Download Chapter 1, Section 1 - Palm Beach State College

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Sec 1.1 - 1
Chapter 1
Review of the Real Number System
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Sec 1.1 - 2
1.1
Basic Concepts
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Sec 1.1 - 3
1.1 Basic Concepts
Objectives
1.
Write sets using set notation.
2.
Use number lines.
3.
Know the common sets of numbers.
4.
Find additive inverses.
5.
Use absolute value.
6.
Use inequality symbols.
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Sec 1.1 - 4
1.1 Basic Concepts
Write Sets Using Set Notation
A set is a collection of objects called the elements, or members, of the set.
• Set braces, { }, are used to enclose the elements.
• For example, 4 is an element of the set, {3, 4, 11, 19}.
• {3, 4, 11, 19} is an example of a finite set since we can
count the number of elements in the set.
• A set containing no elements is called the empty set, or
the null set, denoted by Ø.
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Sec 1.1 - 5
1.1 Basic Concepts
Examples of Sets
Certain sets of numbers have names:
 Natural numbers
 N = {1,2,3,4,5,6,...}
 Whole numbers
 W = {0,1,2,3,4,5,6,…}
 Empty set
 Ø (a set with no elements)
Note: N and W are infinite sets. The three
dots, called an ellipsis, mean “continue on in
the pattern that has been established.”
Caution:
Ø is the empty set; {Ø} is the set with one element, Ø.
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Sec 1.1 - 6
1.1 Basic Concepts
Set-Builder Notation
Sometimes instead of listing the elements of a set,
we use a notation called set-builder notation.
{x | x has property P }
the set of all elements x such that x has a given property P
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Sec 1.1 - 7
1.1 Basic Concepts
Listing the Elements in Sets
a) {x | x is a whole number less than 3}
The whole numbers less than 3 are 0, 1, and
2. This is the set {0, 1, 2}.
b) {x | x is one of the first five odd whole
numbers} = {1, 3, 5, 7, 9}.
c) {z | z is a whole number greater than 11}
This is an infinite set written with three dots as
{12, 13, 14, 15, … }.
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Sec 1.1 - 8
1.1 Basic Concepts
Using Set-Builder Notation to Describe Sets
a) { 0, 1, 2, 3 } can be described as
{m | m is one of the first four whole numbers}.
b) { 7, 14, 21, 28, … } can be described as
{s | s is a multiple of 7 greater than 0}.
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Sec 1.1 - 9
1.1 Basic Concepts
Using Number Lines
A number line is a way to picture a set
of numbers:
–5 –4
–3 –2 –1
0
Negative numbers
1
2
3
4
5
Positive numbers
0 is neither positive
nor negative
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Sec 1.1 - 10
1.1 Basic Concepts
Using Number Lines
The set of numbers identified on this
number line is the set of integers:
I = {…,–3, –2, –1, 0, 1, 2, 3, …}
Graph of 3
–5 –4
–3 –2 –1
0
1
2
3
4
5
Each number on the number line is called a coordinate
of the point it labels.
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Sec 1.1 - 11
1.1 Basic Concepts
Rational Numbers
Rational numbers can be expressed as the
quotient of two integers, with a denominator
that is not 0.
The set of all rational numbers is written:
p

 p and q are integers, q  0 .
q

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Rational
numbers
Sec 1.1 - 12
1.1 Basic Concepts
Rational Numbers
Rational numbers can be written in decimal
form as:
Terminating decimals:
4
14
= .8 and
= 2.8
5
5
Repeating decimals:
Bar means repeating digits.
2
4
 0.66666...  0.66 or
 0.363636...  0.36
3
11
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Sec 1.1 - 13
1.1 Basic Concepts
Irrational Numbers
Irrational numbers have decimals that
neither terminate nor repeat:
3  1.7320508...
 11  3.316624...
  3.141592...
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Sec 1.1 - 14
1.1 Basic Concepts
Graphs of Rational and Irrational Numbers
Irrational
Numbers
 11
–5 –4
Rational
Numbers
3
–3 –2 –1
3

