, Chapter
1 PRELIMINARIES
OVERVIEW This chapter reviews the basic ideas you need to start calculus. The topics in-
clude the real number system, Cartesian coordinates in the plane, straight lines, parabolas,
circles, functions, and trigonometry. We also discuss the use of graphing calculators and
computer graphing software.
1.1 Real Numbers and the Real Line
This section reviews real numbers, inequalities, intervals, and absolute values.
Real Numbers
Much of calculus is based on properties of the real number system. Real numbers are
numbers that can be expressed as decimals, such as
3
- = - 0.75000 Á
4
1
= 0.33333 Á
3
22 = 1.4142 Á
The dots Á in each case indicate that the sequence of decimal digits goes on forever.
Every conceivable decimal expansion represents a real number, although some numbers
have two representations. For instance, the infinite decimals .999 Á and 1.000 Á repre-
sent the same real number 1. A similar statement holds for any number with an infinite tail
of 9’s.
The real numbers can be represented geometrically as points on a number line called
the real line.
–2 –1 – 3 0 1 1 兹2 2 3 4
4 3
The symbol denotes either the real number system or, equivalently, the real line.
The properties of the real number system fall into three categories: algebraic proper-
ties, order properties, and completeness. The algebraic properties say that the real num-
bers can be added, subtracted, multiplied, and divided (except by 0) to produce more real
numbers under the usual rules of arithmetic. You can never divide by 0.
1
,2 Chapter 1: Preliminaries
The order properties of real numbers are given in Appendix 4. The following useful
rules can be derived from them, where the symbol Q means “implies.”
Rules for Inequalities
If a, b, and c are real numbers, then:
1. a 6 b Q a + c 6 b + c
2. a 6 b Q a - c 6 b - c
3. a 6 b and c 7 0 Q ac 6 bc
4. a 6 b and c 6 0 Q bc 6 ac
Special case: a 6 b Q -b 6 - a
1
5. a 7 0 Q a 7 0
1 1
6. If a and b are both positive or both negative, then a 6 b Q 6 a
b
Notice the rules for multiplying an inequality by a number. Multiplying by a positive num-
ber preserves the inequality; multiplying by a negative number reverses the inequality.
Also, reciprocation reverses the inequality for numbers of the same sign. For example,
2 6 5 but - 2 7 - 5 and 1>2 7 1>5.
The completeness property of the real number system is deeper and harder to define
precisely. However, the property is essential to the idea of a limit (Chapter 2). Roughly
speaking, it says that there are enough real numbers to “complete” the real number line, in
the sense that there are no “holes” or “gaps” in it. Many theorems of calculus would fail if
the real number system were not complete. The topic is best saved for a more advanced
course, but Appendix 4 hints about what is involved and how the real numbers are con-
structed.
We distinguish three special subsets of real numbers.
1. The natural numbers, namely 1, 2, 3, 4, Á
2. The integers, namely 0, ; 1, ;2, ; 3, Á
3. The rational numbers, namely the numbers that can be expressed in the form of a
fraction m>n, where m and n are integers and n Z 0. Examples are
1 4 -4 4 200 57
, - = = , , and 57 = .
3 9 9 -9 13 1
The rational numbers are precisely the real numbers with decimal expansions that are
either
(a) terminating (ending in an infinite string of zeros), for example,
3
= 0.75000 Á = 0.75 or
4
(b) eventually repeating (ending with a block of digits that repeats over and over), for
example
The bar indicates the
23
= 2.090909 Á = 2.09 block of repeating
11 digits.
, 1.1 Real Numbers and the Real Line 3
A terminating decimal expansion is a special type of repeating decimal since the ending
zeros repeat.
The set of rational numbers has all the algebraic and order properties of the real num-
bers but lacks the completeness property. For example, there is no rational number whose
square is 2; there is a “hole” in the rational line where 22 should be.
Real numbers that are not rational are called irrational numbers. They are character-
ized by having nonterminating and nonrepeating decimal expansions. Examples are
p, 22, 2 3
5, and log10 3. Since every decimal expansion represents a real number, it
should be clear that there are infinitely many irrational numbers. Both rational and irra-
tional numbers are found arbitrarily close to any point on the real line.
