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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