You have worked extensively with the operations of addition, subtraction, multiplication and division in relation to numerical values. You know that order of operations dictates how, and when, these operations are performed. There are additional properties, such as the distributive property, that show us how to work with certain combinations of these operations. You are most likely familiar with other additional properties relating to these operations whose reference "names" are unknown or forgotten.
Let's take a look at some of these more common additional properties.
We will start with the familiar Distributive Property.
This property shows how to deal with an expression in the form of a (b + c).
This property shows multiplication "over" addition (or subtraction).
4 ( 3 + 2 ) = 4 • 3 + 4 • 2 = 12 + 8 = 20
When working with numerical values, we can show that this result is correct
by applying order of operations: 4 ( 3 + 2 ) = 4 (5) = 20
 Distributive Property |
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over Addition: For real numbers a, b and c, a ( b + c ) = ab + ac.
over Subtraction: For real numbers a, b and c, a ( b - c ) = ab - ac.
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The distributive property allows for a factor to be distributed to each member (term) of a group of numbers being added or subtracted.
The distributive property is commutative: a (b + c) = (b + c) a = ab + ac
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NOTE: The distributive property can make numerical computations easier by allowing for numbers to be broken down into numerical parts that are easier to compute.
Consider: 4(55) = 4(50 + 5) = 200 + 20 = 220
8(19) = 8(20 - 1) = 80(20 + (-1)) = 160 - 8 = 152
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Now, let's take a look at some properties that you most likely know are true,
but whose reference "names" may be unknown, or forgotten.
Commutative Property (ordering) |
We know that the order in which two numbers are added does not affect the answer (the sum).
4 + 7 = 11 and 7 + 4 = 11
The same is true for the order of multiplication. The order in which you multiply does not matter.
3 • 8 = 24 and 8 • 3 = 24
Commutative Property (order) |
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of Addition: For real numbers a and b, a + b = b + a.
of Multiplication: For real numbers a and b, a • b = b • a.
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When adding or multiplying real number values, the order does NOT affect the result.
Addition and multiplication are commutative.
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But what happens when you subtract or divide?
The order of subtracting two number DOES AFFECT the answer.
4 - 7 = -3 but 7 - 4 = 3 NOT the SAME result!
The order of dividing two numbers DOES AFFECT the answer.
 NOT the SAME result! |
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Subtraction and division are NOT commutative. |
NOTE: The commutative property for addition can make addition easier.
Consider: 9 + 3 + 4 + 2 + 5 + 1 + 7 + 6 + 8
The commutative property allows us to "rearrange" the locations of these values.
9 + 1 + 3 + 7 + 4 + 6 + 2 + 8 + 5 = 10 + 10 + 10 + 10 + 5 = 45
(quick and simple without a calculator) |
NOTE: In a similar manner, the commutative property for addition is working when you combine like terms.
Consider: 3a + 5 + 6a - 4
3a + 5 + 6a + (- 4) (change subtraction to addition of a negative)
3a + 6a + 5 + (- 4) (commutative property)
9a + 1 |
Associative Property (grouping) |
We know that changing the grouping of three numbers when adding, does not affect the answer.
(3 + 4) + 6 = 3 + (4 + 6) = 13
Notice: the parentheses moved, but the numbers did not move.
Also, when changing the grouping of three numbers when multiplying does not affect the answer.
(6 • 2) • 4 = 6 • (2 • 4)= 48
Notice: the parentheses moved, but the numbers did not move.
Associative Property (grouping) |
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of Addition: For real numbers a, b and c, (a + b) + c = a + (b + c).
of Multiplication: For real numbers a, b and c, (a • b) • c = a • (b • c).
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Under addition or multiplication, changing the grouping does NOT affect the result.
Addition and multiplication are associative.
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But what happens when you subtract or divide?
Changing grouping under subtraction DOES AFFECT the answer.
(3 - 4) - 6 = -7 but 3 - (4 - 6) = 5 NOT the SAME result!
Notice: the parentheses moved, but the numbers did not move.
Changing the grouping under division DOES AFFECT the answer.
 NOT the SAME result!
Notice: the parentheses moved, but the numbers did not move.
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Subtraction and division are NOT associative. |
Identity Property (no change) |
The Identity Property. under the operations of addition or multiplication, will leave the starting value unchanged. It will return the original value.
Adding 0 to any number, does not change the value of the number.
Zero is called the additive identity
6 + 0 = 6.
Multiplying 1 times any number, does not change the value of the number.
One is called the multiplicative identity.
8 • 1 = 8
Identity Property (no change) |
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for Addition: For any real number a, zero is the additive identity.
a + 0 = a and 0 + a = a.
for Multiplication: For real number a, one is the multiplicative identity.
a • 1 = 1 and 1 • a = a.
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Inverse Property (opposite, inverse) |
The Inverse Property, under the operations of addition or multiplication, will return the identity value of that operation. For addition, zero is returned. For multiplication, one is returned.
What number when added to 7 gives the additive identity of 0?
7 + (-7) = 0 [ -7 is the "opposite" of 7 ]
Notice: a number and its opposite add to zero.
What number multiplied times ¾ gives the multiplicative identity of 1?
 [ 4/3 is the "inverse" (reciprocal) of 3/4]
Notice: a number and its inverse multiply to one. |
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Inverse Property (opposite, inverse) |
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of Addition: For real number a, a + (-a) = 0.
A number and its opposite add to zero.
(-a is the additive inverse of a)
of Multiplication: For real number a,  .
A number and its inverse (reciprocal) multiply to one.
(  is the multiplicative inverse of a)
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We already have stated that when adding zero to any number (Identity Property), we get 0.
And you are most likely already familiar with:
• any number times zero is zero: 3 • 0 = 0
• zero times any number is zero: 0 • 4 = 0
• any number divided by 0 is undefined. 5 ÷ 0 = undefined
• zero divided by any number (not zero) is 0. 0 ÷ 5 = 0
Zero Property of Multiplication: |
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For any real number a, a • 0 = 0 and 0 • a = 0
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Zero Property of Division: |
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For real number a, 0 ÷ a = 0 as long as a does not = 0.
a ÷ 0 is undefined
(division by zero is undefined)
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Division by zero can lead to false conclusions.
Consider this proof that 1 = 2: |
| Start with a and b being equal and not zeros: |
| a = b |
| Multiply by a |
| a2 = ab |
| Subtract b2 |
| a2 - b2 = ab = b2 |
| Factor both sides |
| (a + b)(a - b) = b(a - b) |
| Divide by (a - b) |
a + b = b
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| Substitute a = b |
| b + b = b |
| Combine |
| 2b = b |
| Divide by non-zero b |
| 2 = 1 |
While all of the steps in this proof look like acceptable algebraic maneuvers,
the problem is the fifth step down.
"Divide by (a - b)"
In this particular situation, where a = b, we have (a - b) = 0.
Division by zero is undefined.
And, as seen here, such use can produce false results,
or invalid conclusions.
Particular care must be taken when working with algebraic division.
We might be dividing by zero and not realize it at first glance.
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