1. Special Sets of Numbers
Remember to Silence Your Cell Phone and Put It In Your Bag!

2. Mathematics was invented.
Numbers vs. numerals

3. The Set of Counting Numbers or Natural Numbers
The Counting Process
Say the names of the counting numbers
Name the numerals
Write the numerals
Count a number of objects

4. 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

5. 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, }  {0}
5.1 p. 249

6. The Set of Rational Numbers
Q = { | a, b, ϵ I, b ≠ 0}
This textbook calls a fraction.
Fractions are Rational Numbers!
Integers are Rational Numbers!
Whole Numbers are Rational Numbers!
6.1 p. 302

7. 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

8. Relationships Among these Sets of Numbers
N  W  I  Q Q I W N
6.5 p. 362

9. 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

10. The Set of Irrational Numbers
Real numbers which cannot be expressed as either repeating or terminating decimals.
Examples:
6.5 pp

11. R = {Rational Numbers} ⋃ {Irrational Numbers}
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

12. What numbers are not Real numbers?
Examples: