Classifying Real Numbers Lesson 1 Classifying Real Numbers

Warm Up:

Warm Up Answers:

Example 1: For each number, identify the subsets of real numbers to which it belongs: ½ 2. 5 3. 3√2

Example 1: For each number, identify the subsets of real numbers to which it belongs: ½ 2. 5 3. 3√2 {rational numbers, {natural numbers {irrational numbers real numbers} whole numbers real numbers} integers rational numbers real numbers}

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.

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.

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.

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}

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}

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.

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 9 + 11 = 20 100 + 1000 = 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 = 2 100 - 90 = 10 4 - 6 = -2 The difference of 4 - 6 is not a whole number. This a counterexample that proves the statement false.

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