1.1 & 1.2: Properties of Real Numbers &
Algebraic Expressions
If you would be a real seeker after truth, it is necessary that at
least once in your life you doubt, as far as possible, all things.
Exploring Real Numbers
Real Numbers: Collection of ALL Rational and Irrational Numbers.
Rational Numbers: Any number that can be
written in the form a .
0.25
b
1

Integers
0.3
3
..., 2, 1,0,1,2,...
Whole Numbers
0,1,2,3,4,5,...
2

3
Natural Numbers
1,2,3,4,5,...
0.24
Irrational Numbers:
Can not be written
in the form a .
b
11

0.349579315...
9
100
Ordering and Graphing on a Number Line
Ex1)
2

3
-5
-4
1.5
3.3
 11
6
-3
-2
-1
0
1
2
3
4
5
Opposite and Reciprocal
The opposite of a is –a.
The reciprocal of a is 1 .
a
the opposite of 2 is -2
the reciprocal of 5 is
1
5
1
the reciprocal of
is 2
2
Properties of Real Numbers:
Let a, b, and c represent real numbers
Property
Addition
Multiplication
Commutative
a+b=b+a
ab = ba
Associative
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Identity
a+0=a
a·1 = a
Inverse
a + (-a) = 0
a·(1/a) = 1, a≠ 0
Distributive
a(b +c) = ab +ac
Properties of Real Numbers:
Ex2) 8 + (-8) = 0
Inverse Property of
Addition
Ex3) (2·5) ·(-8) = (2)·(5·8)
Associative Property of
Multiplication
Ex4) (4+1) ·3 = (1+4) ·3
Commutative Property
of Addition
Absolute Value
A number’s distance from zero. REMEMBER: DISTANCE IS
ALWAYS POSITIVE!
Ex5) 9 
Ex6) 2  3 
Ex7) (1)(4) 
Ex8) 0 
Ex9)  2 
Algebraic Expressions: Vocabulary
15x2 + 6y
•Variable: a symbol that represents one or more numbers (x, y)
•Algebraic Expression: an expression that contains one or more
variables (15x2 + 6y)
•Term: a number, variable, or the product of a number and one or
more variables (15x2, 6y)
•Coefficient: The numerical factor in a term (15, 6)
•Evaluate: Substituting numbers for variables in an expression
and following order of operations
Combine Like Terms: Simplify
Ex10) 3k  6  5k  2
Ex11) a 2  2b  4b  6a 2
Evaluate
Ex12) x  4 y
when x  3 and y  2
Ex13) 5 z 2  10 z  8 z 2  z
when z  2
Challenge
Ex14) What is the perimeter of the figure?
2a  b
2
b
a
2a  b
2
b
b
2a
a
1.1 & 1.2: Properties of Real Numbers &
Algebraic Expressions
Homework:
1.1: 1, 2-6 EVEN, 12-16 EVEN, 25-33 ODD, 34, 36 ,37,
48-54 EVEN
1.2: 1-11 odd, 22, 27, 31, 33, 36, 41, 43, 44
If you would be a real seeker after truth, it is necessary that at
least once in your life you doubt, as far as possible, all things.