Special Sets of Numbers
Remember to Silence Your
Cell Phone and Put It In Your
Bag!
Mathematics was invented.
Numbers vs. numerals
The Set of Counting Numbers
or Natural Numbers
N = {1, 2, 3, 4, 5, . . . }
The Counting Process
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


Say the names of the counting numbers
Name the numerals
Write the numerals
Count a number of objects
The Whole Numbers
W = {0, 1, 2, 3, 4, 5, . . . }
A whole number is the unique
characteristic embodied in each finite
set and all the sets equivalent to it.
2.1 p. 65
The Set of Integers
I = { . . . -3, -2, -1, 0, 1, 2, 3, . . . }
For every natural number n, there is a unique
number the opposite of n, denoted by –n, such that
n + -n = 0.
The set of integers, I, is the union of the set of
natural numbers, the set of the opposites of the
natural numbers, and the set that contains zero.
I = {1, 2, 3, …}  {-1, -2, -3 ...}  {0}
5.1 p. 249
The Set of Rational Numbers
Q = { a | a, b, ϵ I, b ≠ 0}
b
This textbook calls
a
b
a fraction.
Fractions are Rational Numbers!
Integers are Rational Numbers!
Whole Numbers are Rational Numbers!
6.1 p. 302
The Set of Rational Numbers
(cont.)
A decimal is a symbol that uses a
base-ten place-value system with tenths
and powers of tenths to represent a
number
A decimal is a rational number!
6.1 p. 207
Relationships Among these
Sets of Numbers
NWIQ
Q
I
W
N
6.5 p. 362
What numbers are not Rational
Numbers?
Every rational number can be expressed
as a terminating or repeating decimal.
Numbers which cannot be expressed
as either repeating or terminating
decimals are not rational numbers.
6.1 p. 310 & 6.5 pp. 359-362
The Set of Irrational Numbers
Real numbers which cannot be
expressed as either repeating or
terminating decimals.
Examples:
6.5 pp. 361-363
The Set of Real Numbers
R = {Rational Numbers} ⋃ {Irrational Numbers}
Note – The set of rational numbers and the
set of irrational numbers are disjoint sets.
(They have no elements in common.)
6.5 pp. 361-363
What numbers are not Real numbers?
_____________ numbers are not real
numbers.
Examples: