1.3 FRACTIONS
REVIEW
Variables-letters that represent numbers
examples: x, y, z, a, b, c
Multiplication-can be shown many ways
examples: ab ab a(b) (a)b
Factors-numbers (or variables) multiplied
together to get a product
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Fractions
a/b
Where a is the numerator (top) and b is the
denominator (bottom)
The fraction bar means to divide
A fraction is said to be reduced or simplified or in
lowest terms when the numerator and
denominator have no common factors except
one.
To simplify a fraction, find the GCF and divide both
numerator and denominator by that number
GCF is the biggest number that goes into the top
and bottom
(see appendix B for more info on GCF)
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a nice trick to remember
Note: you can only cross cancel across a
multiplication sign-never do this across an add,
subtract, or division sign.
Below you can cross cancel the two’s and then
multiply. (like reducing before you multiply)
3 2

2 5
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Forms of fractions
Proper fractions: ½ or a/b when a<b
the numerator is smaller than the
denominator
Improper fractions: 3 or a
when a>b;
2 b
the numerator is larger than the
denominator
Mixed numbers: 3 ½ or A b/c
have two parts – a whole number part and
a fraction part
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Changing forms (circle trick)
A nice trick for changing from a mixed number to
an improper fraction is:
-multiply the denominator and the whole number
-add this to the numerator
1
3
2
23  6
6 1  7
7
2
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Changing forms
The method for
changing from a
improper fraction to a
mixed number is to
divide.
Remember the fraction
bar is a divide sign.
Can you see the 3 ½ ?
3
2 7
6
1
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Multiplying fractions
To multiply fractions:
Mutliply the numerators, multiply the denominators
and reduce
In other words, take top times top; bottom times
bottom and reduce
a c ac
 
b d bd
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Dividing fractions
To divide fractions:
By definition, division is multipying by the
reciprocal. So . . .
-leave the first fraction as is
-flip the second fraction
-multiply (take top times top; bottom times bottom)
-reduce
a c a d ad
   
b d b c bc
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Adding and subtracting fractions
-find a common denominator
(LCD)
-rewrite each fraction with new
denominator
-add or subtract numerators as
indicated
-keep denominator the same
-reduce
See appendix b for more info on
LCD
a c

