Integers – Notes for CBSE Class 7 Maths

Introduction to Integers

Integers  -Mathematics often deals with numbers, and in Class 7, we begin exploring a broader set of numbers called integers. Integers include all whole numbers, both positive and negative, along with zero. This chapter introduces students to integers, their representation, and operations like addition, subtraction, multiplication, and division.

• Definition: Integers are a set of numbers that include:
  • Positive numbers (e.g., 1, 2, 3, …)
  • Negative numbers (e.g., -1, -2, -3, …)
  • Zero (0), which is neither positive nor negative.
• The set of integers is denoted by Z and written as:
  Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
• Real-Life Examples:
  • Temperature: 5°C (positive), -2°C (negative).
  • Altitude: 100 meters above sea level (positive), 50 meters below sea level (negative).
  • Money: Earning ₹50 (positive), owing ₹20 (negative).

Representation of Integers on a Number Line

A number line is a visual tool to represent integers. It is a straight line with:

• Zero (0) at the center.
• Positive integers (1, 2, 3, …) to the right of zero.
• Negative integers (-1, -2, -3, …) to the left of zero.
• Key Points:
  • The distance between consecutive integers is always 1 unit.
  • Moving right increases the value, and moving left decreases it.
  • Example: To represent -3, move 3 units left from 0. To represent 4, move 4 units right from 0.

Properties of Integers

Integers have specific properties that help us perform operations efficiently. These properties are crucial for solving problems.

1. Closure Property:
   • Addition: The sum of two integers is always an integer.
     Example: 5 + (-3) = 2 (an integer).
   • Multiplication: The product of two integers is always an integer.
     Example: (-4) × 2 = -8 (an integer).
2. Commutative Property:
   • Addition: Changing the order of integers does not change the sum.
     Example: 3 + (-5) = -2 and (-5) + 3 = -2.
   • Multiplication: Changing the order does not change the product.
     Example: 2 × (-6) = -12 and (-6) × 2 = -12.
3. Associative Property:
   • Addition: Grouping of integers does not affect the sum.
     Example: (2 + (-3)) + 4 = 2 + ((-3) + 4) = 3.
   • Multiplication: Grouping does not affect the product.
     Example: (2 × (-3)) × 4 = 2 × ((-3) × 4) = -24.
4. Distributive Property:
   • Multiplication distributes over addition.
     Example: 2 × (3 + (-5)) = (2 × 3) + (2 × (-5)) = 6 + (-10) = -4.
5. Identity Property:
   • Additive Identity: 0 is the identity for addition (adding 0 to any integer gives the same integer).
     Example: 7 + 0 = 7, (-4) + 0 = -4.
   • Multiplicative Identity: 1 is the identity for multiplication.
     Example: 5 × 1 = 5, (-3) × 1 = -3.
6. Additive Inverse:
   • For every integer, there exists an opposite integer (negative) such that their sum is 0.
     Example: 5 + (-5) = 0, (-2) + 2 = 0.

Operations on Integers

The chapter focuses on four main operations with integers: addition, subtraction, multiplication, and division. Let’s break them down with rules and examples.

1. Addition of Integers

• Rule 1: Same Signs
  Add the absolute values and keep the common sign.
  • Example: 4 + 6 = 10 (both positive).
  • Example: (-3) + (-5) = -8 (both negative).
• Rule 2: Different Signs
  Subtract the smaller absolute value from the larger one and take the sign of the integer with the larger absolute value.
  • Example: 7 + (-3) = 7 – 3 = 4.
  • Example: (-8) + 5 = -8 + 5 = -3.

2. Subtraction of Integers

• Subtraction is rewritten as addition of the additive inverse.
  Rule: a – b = a + (-b).
  • Example: 5 – 3 = 5 + (-3) = 2.
  • Example: (-4) – 2 = (-4) + (-2) = -6.
  • Example: 6 – (-2) = 6 + 2 = 8.

3. Multiplication of Integers

• Rules:
  • Positive × Positive = Positive.
    Example: 3 × 4 = 12.
  • Negative × Negative = Positive.
    Example: (-2) × (-5) = 10.
  • Positive × Negative = Negative.
    Example: 6 × (-3) = -18.
  • Negative × Positive = Negative.
    Example: (-7) × 2 = -14.
• Key Note: The product of an even number of negative integers is positive, and an odd number is negative.

4. Division of Integers

• Rules:
  • Positive ÷ Positive = Positive.
    Example: 12 ÷ 3 = 4.
  • Negative ÷ Negative = Positive.
    Example: (-15) ÷ (-5) = 3.
  • Positive ÷ Negative = Negative.
    Example: 20 ÷ (-4) = -5.
  • Negative ÷ Positive = Negative.
    Example: (-18) ÷ 6 = -3.
• Note: Division by zero is undefined.

Word Problems Using Integers

Integers are often used to solve real-world problems. Here’s how to approach them:

1. Identify positive and negative quantities.
2. Assign integers accordingly.
3. Apply the appropriate operation.

• Example 1: A shopkeeper earns ₹500 but spends ₹700. What is his net gain or loss?
  Solution: Net = Earnings – Spending = 500 + (-700) = 500 – 700 = -200.
  He incurs a loss of ₹200.
• Example 2: The temperature drops by 3°C from -2°C. What is the new temperature?
  Solution: New temperature = -2 + (-3) = -5°C.

Key Points to Remember

• Negative integers are less than zero, and positive integers are greater than zero.
• Zero is neutral and has no sign.
• On a number line, numbers to the left are smaller, and numbers to the right are larger.
• When performing operations, pay attention to signs to determine the result.

Practice Questions

1. Find: (-15) + 27.
2. Subtract: (-9) – (-4).
3. Multiply: (-6) × (-8).
4. Divide: 24 ÷ (-6).
5. A plane is at 1500 m above sea level. It descends 2000 m. What is its new altitude?

Solutions

1. (-15) + 27 = 12.
2. (-9) – (-4) = (-9) + 4 = -5.
3. (-6) × (-8) = 48.
4. 24 ÷ (-6) = -4.
5. New altitude = 1500 + (-2000) = 1500 – 2000 = -500 m (500 m below sea level).

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Conclusion

Chapter 1: Integers lays the foundation for understanding numbers beyond whole numbers. By mastering the properties and operations of integers, students can solve a variety of mathematical and real-life problems. Practice with number lines, operations, and word problems will strengthen this concept further.

These notes cover the essentials of the chapter as per the CBSE Class 7 syllabus, ensuring clarity and depth for students. Keep practicing to build confidence!

Chapter 1 Integers