Subscribe to our YouTube channel for the latest videos, updates, and tips.


Properties of Adding Integers

The properties of adding integers are discussed here along with the examples.

1. Closure Property: The addition (sum) of any two integers is always an integer.

i.e., 

The sum of integers is always an integer.

Hence, integers are closed under addition. If x and y are two integers, then x + y is always an integer.

For example:

(i) 16 + 48 = 64, which is an integer.

(ii) 12 + (-38) = -26, which is an integer.

(iii) - 24 + (- 14) = - 38, which is an integer.

(iv) 42 + (- 10) = 32, which is an integer.

(v) 5 + 9 = 14 ∈ Z

(vi) (-5) + 9 = 4 ∈ Z

(vii) (-5) + (-9) = -14 ∈ Z

(viii) 5 + (-9) = -4 ∈ Z                    and so on.

2. Commutative Property: Two integers can be added in any order.

Hence, addition is commutative for integers.

For any two integers ‘x’ and ‘y’;

x + y = y + x

For example:

(i) (-7) + 18 = 11 and 18 + (-7) = 11

Therefore, (-7) + 18 = 18 + (-7)


(ii) (-28) + (-5) = - 33 and (-5) + (-28) = -33

Therefore, (-28) + (-5) = (-5) + (-28)


(iii) (+3) + (+8) = (+8) + (+3)

(iv) (-7) + (+3) = (+3) + (-7)

(v) (-9) + (-3) = (-3) + (-9)

(vi) (+5) + (-3) = (+5) + (-3)                    and so on.


3. Associative Property: Three or more integers can be grouped in any order to find their sum. Hence, addition is associative for integers.

For any three integers ‘x’ ‘y’ and ‘z’;

x + (y + z) = (x + y) + z


For example:

(i) [(-5) + (-3)] + 10 = (-8) + 10 = 2 and (-5) + [(-3) + 10] = (-5) + (7) = 2

Therefore, [(-5) + (-3)] + 10 = (-5) + [(-3) + 10]


(ii) [(- 24) + 12] + 6 = (-12) + 6 = -6 and (- 24) + (12 + 6) = - 24 + 18 = -6

Therefore, [(- 24) + 12] + 6 = (- 24) + (12 + 6)


(iii) (+5) + [(-2) + (+3)] = [(+5) + (-2)] + (+3)

(iv) (-3) + [(-4) + (-5)] = [(-3) + (-4)] + (-5)

(v) (+4) + [(+2) + (+3)] = [(+4) + (+2)] + (+3)

(vi) (-2) + [(+3) + (-4)] = [(-2) + (+3)] + (-4)

(vii) (-4) + [(-3) + (+5)] = [(-4) + (-3)] + (+5)

(viii) (+3) + [(+4) + (-2)] = [(+3) + (+4)] + (-2)

(ix) (-3) + [(2) + (7)] = [(-3) + (2)] + (7)

(x) 9 + [(-4) + (-2)] = [9 + (-4)] + (-2)                    and so on.


4. Existence of Additive Identity: The sum of any integer and 0 is the integer itself, 0 is the additive identity for integers.

For any integer ‘x’;

x + 0 = 0 + x = x 

For example:

(i) 100 + 0 = 0 + 100 = 100

(ii) (-45) + 0 = 0 + (-45) = -45

(iii) (+7) + 0 = 0 + (+7) = +7

(iv) (-11) + 0 = 0 + (-11) = -11

(v) 0 + (+9) = (+9) + 0 = +9

(vi) 0 + (-5) = (-5) + 0 = -5                    and so on.


5. Existence of Additive Inverse: For any integer x, there exists its opposite -x such that their sum is zero, i.e., 

x + (-x) = (-x) + x = 0

Integers x and -x are called opposites or negatives or additive inverses of each other. 

For example:

(i) 15 + (-15) = (-15) + 15 = 0.

Thus, the additive inverse of 15 is -15 and 

         the additive inverse of -15 is 15.

(ii) 56 + (-56) = (-56) + 56 = 0.

Thus, the additive inverse of 56 is -56 and 

         the additive inverse of -56 is 56.


(iii) 5 + (-5) = 0

(iv) (-7) + 7 = 0                    and so on.


6. Successor and Predecessor of an Integers: If x is any integer, then (x + 1) is called the successor of x and x - 1 is called the predecessor of x.

For example:

(i) Successor of 6 is 6 + 1 = 7;     Predecessor of 6 is 6 - 1 = 5

(ii) Successor of -5 is -5 + 1 = -4;     Predecessor of -5 is -5 - 1 = -6


Properties of Adding Integers


Solved Examples on Properties of Adding Integers:

1. Fill in the blanks and make each of the following a true statement.

(i) The additive inverse of 17 is __________.

(ii) The additive inverse of -48 is __________.

(iii) The successor of 12 is __________.

(iv) The successor of -90 is __________.

(v) The predecessor of 1000 is __________.

(vi) The predecessor of -10000 is __________.


Solution:

(i) The additive inverse of 17 is -17; [Since, 17 + (-17) = 0]

(ii) The additive inverse of -48 is 48; [Since, (-48) + 48 = 0]

(iii) The successor of 12 is 13; [Since, 12 + 1 = 13] 

(iv) The successor of -90 is -89; [Since, -90 + 1 = -89]

(v) The predecessor of 1000 is 999; [Since, 1000 - 1 = 999]

(vi) The predecessor of -10000 is -10001; [Since, -10000 - 1 = -10001]


2. Example in Find the sum of the following.

(i) (- 15) + (- 18) + 26 + 45

(ii) 42 + (- 4) + (- 78) + (- 7)


Solution:

(i) (- 15) + (- 18) + 26 + 45

  = (- 33) + (71)

  = + (71 - 33)

  = +38

  = 38


(ii) 42 + (- 4) + (- 78) + (- 7)

  = 42 + (-89)

  = - (89 - 42)

  = - (47)

  = - 47


3. Find an integer 'n" such that

(i) 10 + n = 0

(ii) n + (- 7) = 0


Solution:

(i) 10 + n = 0

⟹ (- 10) + 10 + n = (- 10) + 0; [Adding (-10) on both sides]

⟹ [(- 10) + 10] + n = - 10; [Using associative property and property of 0]

⟹ 0 + n = - 10

Hence, n = - 10.


(ii) n + (- 7) = 0

⟹ n + (- 7) + 7 = 0 + 7; [Adding 7 on both sides]

⟹ n + [(- 7) + 7] = 7; [Using associative property and property of 0]

⟹ n + 0 = 7

Hence, n = 7

You might like these




Numbers Page

6th Grade Page

From Properties of Adding Integers to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. How to Find the Average? | What Does Average Mean? | Definition

    May 15, 25 06:05 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  2. Worksheet on Rounding Off Number | Rounding off Number | Nearest 10

    May 15, 25 05:12 PM

    In worksheet on rounding off number we will solve 10 different types of problems. 1. Round off to nearest 10 each of the following numbers: (a) 14 (b) 57 (c) 61 (d) 819 (e) 7729 2. Round off to

    Read More

  3. Worksheet on Rounding Decimals | Questions Related to Round a Decimal

    May 15, 25 11:52 AM

    Worksheet on Rounding Decimals
    The worksheet on rounding decimals would be really good for the students to practice huge number of questions related to round a decimal. This worksheet include questions related

    Read More

  4. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 14, 25 03:01 PM

    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More

  5. Rounding Off to the Nearest Whole Number | Nearest 10, 100, and 1000

    May 13, 25 03:43 PM

    Nearest Ten
    Here we will learn how to rounding off to the nearest whole number?

    Read More