b d
a c

b d
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1.4 REAL NUMBERS
In this section we will be working with set
notation.
A set is a collection of elements listed within
braces
Example {a,b,c,d,e} – this set has 5
elements
{ } 0 -- this set has no elements. It is called
the empty set or null set.
{1,2,3 . . } – this set has an infinite number of
elements
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Natural Numbers {1,2,3, . . .}
This set includes the positive numbers-no
decimals or fractions. Also referred to as
the counting numbers in some books.
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Whole Numbers {0,1,2,3, . . .}
This set includes the natural numbers and
zero; still no decimals or fractions
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Integers { . . . -2,-1,0,1,2, . . .}
This set includes the positive and negative
“whole” numbers; again, no decimals or
fractions
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Rational Numbers
There are a lot of numbers in this set.
This set includes any number that can be
written as a fraction. Fractions; Repeating
and Terminating decimals as well; (1/3 =
0.333333…. Or ½= 0.5) And all the whole
numbers. (put a 1 under them 5 = 5/1)
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Irrational Numbers
We don’t work with these a lot. Common
examples are  and certain square
roots.
This set includes any number that can not
be written as a fraction. These are nonrepeating, non-terminating decimals.
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Real Numbers
This where we spend most of our time.
This set includes natural numbers, whole
numbers, integers, rational numbers,
irrational numbers. Everything we have
talked about so far.
Real numbers are any number that can be
represented on a number line.
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Real Number System
• Venn diagram
Natural
Whole
Integers
Rationals
Reals
Irrationals
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1.5 INEQUALITIES
< less than
 less than or =
> greater than
 greater than or =
= equal to
= not equal to
You can use a number
line to compare
numbers. To the left
things get smaller. To
the right things get
bigger.
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Absolute Value
By definition, absolute value is the distance
away from zero on a number line.
Denoted by straight lines or bars on either
side of a number or an expression
-3 = 3
3 =3
0 =0
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1.6 Addition of Real Numbers
Addition is combining
When adding two numbers with the same
sign, add the absolute values of the
numbers and keep the sign the same.
When adding two numbers with different
signs, find the difference of the absolute
values of the numbers and take the sign of
the number with the larger absolute value.
Note: additive inverse means opposite
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Addition examples:
3 + 2 = 5 the numbers have the same sign, add the
numbers and keep the sign the same
-3 + -2 the numbers have the same sign, add the numbers
and keep the sign the same
-3 + 2 the numbers have different signs, find the difference
of the numbers and take the sign of the
number with the larger absolute value
3 + -2 the numbers have different signs, find the difference
of the numbers and take the sign of the
number with the larger absolute value
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1.7 Subtraction
By definition, subtraction means to
add the opposite
You will rewrite every subtraction problem
into an addition problem. Then use the
rules for addition that we went over in 1.6
a – b = a + -b
3 – 2 = 3 + -2
-3 – 2 = -3 + -2
-3 - -2 = -3 + 2
3 - -2 = 3 + 2
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1.8 Multiplication/Division
Because multiplication and division are so
closely related, the chart below works for
both operations
When multiplying or dividing two numbers:
If the signs are the same, your answer is
positive.
If the signs are different, your answer is
negative.
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In other words
+ + = +
-  - =+
+ - = -  +=-
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1.9 Exponents, Parenthesis, and
Order of Operations
ExponentsAn exponent tells the number of times the
base appears as a factor.
4
2
4
2
2 is the exponent or power
4 is the base
42
is read 4 to the 2nd power or 4
squared
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4
2 means take 4 x 4
4
3
means take 4 x 4 x 4
If no exponent appears, we
assume the exponent is one –
not zero.
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Order of Operations
Order of Operations exists because when
there is more than one operation involved,
if we do not have an agreed upon order to
do things, we will not all come up with the
same answer. The order of operations
ensures that a problem has only one
correct answer.
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Order of Operations
Parenthesis (or grouping symbols)
Exponents
Multiplication or Division from Left to Right
Addition or Subtraction from Left to Right
PEMDAS
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In the parenthesis step, you may encounter
nested parenthesis. Below you will see the
same problem written two ways: once with
nested parenthesis and the other with a
variety of grouping symbols (including
brackets, braces, and parenthesis).
(( 5 x ( 2 + 3 )) + 7 ) – 2
OR
{[ 5 x ( 2 + 3 )] + 7 } - 2
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1.10 Properties of Real Numbers
In general, these properties are things that
you already know to be true. This just puts
a name to the idea that you already
understand.
You will need to memorize these (or think of
tricks to remember the names of them).
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Commutative
Commutative Property says the order does
not matter when you are adding or
multiplying. In other words, you can add or
multiply in any order, it does not affect the
answer.
Commutative Property of Addition
A+B=B+A
Commutative Property of Multiplication
AxB=BxA
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Associative
The Associative Property says when you are
adding or multiplying three or more
numbers, grouping symbols can be placed
around any two adjacent numbers without
changing the result.
Associate Property of Addition
(a+b)+c=a+(b+c)
Associative Property of Multiplication
(axb)xc=ax(bxc)
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The Commutative Property and the
Associative Property do not apply to
Division or Subtraction.
Distributive Property
of Multiplication over Addition
a ( b + c ) = ab + ac
Take something that is out front of the
parenthesis and distribute it through
everything in the parenthesis
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Identity and Inverse
Identity Property
In the Identity Property, whatever you start
with, you end with the same thing.
The additive identity is zero
a+0=a
0+a=a
The multiplicative identity is one
ax1=a
1xa=a
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Identity and Inverse
Inverse Property
With the Inverse Property, you end up with
the IDENTITY as the answer.
The multiplicative inverse means reciprocal
1
1
ax a=1
xa=1
a
The additive inverse means opposite
a + -a = 0
-a + a = 0
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