### 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.