We all know about natural numbers and whole numbers. The natural numbers denoted by N are the set of all positive numbers starting from one up to infinity. It is written N = { 1, 2, 3, 4, 5, 6………… ∞ }. Whole numbers denoted by W are the set of all the natural numbers with the addition of zero. It is written as W = {0, 1, 2, 3, 4, 5, 6………….. ∞ }. But where do the numbers below zero come under? All numbers below zero are negative numbers. They are written as natural numbers with a negative sign, or -N. The set of all numbers consisting of N, 0, and -N is called integers. Integers are basically any and every number without a fractional component. It is represented by the letter Z. The word integer comes from a Latin word meaning whole. Integers include all rational numbers except fractions, decimals, and percentages. If integers are represented on a number line, the positive numbers occupy the right side while the negative numbers occupy the left side. Integers represented by Z are a subset of rational numbers represented by Q. In turn rational numbers Q is a subset of real numbers R. Hence, integers Z are also a subset of real numbers R.

**Symbol Representation :**^{+}, Z_{+}, and Z^{>}are the symbols used to denote positive integers. The symbols Z^{-}, Z_{-}, and Z^{<} are the symbols used to denote negative integers. Also, the symbol Z^{≥ }is used for non-negative integers, Z^{≠} is used for non-zero integers. Z^{*}is the symbol used for non-zero integer.

**Operation of Integers :**

**Addition**** rule of integers**** :**

In case of addition of numbers of the same sign (either positive or negative) simply add the two numbers and put the sign before it.

If the numbers that are supposed to be added have different signs, subtract the smaller number from the larger number ignoring the sign, and then put the sign of the larger number before it.

Example : 2 + 2 = 4, ( -3 ) + (-6) = - 9, ( -8 ) + 4 = - 4

**Subtraction**** rule of integers**** :**

Example : 7 - 3 = 4, ( -4 ) - (-5) = ( -4_{(+) }5 ) = 1, 8 - (-6) = 8_{(+) }6 = 14

**Multiplication rule of integers :**

Example : 9 × 5 = 45, -9 × ( -4 ) = 36, -7 × 5 = ( -35 )

**Division rule of integers:**

Example : 12 ÷ 4 = 3, -16 ÷ 4 = ( -4 ), -36 ÷ ( -12 ) = 3

**Algebraic Properties of Integers:**

**Closure Property of integers :**

p + q is an integer

p – q is an integer

p × q is an integer

Integers are not closed under division, since p/q need not be an integer and can be a fraction. Integers are also not closed under exponentiation as the result can be a fraction if the exponent is negative.

**Associative Property of integers**:

Associative property of integers applies to multiplication and addition. This means for any three integers p, q, and r,

p + ( q + r ) = ( p + q ) + r

p × ( q × r ) = ( p × q ) × r

The associative property does not apply to division and subtraction.

**The existence of Additive Identity and ****Multiplicative Identity of integers :**

The additive identity of integers like the additive identity of any other number is zero.

p + 0 = p

A multiplicative identity is a number which when multiplied to an integer gives the same integer.

The multiplicative identity of integers like the multiplicative identity of any other number is one.

p × 1 = p

**The existence of Additive Inverse and Multiplicative Inverse of integers :**

The additive inverse of a positive number is the negative of the same number, while the additive inverse of a negative number is the positive of the same number.

p + ( -p ) = 0

-p + ( p ) = 0

A multiplicative inverse is a number which when multiplied to an integer gives the answer as one.

The multiplicative inverse of integers like the multiplicative identity of any other number is the reciprocal of the same number.

p × 1/p = 1

-p × ( -1/p ) = 1

**Distributive Property of integers** :

Distributive property of integers applies to multiplication over addition or multiplication over subtraction of integers. This means for any three integers p, q, and r,

p × ( q + r ) = ( p × q ) + ( p × r ) or ( p + q ) × r = ( p × r ) + ( q × r )

p × ( q - r ) = ( p × q ) - ( p × r ) or ( p - q ) × r = ( p × r ) - ( q × r )

In case of addition of numbers of the same sign (either positive or negative) simply add the two numbers and put the sign before it.

If the numbers that are supposed to be added have different signs, subtract the smaller number from the larger number ignoring the sign, and then put the sign of the larger number before it.

Example : 2 + 2 = 4, ( -3 ) + (-6) = - 9, ( -8 ) + 4 = - 4

Example : 7 - 3 = 4, ( -4 ) - (-5) = ( -4

Example : 9 × 5 = 45, -9 × ( -4 ) = 36, -7 × 5 = ( -35 )

Example : 12 ÷ 4 = 3, -16 ÷ 4 = ( -4 ), -36 ÷ ( -12 ) = 3

p + q is an integer

p – q is an integer

p × q is an integer

Integers are not closed under division, since p/q need not be an integer and can be a fraction. Integers are also not closed under exponentiation as the result can be a fraction if the exponent is negative.

Associative property of integers applies to multiplication and addition. This means for any three integers p, q, and r,

p + ( q + r ) = ( p + q ) + r

p × ( q × r ) = ( p × q ) × r

The associative property does not apply to division and subtraction.

The additive identity of integers like the additive identity of any other number is zero.

p + 0 = p

A multiplicative identity is a number which when multiplied to an integer gives the same integer.

The multiplicative identity of integers like the multiplicative identity of any other number is one.

p × 1 = p

The additive inverse of a positive number is the negative of the same number, while the additive inverse of a negative number is the positive of the same number.

p + ( -p ) = 0

-p + ( p ) = 0

A multiplicative inverse is a number which when multiplied to an integer gives the answer as one.

The multiplicative inverse of integers like the multiplicative identity of any other number is the reciprocal of the same number.

p × 1/p = 1

-p × ( -1/p ) = 1

Distributive property of integers applies to multiplication over addition or multiplication over subtraction of integers. This means for any three integers p, q, and r,

p × ( q + r ) = ( p × q ) + ( p × r ) or ( p + q ) × r = ( p × r ) + ( q × r )

p × ( q - r ) = ( p × q ) - ( p × r ) or ( p - q ) × r = ( p × r ) - ( q × r )