1. Classifying Real Numbers
Lesson 1
Classifying Real Numbers

2. Warm Up:

3. Warm Up Answers:

4. Example 1:
For each number, identify the subsets of real numbers to which it belongs:
½ √2

5. Example 1:
For each number, identify the subsets of real numbers to which it belongs:
½ √2
{rational numbers, {natural numbers {irrational numbers
real numbers} whole numbers real numbers}
integers
rational numbers
real numbers}

6. Example 2:
The value of the bills in a person’s wallet.
The balance of a checking account.
The circumference of a circular table when the diameter is a rational number.

7. Example 2:
The value of the bills in a person’s wallet.
The set of whole numbers--the wallet can contain any number
of bills or zero bills.
The balance of a checking account.
The set of rational numbers best describes the situation. The
balance could be positive or negative and could contain
decimal amounts.
The circumference of a circular table when the diameter is a rational number.
The set of irrational numbers best describes this situation.
Since the formula for circumference contains pi, the answer
will be irrational.

8. Vocabulary:
Intersection of sets: Elements that are contained in BOTH original sets.
Symbol: ⋂
Union of sets: Joining all the elements of sets together.
Symbol: ⋃
Closure (of a set): a set is closed under an operation if the outcome of the operation on any two members of the set is also a member of that set. For example, the sum of any two natural numbers is also a natural number. Therefore, the set of natural numbers is closed under addition.
Counterexample: one example that proves a statement false.

9. Example 3:
Find A ∩ B and A ∪ B
A = {2,4,6,8,10,12}; B = {3,6,9,12}
A = {11,13,15,17}; B = {12,14,16,18}

10. Example 3:
Find A ∩ B and A ∪ B
A = {2,4,6,8,10,12}; B = {3,6,9,12}
Solution: A ∩ B = {6,12}; A ∪ B = {2,3,4,6,8,9,10,12}
A = {11,13,15,17}; B = {12,14,16,18}
Solution: A ∩ B = {11,13,15,17};
A ∪ B = { 11,12,13,14,15,16,17,18}

11. Example 4:
Determine whether each statement is true or false. Give a counterexample for false statements.
The set of whole numbers is closed under addition.
The set of whole numbers is closed under subtraction.

12. Example 4:
Determine whether each statement is true or false. Give a counterexample for false statements.
The set of whole numbers is closed under addition.
Verify the statement by adding two whole numbers:
2 + 3 = 5
= 20
= 1100
The sum is always a whole number so the statement is true.
The set of whole numbers is closed under subtraction.
Verify the statement by subtracting two whole numbers:
6 - 4 = 2
= 10
4 - 6 = -2
The difference of is not a whole number. This a counterexample that proves the statement false.

13. Practice Practice Practice!!
Lesson Practice Page 5