# NCERT Class 7 Mathematics Chapter 10 Operations with Integers: Summary, Concepts, and Revision Notes | SwaVid

The chapter "Operations with Integers" builds upon your understanding of integers from Class 6. It systematically introduces and explains how to perform...

Canonical: https://swavid.com/maths/class/7/chapter/operations-with-integers

Source: https://swavid.com/maths/class/7/chapter/operations-with-integers

# Operations with Integers

## Concepts in this chapter

## The ideas this chapter keeps returning to

## Where this chapter usually trips people up

## Find the gaps before this chapter finds them for you.

## Related chapters

### Addition of Integers

Chapter 10 · Class 7 Mathematics

The chapter "Operations with Integers" builds upon your understanding of integers from Class 6. It systematically introduces and explains how to perform addition, subtraction, multiplication, and division with positive and negative numbers. You will learn the rules governing these operations and explore important properties that integers follow, or sometimes do not follow, under these operations. This knowledge is fundamental for advanced mathematics and for solving real-world problems involving quantities that can be both positive and negative.

Your study route

Key topics

This topic explains how to combine positive and negative integers. When adding integers with the same sign, add their absolute values and keep the common sign. When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the integer with the larger absolute value.

Example

Find the sum: (a) (-9) + (+4) = -5 (b) (-3) + (-5) = -8

Watch out

Students often confuse the operation sign with the integer's sign, especially when dealing with negative numbers, leading to errors like treating (-5) + (-3) as 5-3.

This topic explains how to combine positive and negative integers. When adding integers with the same sign, add their absolute values and keep the common sign. When adding integers with different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the integer with the larger absolute value.

Tap the card for an example

Example

Find the sum: (a) (-9) + (+4) = -5 (b) (-3) + (-5) = -8

Why it matters

Essential for calculating net changes in quantities like temperature, bank balances, or elevation, where values can increase or decrease.

Watch out

Students often confuse the operation sign with the integer's sign, especially when dealing with negative numbers, leading to errors like treating (-5) + (-3) as 5-3.

Ask at home

Ask your child to calculate sums like (-7) + 3, 5 + (-9), and (-4) + (-6). Check if they consistently apply the sign rules.

This chapter, "Operations with Integers," comprehensively covers the four fundamental arithmetic operations: addition, subtraction, multiplication, and division, specifically applied to integers. It begins by revisiting integer addition and subtraction, emphasizing the rules for combining numbers with same or different signs, and introduces subtraction as adding the additive inverse. The chapter then delves into multiplication, explaining how the product's sign is determined by the signs of the factors, and similarly for division. Crucially, it explores the properties of integers under each operation, including closure, commutativity, associativity, and the distributive property, along with the roles of additive and multiplicative identities and multiplication by zero. It highlights that while integers are closed under addition, subtraction, and multiplication, they are not always closed under division. Understanding these rules and properties is essential for accurate calculations and problem-solving involving integers.

Chapter summary

This chapter, "Operations with Integers," comprehensively covers the four fundamental arithmetic operations: addition, subtraction, multiplication, and division, specifically applied to integers. It begins by revisiting integer addition and subtraction, emphasizing the rules for combining numbers with same or different signs, and introduces subtraction as adding the additive inverse. The chapter then delves into multiplication, explaining how the product's sign is determined by the signs of the factors, and similarly for division. Crucially, it explores the properties of integers under each operation, including closure, commutativity, associativity, and the distributive property, along with the roles of additive and multiplicative identities and multiplication by zero. It highlights that while integers are closed under addition, subtraction, and multiplication, they are not always closed under division. Understanding these rules and properties is essential for accurate calculations and problem-solving involving integers.

What you should learn

Keep these close

When adding integers with the same sign, add their absolute values and keep the common sign.

When adding integers with different signs, subtract the smaller absolute value from the larger, and take the sign of the integer with the larger absolute value.

To subtract an integer, add its additive inverse (change the sign of the number being subtracted and then add).

The product or quotient of two integers with the same sign is always positive.

The product or quotient of two integers with different signs is always negative.

