In Mathematics, we have come across different types of numbers, such as whole numbers, natural numbers, complex numbers, imaginary numbers, etc. Each number is different from other numbers, even if they share some common properties. To understand the differences and similarities between the numbers, Mathematicians have developed a grouping system that categorizes the numbers based on the characteristics.

A number is an arithmetic value that represents the quantity and helps in making the calculations. People have been using numbers for thousands of years to count something in their real life. Here, we are going to learn the definition of integers, properties and operations on integers.

## Integers in Mathematics

In Math, integers are the numbers that include positive integers, negative integers and zero. Integers are the special group of numbers that consists of a set of numbers, such as {…., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}. All the positive and negative whole numbers are the integers, excluding the fractional and decimal numbers. The set of integers are represented using the symbol “Z”. The integers that follow each other in an order are called the ** consecutive integers**.

If ” n” can be any integer, then.

The general form of consecutive odd integer is 2n+1

The general form of consecutive even integers is 2n.

Some of the examples of integers include -9, -7, 4, 0, -2, and so on.

The numbers like ½, 3.14, 8/9, √2, √5, etc., are not integers as the numbers have the decimal part in them.

## Integer Operations

The various arithmetic operations in Maths are addition, subtraction, multiplication and division. We can perform these various operations using the integers. Integers use certain rules to perform the arithmetic operations. Now let us discuss the rules and operations on integers one by one.

**Addition Rule:**

The addition of two positive integers will result in the positive integer. Similarly, the addition of two negative integers will result in the addition operation with a negative sign.

For example,

- 3 + 2 = 5
- (-3) + (-2) = -5

In case if one number is positive and the other number is negative, the addition operation becomes a subtraction operation, and we have to consider the sign of the greatest integer.

- (-3) + 2 = -1

**Subtraction Rule:**

While subtracting two numbers, change the sign of the number which is being subtracted, and follow the same procedure of addition operation.

For example,

- (+5) – (+8) = +5 – 8 = -3
- (-5) – (-8) = -5 + 8 = 3
- (-5) – (+8) = -5 -8 = -13

**Multiplication and Division Rule:**

The rule for the multiplication of integers is the same as the division of the integers.

While multiplying or dividing two integers, follow these two rules.

If the sign of the two numbers is the same, then the result is positive.

If the sign of the two numbers is different, then the result is negative.

For example:

- (2) × (3) = 6
- (-2) × (-3) = 6
- (-2) × (4) = -8
- (2) × (-4) = -8

## Properties of Integers

As discussed above, each classification of numbers is based on the ** properties of numbers**. Now, let us discuss the properties of integer numbers.

Let a, b, c be any integers, and “b” is a non-zero integer, then the following properties hold for all integers.

**Closure Property:**

- Addition: a+b ∈ Z
- Subtraction: a-b ∈ Z
- Multiplication: a × b ∈ Z
- Division: a ÷ b ∉ Z

**Identity Property:**

- Addition: a + 0 = 0 + a = a
- Multiplication: a × 1 = 1 × a = a

**Commutative Property:**

- Addition: a+ b = b+a
- Multiplication: a × b = b × a

**Associative Property:**

- Addition: a + (b + c) = (a + b) +c
- Multiplication: a × (b × c) = (a × b) × c

**Distributive Property:**

- Addition: a × (b + c) = (a × b) + (a× c)
- Subtraction: a × (b − c) = (a × b) − (a × c)

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