Chapter 14.2a Properties of Real Numbers.notebook
August 16, 2013
Bellwork:
Tell if the following statements are true or false. If false give a counterexample.
1) Integers are closed under subtraction.
2) The irrational numbers are closed under addition.
3) The whole number set is closed under division.
Homework Questions
Convert the decimal to a fraction.
4) 0.867867...
5) 0.444... Aug 13­4:07 PM
Aug 13­4:30 PM
Before we begin we need to define some symbols and how they are read in mathematics.
14.2a Properties of Real Numbers
∀ ­ for all
Be able to define the properties of the real number system and use them to evaluate expressions.
∈ ­ are elements of
R ­ the set of real numbers
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Aug 13­4:32 PM
Commutative Property
What does the word commute mean? How might that relate to the commutative property?
Commutative property examples:
Commutative Property: numbers can be added or multiplied in any order and the sum or product will be the same.
Addition
5 + 6 = 11
6 + 5 = 11
Commutative Property of Addition
∀ a, b ∈ R, a + b = b + a
Can you come up with one for multiplication?
"For all numbers a, b that are elements of the set of real numbers, a plus b = b plus a."
Commutative Property of Multiplication
∀ a, b ∈ R, ab = ba
Rewrite the statement in words. Aug 13­4:35 PM
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Chapter 14.2a Properties of Real Numbers.notebook
August 16, 2013
Associative Properties
What does the word associate mean? How might this relate to the associative properties of algebra?
Are subtraction and division commutative? If yes, write the properties in set notation. If no, give a counterexample.
Associative Properties: Numbers added or multiplied can be grouped in any way and the sum or product is the same.
Associative Property of Addition:
Subtraction:
∀ a, b, c ∈ R, (a + b) + c = a + (b + c) For all numbers a, b, c that are elements of the set of real numbers, the sum of a and b plus c is equal to a plus the sum of b and c.
Division:
Associative Property of Multiplication:
Write in set notation.
For all numbers a, b, c that are elements of the set of real numbers, the product of a and b times c equals a times the product of b and c.
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Examples of Associative Properties
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Are division and subtraction associative? If so, write the properties in similar set notation. If not, give counterexamples.
Multiplication:
Subtraction:
3(5*6) = (3*5)6
Can you write one for Addition?
Division:
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Aug 13­4:51 PM
Distributive Property
Multiplication is Distributive over Addition:
Distributive Property
Is division distributive over addition? subtraction?
∀ a, b, c ∈ R a(b + c) = ab + ac
Is Multiplication Distributive over Subtraction? If yes, in set notation. If not, give an counterexample.
Addition:
Subtraction:
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Aug 13­5:00 PM
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Chapter 14.2a Properties of Real Numbers.notebook
August 16, 2013
Symmetric Property
Reflexive Property
If a = b, then b = a.
States any real number is equal to itself
∀ a ∈ R, a = a
ex. If 3 + x = 7 + y, then 7 + y = 3 + x
Substitution Property
Transitive Property
If x = y, then y can be substituted for x in any expression.
If a =b and b = c, then a = c
ex. x = y and 3x + 9 = 35, then 3y + 9 = 35. ex. 2 + x = 8, 8 = 4y, then 2 + x = 4y
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Aug 13­5:09 PM
Identify what property each statement is illustrating.
Substitution Property vs. Transitive Property.
1) 5(3 + n) = 15 + 5n
2) 4 + x = y, then y = 4 + x
3) 5(7 + x) = 5(x + 7)
4) 2 + y = 2 + y
5) x = 4, 2x + 7 = y, then 2(4) + 7 = y
6) 7(16 * 4) = (7 * 16)4
7) 4 + 5 = 9, 9 = 32, 4 + 5 = 32
8) 5x + 8 = x5 + 8
9) 4x + (3x + 2) = (4x + 3x) + 2
Aug 13­5:11 PM
Using the distributive property
How do you distribute? Do you need to multiply/divide to all numbers in the parenthesis?
Aug 14­8:50 AM
You Try:
1) 4(2x + y)
2)
3)
4) n(5 + 4m) 8(12 + a)
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Chapter 14.2a Properties of Real Numbers.notebook
What about going backwards
August 16, 2013
You Try:
1) 6a + 6b =
3x + 3y = __ (__ + __)
2) ¾x + ¾y =
¼r ­ ¼s + ¼t = ___(___ ___ ___)
3) ab ­ bc = 4) 5kx + 12ky = Aug 14­9:04 AM
Homework
Why do Sky Divers Use Decimals? WS
Aug 13­5:12 PM
Aug 14­9:06 AM
Homework Answers
When do sky divers use decimals?
1) commutative multi.
2) commutative add.
3) associative add.
4) associative multi.
5) commutative add.
6) commutative multi.
7) associative multi.
8) commutative add.
9) associative add.
10) commutative add.
11) commutative multi.
12) associative multi.
13) associative add.
14) commutative add.
15) commutative multi.
16) associative add.
17) commutative multi.
18) associative multi.
Why are Handcuffs like souvenirs:
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M 3/4
R 4x
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W 4
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T k
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They are made for two wrists.
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