Properties of Addition of Integers:
Property 1: (Closure Property):
The sum of two integers is always an integer.
That is, for any two integers a and b, a + b is always an integer.
Verification: We have,
1) 8 + 54 = 62 is an integer
2) 5 + (-9) = -4 and -4 is an integer.
Property 2: (Commutativity of addition):
The addition of integers is commutative, i.e, for any two integers a and b, we have a + b = b + a
Verification: We have,
15 + (-7) = -8 and (-7) + 15 = -8
Therefore, 15 + (-7) = (-7) + 15.
Property 3: (Associative Law of Addition):
If a, b, c are any three integers then (a + b) + c = a + (b + c).
Verification: Consider the Integers -5, —7 and 3.
We have,
((—5) + (—7)) + 3 = (—12) + 3 = -9.
And, (—5) + ((—7) + 3) = (-5) + (-4) = —9.
Therefore, ((—5) + (—7)) + 3 = (—5) + ((—7) + 3)
Property 4:
If a is any integer then a + 0 = a and 0 + a = a.
Verification: We have,
(i) 8 + 0 = 8
(ii) (-9) + 0 = -9
Remark: 0 is called the additive identity.
Property 5:
The sum of an integer and its opposite is 0.
Thus, If a is an integer then a + (-a) = 0.
a and —a are called opposites or negatives or additive inverses of each other.
Verification: We have,
3 + (-3) = 0 and (-3) + 3 = 0.
Thus, the additive Inverse of 3 is —3.
And, the additive inverse of -3 is 3.
Remark: Clearly, the additive inverse of 0 is 0.
Successor and Predecessor of an Integer:
Let a be an integer.
Then, (a + 1) is called the successor of a.
And, (a — 1) is called the predecessor of a.
Examples:
(1) The successor of 27 is (27 + l) = 28.
(2) The successor of —12 is (—12 + 1) = —11.
(3) The predecessor of 32 is (32 — 1) = 31.
(4) The predecessor of —45 is (—45 —1) = —46.