Transcript Real Numbers and Their Graphs

Real Numbers
and Their Basic Properties
Copyright © Cengage Learning. All rights reserved.
1
Section
1.1
Real Numbers and Their Graphs
Copyright © Cengage Learning. All rights reserved.
What Is a Number?
Can animals count?
Do animals understand numbers?
Imagine 2 people; 2 cars; 2 books; 2 chocolate bars.
Now, imagine 2.
Numbers and Numerals
3
Objectives
1
List the numbers in a set of real numbers that
are natural, whole, integers, rational,
irrational, composite, prime, even, or odd.
2
Relate two rational numbers with a symbol <,
>, or = .
3
Graph a real number or a subset of real
numbers on the number line.
4
Find the absolute value of a real number.
4
Set of Natural Numbers
A set is a collection of objects.
For example, the set
{1, 2, 3, 4, 5}
contains the elements 1, 2, 3, 4, and 5.
Two basic sets of numbers
•natural numbers (often called the positive integers)
•whole numbers.
The Set of Natural Numbers (Positive Integers)
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
5
Set of Whole Numbers
The Set of Whole Numbers
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
Ellipses: … to indicate that each list of numbers continues
on forever.
Whole numbers to represent:
•30 miles per gallon
•$16,500 price for a used car
6
Set of Negative Numbers
N = {-1, -2, -3, -4, …}
Negative numbers to represent
•-$15,000 for a debt
•-20 for temperature below zero degree
The negatives of the natural numbers and the whole
numbers together form the set of integers.
7
Set of Integers
The Set of Integers
I = {. . . , –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, . . .}
Subsets:
•W = {0, 1, 2, 3, 4, 5, …} is a Subset of set I
•N = {-1, -2, -3, -4, -5, …} is Subset of set I
B is a subset of A means:
•Every element of B is an element of A.
8
Set of Rational Numbers
R = {1/2, 5/18, -1/3, 21/37, etc}
Rational Numbers: can be expressed as a ratio of two
whole numbers
• Integers
• Fractions, that have an integer numerator
and a nonzero integer denominator.
Set-Builder Notation
R=
Read: R is a set of a/b such that a is an integer and b is a nonzero integer.
9
Examples of Rational Numbers
Some examples of rational numbers are
•0.25 = ¼
•0.125 = 1/8
•0.66666… = 2/3
•0.94444…=17/18
•5 = 5/1
(repeating decimals)
(integer)
but not
•3.141592653589793238462643383279502884197169399
37510…
(irrational)
10
Irrational Numbers
  3.141592654
Using a scientific calculator, press . Using a
graphing calculator, press
.
Read  as “is approximately equal to.”
 1.414213562
Using a scientific calculator, press 2
Using a graphing calculator, press
.
.
If we combine the rational and the irrational numbers, we
have the set of real numbers.
11
Set of Real Numbers
The Set of Real Numbers
R = {x | x is either a rational number or an irrational number.}
12
Real Number System
Figure 1-1
13
Your Turn 1
Which numbers in the set {–3, 0, , 1.25,
a. natural numbers
c. negative integers
e. irrational numbers
, 5} are
b. whole numbers
d. rational numbers
f. real numbers?
Solution:
a. The only natural number is 5.
b. The whole numbers are 0 and 5.
c. The only negative integer is –3.
14
Example 1 – Solution
cont’d
d. The rational numbers are –3, 0, , 1.25, and 5. (1.25 is
rational, because 1.25 can be written in the form
.)
e. The only irrational number is
.
f. All of the numbers are real numbers.
15
Prime Numbers
Prime Number: A natural number greater than 1 that can
be divided evenly only by 1 and itself
The set of prime numbers:
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, . . .}
A nonprime natural number greater than 1 is called a
composite number.
The set of composite numbers:
{4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, . . .}
16
Even and Odd Numbers
An integer that can be divided evenly by 2 is called an
even integer. An integer that cannot be divided evenly by
2 is called an odd integer.
The set of even integers:
{. . . , –10, –8, –6, –4, –2, 0, 2, 4, 6, 8, 10, . . .}
The set of odd integers:
{. . . , –9, –7, –5, –3, –1, 1, 3, 5, 7, 9, . . .}
17
Equality Symbol
To show that two expressions represent the same number,
we use an = sign. Since 4 + 5 and 9 represent the same
number, we can write
4+5=9
5–3=2
4  5 = 20
30  6 = 5
18
In Equality Symbols
We can use inequality symbols to show that expressions
are not equal.
19
Example – Inequality symbols
a.   3.14
Read as “pi is approximately equal to 3.14.”
b. 6  9
Read as “6 is not equal to 9.”
c. 8  10
Read as “8 is less than 10.”
d. 12  1
Read as “12 is greater than 1.”
e. 5  5
Read as “5 is less than or equal to 5.”
