Transcript Notes: Properties of Real Numbers (ppt)

SOL 7.16
Addend: a number that is added to another
Sum: The answer to an addition problem
Factor: a number that is being multiplied
Product: The answer to a multiplication problem
Difference: The answer to a subtraction problem
Reciprocal: the multiplicative inverse of a number
Identity Element: numbers that combine with other numbers without
changing the other numbers
Inverse: numbers that combine with other numbers and result in identity
elements
Make sure to learn these important terms…we
will use them from now on! 
Addition
Changing the order of the addends does not change the sum.
For Example:
5+2=2+5
m+𝑛 =𝑛+𝑚
3 + (2 ∙ 3) = 2 ∙ 3 + 3
Multiplication
Changing the order of the factors does not change the product.
For Example:
3∙6=6∙3
a𝑏 = 𝑏𝑎
What Changed from one side of the equal sign to the other?
What should we look for to help us identify this property?
The order of the addends or factors changed.
I think of commuters when I think of the commutative property. You start one
way and end, but to start again you must leave where you are and go back to
where you started.
Write down what you think of!
Complete the examples using the commutative property.
a) 4 + 5 = 5 + 4
b) 8 (3 - 2) = (3 - 2)8
c) 6- ( 5+ 9) = 6-(9 + 5)
d) fg= gf
Addition
Regrouping the addends does not change the sum.
For Example:
5 + 3 + 2 = 5 + (3 + 2)
𝑥 + (m + 𝑛) = (𝑥 + 𝑚) + 𝑛
Multiplication
Regrouping the factors does not change the product.
For Example:
2 3∙6 = 2∙3 6
a(𝑏 ∙ 𝑐) = 𝑎 ∙ 𝑏 𝑐
What Changed from one side of the equal sign to the other?
How is the associative property different from the commutative?
The number inside of the grouping symbols or parentheses changed.
In the commutative property the order of the numbers changed, but the
numbers that were in the parentheses changed with the associative.
How will you remember this property?
Complete the examples using the associative property.
a) 2+(4 + 5) = (2 + 4) + 5
b) 8(3 • 2) = (8 •3) •2
c) 6+( 5+ 9) =(6+5) + 9
d) a(b•c) = (a•b)c
Subtraction?
Is subtraction commutative? Associative?
For Example:
≠ 2−5
5 − 2__
≠ 2−3 −4
2 − 3 − 4 __
Division?
Is division commutative? Associative?
For Example:
≠
3 ÷ 6 ___6
÷3
10
20 ÷ 4
≠
___
20 ÷ 4
10
The product of a number and the sum (or difference) of two
other numbers equals the sum (or difference) of the
products of the number and each other number.
For Example:
3 5 + 2 = 3 ∙ 5 + (3 ∙ 2)
3 5 − 2 = 3 ∙ 5 − (3 ∙ 2)
3(6 ∙ 4) = 72
What Changed from one side of the equal sign to the other?
What should we look for to help us identify this property?
The number on the outside of the parentheses was passed out to the
numbers inside the parentheses.
I think of distributing flyers to the class. You start with one stack of flyers, then
once you distribute the stack to the class each person has that same flyer.
Write down what you think of!
Complete the examples using the distributive property.
a) 2(4 + 5) = (2•4) + (2•5)
b) 8 (3 - 2) = (8 •3)- (8•2)
c) 6( 5+ 9) = (6•5) + (6•9)
d) e(f-g)= (e•f) – (e•g)
Identity
The sum of any real number and zero is equal to the given
real number.
For Example:
5+0=5
0+𝑛 =𝑛
Inverse
The sum of a number and its additive inverse (opposite) always
equals zero.
For Example:
3 + (−3) = 0
−a + 𝑎 = 0
Identity
The product of any real number and one is equal to that given
number.
For Example:
5×1=5
1∙𝑛 =𝑛
Inverse
The product of a number and its multiplicative inverse
(reciprocal) always equals one.
For Example:
1
3∙ =1
3
1
𝑎∙ =1
𝑎
What are the identity elements? How can we tell the identity
properties from the inverse properties?
The additive identity element is 0 and the multiplicative identity
element is one.
The identity properties result in the same real number that was in the problem.
Also, in the identity properties 0 and 1 are being added or multiplied. They are
the answers in the inverse properties!
Complete the examples using the identity properties.
a) 4 + 0 = 4
b) 8 (1) = 8
c) 0+( 5+ 9) = (5+9)
Is there a number with no additive inverse? What is the
difference between the additive inverse and the
multiplicative inverse?
Zero does not have an opposite. It is neither positive or negative to have
an additive inverse.
With the additive inverse you are adding the opposite, but with multiplicative
inverse you are multiplying the reciprocal, not the opposite.
Complete the examples using the inverse properties.
-4 = 0
a) 4 +__
1
b) 8 (__)
=1
8
1
c) ∙ ___
2 =1
2
The product of any real number and zero is zero!
For Example:
5+3 ∙0=0
a∙0=0
0=2∙0
0 = 0 ∙ 197
Both the Multiplicative Property of Zero and the Additive
Identity both have an answer of zero. How will you remember
which property is which?
The multiplicative property of zero has a product of zero and there is a
zero both on the right and the left of the equal sign.
The additive identity has a sum of zero and there is only a zero on one side of
the equal sign.
Complete the examples using the multiplicative property of zero.
a) 2(0) = 0
b) 0 = 0 • any real number
0 =0
c)( 5- 9)• ___
Let’s see if you can identify these
properties!
3∙2=2∙3
1
8∙ =1
8
3 2+3 =3 3+2
4(3 ∙ 2) = 4 ∙ 3 2
3∙0=0
3=1∙3
4 + 2 + 1 = 4 + (2 + 1)
3(2 + 1) = 9
8 + (−8) = 0
2 1+3 =2∙1+2∙3
5 2 + 3 = (2 + 3) ∙ 5
1
1= ∙2
2
(5𝑥 + 1) + 0 = (5𝑥 + 1)
9 + 1 + 8 = 9 + (1 + 8)
3 𝑚 + 𝑛 = 3𝑚 + 3𝑛