Hi Students, welcome to **Amans Maths Blogs (AMB)**. In this article, we will study about **CBSE Class 7 Maths Chapter Integers**. It is the notes of **Integers **chapter which help for the students who study in class 7 of CBSE or any affiliated school.

In this notes, we discuss about **integers numbers**, **integers definition**, **positive integers**, **negative integers,** **natural numbers** and **whole numbers** etc.

We also cover some **properties of integers** like **commutative**, **associative** and **distributive** on the mathematical operation of integers as **addition of integers**, **subtractions of integers**, **multiplication of integers** and **division of integers** etc.

So let discuss one by one.

# Integers

**INTEGERS** is the set of **natural numbers** (1, 2, 3, …), **zero** (0) and the **negative** of natural numbers (-1, -2, -3, …).

The integers are denoted by Z. Thus, **Z = {…, -3, -2, -1, 0, 1, 2, 3, …}**

*ZAHLEN*, a German word, is used to denote integers.

Thus, integers is the set of

## Positive Integers

Positive integers is the natural numbers (1, 2, 3, …) which is located on the right hand side of zero on the number line.

## Negative Integers

Negative integers is the negative of the natural numbers (-1, -2, -3, …) which is located on the left hand side of zero on the number line. The negative integers are less than zero.

## Zero

Zero (0) is neither positive nor negative and it is located in between positive and negative integers on the number line.

# Operation of Integers

## Addition of Integers

In the **addition of integers**, we use the following rules:

Rule 1 : To add two positive integers, we add the numbers and assign positive sign to the addition. It means, **(+a) + (+b) = +(a + b)**.

For example: (+10) + (+15) = +25

Rule 2 : To add two negative integers, we add the numbers and assign negative sign to the addition. It means, **(-a) + (-b) = -(a + b)**.

For example: (-10) + (-15) = -25

Rule 3 : To add one positive and one negative integers, we subtract the numbers and assign the sign of larger number to the difference. It means, **(-a) + (+b) = -(a – b) if a > b and (-a) + (+b) = +(b – a) if b > a**.

For example: (-10) + (+15) = +5, (-25) + (+23) = -2.

The summary of **addition of integers** as below:

## Subtraction of Integers

In the **subtraction of integers**, we use the following rules:

Rule 1 : To subtract two positive integers, we subtract the numbers and assign the sign of larger number to the difference. It means, **(+a) – (+b) = +(a – b) if a > b and (+a) – (+b) = -(b – a) if b > a**.

For example: (+10) – (+15) = -5, (+20) – (+15) = +5

Rule 2 : To subtract two negative integers, we subtract the numbers and assign the sign of larger number to the difference. It means, **(-a) – (-b) = -(a – b) if a > b and (-a) – (-b) = +(b – a) if b > a**.

For example: (-10) – (-15) = +5, (-20) – (-15) = +5

Rule 3 : To subtract one positive and one negative integers, we add the numbers and assign the sign. It means, **(+a) – (-b) = +(a + b) and (-a) – (+b) = -(a + b)**.

For example: (+10) – (-15) = +25, (-20) – (+15) = -35

The summary of **subtraction of integers** as below:

## Multiplication of Integers

In the **multiplication of integers**, we use the following rules:

Rule 1 : To multiply two positive integers, we multiply the numbers and assign positive sign to the product. It means, **(+a) x (+b) = +(a x b)**.

For example: (+10) x (+15) = +150

Rule 2 : To multiply two negative integers, we multiply the numbers and assign positive sign to the product. It means, **(-a) x (-b) = +(a x b)**.

For example: (-10) x (-15) = +150

Rule 3 : To multiply one positive and one negative integers, we multiply the numbers and assign negative sign to the product. It means, **(-a) x (+b) = -(a x b) **and **(+a) x (-b) = -(a x b).**

For example: (-10) x (+15) = -150, (+20) x (-5) = -100

The summary of **multiplication of integers** as below:

## Division of Integers

In the **division of integers**, we use the following rules:

Rule 1 : To divide two positive integers, we divide the numbers and assign positive sign to the quotient. It means, **(+a) ÷ (+b) = +(a ÷ b)**.

For example: (+100) ÷ (+5) = +20

Rule 2 : To divide two negative integers, we divide the numbers and assign positive sign to the quotient. It means, **(-a) ÷ (-b) = +(a ÷ b)**.

