Integers Class 7 Notes: Chapter 1

Chapter 1: Integers Class 7 Notes by Team Lawtantra

Here are easy-to-understand notes for Class 7 Maths Chapter 1: Integers. These notes are helpful for a quick look at the chapter and make learning fun and simple. The NCERT Notes are great for studying on your own using your textbook. These notes will help you understand difficult topics easily and prepare better for exams. Experts at Lawtantra, who know Maths really well, have created these notes just for you!

Introduction to Integers

Integers are a group of numbers that include:

• Whole numbers (like 0, 1, 2, 3, ...)
• Negative numbers (like -1, -2, -3, ...).

Key Points About Integers:

• Positive integers (1, 2, 3, ...) are greater than negative integers (-1, -2, -3, ...).
• Zero is special! It is smaller than every positive number but bigger than every negative number.

Number Line Basics:
A number line helps us understand integers better.

• Add a positive number → Move right.
• Add a negative number → Move left.
• Subtract a positive number → Move left.
• Subtract a negative number → Move right.

Introduction to Numbers

Natural numbers are the numbers we use for counting things.

Examples: 1, 2, 3, 4, 5, 6, ... and so on.

Key Points:

• Smallest natural number is 1.
• Natural numbers do not include zero (0) or any negative numbers (-1, -2, ...).
• They are sometimes called "counting numbers" because we use them to count objects.

Fun Fact:
Natural numbers keep going forever—they never stop! 😊

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Whole numbers are all the natural numbers (1, 2, 3, ...) plus zero (0).

Examples: 0, 1, 2, 3, 4, 5, ... and so on.

Key Points:

• The smallest whole number is 0.
• Whole numbers do not include negative numbers (-1, -2, ...) or fractions (like 1/2).
• Zero is the only whole number that is not a natural number.

Easy Way to Remember:
Whole numbers = Natural numbers + 0

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Integers Operations

Addition/Subtraction of Integers

• Adding Positive Numbers:
  When two positive integers are added, the result is positive.
  Example: 3 + 3 = 6


• Adding Negative Numbers:
  When two negative integers are added, the result is negative.
  Example: (-2) + (-1) = -3


• Adding Positive and Negative Numbers:
  Subtract the smaller number from the bigger number and keep the sign of the bigger number.
  Example: (-3) + 1 = -2 (since 3 is bigger, the answer is negative)


• Additive Inverse:
  The additive inverse of any integer is the number that makes the sum zero.
  Example: The additive inverse of 5 is -5, and the additive inverse of -5 is 5.

First No.	Second No.	Result
Positive	Positive	Positive
Positive	Negative	Negative
Negative	Positive	Negative
Negative	Negative	Positive

Note: In addition or subtraction of integers, the sign of the result depends on the bigger number (in absolute value).

Properties of Addition:

• Closure Property: Adding any two integers always gives an integer.
  Example: 20 + 10 = 30, which is an integer.


• Commutative Property: Changing the order of numbers does not change the result.
  Example: 7 + (-6) = -1 and (-6) + 7 = -1


• Associative Property: Grouping of numbers does not change the result.
  Example: (-7) + [(-2) + (-1)] = [(-7) + (-2)] + (-1) = -10


• Additive Identity: Adding zero to any integer does not change its value.
  Example: 5 + 0 = 5


• Repeated Addition: is a way to add the same number multiple times, and it is the basis of multiplication.
  Example: 1 + 1 + 1 + 1 + 1 = 5 × 1 = 5

Properties of Subtraction:

• Closure Property: Subtracting any two integers always gives an integer.
  Example: 20 - 10 = 10, which is an integer.


• Subtraction is Not Commutative: The order of subtraction matters.
  Example: 20 - 30 ≠ 30 - 20


• Subtraction is Not Associative: Changing the grouping in subtraction changes the result.
  

Multiplication of Integers:

• Positive × Negative = Negative
  Example: 2 × (-4) = -8


• Negative × Negative = Positive
  Example: (-2) × (-4) = 8


• Even Number of Negatives = Positive
  Example: (-1) × (-2) × (-3) × (-4) = 24


• Odd Number of Negatives = Negative
  Example: (-1) × (-2) × (-3) = -6

Properties of Multiplication:

• Closure Property: Multiplying any two integers gives an integer.
  Example: 2 × (-4) = -8, which is an integer.


• Commutative Property: Changing the order of multiplication does not change the result.
  Example: (-2) × (-4) = (-4) × (-2)


• Associative Property: Grouping of numbers does not change the result.
  Example: (-1) × [(-2) × (-3)] = [(-1) × (-2)] × (-3)


• Distributive Property: Multiplication distributes over addition.
  Example: (-2) × (3 + 4) = [(-2) × 3] + [(-2) × 4]


• Multiplicative Identity: Any number multiplied by 1 stays the same.
  Example: 1 × 2 = 2


• Multiplication by Zero: Any number multiplied by zero gives zero.
  Example: 2 × 0 = 0

Division of Integers:

• Positive ÷ Negative = Negative
  Example: 4 ÷ (-2) = -2


• Negative ÷ Negative = Positive
  Example: (-4) ÷ (-2) = 2


• Dividing by 1: Any number divided by 1 stays the same.
  Example: 2 ÷ 1 = 2


• Division by Zero: Division by zero is not defined.