©
:
Pre

Calculus

Chapter 1A
Chapter 1A

Real Numbers
Properties of Real Numbers
Real numbers are used in almost every human endeavor. Whenever we need to quantify objects, we use
numbers. Cooking recipes, prices, interest rates, blood pressure, height, age, voltage, and wind velocity
are a few of the everyday objects that are quantified by real numbers. As we know, the two operations
of addition
!
!
"
and multiplication
!#"
are defined for real numbers. In other words, for any two real
numbers
a
and
b
, the sum
a
!
b
and the product
a
#
b
are uniquely defined real numbers. Two special
real numbers are zero (0) and one (1). These operations satisfy:
Properties of real numbers:
Commutative:
a
!
b
"
b
!
a
ab
"
ba
Example
:7
!
3
"
3
!
7
"
10
5
#
6
"
6
#
5
"
30
Associative:
a
!
!
b
!
c
"
"
!
a
!
b
"
!
ca
!
bc
"
"
!
ab
"
c
Example
:3
!
!
4
!
7
"
"
!
3
!
4
"
!
72
# !
5
#
3
"
"
!
2
#
5
" #
3
3
!
11
"
7
!
7
"
14
2
#
15
"
10
#
3
"
30
Identity:
a
!
0
"
0
!
a
"
aa
#
1
"
1
#
a
"
a
Example
:8
!
0
"
0
!
8
"
8
11
#
1
"
1
#
11
"
11
For each real number
a
, there is a real number, denoted by
!
a
, called the negative of
a
, for which
Inverse:
a
!
!
!
a
"
"
0
"
!
!
a
"
!
a
Example
!
!
!
7
"
"
0
Subtraction, denoted by
a
!
b
, is defined as follows:
a
!
b
"
a
!
!
!
b
"
For each
a
"
0, there is a real number, denoted by
1
a
or 1/
a
or
a
!
1
, called the reciprocal of
a
, for which
Inverse:
a
1
a
"
1
"
1
a
a
Example:
7
#
1
7
"
1
7
#
7
"
1
Division, denoted by
a
!
b
or
a
b
or
a
/
b
, where
b
"
0, is defined as follows:
a
!
b
"
a
b
"
a
/
b
"
a
!
1
b
"
"
ab
!
1
Finally, there is a property which relates addition and multiplication:
Distributive:
a
!
b
!
c
"
"
ab
!
ac
!
a
!
b
"
c
"
ac
!
bc
Example:
!
4
# !
3
!
2
"
"
!
!
4
#
3
"
!
!
!
4
#
2
"
!
4
#
5
"
!
12
!
!
!
8
"
"
!
20
©
:
Pre

Calculus