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Chapter 1
Set of Real Numbers
1.1 Sets of Numbers and the Real
Number Line
A set is a collection of objects. We are
interested in different sets of numbers.
Notation: the set containing 1, 3, and 5
is written
{1, 3, 5}
Natural numbers {1, 2, 3, . . .}
Whole numbers {0, 1, 2, 3, . . .}
Note that to get the set of whole
numbers, we took the set of natural
numbers and included 0. Thus every
natural number is also in the set of whole
numbers.
We say that the set of natural numbers is
contained in or is a subset of the whole
numbers.
Integers {. . . –3, –2, –1, 0, 1, 2, 3, . . .}
Note that to get the set of integers, we
took the set of whole numbers and
included the negatives of each of them.
The set of whole numbers is a subset of
the set of integers.
Rational numbers: Any number that
can be put in the form a where a and b
b
are integers with b ≠ 0.
Are the following rational numbers?
2
3
5
-3
0
Irrational numbers: Any real number
that is not rational. That is, it cannot be
a
b
put in the form .
Examples
2
π
When rational and irrational numbers are
written as decimal numbers, we see this
difference:
1) When a rational number is written as
a decimal number, it will either
terminate or have a repeating block of
digits.
2) When an irrational number is written
as a decimal number, it will continue
forever without a repeating block of
digits.
Examples
Rational:
1
 0.25
4
1
 0.33333 3
3
1
 0.142857142857
7
Irrational:
π = 3.14159265359. . .
5  2.2360679775
Real numbers: The set of rational
numbers combined with the set of
irrational numbers.
Example
Classify: 0.569
Yes
Rational
Real
___
No____________
Natural
Whole
Integer
Irrational
Classify: 2
Yes
___
Irrational
Real
No____________
Natural
Whole
Integer
Rational
All the numbers we will be using will be
real numbers. An example of a nonreal
number would be  2 .
How do the sets of numbers fit together?
Real numbers:
Rational numbers
Integers
Whole
numbers Natural
numbers
Irrational numbers
The set of real numbers can also be
defined as all numbers corresponding to
points on the number line. Thus, the real
numbers are “ordered”.
A number a is less than the number b if a
is to the left of b on the number line.
Order symbols
<
less than:
4 < 20
≤ less than or equal to:
4 ≤ 20
5 ≤ 5 is true but 5 < 5 is not true
>
greater than:
16 > 5
≥ greater than or equal to: 16 ≥ 5
16 ≥ 16 is true but 16 > 16 is not
≠
not equal to
5 ≠7
Put the appropriate sign(s) between the
two numbers:
5
87
Signs that work: ≠
<
12
12
Signs that work: =
≤
≥
-8
- 12
Signs that work: >
≥
≠
≤
Note that the inequality signs always
“point to” the smallest number.
The absolute value of a number is its
distance from 0 on the number line.
Note that this definition says nothing
about the direction, only the distance.
Notation: The absolute value of a: | a |
| |
| | |
-4 -3 -2 -1 0
|
1
|
2
|
|
3 4
| - 3| = | 3 | = 3
since 3 and – 3 are both 3 steps from 0
Note that since absolute value is a
distance, | a | ≥ 0 for any number a.
Two numbers are opposites if they have
the same absolute value but lie in
opposite directions from 0 on the number
line.
−3 and 3 are opposites.
The sum of opposites is 0.
Example
Find the opposite of 2.
The opposite of 2 is -2.
Find the opposite of -3.
The opposite of -3 is 3
-(-3) = 3
* KEY concept that will be used later!!!*
| 32| = 32
| - 45| = 45
|0|=0
−|−6|=6
1.2 Order of Operations
A variable is a letter that is used to hold
the place of a number. It is called a
variable because it can take on different
values.
Example
Suppose your car gets 20 miles per
gallon of gas. We can express how far
you can drive your car as 20n where n
represents how much gas you have.
The 20 is called a constant since its
value does not change.
20n is called an algebraic expression.
When we put in some number for n and
determine the distance you can drive, we
are evaluating the expression.
An algebraic expression consists of
variables and/or numbers combined with
operation symbols and/or grouping
symbols.
Example
x + 25
3(n – 4)
2x2 – 6x
28 ÷ 3y
An equation is a mathematical statement
that sets two expressions as being equal.
Example
3(n – 4) = 32
Note the difference between expressions
and equations.
Example
evaluate 2(x – 3y) for x = 10 and y = 2
2(10 − 3·2) = 2(10 – 6) = 2(4) = 8
Exponents
Exponent
23
Base
The exponent tells how many times the
base is used as a factor.
Thus, 23 = 2 · 2 · 2
Example
24
2
 
