**Integers** are the numbers that can be positive, negative or zero, but cannot be a fraction. These numbers are used to perform various arithmetic operations, like addition, subtraction, multiplication, and division. The symbol of integers is “**Z**“.

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

The word integer originated from the Latin word “Integer” which means whole. It is a special set of whole numbers comprised of zero, positive numbers, and negative numbers and denoted by the letter Z.

### Types of Integers

There are different types of numbers in Mathematics, They are:

- Real Numbers
- Natural Numbers
- Whole Numbers
- Rational numbers
- Irrational numbers
- Even and Odd Numbers, etc.

**Rules of Integers**

- The Sum of two positive integrals is an integer
- The Sum of two negative integers is an integer
- The product of two positive integrals is an integer
- The product of two negative integers is an integer
- The Sum of an integer and its inverse is equal to zero
- Product of an integer and its reciprocal is equal to 1

### Properties

**Closure Property:**

According to the closure property of numbers, when two integers are added or multiplied together, it results in an integer only. If a and b are integers, then:

- a + b = integer
- a x b = integer

**Examples:**

2 + 5 = 7 (is an integer)

2 x 5 = 10 (is an integer)

**Commutative Property:**

According to the commutative property of integers, if a and b are two integers, then:

- a + b = b + a
- a x b = b x a

**Examples:**

3 + 8 = 8 + 3 = 11

3 x 8 = 8 x 3 = 24

But for subtraction and division, the commutative property does not obey.

**Associative Property:**

As per the associative property , if a, b and c are integers, then:

- a+(b+c) = (a+b)+c
- ax(bxc) = (axb)xc

**Examples:**

2+(3+4) = (2+3)+4 = 9

2x(3×4) = (2×3)x4 = 24

This property is applicable for addition and multiplication operations only.

#### Distributive property

According to the distributive property of integers, if a, b and c are integers, then:

a x (b + c) = a x b + a x c

Example: Prove that: 3 x (5 + 1) = 3 x 5 + 3 x 1

LHS = 3 x (5 + 1) = 3 x 6 = 18

RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18

Since, LHS = RHS

Hence, proved.

**Additive Inverse Property:**

If a is an integer, then as per the additive inverse property of numbers,

a + (-a) = 0

Hence, -a is the additive inverse of integer a.

**Multiplicative inverse Property:**

If a is an integer, then as per the multiplicative inverse property of numbers,

a x (1/a) = 1

Hence, 1/a is the multiplicative inverse of integer a.

**Identity Property:**

a+0 = a

a x 1 = a

Example: -100,-12,-1, 0, 2, 1000, 989 etc…

As a set, it can be represented as follows:

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}