The Commutative Property of Addition states that the sum of two numbers does not depend on the order in which they are added. In the example below, adding 35 and 50 in either order does not change the sum.
35 + 50 = 85
50 + 35 = 85
This example illustrates the Commutative Property of Addition
|
Commutative |
Words: The
order in which two numbers are added does not change their
sum. |
Likewise, the order in which you multiply numbers does not
matter.
|
Commutative |
Words: The order in which two numbers are multiplied does not
change their product. |
Some expressions are easier to evaluate if you group or associate
certain numbers. Look at the expression below.
16 + 7 + 3 = 16 + (7 + 3) Group
7 and 3.
= 16 + 10 Add 7 and 3.
= 26 Add
16 and 10.
This is an application of the Associative Property of Addition.
|
Associative |
Words: The way in which three numbers are grouped when they are
added does not change their sum. Symbols: For any numbers a, b, and c, (a + b) + c = a + (b + c). Numbers: (24 + 8) + 2 = 24 + (8 + 2) |
The Associative Property also holds true for multiplication.
|
Associative |
Words: The way in which three numbers are grouped when they are
multiplied does Symbols: For any numbers a, b, and c, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c). Numbers: (9 ⋅ 4) ⋅ 25 = 9 ⋅ (4 ⋅ 25) |
Example
Name the property shown by each statement.
1.
4
⋅ 11 ⋅ 2 = 11 ⋅ 4 ⋅ 2 Commutative Property of
Multiplication
2.
(n + 12) + 5 = n + (12 + 5) Associative Property of Addition
Example
3. Simplify the expression 15 + (3x + 8). Identify the properties used in each step.
15
+ (3x + 8) = 15 + (8 + 3x) Commutative
Property of Addition
= (15 + 8) + 3x Associative
Property of Addition
= 23 + 3x Substitution
Property
Example
Geometry Link
4. The
volume of a box can be found using the expression l × w × h, where l is the length, w is
the width, and h is the height. Find the volume of a box whose length is 30
inches, width is 6 inches, and height is 5 inches.
Alternative Solutions :
l × w × h = 30 × 6 × 5 Replace l with 30, w with 6, and h with 5.
=
30 × (6 × 5) Associative
Property of Multiplication
=
30 × 30 Substitution Property
=
900 Substitution
Property
The volume
of the box is 900 cubic inches.
Whole numbers are the numbers 0, 1, 2, 3, 4, and so on. When you add
whole numbers, the sum is always a whole number. Likewise, when you multiply
whole numbers, the product is a whole number. This is an example of the Closure Property. We say
that the whole numbers are closed under addition and multiplication.
|
Closure |
Words: Because the sum or product of two whole numbers is also a whole number, the set of whole
numbers is closed under addition and multiplication. Numbers: 2 + 5 = 7, and 7 is a whole number. 2 ⋅ 5 = 10, and 10 is a whole number. |
Are the whole numbers closed under
division? Study these examples.
2 ÷ 1 = 2 whole
number
28 ÷ 4 = 7 whole number
fraction
It is impossible to list every possible division expression
to prove that the Closure Property holds true. However, we can easily show that
the statement is false by finding one counterexample. A counterexample is an
example that shows the statement is not true. Consider 
.
While 5 and 3 are whole numbers, 
is not. So, the
statement The whole numbers are closed
under division is false.
Example
5. State whether the statement Division
of whole numbers is commutative is true or false. If false, provide a
counterexample.
Alternative Solutions :
Write
two division expressions using the Commutative Property and check to see
whether they are equal.
We found a
counterexample, so the statement is false. Division of whole numbers is not
commutative.
Sumber
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