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Properties of Addition

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses properties of addition. By the end of the module students should be able to understand the commutative and associative properties of addition and understand why 0 is the additive identity.

Section Overview

  • The Commutative Property of Addition
  • The Associative Property of Addition
  • The Additive Identity

We now consider three simple but very important properties of addition.

The Commutative Property of Addition

Commutative Property of Addition

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

Sample Set A

Example 1

Add the whole numbers

8 and 5.


 8 + 5 = 13 8 + 5 = 13 size 12{8+5="13"} {}
5 + 8 = 13 5 + 8 = 13 size 12{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

Exercise 1

Use the commutative property of addition to find the sum of 12 and 41 in two different ways.

41 and 12.
Solution

12+41=5312+41=53 size 12{"12"+"41"="53"} {} and 41+12=5341+12=53 size 12{"41"+"12"="53"} {}

Exercise 2

Add the whole numbers

1,958 and 837.
Solution

837+1,958=2,795837+1,958=2,795 size 12{"837"+1,"958"=2,"795"} {} and 1,958+837=2,7951,958+837=2,795 size 12{1,"958"+"837"=2,"795"} {}

The Associative Property of Addition

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

Example 2

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

Exercise 3

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

17, 32, and 25.
Solution

(17+32)+25=49+25=74(17+32)+25=49+25=74 size 12{ \( "17"+"32" \) +"25"="49"+"25"="74"} {} and 17+(32+25)=17+57=7417+(32+25)=17+57=74 size 12{"17"+ \( "32"+"25" \) ="17"+"57"="74"} {}

Exercise 4

1,629, 806, and 429.
Solution

( 1, 629 + 806 ) + 429 = 2, 435 + 429 = 2, 864 ( 1, 629 + 806 ) + 429 = 2, 435 + 429 = 2, 864 size 12{ \( 1,"629"+"806" \) +"429"=2,"435"+"429"=2,"864"} {}

1, 629 + ( 806 + 429 ) = 1, 629 + 1, 235 = 2, 864 1, 629 + ( 806 + 429 ) = 1, 629 + 1, 235 = 2, 864 size 12{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

Example 3

Add the whole numbers.

29 and 0.


 29 + 0 = 29 29 + 0 = 29 size 12{"29"+0="29"} {}
0 + 29 = 29 0 + 29 = 29 size 12{0+"29"="29"} {}

Zero added to 29 does not change the identity of 29.

Practice Set C

Add the following whole numbers.

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.

Exercises

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

Exercise 11

36 and 12.

Exercise 13

117 and 26.

Exercise 15

4,251 and 1,096.

Exercise 16

73,205 and 49,118.

Solution

122,323

Exercise 17

265,094 and 32,508.

Exercise 19

18, 16, and 14.

Exercise 21

36, 84, and 7.

Exercise 23

11, 1019, and 586.

Exercise 24

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

Solution

43,461

For the following problems, show that the pairs of quantities yield the same sum.

Exercise 25

(11+27)+9(11+27)+9 size 12{ \( "11"+"27" \) +9} {} and 11+(27+9)11+(27+9) size 12{"11"+ \( "27"+9 \) } {}

Exercise 26

(80+52)+6(80+52)+6 size 12{ \( "80"+"52" \) +6} {} and 80+(52+6)80+(52+6) size 12{"80"+ \( "52"+6 \) } {}

Solution

132+6=132+6= size 12{"132"+6="138"} {}80+58=13880+58=138 size 12{"80"+"58"="138"} {}

Exercise 27

(114+226)+108(114+226)+108 size 12{ \( "114"+"226" \) +"108"} {} and 114+(226+108)114+(226+108) size 12{"114"+ \( "226"+"108" \) } {}

Exercise 28

(731+256)+171(731+256)+171 size 12{ \( "731"+"256" \) +"171"} {} and 731+(256+171)731+(256+171) size 12{"731"+ \( "256"+"171" \) } {}

Solution

987+171=987+171= size 12{"987"+"171"=1,"158"} {}731+427=1,158731+427=1,158 size 12{"731"+"427"=1,"158"} {}

Exercise 29

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 30

The fact that 0 + any number = that particular number is an example of the
property of addi­tion.

Solution

Identity

Exercise 31

The fact that a first number + a second number = a second number + a first number is an example of the
property of addi­tion.

Exercise 32

Use the numbers 15 and 8 to illustrate the com­mutative property of addition.

Solution

15+8=8+15=2315+8=8+15=23 size 12{5+8=8+"15"="23"} {}

Exercise 33

Use the numbers 6, 5, and 11 to illustrate the associative property of addition.

Exercise 34

The number zero is called the additive identity. Why is the term identity so appropriate?

Solution

…because its partner in addition remains identically the same after that addition

Exercises for Review

Exercise 35

((Reference)) How many hundreds in 46,581?

Exercise 36

((Reference)) Write 2,218 as you would read it.

Solution

Two thousand, two hundred eighteen.

Exercise 37

((Reference)) Round 506,207 to the nearest thousand.

Exercise 38

((Reference)) Find the sum of 482 +   68 ̲ 482 +   68 ̲

Solution

550

Exercise 39

((Reference)) Find the difference: 3,318 -   429 ̲ 3,318 -   429 ̲

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