Download Real Numbers and Their Properties

Thinking
Mathematically
Number Theory and the Real Number
System
5.5 Real Numbers and Their Properties
Real Numbers
Real numbers are the union of the rational
numbers and the irrational numbers.
Subsets of the Real Numbers
Natural Numbers: {1, 2, 3, 4, 5, …} These
numbers are used for counting.
Whole Numbers: {0, 1, 2, 3, 4, 5, …} The
whole numbers add 0 to the set of natural
numbers.
Integers: {…, -3, -2, -1, 0, 1, 2, 3,…} The
integers add the negatives of the natural
numbers to the set of whole numbers.
Subsets of the Real Numbers
Rational Numbers: These numbers can be
expressed as an integer divided by a nonzero
integer: a/b: a and b are integers: b does not equal
zero. Rational numbers can be expressed as
terminating or repeating decimals.
Irrational Numbers: This is the set of numbers
whose decimal representations are neither
terminating nor repeating. Irrational numbers
cannot be expressed as a quotient of integers.
Examples: Classifying Numbers
Exercise Set 5.5 #3

5


11
,

,
0
,
0
.
75
,
5
,

,
64


6


Which elements of this set are:
For the set
•Natural numbers
•Whole numbers
•Integers
•Rational numbers
•Irrational numbers
•Real numbers
Properties of Real Numbers
• Closure under addition, multiplication
• Addition and multiplication are
commutative.
• Addition and multiplication are associative
• Multiplication distributes over addition
“Closure” of the Real Numbers
The sum or the difference of any two real
numbers is another real number. This is
called the “closure” property of addition.
The product or the quotient of any two real
numbers (the denominator cannot be zero)
is another real number. This is called the
“closure” property of multiplication.
“Commutative” Property of Addition
Two real numbers can be added in any order. This
is called the “commutative” property of addition.
a+b=b+a
Example:
2+3=3+2
“Commutative” Property of
Multiplication
Two real numbers can be multiplied in any
order. This is called the “commutative”
property of multiplication.
axb=bxa
Example: 12 x 5 = 5 x 12 = 60
“Associative” Property of Addition
When three real numbers are added, it makes
no difference which two are added first. This
is called the “associative” property of addition.
(a + b) + c = a + (b + c)
Example:
(12 + 5) + 3 = 12 + (5 + 3) = 17 + 3 = 12 + 8 = 20
“Associative” Property of Multiplication
When three real numbers are multiplied, it
makes no difference which two are multiplied
first. This is called the “associative” property
of multiplication.
(a x b) x c = a x (b x c)
Example:
(12 x 5) x 3 = 12 x (5 x 3) = 60 x 3 = 12 x 15 = 180
Multiplication “Distributes” over Addition
The product of a number with a sum is the
sum of the individual products. This is called
the “distributive” property of multiplication
over addition.
a x (b + c) = (a x b) + (a x c)
Example:
12 x (5 + 3) = (12 x 5) + (12 x 3)
= 12 x 8 = 60 + 36 = 96
Exercises – Identifying Properties
Exercise Set 5.5 #31, 33, 35
Name the property illustrated
6 + (2 + 7) = (6 + 2) + 7
(2 + 3) + (4 + 5) = (4 + 5) + (2 + 3)
2 (-8 + 6) = -16 + 12
Thinking
Mathematically
Number Theory and the Real Number
System
5.5 Real Numbers and Their Properties