properties of real numbers 2 - staff.pausd.orgstaff.pausd.org/~pjorgens/papers/16notes.pdfProperties of Real Numbers Real Numbers - The set of numbers consisting of positive numbers,

Intro to Algebra Today

Homework Next Week

We will learn names for the properties of real numbers.

Homework Next Week

Due Tuesdayy45-47/ 15-20, 32-35, 40-41, *28,29,38

Due ThursdayPages 51-53/ 19-24, 29-36, *48-50, 60-65

Due FridayIntegers and Expressions QuizIntegers and Expressions Quiz

23-24/ 14-33 or 63, 68-71, 76-88

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

a + b = b + a

Commutative Property of Addition

a + b = b + a

When adding two numbers the order of theWhen adding two numbers, the order of the numbers does not matter.

Examples 2 + 3 = 3 + 2 ( 5) + 4 = 4 + ( 5)2 + 3 = 3 + 2 (-5) + 4 = 4 + (-5)

Which of the following operations are also commutative?

Subtraction

Multiplication

DivisionDivision

Exponentsp

Commutative Property of Multiplication

a b = b a

When multiplying two numbers, the order of the numbers does not matterof the numbers does not matter.

ExamplesExamples 2 3 = 3 2 (-3) 24 = 24 (-3)

Associative Property of Addition

a + (b + c) = (a + b) + c

When three numbers are added, changing the grouping does not change the answer.

ExamplesExamples 2 + (3 + 5) = (2 + 3) + 5(4 + 2) + 6 = 4 + (2 + 6)(4 + 2) + 6 = 4 + (2 + 6)

Which of the following operations are also i ti ?associative?

Subtraction

Multiplication

Division

E ponentsExponents

Associative Property of Multiplication

a(bc) = (ab)c

When three numbers are multiplied, it makes no difference which two numbers are multiplied first.p

Examples p2 (3 5) = (2 3) 5(4 2) 6 = 4 (2 6)(4 2) 6 4 (2 6)

Name the property that is illustrated in each equation.

A. 7(mn) = (7m)n


The grouping is different.

B. (a + 3) + b = a + (3 + b) Associative Property of Addition


C. x + (y + z) = x + (z + y)


The order is different.



Th d ia. n + (–7) = –7 + n



b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3



c. (xy)z = (yx)z




Distributive Property

a(b + c) = ab + ac

Multiplication distributes over addition.

Examples 2 (3 5) (2 3) (2 5)2 (3 + 5) = (2 3) + (2 5)

(4 + 2) 6 = (4 6) + (2 6)

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 numberreal numbers, the answer is also a real number.


1. 6(rs) = (6r)s Associative Property of Multiplication

2 (3 + n) + p (n + 3) + p2. (3 + n) + p = (n + 3) + pCommutative Property of Addition

3 (3 + n) + p = 3 + (n + p)3. (3 + n) + p = 3 + (n + p)Associative Property of Addition

4. Find a counterexample to disprove the statement p p“The Commutative Property is true for division.”Possible answer: 3 ÷ 6 ≠ 6 ÷ 3

Write each product using the Distributive Property. Then simplify.

5. 8(21)

6. 5(97)

8(20) + 8(1) = 168

5(100) – 5(3) = 4856. 5(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 which is not natural –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.

Additive Identity Property

a + 0 = a

The additive identity property states that if 0 is added to a number the result is thatis added to a number, the result is that number.

Example: 3 + 0 = 0 + 3 = 3

Multiplicative Identity Property

a 1 = aThe multiplicative identity property states thatThe multiplicative identity property states that

if a number is multiplied by 1, the result is that numberthat number.

E l 5 1 1 5 5Example: 5 1 = 1 5 = 5

Additive Inverse Property

a + (-a) = 0

The additive inverse property states that opposites add to zeroopposites add to zero.

7 + (-7) = 0 and -4 + 4 = 0

Multiplicative Inverse Property

ab b

a1

The multiplicative inverse property states that reciprocals multiply to 1

b a

reciprocals multiply to 1.

515

1 5

23

32

1 3 2

Zero Product Property0 0a x 0 = 0

The product of any real number and 0 is 0.

lexamples

3 x 0 = 0 0 x (-7) = 0( )

Opposites

Two real numbers that are the same distance from the origin of the real number line are gopposites of each other.

Examples of opposites:2 d 2 100 d 100 d15 152 and -2 -100 and 100 and 15 15

Reciprocals

Two numbers whose product is 1 are reciprocals of each other.p

Examples of Reciprocals:Examples of Reciprocals:and 5 -3 and 1

354

45

and3 4 5

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:3 3 5 5 37

37

Identify which property thatIdentify which property that justifies each of the following.