Set notation is very useful for specifying a particular subset of real numbers. A set is a
collection of objects, and these objects are the elements of the set. If S is a set, the notation
a H S means that a is an element of S, and a x S means that a is not an element of S. If S
and T are sets, then S ´ T is their union and consists of all elements belonging either to S
or T (or to both S and T). The intersection S ¨ T consists of all elements belonging to both
S and T. The empty set ¤ is the set that contains no elements. For example, the intersec-
tion of the rational numbers and the irrational numbers is the empty set.
Some sets can be described by listing their elements in braces. For instance, the set A
consisting of the natural numbers (or positive integers) less than 6 can be expressed as
A = 51, 2, 3, 4, 56.
The entire set of integers is written as
50, ;1, ; 2, ;3, Á 6.
Another way to describe a set is to enclose in braces a rule that generates all the ele-
ments of the set. For instance, the set
A = 5x ƒ x is an integer and 0 6 x 6 66
is the set of positive integers less than 6.
Intervals
A subset of the real line is called an interval if it contains at least two numbers and con-
tains all the real numbers lying between any two of its elements. For example, the set of all
real numbers x such that x 7 6 is an interval, as is the set of all x such that -2 … x … 5.
The set of all nonzero real numbers is not an interval; since 0 is absent, the set fails to con-
tain every real number between -1 and 1 (for example).
Geometrically, intervals correspond to rays and line segments on the real line, along
with the real line itself. Intervals of numbers corresponding to line segments are finite in-
tervals; intervals corresponding to rays and the real line are infinite intervals.
A finite interval is said to be closed if it contains both of its endpoints, half-open if it
contains one endpoint but not the other, and open if it contains neither endpoint. The end-
points are also called boundary points; they make up the interval’s boundary. The re-
maining points of the interval are interior points and together comprise the interval’s in-
terior. Infinite intervals are closed if they contain a finite endpoint, and open otherwise.
The entire real line is an infinite interval that is both open and closed.
Solving Inequalities
The process of finding the interval or intervals of numbers that satisfy an inequality in x is
called solving the inequality.
1 PRELIMINARIES
OVERVIEW This chapter reviews the basic ideas you need to start calculus. The topics in-
clude the real number system, Cartesian coordinates in the plane, straight lines, parabolas,
circles, functions, and trigonometry. We also discuss the use of graphing calculators and
computer graphing software.
1.1 Real Numbers and the Real Line
This section reviews real numbers, inequalities, intervals, and absolute values.
Real Numbers
Much of calculus is based on properties of the real number system. Real numbers are
numbers that can be expressed as decimals, such as
3
- = - 0.75000 Á
4
1
= 0.33333 Á
3
22 = 1.4142 Á
The dots Á in each case indicate that the sequence of decimal digits goes on forever.
Every conceivable decimal expansion represents a real number, although some numbers
have two representations. For instance, the infinite decimals .999 Á and 1.000 Á repre-
sent the same real number 1. A similar statement holds for any number with an infinite tail
of 9’s.
The real numbers can be represented geometrically as points on a number line called
the real line.
–2 –1 – 3 0 1 1 兹2 2 3 4
4 3
The symbol denotes either the real number system or, equivalently, the real line.
The properties of the real number system fall into three categories: algebraic proper-
ties, order properties, and completeness. The algebraic properties say that the real num-
bers can be added, subtracted, multiplied, and divided (except by 0) to produce more real
numbers under the usual rules of arithmetic. You can never divide by 0.
1
,2 Chapter 1: Preliminaries
The order properties of real numbers are given in Appendix 4. The following useful
rules can be derived from them, where the symbol Q means “implies.”
Rules for Inequalities
If a, b, and c are real numbers, then:
1. a 6 b Q a + c 6 b + c
2. a 6 b Q a - c 6 b - c
3. a 6 b and c 7 0 Q ac 6 bc
4. a 6 b and c 6 0 Q bc 6 ac
Special case: a 6 b Q -b 6 - a
1
5. a 7 0 Q a 7 0
1 1
6. If a and b are both positive or both negative, then a 6 b Q 6 a
b
Notice the rules for multiplying an inequality by a number. Multiplying by a positive num-
ber preserves the inequality; multiplying by a negative number reverses the inequality.
