1-6 Properties of Real Numbers

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California Standards

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1-6 Properties of Real Numbers

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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

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

1-6 Properties of Real Numbers

counterexampleclosure

Vocabulary

1-6 Properties of Real Numbers

The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression.

1-6 Properties of Real Numbers

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.

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.

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.

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.

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

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 a³ = b and c² = d, so a³ ≠ c².

Try a³ = 2³, and c² = 3².

a³ = b2³ = 8

c² = d3² = 9

Since 2³ ≠ 3², this is a counterexample. The statement is false.

1-6 Properties of Real NumbersCheck 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.

1-6 Properties of Real Numbers

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

1-6 Properties of Real NumbersAdditional Example 3: Using the Distributive Property

with Mental MathWrite 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).

1-6 Properties of Real NumbersCheck 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).

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.

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.

1-6 Properties of Real Numbers

Closure Property of Real Numbers

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.

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.

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.

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.

1-6 Properties of Real NumbersLesson Quiz: Part I

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: 3 ÷ 6 ≠ 6 ÷ 3

1-6 Properties of Real NumbersLesson 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.