2
0
1
2
p
3
4
5
0.36
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Sec 1.1 - 15
1.1 Basic Concepts
Real Numbers
Rational numbers
4 5
, - , 0.6, 1.75
9 8
Integers
11,  6,  4
Irrational numbers
 8,
15,  ,

4
Whole numbers
0
Natural
numbers
1, 2, 3, 4,
5, 27, 45
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Sec 1.1 - 16
1.1 Basic Concepts
Relationships Between Sets of Numbers
Irrational numbers
Positive integers
Real
numbers
Integers
Rational
numbers
Zero
Negative integers
Noninteger rational numbers
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Sec 1.1 - 17
1.1 Basic Concepts
Sets of Numbers
{1, 2, 3, 4, 5, 6, … }
Natural numbers or
{0, 1, 2, 3, 4, 5, 6, … }
Whole numbers
{…,–3, –2, –1, 0, 1, 2, 3, … }
Integers
Rational numbers
p

 p and q are integers, q  0
q

Irrational numbers
x | x is a real number that is not rational
Real numbers
x | x is represented by a point on the number line
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Sec 1.1 - 18
1.1 Basic Concepts
Relationships Between Sets of Numbers
Decide whether each statement is true or false.
a) All natural numbers are integers.
b) Zero is an irrational number.
True
False
c) Every integer is a rational number.
True
d) The square root of 9 is an irrational number.
p
e) - 3 is an irrational number.
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False
False
Sec 1.1 - 19
1.1 Basic Concepts
Additive Inverse
Additive Inverse
For any real number a, the number –a is the additive inverse of a.
–4 units from zero
–5 –4
–3 –2 –1
4 units from zero
0
1
2
3
4
5
The number –4 is the additive inverse of 4, and the
number 4 is the additive inverse of –4.
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Sec 1.1 - 20
1.1 Basic Concepts
The Minus Sign
The symbol “−” can be used to indicate
any of the following:
1. a negative number, such as –13 or –121;
2. the additive inverse of a number, as in
“ –7 is the additive inverse of 7”.
3. subtraction, as in 19 – 7.
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Sec 1.1 - 21
1.1 Basic Concepts
Signed Numbers / Additive Inverses
• The sum of a number and its additive
inverse is always zero.
4 + (–4) = 0
or
–16 + 16 = 0
• For any real number a,
–(–a) = a.
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Sec 1.1 - 22
1.1 Basic Concepts
Absolute Value
Geometrically, the absolute value of a number,
a, written |a| is the distance on the number line
from 0 to a.
Distance is 4,
so |–4| = 4.
–5 –4
–3 –2 –1
Distance is 4,
so |4| = 4.
0
1
2
3
4
5
Absolute value is always positive.
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Sec 1.1 - 23
1.1 Basic Concepts
Formal Definition of Absolute Value
if a is positive or zero
a
a 
a if a is negative
Evaluate the following absolute value expressions:
|–14|
–|9|
|0|
|14| + |–7|
–|6–3|
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–|–13|
–|8–8|
Sec 1.1 - 24
1.1 Basic Concepts
Equality vs. Inequality
 An equation is a statement that two quantities are
equal.
8 + 3 = 11
19 – 12 = 7
 An inequality is a statement that two quantities
are not equal. One must be less than the other.
9 < 12
–7 > – 10
This means that 9 is less than 12.
This means that –7 is greater than –10.
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Sec 1.1 - 25
1.1 Basic Concepts
Inequalities on the Number Line
On the number line, a < b if a is to the left of b;
a > b if a is to the right of b.
–2 < 3
–5 –4
–3 –2 –1
0
1 > –4
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1
2
3
4
5
The inequality symbol
always points to the
smaller number.
Sec 1.1 - 26
1.1 Basic Concepts
Inequality Symbols
Symbol
Meaning
Example

is not equal to
–6  10
<
is less than
–9 < –3
>
is greater than
8 > –2

is less than or equal to
–8  –8

is greater than or equal to
–2  –7
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Sec 1.1 - 27