For example: (-100) ÷ (-5) = +20

Rule 3 : To divide one positive and one negative integers, we divide the numbers and assign negative sign to the quotient. It means, **(-a) ÷ (+b) = -(a ÷ b) **and **(+a) ÷ (-b) = -(a ÷ b).**

For example: (-150) ÷ (+15) = -10, (+20) ÷ (-5) = -4

The summary of **division of integers** as below:

# Properties of Integers

## Properties of Addition of Integers:

**Closure Property of Addition**: Natural numbers, whole numbers and integers are closed under addition. It means, if we add two natural numbers, then we get the sum as a natural number, similarly for whole numbers and integers.

For example: 23 + 15 = 38; here 23 and 15 are natural numbers and their sum 38 is also a natural number.

**Commutative Property of Addition** : For two integers a and b, we have **a + b = b + a**. It means, the order of operation of addition on integers does not change the sum.

For example: 23 + 15 = 15 + 23 = 38

**Associative Property of Addition** : For three integers a, b and c, we have **a + (b + c) = (a + b) + c**. It means, the grouping of numbers does not change the sum

For example: 23 + (15 + 10) = (23 + 15) + 10 = 48

**Distributive Over Addition Property **: For three integers a, b and c, we have **a(b + c) = ab + ac.** It means, each integer inside the parenthesis is multiplied by the integer outside the parenthesis, then the resulting products are added together.

For example: 5(15 + 10) = (5 x 15) + (5 x 10) = 75 + 50 = 125

The summary of **properties of addition of integers** as below:

## Properties of Subtraction of Integers:

**Closure Property of Subtraction**: Natural numbers and whole numbers are **NOT** closed under subtraction. It means, if we subtract two natural numbers, then the difference is not always natural number similarly for whole numbers. **BUT**, the integers are closed under subtraction. It means, if we subtract two integers, then the difference is always integers.

For example: 23 – 43 = -20; here 23 and 43 are integers and their difference -20 is also an integer.

**Commutative Property of Subtraction** : For two integers a and b, we have **a – b ≠ b – a**. It means, the subtraction does **NOT** hold the commutative.

For example: 23 – 15 ≠ 15 – 23

**Associative Property of Addition** : For three integers a, b and c, we have **a – (b – c) = (a – b) – c**. It means, the grouping of numbers changes the difference.

For example: 23 – (15 – 10) ≠ (23 – 15) – 10

**Distributive Over ****Subtraction**** Property **: For three integers a, b and c, we have **a(b – c) = ab – ac.** It means, each integer inside the parenthesis is multiplied by the integer outside the parenthesis, then find the difference of the resulting products.

For example: 5(15 – 10) = (5 x 15) – (5 x 10) = 75 – 50 = 25

The summary of **properties of subtraction of integers** as below:

## Properties of Multiplication of Integers:

**Closure Property of Multiplication**: Natural numbers, whole numbers and integers are closed under multiplication. It means, if we add two natural numbers, then we get the product as a natural number, similarly for whole numbers and integers.

For example: 23 x 10 = 230; here 23 and 10 are natural numbers and their product 230 is also a natural number.

**Commutative Property of Multiplication** : For two integers a and b, we have **a x b = b x a**. It means, the order of operation of multiplication on integers does not change the product.

For example: 23 x 10 = 10 x 23 = 230

**Associative Property of Multiplication** : For three integers a, b and c, we have **a x (b x c) = (a x b) x c**. It means, the grouping of numbers does not change the product.

For example: 3 x (15 x 10) = (3 x 15) x 10 = 450

The summary of **properties of multiplication of integers** as below:

## Additive Identity

If we add zero (0) to any integers, then the value of the integers does not change.

It means **a + 0 = 0 + a = a**.

For example: 23 + 0 = 0 + 23 = 23

## Additive Inverse

If the sum of two numbers is zero (0), then each of the numbers is additive inverse of each other.

It means **a + b = 0 ⇒ b = -a**.

For example: 23 + (-23) = 0

## Multiplicative Identity

If we multiply one (1) to any integers, then the value of the integers does not change.

It means **a x 1 = 1 x a = a**.

For example: 23 x 1 = 1 x 23 = 23

## Multiplicative Inverse

If the product of two numbers is one (1), then each of the numbers is multiplicative inverse of each other.

It means **a x b = 1 ⇒ b = 1/a**.

For example: 23 x (1/23) = 1

# BODMAS Rule

To learn about **BODMAS** rule, **Click Here**.