Multiplication by zero always results in zero.

Division by zero is undefined.

Integers are closed under addition, subtraction, and multiplication.

Addition and multiplication of integers are commutative and associative.

Subtraction and division of integers are neither commutative nor associative.

Zero is the additive identity for integers (a + 0 = a).

One is the multiplicative identity for integers (a × 1 = a).

Common confusions

It is easy to think

Subtracting a negative number is the same as subtracting a positive number.

The clearer idea

Subtracting a negative number is equivalent to adding its positive counterpart. For example, 5 - (-3) is 5 + 3 = 8, not 5 - 3 = 2.

It is easy to think

The product of two negative integers is always negative.

The clearer idea

The product of two negative integers is always positive. For example, (-4) × (-5) = 20, not -20.

It is easy to think

Division of integers always results in an integer.

The clearer idea

Division of integers does not always result in an integer. For example, 5 ÷ 2 is not an integer. This means integers are not closed under division.

It is easy to think

The order of numbers doesn't matter for subtraction or division.

The clearer idea

Subtraction and division are not commutative. For example, 10 - 5 is not the same as 5 - 10, and 10 ÷ 2 is not the same as 2 ÷ 10.

Before you push ahead

Most stuck chapters trace back to one earlier idea. Check the prerequisites first, or let SwaVid adapt this chapter to the way your child learns.

Keep exploring Class 7

Source

- Perform addition and subtraction of integers accurately, applying the correct sign rules.
- Multiply and divide integers, determining the sign of the product or quotient correctly.
- Identify and apply the properties of integers (closure, commutativity, associativity, distributivity, identities) for addition and multiplication.
- Recognize when properties like commutativity or associativity do not hold for subtraction and division of integers.
- Solve simple word problems involving operations with integers.
- Perform addition and subtraction of integers accurately, applying the correct sign rules.
- Multiply and divide integers, determining the sign of the product or quotient correctly.
- Identify and apply the properties of integers (closure, commutativity, associativity, distributivity, identities) for addition and multiplication.
- Recognize when properties like commutativity or associativity do not hold for subtraction and division of integers.
- Solve simple word problems involving operations with integers.
- NCERT Class 7 Mathematics textbook: Ganita Prakash-II : Chapter 10: Operations with Integers

## Key Links

- [NCERT Mathematics](https://swavid.com/maths)
- [Class 7](https://swavid.com/maths/class/7)
- [Start with concepts](https://swavid.com/maths/class/7/chapter/operations-with-integers)
- [Check a prerequisite](https://swavid.com/learning-debt-identifier)
- [Concepts](https://swavid.com/maths/class/7/chapter/operations-with-integers)
- [Practice](https://swavid.com/maths/class/7/chapter/operations-with-integers/practice-questions)
- [NCERT Solutions](https://swavid.com/maths/class/7/chapter/operations-with-integers/ncert-solutions)
- [Find the gaps before this chapter](https://swavid.com/learning-debt-identifier)
- [Learn this chapter adapted to you](https://swavid.com/learning-style-test)
- [Previous chapter 9 . Geometric Twins](https://swavid.com/maths/class/7/chapter/geometric-twins)
- [Next chapter 11 . Finding Common Ground](https://swavid.com/maths/class/7/chapter/finding-common-ground)
- [All Class 7 chapters](https://swavid.com/maths/class/7)
- [9 Geometric Twins Congruence of Plane Figures · Congruence of Line Segments Open chapter](https://swavid.com/maths/class/7/chapter/geometric-twins)
- [11 Finding Common Ground Reflectional Symmetry · Rotational Symmetry Open chapter](https://swavid.com/maths/class/7/chapter/finding-common-ground)
- [8 Working with Fractions Multiplying a Fraction by a Whole Number · Multiplying a Fraction by Another Fraction Open chapter](https://swavid.com/maths/class/7/chapter/working-with-fractions)
- [NCERT Class 7 Mathematics textbook: Ganita Prakash-II](https://ncert.nic.in/textbook/pdf/gegp202.pdf)