Chapter 01 – Section 06
Identity and Equality Properties
To recognize and use the properties of identity and
equality, and to determine the multiplicative inverse of a number.
This section is all about mathematical properties.
You do NOT need to memorize these properties by name.
You have been using these properties for years without knowing their
names.
The only time I have ever needed to know the names was in Advanced
Modern Algebra 1, 2, & 3 – the hardest math course you could take as an
undergraduate at UGA.
Focus on understanding the meaning of the properties.
© William James Calhoun
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Question: What is 0 plus any number?
Answer: 0 plus any number always gives you that number.
• additive identity- a number that can add to other numbers without
changing their values
Zero is the additive identity since it can add without changing values.
1.6.1 ADDITIVE IDENTITY PROPERTY
For any number a, a + 0 = 0 + a = a.
Think about this…
0 + 5 is the same as 5 + 0 is the same as 5.
© William James Calhoun
Question: What is 1 times any number?
Answer: 1 times any number always gives you that number.
• multiplicative identity- a number that can multiply with other numbers
without changing their values
The number one is the multiplicative identity since it can multiply
without changing values.
1.6.2 MULTIPLICATIVE IDENTITY PROPERTY
For any number a, a * 1 = 1 * a = a.
Think about this…
1 * 8 is the same as 8 * 1 is the same as 8.
© William James Calhoun
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Question: What is 0 times any number?
Answer: 0 times any number always gives you zero.
This property of multiplying by zero is known as the:
• multiplicative property of zero- a property of zero that when it
multiplies by any number, it yields zero
1.6.3 MULTIPLICATIVE PROPERTY OF ZERO
For any number a, a * 0 = 0 * a = 0.
Think about this…
0 * 14 is the same as 14 * 0 is the same as 0.
© William James Calhoun
Two more terms:
• multiplicative inverses- two numbers whose product is 1
• reciprocals - two numbers whose product is 1
You need to remember these two terms – they will be used in many other
sections!
Since any number times 0 always gives you 0,
0 does NOT have a multiplicative inverse.
You can not multiply it by anything to get to 1- you would have to add.
© William James Calhoun
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1.6.4 MULTIPLICATIVE INVERSE PROPERTY
a
For every nonzero number b , where a and b ≠ 0, there is exactly one
b
a b
number
such that ⋅ = 1 .
a
b a
What this property says is that you can
1) Take any non- zero, rational number,
2) Write that number as a fraction,
3) Find its reciprocal (flip it over),
4) The two numbers multiplied will always yield one.
© William James Calhoun
EXAMPLE 1α: Name the multiplicative inverse of each number or
variable. Assume no variable equals zero.
a. 5
b. x
c. 2/3
Write 5 as a fraction
5/
1
Flip it over to get the
multiplicative inverse.
Write x as a fraction
x/
1
Flip it over to get the
multiplicative inverse.
Solution is 1/5.
Solution is 1/x.
This is already written as a
fraction so just flip it over
to get the multiplicative
inverse.
Solution is 3/2.
EXAMPLE 1β: Name the multiplicative inverse of each number or
variable. Assume no variable equals zero.
a. 3
b. x
c. xy
3
© William James Calhoun
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For Example 1, the instructions have “Assume no variable equals zero”?
Why?
Look at question 1α, part b.
b. x
The solution was 1/x and if x = 0, then the solution would be 1/0.
Try putting 1/0 into your calculator.
You get an error because YOU CAN NEVER DIVIDE BY ZERO.
Dividing is breaking something up into groups. Divide by 2 means
divide into 2 groups.
You can not take something and divide it into nothing!
© William James Calhoun
Question: Three students have 12
- inch rulers.
One student cuts a -3 inch piece off. The second cuts a -5 inch
piece. The third keeps their piece whole.
How much does each student have?
Answer: Each student still has 12
- inches of ruler.
No matter how divided, the pieces of the ruler still add to 12.
3 inch + 9 inch = 9 inch + 3 inch = 12 inch
• reflexive property of equality- any quantity is equal to itself
1.6.5 REFLEXIVE PROPERTY OF EQUALITY
For any number a, a = a.
Any number is equal to itself.
© William James Calhoun
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• symmetric property of equality- if one quantity equals a second
quantity, then the second quantity also equals the first.
1.6.6 SYMMETRIC PROPERTY OF EQUALITY
For any numbers a and b, if a = b, then b = a.
If Mike is as old as Mary, then Mary is as old as Mike.
• transitive property of equality- if one quantity equals a second quantity
and the second quantity equals a third quantity, then the third quantity
equals the first quantity.
1.6.7 TRANSITIVE PROPERTY OF EQUALITY
For any numbers a, b, and c, if a = b and b = c, then a = c.
If Mike is as old as Mary, and Mary is as old as Chuck, then Mike and
Chuck are also the same age.
© William James Calhoun
• substitution property of equality- a quantity may be substituted for its
equal in any expression
1.6.8 SUBSTITUTION PROPERTY OF EQUALITY
If a = b, then a may be replaced by b in any expression.
If Mike and Mitch are tailbacks in football, Mike can substitute for Mitch
and Mitch can substitute for Mike.
Is that enough properties for you?!?
© William James Calhoun
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Please keep this in mind:
I know these are a lot of properties to remember.
You have been using most of these properties for years in your math
classes.
We are just giving them official names now.
Do NOT let the names intimidate you!
You only need to apply these properties- no need to memorize!
The next slide is a summary of the properties in layman’s terms (where
possible.)
© William James Calhoun
Additive Identity
Multiplicative Identity
the number 0 because adding zero
changes nothing
the number 1 because multiplying by
zero changes nothing
Multiplicative Property of Zero zero times anything yields zero
Multiplicative Inverse Property a number times its flip always equals 1
Reflexive Property
a number is always equal to itself
Symmetric Property
if a = b then b = a
Transitive Property
if a = b and b = c then a = c
Substitution Property
if a = b, you can exchange them any
time
© William James Calhoun
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PAGE 42
#21 – 25 odd
#41 – 45 odd – no properties
#49 – 57 odd
© William James Calhoun
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