4 (8 2) (4 8) 24 (8 2) = (4 8) 2



6 8 8 66 + 8 = 8 + 6



12 0 1212 + 0 = 12

Additive Identity Property


5(2 9) (5 2) (5 9)5(2 + 9) = (5 2) + (5 9)

Distributive Property


5 (2 8) (5 2) 85 + (2 + 8) = (5 + 2) + 8



5 959

95

1

Multiplicative Inverse Property


5 24 24 55 24 = 24 5



18 18 018 + -18 = 0

Additive Inverse Property


34 1 34-34 1 = -34

Multiplicative Identity Property

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

Additional Example 3: Using the Distributive Property with Mental Math

Write each product using the Distributive Property. Then simplify.p y p yA. 5(71)

5(71) = 5(70 + 1)( ) ( )

Rewrite 71 as 70 + 1.= 5(70) + 5(1)= 350 + 5= 355

Use the Distributive Property.Multiply (mentally).Add (mentally). 355

B. 4(38)4(38) = 4(40 – 2)

Add (mentally).

Rewrite 38 as 40 – 2.= 4(40) – 4(2)= 160 – 8= 152

Use the Distributive Property.Multiply (mentally).Subtract (mentally)= 152 Subtract (mentally).

Check It Out! Example 3Check It Out! Example 3Write each product using the Distributive Property. Then simplify.

9(52)a. 9(52)9(52) = 9(50 + 2)

= 9(50) + 9(2)Rewrite 52 as 50 + 2.Use the Distributive Property= 9(50) + 9(2)

= 450 + 18= 468

Use the Distributive Property.Multiply (mentally).Add (mentally).

b. 12(98)12(98) = 12(100 – 2)

12(100) 12(2)Rewrite 98 as 100 – 2.U th Di t ib ti P t= 12(100) – 12(2)

= 1200 – 24= 1176

Use the Distributive Property.Multiply (mentally).Subtract (mentally). 1176 Subtract (mentally).

Ch k It O t! E l 3Check It Out! Example 3Write each product using the Distributive Property. Then simplify.

c. 7(34)

( ) ( )

p y p y

7(34) = 7(30 + 4)

= 7(30) + 7(4)

Rewrite 34 as 30 + 4.

Use the Distributive Property.

= 210 + 28

= 238

Multiply (mentally).

Subtract (mentally).

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 operation on any two numbers in the set is also in the set.

Closure Property of Real Numbers

Additi l E l 4 Fi di C t l t Additional Example 4: Finding Counterexamples to Statements About Closure

Find a counterexample to show that each Find a counterexample to show that each statement is false.A. The prime numbers are closed under addition.A. The prime numbers are closed under addition.

Find two prime numbers, a and b, such that their sum is not a prime number. their sum is not a prime number.

Try a = 3 and b = 5.

a + b = 3 + 5 = 8a + b = 3 + 5 = 8

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

Additional Example 4: Finding Counterexamples to Additional Example 4: Finding Counterexamples to Statements About Closure

Find a counterexample to show that each pstatement is false.B. The set of odd numbers is closed under

subtractionsubtraction.Find two odd numbers, a and b, such that the difference a – b is not an odd number. difference a b is not an odd number.

Try a = 11 and b = 9.

a b = 11 9 = 2a – b = 11 – 9 = 2

11 and 9 are odd numbers, but 11 – 9 = 2, which is not an odd number. The statement is false.is not an odd number. The statement is false.

Ch k It O t! E l 4Check It Out! Example 4

Find a counterexample to show that each statement is falsestatement is false.a. The set of negative integers is closed

under multiplication.p

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) = 2a b = –2(–1) = 2

Since 2 is not a negative integer, this is a counterexample. The statement is false.counterexample. The statement is false.

Check It Out! Example 4

Find a counterexample to show that each statement is false.

b. The whole numbers are closed under the ti f t ki toperation of taking a square root.

Find a whole number, a, such that is not a whole number

Try a = 15.

whole number.

Since is not a whole number, this is a counterexample. The statement is false.

The Commutative and Associative Properties are true for addition and multiplication. They may not be true for other operations other operations.

A counterexample is an example that disproves a statement, or shows that it is false. One counterexample statement, or shows that it is false. One counterexample is enough to disprove a statement.

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

Additional Example 2: Finding Counterexamples to Additional Example 2: Finding Counterexamples to Statements About Properties

Find a counterexample to disprove the statement p p“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.

T 2 d 3Try a³ = 2³, and c² = 3².a³ = b2³ = 8

c² = d3² = 92 = 8 3 = 9

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

Check It Out! Example 2pFind a counterexample to disprove the statement “The Commutative Property is true for division ”for division.

Find two real numbers a and b, such thatTry a = 4 and b = 8Try a = 4 and b = 8.

Since , this is a counterexample.

The statement is falseThe statement is false.

Documents

properties of real numbers 2 - staff.pausd.orgstaff.pausd.org/~pjorgens/papers/16notes.pdfProperties of Real Numbers Real Numbers - The set of numbers consisting of positive numbers,