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College Algebra
June 4, 2007
Despina Stavri
Section P.1: Review of Real Numbers and Their Properties

You should know the following sets:
Set of integers
Set of natural numbers
Set of whole numbers
Set of rational numbers
Set of irrational numbers
A real number is rational if it can be written as the ratio
p
of two integers where q  0 . The decimal
q
representation of a rational number either repeats or
terminates.
A real number is irrational if it cannot be written as the
ratio of two integers. Irrational numbers have infinite nonrepeating decimal representation.
Set of real numbers


An integer that has exactly two positive factors, the integer itself and 1, is called a
________________ number.
Real numbers are represented graphically by a _________________________.
The point zero on the real number line is the _________________. The numbers
to the left of zero are _______________. The numbers to the right of zero are
________________. Every point on the real line corresponds to exactly
_________ real number.
The term _______________ describes a number that is either positive or zero.
 Know the inequality symbols.
If a and b are real numbers, a is less than b if _________________________.
(a) a < b means ________________________________________________.
(b) a > b means ________________________________________________.
(c) a  b means ________________________________________________.
(d) a  b means ________________________________________________.
Example:
Place the correct6 symbol (< or >) between the numbers:
-5 _________ - 16
Inequalities can be used to describe subsets of real numbers called _____________.
Interval
Notation
Inequality Notation
Graph
[a,b]
(a,b)
[a,b)
(a,b]
[a,  )
(a,  )
(-  ,b]
(-  ,b)
(-  ,  )
Example:
Use inequality notation to describe each of the following.
(a) d is at most 3
(b) c is at least –3
(c) All x in the interval [-3,5].
Type

Absolute Value
Definition of Absolute Value:
If c is a real number, then the absolute value of c is
c
{
 Algebraic Expressions
An algebraic expression is a collection of letters (variables) and real numbers (constants)
combined using the operations of addition, subtraction, multiplication, division, and
exponentiation.
Examples:
b0 ,
x2  5x  8
,
5
x 3
3
The terms of an algebraic expression are those parts that are separated by ____________.
A term that contains variables is called a ______________ term, and a term that consists
of a number alone is called a _____________ term. The numerical factor of a variable
term is the _______________ of the variable term.
 Basic Rules of Algebra
Let a, b and c be real numbers, variables, or algebraic expressions.
Property
Commutative Property of
Addition
Commutative Property of
Multiplication
Associative Property of
Addition
Associative Property of
Multiplication
Distributive Properties
Additive Inverse Property
Multiplicative Identity
Property
Additive Inverse Property
Multiplicative Inverse
Property
a+b=b+a
ab = ba
(a + b) + c = a + ( b + c )
(ab)c = a(bc)
a( b + c ) = ab + ac
(a + b )c = ac + bc
a+0=a
a1=a
a + (-a) = 0
1
a   1, a  0
a
 Properties of Negation and Equality
Let a and b be real numbers, variables, or algebraic expressions.
Property
Example
(-1) a = -a
-(-a) = a
(-a)b = -(ab) = a(-b)
(-a)(-b) = ab
-(a+b) = (-a) + (-b)
If a = b then a  c = b  c
If a = b, then ac = bc
If a  c = b  c, then a = b.
If ac = bc and c  0 then a = b.
 Properties of Zero
Let a and b be real numbers, variables, or algebraic expressions.
1. a  0  a and a  0  a
2. a 0  0
0
 0, a  0
3.
a
a
 undefined
4.
0
5. If ab  0 , then a  0 or b  0 .

Properties and Operations of Fractions
Let a, b, c and d be real numbers, variables, or algebraic expressions such that b  0 and
d  0.
Property
1. Equivalent Fractions
2. Rules of Signs:
3. Generate Equivalent
Fractions:
4. Add or Subtract with
Like Denominators:
5. Add or Subtract with
Unlike Denominators:
6. Multiply Fractions:
7. Divide Fractions:
a c
 if and only if ad  bc
b d
a a
a
a a
 


and
b
b
b
b b
a ac
 , c0
b bc
a c ac
 
b b
b
a c ad  bc
 
b d
bd
a c ac
 
b d bd
a c a d ad
   
, c0
b d b c bc
Example