 2. 5-3(-4)
3. 6+7(-8+6) 4. (8-5) (4-7)
5, 6. 1- [1- (1-2)]
7. 3-4(-l-3) 8. -[3- (11- 20)(-2)]
9 10.
LI 12.
13, 14.
15. 16.
17. 18,
19. 20.
21. 22.
23. 24
25. 26
27. Three people decide to rent a house together and contri­
bute to the rent according to their income. It is decided
1 3
that David will pay ^ and Jim will pay -j of the rent. What
fraction will the third person pay?
28. At the beginning of the first day of September, the price of
common stock in the Podunk Power Company was $21 per share.
Over the next 30 days the price changed as follows: it went
4.
up T (dollar) each day for the first 11 days, stayed the
4 1
same for 15 days, dropped -r each day for the next three
1 z
days, and went up 2 -g on the last day. What was the price
of Podunk stock at the close of September?
1.2 THE NUMBER LINE
Numbers can be represented geometrically by points on a
straight line. To start, take a line extending infinitely far
in both directions. Select a point on this line to serve as a
reference point, and call it the origin. Next, choose one di­
rection from the origin as the Positive direction and let the
other direction be called the negative direction; it is usual
to mark the positive direction with an arrowhead as shown in
Figure 1.1a. Then, select a unit of length for measuring dis­
tances. With each number, we can now associate a point on the
number line in the following way:
(a) With each positive number r associate the point that is a
distance of r units from the origin in the positive direc­
tion.
(b) With each negative number -r associate the point that is
a distance of r units from the origin in the negative di­
rection.
(c) Finally, associate the origin with the number 0.
-3-2-10 1 2 3
(a) (b)
Figure 1.1
In Figure lb we have marked on the line the points that are asso­
ciated with some of the integers.
5.
The number corresponding to a point on the number line is
called the coordinate of the point. In Figure 1.2 we have marked
the points whose coordinates are
1.25 , and
13 1 25
3 2 1.25 7
-M 1 1 1 • 1 1 1 "
- 4 - 3 - 2 - 10 1 2 3 4
Figure 1.2
To meet the requirements for physical measurement it is im­
portant that our number system provide a label for every point
along the line. Unfortunately, the rational number system does
not meet this requirement. Indeed, mathematicians of ancient
Greece discovered that some very common geometric quantities,
such as the diagonal of a one-by-one square, cannot be represented
by a rational number. In high school geometry we learn the Pyth­
agorean theorem:
.c
2.2 2
a ±+b = c
a
6.
This implies that the unit square has a diagonal whose length
must be /T. But it can be proved that /? is not a rational num­
ber. The number IT, frequently encountered in geometry, is also
not rational. Thus the rational number system is incomplete for
purposes of physical measurement.
The real number system provides one number for every point
along a straight line and includes all the rational numbers (and
hence all the integers); it fills all holes in the line left by
the rational number system. The real numbers that are not ration­
al are called irrational numbers.
The real number system obeys all the rules of arithmetic
satisfied by the rational number system, such as those described
in Tables 1 and 2.
EXERCISE SET 1.2
Locate each of the following numbers on a number line.
1. 2.
3. 4.
7.
5. Which of the following are natural numbers? Integers?
Rational numbers? Real numbers?
(a) (b) (c) (d)
(e) TT (f) (g) (h) -100
6. True or False .
(a) Every integer is a real number.
(b) Every point on the number line corresponds to a rational
number.
(c) ir has a location on the number line.
(d) -5 is a rational number.
(e) Every real number is either rational or irrational, but
not both.
1.3 DECIMAL REPRESENTATION
Decimals provide an alternative way of writing familiar
numbers; for example>
.25 = 2.375 -.003
We say that the decimal representation of T is .25, and so on.
1 3
Some decimals, like • .3333-«» and = .212121*** are nontermi-
nating, in the sense that they require infinitely many decimal
places for their complete expression.
The relevant fact here is that every real number has a deci­
mal re-presentation. Thus the decimals entirely fill the line,
leaving no holes. In fact, decimals and real numbers are synony­
mous. This fact is not obvious, but it can be proved.
If a nonterminating decimal has a block of one or more
digits that repeats, then that decimal is called a repeating deci-
maI. For example,