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1.6: Properties of Addition

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    49570

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    Learning Objectives

    • 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

    Definition: Commutative Property of Addition

    If two whole numbers are added in any order, the sum will not change.

    Sample Set A

    Add the whole numbers

    8 and 5.

    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.

    41 and 12.

    Answer

    12 + 41 = 53 and 41 + 12 = 53

    Practice Set A

    Add the whole numbers

    1,958 and 837.

    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.

    Sample Set B

    Add the whole numbers.

    43, 16, and 27. Two equations are displayed. (43 + 16) + 27 = 59 + 27 = 86. 43 + (16 + 27) = 43 +43 = 86. Arrows point to the two groupings of numbers in parenthesis to show that they are associated.

    Practice Set B

    Use the associative property of addition to add the following whole numbers two different ways.

    17, 32, and 25.

    Answer

    (17 + 32) + 25 = 49 + 25 = 74 and 17 + (32 + 25) = 17 + 57 = 74

    Practice Set B

    1,629, 806, and 429.

    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.

    Sample Set C

    Add the whole numbers.

    29 and 0.

    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.

    0 and 8.

    Answer

    8

    Practice Set C

    0 and 5.

    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

    x and 0.

    Answer

    x

    Practice Set C

    x and 0.

    Answer

    x

    Practice Set C

    0 and x.

    Answer

    17

    Exercises

    For the following problems, add the numbers in two ways.

    Exercise \(\PageIndex{1}\)

    8 and 29.

    Answer

    37

    Exercise \(\PageIndex{2}\)

    36 and 12.

    Exercise \(\PageIndex{3}\)

    48 and 36.

    Answer

    45

    Exercise \(\PageIndex{4}\)

    117 and 26.

    Exercise \(\PageIndex{5}\)

    456 and 112.

    Answer

    568

    Exercise \(\PageIndex{6}\)

    4,251 and 1,096.

    Exercise \(\PageIndex{7}\)

    73,205 and 49,118.

    Answer

    122,323

    Exercise \(\PageIndex{8}\)

    265,094 and 32,508.

    Exercise \(\PageIndex{9}\)

    5, 32, and 8.

    Answer

    45

    Exercise \(\PageIndex{10}\)

    18, 16, and 14.

    Exercise \(\PageIndex{11}\)

    10, 52, and 38.

    Answer

    100

    Exercise \(\PageIndex{12}\)

    36, 84, and 7.

    Exercise \(\PageIndex{13}\)

    17, 114, and 425.

    Answer

    556

    Exercise \(\PageIndex{14}\)

    11, 1019, and 586.

    Exercise \(\PageIndex{15}\)

    37,728, 4,472, and 1,261.

    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: \(\begin{array} {r} {3,318} \\ {\underline{-\ \ 429}} \end{array}\)

    This page titled 1.6: Properties of Addition is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .