Intro to Algebra Today. We will learn names for the properties of real numbers. Homework Next Week. Due Tuesday 45-47/ 15-20, 32-35, 40-41, *28,29,38

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1 Intro to Algebra Today We will learn names for the properties of real numbers. Homework Next Week Due Tuesday 45-47/ 15-20, 32-35, 40-41, *28,29,38 Due Thursday Pages 51-53/ 19-24, 29-36, *48-50, Due Friday Integers and Expressions Quiz

2 23-24/ or 63, 68-71, 76-88

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6 Properties of Real Numbers Real Numbers - The set of numbers consisting of positive numbers, negative numbers and zero. The set includes decimals, fractions and irrational numbers like or 2

7 Commutative Property of Addition a+b=b+a b + a When adding two numbers, the order of the numbers does not matter. Examples 2+3= (-5) 5)+4=4+(5) 4 + (-5)

8 Which of the following operations are also commutative? Subtraction Multiplication Division Exponentsp

9 Commutative Property of Multiplication a b = b a When multiplying two numbers, the order of the numbers does not matter. Examples 2 3 = 3 2 (-3) 24 = 24 (-3)

10 Associative Property of Addition a + (b + c) = (a + b) + c When three numbers are added, changing the grouping does not change the answer. Examples 2 + (3 + 5) = (2 + 3) + 5 (4+2)+6=4+(2+6) (2 + 6)

11 Which of the following operations are also associative? Subtraction Multiplication Division Exponents

12 Associative Property of Multiplication a(bc) = (ab)c When three numbers are multiplied, it makes no difference which two numbers are multiplied first. Examples 2 (3 5) = (2 3) 5 (4 2) 6 = 4 (2 6)

13 Name the property that is illustrated in each equation. A. 7(mn) = (7m)n The grouping is different. Associative Property of Multiplication B. (a + 3) + b = a + (3 + b) The grouping is different. Associative Property of Addition C. x + (y + z) = x + (z + y) The order is different. Commutative Property of Addition

14 Name the property that is illustrated in each equation. a. n + ( 7) = 7 + n Commutative Property of Addition b (g + 2.3) = (1.5 + g) Associative Property of Addition The order is different. The grouping is different. c. (xy)z = (yx)z The order is different. Commutative Property of Multiplication

15 Distributive Property a(b + c) = ab + ac Multiplication distributes over addition. Examples 2(3 + 5) = (2 3) + (2 5) (4 + 2) 6 = (4 6) + (2 6)

16 Closure Property The real numbers are closed for addition, subtraction and multiplication. Closure Whenever you add, subtract or multiply two real numbers, the answer is also a real number.

17 Name the property that is illustrated in each equation. 1. 6(rs) = (6r)s Associative 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:

18 Write each product using the Distributive Property. Then simplify. 5. 8(21) 8(20) + 8(1) = (97) 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, 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.

19 Additive Identity Property a + 0 = a The additive identity property states that if 0 is added to a number, the result is that number. Example: = = 3

20 Multiplicative Identity Property a 1 = a The multiplicative identity property states that if a number is multiplied by 1, the result is that number. Example: 5 1 = 1 5 = 5

21 Additive Inverse Property a + (-a) = 0 The additive inverse property states that opposites add to zero. 7 + (-7) = 0 and = 0

22 Multiplicative Inverse Property a b b a 1 The multiplicative inverse property states that reciprocals multiply to

23 Zero Product Property a x 0 = 0 The product of any real number and 0 is 0. examples 3 x 0 = 0 0 x (-7) = 0

24 Opposites Two real numbers that are the same distance from the origin of the real number line are opposites of each other. Examples of opposites: 2 and and 100 and 15 15

25 Reciprocals Two numbers whose product is 1 are reciprocals of each other. Examples of Reciprocals: and 5-3 and and 3 4 5

26 Absolute Value The absolute value of a number is its distance from 0 on the number line. The absolute value of x is written x. Examples of absolute value:

27 Identify which property that justifies each of the following. 4 (8 2) = (4 8) 2 Associative Property of Multiplication

28 Identify which property that justifies each of the following = Commutative Property of Addition

29 Identify which property that justifies each of the following = 12 Additive Identity Property

30 Identify which property that justifies each of the following. 5(2 + 9) = (5 2) + (5 9) Distributive Property

31 Identify which property that justifies each of the following. 5 + (2 + 8) = (5 + 2) + 8 Associative Property of Addition

32 Identify which property that justifies each of the following Multiplicative Inverse Property

33 Identify which property that justifies each of the following = 24 5 Commutative Property of Multiplication

34 Identify which property that justifies each of the following = 0 Additive Inverse Property

35 Identify which property that justifies each of the following = -34 Multiplicative Identity Property

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37 The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

38 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) Rewrite 71 as = 5(70) + 5(1) Use the Distributive Property. = Multiply (mentally). = 355 B. 4(38) 4(38) = 4(40 2) Add (mentally). Rewrite 38 as = 4(40) 4(2) Use the Distributive Property. = Multiply (mentally). = 152 Subtract (mentally).

39 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) = = 468 b. 12(98) 12(98) = 12(100 2) Rewrite 52 as Use the Distributive Property. Multiply (mentally). Add (mentally). Rewrite 98 as = 12(100) 12(2) Use the Distributive ib ti Property. = Multiply (mentally). = 1176 Subtract (mentally).

40 Check It Out! Example 3 Write each product using the Distributive Property. Then simplify. c. 7(34) 7(34) = 7(30 + 4) = 7(30) + 7(4) = = 238 Rewrite 34 as Use the Distributive Property. Multiply (mentally). Subtract (mentally).

41 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.

42 Closure Property of Real Numbers

43 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 = = 8 Since 8 is not a prime number, this is a counterexample. The statement is false.

44 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.

45 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.

46 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. Find a whole number, a, such that is not a whole number. Try a = 15. Since is not a whole number, this is a counterexample. The statement is false.

47 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. Caution! One counterexample is enough to disprove a statement, but one example is not enough to prove a statement.

48 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 a³ = b and c² = d, so a³ c². Try a³ = 2³, and c² = 3². a³ = b c² = d 2³ = 8 3² = 9 Since 2³ 3², this is a counterexample. The statement is false.

49 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.

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