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
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
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.
Commutative Property of Addition
Name the property that is illustrated in each equation.
Th d ia. n + (–7) = –7 + n
Commutative Property of Addition
The order is different.
b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3
Associative Property of Addition
The grouping is different.
c. (xy)z = (yx)z
Commutative Property of Multiplication
The order is different.
Commutative Property of Multiplication
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.
Name the property that is illustrated in each equation.
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
Associative Property of Multiplication
Identify which property thatIdentify which property that justifies each of the following.
6 8 8 66 + 8 = 8 + 6
Commutative Property of Addition
Identify which property thatIdentify which property that justifies each of the following.
12 0 1212 + 0 = 12
Additive Identity Property
Identify which property thatIdentify which property that justifies each of the following.
5(2 9) (5 2) (5 9)5(2 + 9) = (5 2) + (5 9)
Distributive Property
Identify which property thatIdentify which property that justifies each of the following.
5 (2 8) (5 2) 85 + (2 + 8) = (5 + 2) + 8
Associative Property of Addition
Identify which property thatIdentify which property that justifies each of the following.
5 959
95
1
Multiplicative Inverse Property
Identify which property thatIdentify which property that justifies each of the following.
5 24 24 55 24 = 24 5
Commutative Property of Multiplication
Identify which property thatIdentify which property that justifies each of the following.
18 18 018 + -18 = 0
Additive Inverse Property
Identify which property thatIdentify which property that justifies each of the following.
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.
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.
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