• Natural #’s: N = {1, 2, 3, 4, ...}
• Whole #’s: W = {0, 1, 2, 3, 4, ...}
• Integers: I = {...,−3,−2,−1, 0, 1, 2, 3, ...}
• Rational #’s: 
• Irrational #’s: non-repeating, non- terminating decimal numbers
ALL of these sets, together, make up the set of Real
Numbers ,
To change repeating decimals into rational form:
• multiply by 10n where n is the number of digits that repeat
• ie: Suppose x = 
• Write x in fraction form .
PROPERTIES OF REAL NUMBERS:
1. The Commutative Properties
• addition: a + b = b + a
• multiplication: ab = ba
2. The Associative Properties
• addition: a + (b + c) = (a + b) + c
• multiplication: a(bc) = (ab)c
3. The Distributive Property
• a(b + c) = ab + ac
4. Identities - an identity is a unique number where the
answer for a particular operation
is the original number
is the original number
• for addition the identity element is 0: a + 0 = 0 + a = a
• for multiplication the identity element is 1: a · 1 = 1 · a = a
5. Inverses - an inverse is a number which when acted with
the original number, results
in the identity
• Each real number a has a unique additive inverse
represented by −a:
a + (−a) = (−a) + a = 0
• Each non zero real number a has a unique multiplicative
inverse represented
by 1/a:
Working With Fractions
• Multiplying Fractions
• Dividing Fractions
• Adding Fractions, Same Denominator
• Adding fractions if denominators are different :
• Reducing Fractions
• Cross Products
• Using the LCD to add fractions
The Real Line
• Always label units on the graph.
Inequalities For a , b ∈ R, one and only one of the
following holds:
a < b, a > b, or a = b
Sets and Intervals: A set is a well defined collection of
objects. The objects in the set
are called elements. Sets are described by one of the following methods:
• roster method
• set builder notation
• graph
Symbols
• ∈: is an element of
• 
is not an element of
• Union:
A ∪ B is the set which contains all elements in A or B or
both.
• Intersection:
A∩B is the set which contains all elements in common , elements in both A and B.
• The empty set,
contains no elements.
Example: Given Q = {a, b, c, d, e, f} and R = {a, e, i, o,
u}
• find: Q ∪ R
• find: Q ∩ R
Interval Notation: the following notation represents
different intervals
| |
Interval Notation |
Inequality Notation |
Graph |
| open interval |
(a, b) |
a < x < b |
 |
| closed interval |
[a, b] |
a ≤ x ≤ b |
 |
| half-open interval |
(a, b] |
a < x ≤ b |
 |
| undbounded interval |
(a,∞) |
x > a |
 |
| u |
|
|
|
Example: Graph each of the following given A = (−2, 8] , B
= (−∞,−1), and C = [5,∞)
• graph A
• graph B
• graph C
• find A ∩ B
• find A ∪ B
• find A ∩ C
• find A ∪ C
• find B ∩ C
• find B ∪ C
Absolute Value: the distance from a number to the origin;
always positive or 0. Mathe -
matically,
• Find the absolute value of each of the following:
Distance between two points on a number line
• The distance between a and b is given by | a − b | or | b − a |
d(a, b) =| b − a |=| a − b |