Also, reciprocation reverses the inequality for numbers of the same sign. For example,
2 6 5 but - 2 7 - 5 and 1>2 7 1>5.
The completeness property of the real number system is deeper and harder to define
precisely. However, the property is essential to the idea of a limit (Chapter 2). Roughly
speaking, it says that there are enough real numbers to “complete” the real number line, in
the sense that there are no “holes” or “gaps” in it. Many theorems of calculus would fail if
the real number system were not complete. The topic is best saved for a more advanced
course, but Appendix 4 hints about what is involved and how the real numbers are con-
structed.
We distinguish three special subsets of real numbers.
1. The natural numbers, namely 1, 2, 3, 4, Á
2. The integers, namely 0, ; 1, ;2, ; 3, Á
3. The rational numbers, namely the numbers that can be expressed in the form of a
fraction m>n, where m and n are integers and n Z 0. Examples are
1 4 -4 4 200 57
, - = = , , and 57 = .
3 9 9 -9 13 1
The rational numbers are precisely the real numbers with decimal expansions that are
either
(a) terminating (ending in an infinite string of zeros), for example,
3
= 0.75000 Á = 0.75 or
4
(b) eventually repeating (ending with a block of digits that repeats over and over), for
example
The bar indicates the
23
= 2.090909 Á = 2.09 block of repeating
11 digits.
, 1.1 Real Numbers and the Real Line 3
A terminating decimal expansion is a special type of repeating decimal since the ending
zeros repeat.
The set of rational numbers has all the algebraic and order properties of the real num-
bers but lacks the completeness property. For example, there is no rational number whose
square is 2; there is a “hole” in the rational line where 22 should be.
Real numbers that are not rational are called irrational numbers. They are character-
ized by having nonterminating and nonrepeating decimal expansions. Examples are
p, 22, 2 3
5, and log10 3. Since every decimal expansion represents a real number, it
should be clear that there are infinitely many irrational numbers. Both rational and irra-
tional numbers are found arbitrarily close to any point on the real line.
Set notation is very useful for specifying a particular subset of real numbers. A set is a
collection of objects, and these objects are the elements of the set. If S is a set, the notation
a H S means that a is an element of S, and a x S means that a is not an element of S. If S
and T are sets, then S ´ T is their union and consists of all elements belonging either to S
or T (or to both S and T). The intersection S ¨ T consists of all elements belonging to both
S and T. The empty set ¤ is the set that contains no elements. For example, the intersec-
tion of the rational numbers and the irrational numbers is the empty set.
Some sets can be described by listing their elements in braces. For instance, the set A
consisting of the natural numbers (or positive integers) less than 6 can be expressed as
A = 51, 2, 3, 4, 56.
The entire set of integers is written as
50, ;1, ; 2, ;3, Á 6.
Another way to describe a set is to enclose in braces a rule that generates all the ele-
ments of the set. For instance, the set
A = 5x ƒ x is an integer and 0 6 x 6 66
is the set of positive integers less than 6.
Intervals
A subset of the real line is called an interval if it contains at least two numbers and con-
tains all the real numbers lying between any two of its elements. For example, the set of all
real numbers x such that x 7 6 is an interval, as is the set of all x such that -2 … x … 5.
The set of all nonzero real numbers is not an interval; since 0 is absent, the set fails to con-
tain every real number between -1 and 1 (for example).
Geometrically, intervals correspond to rays and line segments on the real line, along
with the real line itself. Intervals of numbers corresponding to line segments are finite in-
tervals; intervals corresponding to rays and the real line are infinite intervals.
A finite interval is said to be closed if it contains both of its endpoints, half-open if it
contains one endpoint but not the other, and open if it contains neither endpoint. The end-
points are also called boundary points; they make up the interval’s boundary. The re-
maining points of the interval are interior points and together comprise the interval’s in-
terior. Infinite intervals are closed if they contain a finite endpoint, and open otherwise.
The entire real line is an infinite interval that is both open and closed.
Solving Inequalities
The process of finding the interval or intervals of numbers that satisfy an inequality in x is
called solving the inequality.
