Identity Properties
Identity properties tell us how we can
add or multiply and get an answer that
is identical to the number
we started with.
Identity Property of Addition
The identity property of addition tells us that we can add zero to any number and get that identical
number as the answer.
a + 0 = a
That makes the identity element for addition:
Identity Property of Multiplication
The identity property of multiplication tells us that we can multiply any number by 1 and get that
identical number as the answer.
That makes the identity element for multiplication:
1a a
Inverse Properties
The inverse properties tell us how we can create the identity elements out of other numbers.
Sort of like building a sand castle!
Additive Inverse Property
The identity element for addition is
How can we add and get an answer equal to zero?
Ah ha! - 5 + 5 = 0
In fact: - a + a = 0
Additive Inverse Property
-a + a = 0
Numbers that have the same magnitude but different signs are called opposites.
Another term for opposite is additive inverse.
Additive Inverse Property
The additive inverse property tells us that when we add any number and its opposite the answer
will be zero.
- a + a = 0
Double Negative Property
Closely related to the concept of opposite is the double negative property.
- ( - a ) = a
We can think if that as two negative signs in a row convert to one positive sign.
(signs are the same, replace with +)
Double Negative Property
Closely related to the concept of opposite is the double negative property.
- ( - a ) = a
Or we can think if it as ‘taking the opposite of an opposite gets you back where you started’!
Multiplicative Inverse Property
The identity element for multiplication is
How can we multiply and get an answer equal to one?
Ah ha!
In fact:
12 1
21
1aa
Reciprocals
Reciprocals are the gymnasts of
algebra.
They just love to do
headstands!
Watch them flip!
2 3 flip
3 21 5
flip or 55 1
16 flip
6
Multiplicative Inverse Property
The multiplicative inverse property tells us that when we multiply any number by its reciprocal
the answer will be one.
Another term for reciprocal is multiplicative inverse.
Multiplication Property of Zero
Zero has a unique property that we will use a lot later in the semester.
Any number multiplied by zero equals zero.
0 0a
Reduction Property
When we reduce fractions, we are using the reduction property.
In other words, as long as b and c are not equal to zero, we can cancel and reduce.
ac ac a
bc bc b