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

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

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