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Natural Numbers
Natural numbers are
counting numbers.
= {1, 2, 3, 4, 5…}
Whole Numbers
Whole numbers are natural
numbers and zero.
= {0, 1, 2, 3, 4, 5…}
N is a subset of W.
Integers
Integers are whole numbers
and opposites of naturals.
= {...−3, −2, −1, 0, 1, 2, 3…}
N and W are subsets
of Z.
Rational Numbers
Rational numbers are
integers and all fractions.
a
= {b a
,b
, & b ≠ 0}
Irrational Numbers
Irrational numbers are
totally different from
rational numbers. The two
have nothing in common.
Rationals and irrationals
are disjoint sets.
In other words, they have
no common element.
Irrationals
2, p , 5 7, 1.305276...
Real Numbers
Real numbers are rational
and irrational.
=
& irrationals
There are an infinite
number of rational numbers
between each pair of
integers. This is called the
density of numbers.
Rational Numbers
A rational number is any
number that can be written
a
in the form b , where a and
b are integers and b ≠ 0.
Lowest Terms
a
A rational fraction b is in
lowest terms if the GCF of
a and b is one.
Example 1
12
Rename
in lowest terms.
18
12 = 2 • 2 • 3
18 = 2 • 3 • 3
GCF = 2 • 3 = 6
12 = 2 x 6 = 2
3
18 3 x 6
Example 2
24
Rename
in lowest terms.
90
24 = 2 x 2 x 2 x 3
90 2 x 3 x 3 x 5
2
x
2
x
2
x
3
=
2x3x3x5
4
=
15
Example
Rename in lowest terms.
30 = 5
42
7
Example
Rename in lowest terms.
3,000 = 5
4,200 7
Example
Rename in lowest terms.
72
4
=
90
5
A proper fraction is one
whose numerator is less
than its denominator.
If the numerator is greater
than or equal to the
denominator, the fraction is
greater than or equal to one
and is called an improper
fraction.
A mixed number is actually
the sum of a whole number
and a fraction.
Renaming Improper Fractions
as Mixed Numbers
1. Divide the numerator by the
denominator.
2. Write the quotient as the
whole number.
3. Write the remainder over the
divisor as a fraction.
4. If possible, reduce the
fraction to lowest terms.
Example 3
19
Rename
as a mixed
7
number.
2
5
7 19
= 2
7
- 14
5
Example 3
12
Rename
as a mixed
8
number.
1
1
4
1
8 12
=1
2
8
- 8
4
Example
Rename the improper
fraction as a mixed number.
78 = 2 1
6
36
Example
Rename the improper
fraction as a mixed number.
5
93
−
= −11 8
8
Example 4
y
Evaluate the expression 3z
when y = 38 and z = 2. Write
the answer as a mixed number
in lowest terms.
1
6
=6
38 = 38
3
3
19
6
3(2)
- 18
19
x
2
19
=
=
1
3x2
3
Example
Evaluate when x = 2, y = – 3,
and z = 5.
y
3
=6
3x –
5
z
Example
Evaluate when x = 2, y = – 3,
and z = 5.
3x2 – y = 5 2
z
5
Example
Evaluate when x = 2, y = – 3,
and z = 5.
(3x)2
4
=−
2
25
3yz
Renaming Mixed Numbers as
Improper Fractions
1. Multiply the whole number
by the denominator.
2. Add the numerator to the
product.
3. Write the sum over the
denominator.
4. If possible, reduce the
fraction to lowest terms.
Example 5
1
Rename 3 5 as an improper
fraction in lowest terms.
1 = 5(3) + 1
35
5
15
+
1
16
=
=
5
5
Example 5
6
Rename 7 8 as an improper
fraction in lowest terms.
6 = 8(7) + 6 = 62 = 31 x 2
78
8
4x2
8
56
+
6
=
= 31
4
8
Example
Rename the mixed number as
an improper fraction.
9
31
2
=
11
11
Example
Rename the mixed number as
an improper fraction.
4
64
− 12 = −
5
5