(Since 5 = 5, this is a true statement.)
f. 9  7
Read as “9 is greater than or equal to 7.”
(Since 9 > 7, this is a true statement.)
20
Insert a symbol , , or = to define the relationship
between two rational numbers
Inequality statements can be written so that the inequality
symbol points in the opposite direction.
For example,
5  7 and 7  5
both indicate that 5 is less than 7. Likewise,
12  3 and 3  12
both indicate that 12 is greater than or equal to 3.
21
Insert a symbol , , or = to define the relationship
between two rational numbers
In algebra, we use letters, called variables, to represent
real numbers. For example,
• If x represents 4, then x = 4.
• If y represents any number greater than 3, then y  3.
• If z represents any number less than or equal to –4, then
z  –4.
22
3.
Graph a real number or a subset
of real numbers on the number line
23
Number Line
We can use the number line shown in Figure 1-2 to
represent sets of numbers.
Figure 1-2
The number line continues forever to the left and to the
right. Numbers to the left of 0 (the origin) are negative, and
numbers to the right of 0 are positive.
The number that corresponds to a point on the number line
is called the coordinate of that point. For example, the
coordinate of the origin is 0.
24
Number Line
Comment
The number 0 is neither positive nor negative.
Many points on the number line do not have integer
coordinates.
For example, the point midway between 0 and 1 has the
coordinate , and the point midway between –3 and –2 has
the coordinate
(see Figure 1-3).
Figure 1-3
25
Number Line
Numbers represented by points that lie on opposite sides of
the origin and at equal distances from the origin are called
negatives (or opposites) of each other.
For example, 5 and –5 are negatives (or opposites).
We need parentheses to express the opposite of a
negative number. For example –(–5), represents the
opposite of –5, which we know to be 5. Thus,
–(–5) = 5
26
Number Line
Double Negative Rule
If x represents a real number, then
–(–x) = x
If one point lies to the right of a second point on a number
line, its coordinate is the greater.
Since the point with coordinate 1 lies to the right of the
point with coordinate –2 (see Figure 1-4(a)), it follows that
1  –2.
Figure 1-4(a)
27
Graph a real number or a subset of real numbers
on the number line
If one point lies to the left of another, its coordinate is the
smaller (see Figure 1-4(b)).
The point with coordinate –6 lies to the left of the point with
coordinate –3 so it follows that –6  –3.
Figure 1-4(b)
28
Graph a real number or a subset of real numbers
on the number line
Figure 1-5 shows the graph of the natural numbers from
2 to 8. The points on the line are called graphs of their
corresponding coordinates.
Figure 1-5
29
Example 4
Graph the set of integers between –3 and 3.
Solution:
The integers between –3 and 3 are –2, –1, 0, 1, and 2.
The graph is shown in Figure 1-6.
Figure 1-6
30
Graph a real number or a subset of real numbers
on the number line
Graphs of many sets of real numbers are intervals on the
number line.
For example, two graphs of all real numbers x such that
x  –2 are shown in Figure 1-7.
Figure 1-7
31
Graph a real number or a subset of real numbers
on the number line
The parenthesis and the open circle at –2 show that this
point is not included in the graph.
The arrow pointing to the right shows that all numbers to
the right of –2 are included.
Figure 1-8 shows two graphs of the set of real numbers
x between –2 and 4.
Figure 1-8
32
Graph a real number or a subset of real numbers
on the number line
This is the graph of all real numbers x such that x  –2 and
x  4. The parentheses or open circles at –2 and 4 show
that these points are not included in the graph.
However, all the numbers between –2 and 4 are included.
33
4.
Find the absolute value of a
real number
34
Find the absolute value of a real number
On a number line, the distance between a number x and 0
is called the absolute value of x.
For example, the distance between 5 and 0 is 5 units
(see Figure 1-11). Thus, the absolute value of 5 is 5:
|5| = 5
Read as “The absolute value of 5 is 5.”
Figure 1-11
35
Find the absolute value of a real number
Since the distance between –6 and 0 is 6,
|–6| = 6
Read as “The absolute value of –6 is 6.”
Because the absolute value of a real number represents
that number’s distance from 0 on the number line, the
absolute value of every real number x is either positive
or 0.
In symbols, we say
|x|  0
for every real number x
36
Example 7
Evaluate: a. |6| b. |–3| c. |0| d. –|2 + 3|
Solution:
a. |6| = 6, because 6 is six units from 0.
b. |–3| = 3, because –3 is three units from 0.
c. |0| = 0, because 0 is zero units from 0.
d. –| 2 + 3| = –|5| = –5, because the opposite of the
absolute value of 5 is –5.
37