1. 1-6 Properties of Real Numbers Warm Up Warm Up Lesson Presentation Lesson Presentation California Standards California StandardsPreview

2. 1-6 Properties of Real Numbers Warm Up Add. 1. –6 + (–4) 2. 17 + (–5) 3. (–9) + 7 Subtract. 4. 12 – (–4) 5. –3 – (–5) 6. –7 – 15 –10 12 16 –2 2 – 22

3. 1-6 Properties of Real Numbers 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable. 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. Also covered: 25.1 California Standards

4. 1-6 Properties of Real Numbers counterexample closure Vocabulary

5. 1-6 Properties of Real Numbers The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression.

6. 1-6 Properties of Real Numbers

7. 1-6 Properties of Real Numbers Name the property that is illustrated in each equation. Additional Example 1: Identifying Properties A. 7(mn) = (7m)n Associative Property of Multiplication The grouping is different. B. (a + 3) + b = a + (3 + b) Associative Property of Addition The grouping is different. C. x + (y + z) = x + (z + y) Commutative Property of Addition The order is different.

8. 1-6 Properties of Real Numbers Check It Out! Example 1 Name the property that is illustrated in each equation. a. n + ( – 7) = – 7 + n b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3 c. (xy)z = (yx)z Commutative Property of Addition Commutative Property of Multiplication Associative Property of Addition The order is different. The grouping is different. The order is different.

9. 1-6 Properties of Real Numbers The Commutative and Associative Properties are true for addition and multiplication. They may not be true for other operations. A counterexample is an example that disproves a statement, or shows that it is false. One counterexample is enough to disprove a statement.

10. 1-6 Properties of Real Numbers Caution! One counterexample is enough to disprove a statement, but one example is not enough to prove a statement.

11. 1-6 Properties of Real Numbers Statement Counterexample No month has fewer than 30 days. February has fewer than 30 days, so the statement is false. Every integer that is divisible by 2 is also divisible by 4. The integer 18 is divisible by 2 but is not by 4, so the statement is false. Counterexamples

12. 1-6 Properties of Real Numbers Additional Example 2: Finding Counterexamples to Statements About Properties Find a counterexample to disprove the statement “ The Commutative Property is true for raising to a power. ” Find four real numbers a, b, c, and d such that a ³ = b and c ² = d, so a ³ ≠ c ². Try a ³ = 2 ³, and c ² = 3 ². a ³ = b 2 ³ = 8 c ² = d 3 ² = 9 Since 2 ³ ≠ 3 ², this is a counterexample. The statement is false.

13. 1-6 Properties of Real Numbers Check It Out! Example 2 Find a counterexample to disprove the statement “ The Commutative Property is true for division. ” Find two real numbers a and b, such that Try a = 4 and b = 8. Since, this is a counterexample. The statement is false.

14. 1-6 Properties of Real Numbers The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

15. 1-6 Properties of Real Numbers Additional Example 3: Using the Distributive Property with Mental Math Write each product using the Distributive Property. Then simplify. A. 5(71) 5(71) = 5(70 + 1) = 5(70) + 5(1) = 350 + 5 = 355 B. 4(38) 4(38) = 4(40 – 2) = 4(40) – 4(2) = 160 – 8 = 152 Rewrite 71 as 70 + 1. Use the Distributive Property. Multiply (mentally). Add (mentally). Rewrite 38 as 40 – 2. Use the Distributive Property. Multiply (mentally). Subtract (mentally).

16. 1-6 Properties of Real Numbers Check It Out! Example 3 Write each product using the Distributive Property. Then simplify. a. 9(52) 9(52) = 9(50 + 2) = 9(50) + 9(2) = 450 + 18 = 468 Rewrite 52 as 50 + 2. Use the Distributive Property. Multiply (mentally). Add (mentally). b. 12(98) 12(98) = 12(100 – 2) = 12(100) – 12(2) = 1200 – 24 = 1176 Rewrite 98 as 100 – 2. Use the Distributive Property. Multiply (mentally). Subtract (mentally).

17. 1-6 Properties of Real Numbers Check It Out! Example 3 c. 7(34) 7(34) = 7(30 + 4) = 7(30) + 7(4) = 210 + 28 = 238 Rewrite 34 as 30 + 4. Use the Distributive Property. Multiply (mentally). Subtract (mentally). Write each product using the Distributive Property. Then simplify.

18. 1-6 Properties of Real Numbers A set of numbers is said to be closed, or to have closure, under an operation if the result of the operation on any two numbers in the set is also in the set.

19. 1-6 Properties of Real Numbers Closure Property of Real Numbers

20. 1-6 Properties of Real Numbers Additional Example 4: Finding Counterexamples to Statements About Closure Find a counterexample to show that each statement is false. A. The prime numbers are closed under addition. Find two prime numbers, a and b, such that their sum is not a prime number. Try a = 3 and b = 5. a + b = 3 + 5 = 8 Since 8 is not a prime number, this is a counterexample. The statement is false.

21. 1-6 Properties of Real Numbers Additional Example 4: Finding Counterexamples to Statements About Closure Find a counterexample to show that each statement is false. B. The set of odd numbers is closed under subtraction. Find two odd numbers, a and b, such that the difference a – b is not an odd number. Try a = 11 and b = 9. a – b = 11 – 9 = 2 11 and 9 are odd numbers, but 11 – 9 = 2, which is not an odd number. The statement is false.

22. 1-6 Properties of Real Numbers Check It Out! Example 4 Find a counterexample to show that each statement is false. a. The set of negative integers is closed under multiplication. Find two negative integers, a and b, such that the product a  b is not a negative integer. Try a = – 2 and b = – 1. a  b = – 2( – 1) = 2 Since 2 is not a negative integer, this is a counterexample. The statement is false.

23. 1-6 Properties of Real Numbers Check It Out! Example 4 Find a counterexample to show that each statement is false. b. The whole numbers are closed under the operation of taking a square root. Try a = 15. Since is not a whole number, this is a counterexample. The statement is false. Find a whole number, a, such that is not a whole number.

24. 1-6 Properties of Real Numbers Lesson Quiz: Part I Name the property that is illustrated in each equation. 1. 6(rs) = (6r)sAssociative Property of Multiplication 2. (3 + n) + p = (n + 3) + p Commutative Property of Addition 3. (3 + n) + p = 3 + (n + p) Associative Property of Addition 4. Find a counterexample to disprove the statement “ The Commutative Property is true for division. ” Possible answer: 3 ÷ 6 ≠ 6 ÷ 3

25. 1-6 Properties of Real Numbers Lesson Quiz: Part II Write each product using the Distributive Property. Then simplify. 5. 8(21) 6. 5(97) 8(20) + 8(1) = 168 5(100) – 5(3) = 485 Find a counterexample to show that each statement is false. 7. The natural numbers are closed under subtraction. Possible answer: 6 and 8 are natural, but 6 – 8 = – 2, which is not natural. 8. The set of even numbers is closed under division. Possible answer: 12 and 4 are even, but 12 ÷ 4 = 3, which is not even.