1.6: Properties of Addition
- understand the commutative and associative properties of addition
- understand why 0 is the additive identity
We now consider three simple but very important properties of addition.
The Commutative Property of Addition
If two whole numbers are added in any order, the sum will not change.
Add the whole numbers
8 + 5 = 13
5 + 8 = 13
The numbers 8 and 5 can be added in any order. Regardless of the order they are added, the sum is 13.
Practice Set A
Use the commutative property of addition to find the sum of 12 and 41 in two different ways.
- Answer
-
12 + 41 = 53 and 41 + 12 = 53
Practice Set A
Add the whole numbers
- Answer
-
837 + 1,958 = 2,795 and 1,958 + 837 = 2,795
Associative Property of Addition
If three whole numbers are to be added, the sum will be the same if the first two are added first, then that sum is added to the third, or, the second two are added first, and that sum is added to the first.
Using Parentheses
It is a common mathematical practice to use parentheses to show which pair of numbers we wish to combine first.
Add the whole numbers.
Practice Set B
Use the associative property of addition to add the following whole numbers two different ways.
- Answer
-
(17 + 32) + 25 = 49 + 25 = 74 and 17 + (32 + 25) = 17 + 57 = 74
Practice Set B
- Answer
-
(1,629 + 806) + 429 = 2,435 + 429 = 2,864
1,629 + (806 + 429) = 1,629 + 1,235 = 2,864
The Additive Identity
0 Is the Additive Identity
The whole number 0 is called the additive identity , since when it is added to any whole number, the sum is identical to that whole number.
Add the whole numbers.
29 + 0 = 29
0 + 29 = 29
Zero added to 29 does not change the identity of 29.
Practice Set C
Add the following whole numbers.
- Answer
-
8
Practice Set C
- Answer
-
5
Suppose we let the letter x represent a choice for some whole number. For the first two problems, find the sums. For the third problem, find the sum provided we now know that x represents the whole number 17.
Practice Set C
- Answer
-
x
Practice Set C
- Answer
-
x
Practice Set C
- Answer
-
17
Exercises
For the following problems, add the numbers in two ways.
Exercise \(\PageIndex{1}\)
- Answer
-
37
Exercise \(\PageIndex{2}\)
Exercise \(\PageIndex{3}\)
- Answer
-
45
Exercise \(\PageIndex{4}\)
Exercise \(\PageIndex{5}\)
- Answer
-
568
Exercise \(\PageIndex{6}\)
Exercise \(\PageIndex{7}\)
- Answer
-
122,323
Exercise \(\PageIndex{8}\)
Exercise \(\PageIndex{9}\)
- Answer
-
45
Exercise \(\PageIndex{10}\)
Exercise \(\PageIndex{11}\)
- Answer
-
100
Exercise \(\PageIndex{12}\)
Exercise \(\PageIndex{13}\)
- Answer
-
556
Exercise \(\PageIndex{14}\)
Exercise \(\PageIndex{15}\)
- Answer
-
43,461
For the following problems, show that the pairs of quantities yield the same sum.
Exercise \(\PageIndex{16}\)
(11 + 27) + 9 and 11 + (27 + 9)
Exercise \(\PageIndex{17}\)
(80 + 52) + 6 and 80 + (52 + 6)
- Answer
-
132 + 6 = 80 + 58 = 138
Exercise \(\PageIndex{18}\)
(114 + 226) + 108 and 114 + (226 + 108)
Exercise \(\PageIndex{19}\)
(731 + 256) + 171 and 731 + (256 + 171)
- Answer
-
987 + 171 = 731 + 427 = 1,158
Exercise \(\PageIndex{20}\)
The fact that (a first number + a second number) + third number = a first number + (a second number + a third number) is an example of the property of addition.
Exercise \(\PageIndex{21}\)
The fact that 0 + any number = that particular number is an example of the property of addition.
- Answer
-
Identity
Exercise \(\PageIndex{22}\)
The fact that a first number + a second number = a second number + a first number is an example of the property of addition.
Exercise \(\PageIndex{23}\)
Use the numbers 15 and 8 to illustrate the commutative property of addition.
- Answer
-
15 + 8 = 8 + 15 = 23
Exercise \(\PageIndex{22}\)
Use the numbers 6, 5, and 11 to illustrate the associative property of addition.
Exercise \(\PageIndex{23}\)
The number zero is called the additive identity. Why is the term identity so appropriate?
- Answer
-
…because its partner in addition remains identically the same after that addition
Exercises for Review
Exercise \(\PageIndex{24}\)
How many hundreds in 46,581?
Exercise \(\PageIndex{25}\)
Write 2,218 as you would read it.
- Answer
-
Two thousand, two hundred eighteen.
Exercise \(\PageIndex{26}\)
Round 506,207 to the nearest thousand.
Exercise \(\PageIndex{27}\)
Find the sum of \(\begin{array} {r} {482} \\ {\underline{+\ \ 68}} \end{array}\)
- Answer
-
550
Exercise \(\PageIndex{28}\)
Find the difference: \(\begin{array} {r} {3,318} \\ {\underline{-\ \ 429}} \end{array}\)