3
3
32
(− 3)2
81
− 32
The square root of a number is the
number we would have to square to get
that number. The radical sign
is
used to find the principal square root
(the positive root).
The square root is the inverse operation
to squaring a number.
25 = 5 since 52 = 25
Example
16
100
64
0
4
Order of Operations
1. Grouping Symbols: Do all
operations within grouping symbols.
Grouping symbols include: parenthesis,
brackets, braces, radicals, absolute value
bars and fraction bars.
2. Exponents, Radicals and Absolute
Values: Evaluate all powers, radicals
and absolute values.
3. Multiplications and Divisions:
Perform multiplication and division from
left to right in the order they appear in
the problem.
4. Additions and Subtractions:
Perform addition and subtraction from
left to right in the order they appear in
the problem.
Example
Simplify: – 2 + [(8 – 11) – (– 2 – 9)]
– 2 + [– 3 – (– 11)]
– 2 + [– 3 + 11]
–2+8=6
Simplify: – 3 + [(– 2 – 5) – 2]
– 3 + [– 7 – 2]
– 3 + (– 9) = – 12
Simplify: 2[5 + 2(8 – 3)]
2[5 + 2(5)]
2[5 + 10]
2[15] = 30
Simplify: 3(10)2  8  22
= 3(100) – 8 ÷ 4
= 300 – 8 ÷ 4
= 300 – 2
= 298
Simplify:
6  2  8  32
18  3
6  6  32

18  3
669

18  3
21

15
7

5
In order to apply math to real life
situations, we need to be able to translate
English phrases and sentences into
algebraic expressions and equations.
Words for addition: add, the sum of,
plus, more than, increased by
Words for subtraction: subtract, minus,
the difference of, less than, decreased by
Words for multiplication: multiply, the
product of, times, twice, of (with a
fraction, like ½ of 8 means ½ times 8)
Word for division: divide, the quotient
of, ratio of, per
Words for variables: For a phrase like
“some number”, “a number”, “what
number”, we use a variable.
Words for equal: is, equals, results in
Example
The sum of 2 and 3:
2+3
The product of 6 and a number:
6n
The quotient of 24 and 4:
24 ÷ 4
2 times the sum of a number and 5:
2(x + 5)
The sum of 2 times a number and 5:
2x + 5
8 subtracted from twice a number:
2n – 8
The difference of 8 and twice a number:
8 – 2n
The quotient of 18 and a number is 9:
18 ÷ n = 9
5 subtracted from 3 times a number is
equal to 21:
3x – 5 = 21
Section 1.3 Addition of Real Numbers
Adding real numbers can be visualized
on the number line. To add a positive
number, move right on the number line
and to add a negative number, move left
on the number line.
To add numbers with the same sign:
1. Add their absolute values
2. Give the answer the common sign.
To add numbers with different signs:
1. Subtract the smaller absolute value
from the larger
2. Give the answer the sign of the larger.
Note: We can think of – as meaning
“negative”, “minus” or “the opposite of”
Section 1.4 Subtraction of Real Numbers
The opposite or additive inverse of a
number a is written − a.
Subtracting a number is the same as
adding its opposite.
1. Change the subtraction symbol to the
additional symbol
2. Change the sign of the number being
subtracted
3. Add using the rules of addition
So 5 – 7 is the same as 5 + (-7)
6 – (-2) is the same as 6 + 2
Section 1.5 Multiplication and Division
of Real Numbers
Multiplying and Dividing Real Numbers
1. Multiply, divide and absolute values.
2. If the two numbers have the same
sign, the answer is positive.
If the two numbers have different
signs, the answer is negative.
Notation for multiplication:
3·2
3x2 (3)2 3(2)
(3)(2)
Notation for division:
25
5
25  5
5 25
Zero property
a·0=0
0 ÷ a = 0 for any real
number a.
Division by 0 is undefined
Example
− 9 · (− 2)
−3 · (− 5) · (− 2) · (−1)
(− 1)(− 1)(− 1)(− 1)(− 1)(− 1)(− 1)
 28 28
28
=
=
7
7
7
Property:
a a
a


b
b
b
 50
=
5
50
5
Property:
a a

b b
Section 1.6 Properties of Real Numbers
and Simplifying Expressions
Expressions that represent the same
number are called equivalent
expressions. The expressions 3 + 8
and 8 + 3 are equivalent expressions.
To solve equations, we use mathematical
laws and properties to write equivalent
expressions. For the following
properties, we let a, b, c represent any
real number.
The Commutative Properties
For addition: a + b = b + a
For multiplication: ab = ba
When we add or multiply numbers, the
order of the numbers does not matter.
The Associative Properties
For addition: a + (b + c) = (a + b) + c
For multiplication: a(bc) = (ab)c
The associative property allows us to
change the grouping of numbers when
we add or multiply.
The Distributive Property
a(b + c) = ab + ac
a(b – c) = ab – ac
Identities
The additive identity is 0. For any real
number a: a  0  a
The multiplicative identity is 1.
any real number a: a 1  a
For
Inverse Properties
Inverse property of addition:
a  (a)  0
Inverse property of multiplication:
1
b 1
b
Combining like terms
A term is a number or the product of
numbers and variables.
Examples 3
5x
7xy
2x3
When two terms have the same variable
parts, the terms are called similar terms
or like terms.
The distributive property helps us
combine similar terms.
2x + 3x = (2 + 3)x
like 2 marbles + 3 marbles give us 5
marbles
7ab – 4ab = 3ab
What about 4x + 7x2?
They are not like terms, so they cannot
be combined.
Examples
3(2x − 5) + 3x
5 + 7y(2 − 3y) − 2y(4y − 3)
5 – (2y – 8)
4a + 2a2 - 6a + 5 + 3a2 - 2
9k – 6 – 3(2 – 5k)