Class 7 Mathematics Notes Chapter 1 (Integers) – Mathematics Book

Alright class, let's dive into Chapter 1: Integers. This is a fundamental chapter, and understanding it well will be crucial not just for Class 7, but for all your future mathematics and competitive exams. Pay close attention!

Chapter 1: Integers - Detailed Notes for Government Exam Preparation

1. What are Integers?

• Definition: Integers are a collection of whole numbers (0, 1, 2, 3, ...) and their negative counterparts (-1, -2, -3, ...).
• Set Notation: The set of integers is usually denoted by Z (from the German word 'Zahlen' meaning 'numbers').
  Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
• Types:
  • Positive Integers: 1, 2, 3, 4, ... (These are also the natural numbers)
  • Negative Integers: -1, -2, -3, -4, ...
  • Zero (0): Zero is an integer which is neither positive nor negative.
• Number Line Representation: Integers can be visualized on a number line.
  • Zero is at the center.
  • Positive integers are to the right of zero.
  • Negative integers are to the left of zero.
  • Numbers increase as we move to the right and decrease as we move to the left.
    <--|---|---|---|---|---|---|---|---|---|---|---|-->
    ... -5 -4 -3 -2 -1 0 1 2 3 4 5 ...

2. Operations on Integers: Recap and Rules

• Addition:
  • Positive + Positive = Positive (Sum the numbers) e.g., 5 + 3 = 8
  • Negative + Negative = Negative (Sum the absolute values, keep the negative sign) e.g., (-5) + (-3) = -8
  • Positive + Negative (or Negative + Positive): Subtract the smaller absolute value from the larger absolute value. Keep the sign of the number with the larger absolute value.
    • e.g., (-5) + 3 = -(5-3) = -2
    • e.g., 5 + (-3) = +(5-3) = 2
• Subtraction: Subtracting an integer is the same as adding its additive inverse (opposite). a - b = a + (-b)
  • e.g., 5 - 3 = 5 + (-3) = 2
  • e.g., 5 - (-3) = 5 + 3 = 8
  • e.g., (-5) - 3 = (-5) + (-3) = -8
  • e.g., (-5) - (-3) = (-5) + 3 = -2

3. Properties of Addition and Subtraction of Integers

• a) Closure Property:
  • Addition: Integers are closed under addition. If 'a' and 'b' are integers, then a + b is always an integer. (e.g., 2 + (-5) = -3, which is an integer). Holds True.
  • Subtraction: Integers are closed under subtraction. If 'a' and 'b' are integers, then a - b is always an integer. (e.g., 2 - 5 = -3, which is an integer). Holds True.
• b) Commutative Property:
  • Addition: Addition is commutative for integers. a + b = b + a for any integers 'a' and 'b'. (e.g., (-3) + 5 = 2 and 5 + (-3) = 2). Holds True.
  • Subtraction: Subtraction is not commutative for integers. a - b ≠ b - a generally. (e.g., 5 - 3 = 2, but 3 - 5 = -2). Does NOT Hold True.
• c) Associative Property:
  • Addition: Addition is associative for integers. (a + b) + c = a + (b + c) for any integers 'a', 'b', and 'c'. (e.g., ((-2) + 3) + 5 = 1 + 5 = 6 and (-2) + (3 + 5) = (-2) + 8 = 6). Holds True.
  • Subtraction: Subtraction is not associative for integers. (a - b) - c ≠ a - (b - c) generally. (e.g., (5 - 3) - 2 = 2 - 2 = 0, but 5 - (3 - 2) = 5 - 1 = 4). Does NOT Hold True.
• d) Additive Identity:
  • Zero (0) is the additive identity for integers. Adding 0 to any integer gives the integer itself. a + 0 = 0 + a = a for any integer 'a'. (e.g., -7 + 0 = -7).
• e) Additive Inverse:
  • For every integer 'a', there exists an integer '-a' such that a + (-a) = 0. '-a' is called the additive inverse or opposite of 'a'. (e.g., The additive inverse of 5 is -5; the additive inverse of -8 is 8).

4. Multiplication of Integers

• Sign Rules:
  • Positive × Positive = Positive (+) × (+) = + (e.g., 3 × 5 = 15)
  • Positive × Negative = Negative (+) × (-) = - (e.g., 3 × (-5) = -15)
  • Negative × Positive = Negative (-) × (+) = - (e.g., (-3) × 5 = -15)
  • Negative × Negative = Positive (-) × (-) = + (e.g., (-3) × (-5) = 15)
• Multiplication by Zero: Any integer multiplied by zero is zero. a × 0 = 0 × a = 0.
• Multiplication by One: One (1) is the multiplicative identity. Any integer multiplied by 1 is the integer itself. a × 1 = 1 × a = a.
• Multiplication by Minus One: a × (-1) = -a (gives the additive inverse).

5. Properties of Multiplication of Integers

• a) Closure Property: Integers are closed under multiplication. If 'a' and 'b' are integers, then a × b is always an integer. (e.g., -2 × 7 = -14, which is an integer). Holds True.
• b) Commutative Property: Multiplication is commutative for integers. a × b = b × a for any integers 'a' and 'b'. (e.g., (-4) × 6 = -24 and 6 × (-4) = -24). Holds True.
• c) Associative Property: Multiplication is associative for integers. (a × b) × c = a × (b × c) for any integers 'a', 'b', and 'c'. (e.g., ((-2) × 3) × (-4) = (-6) × (-4) = 24 and (-2) × (3 × (-4)) = (-2) × (-12) = 24). Holds True.
• d) Multiplicative Identity: One (1) is the multiplicative identity for integers. a × 1 = 1 × a = a.
• e) Distributive Property of Multiplication over Addition: This is very important! Multiplication distributes over addition. a × (b + c) = (a × b) + (a × c) for any integers 'a', 'b', and 'c'.
  • e.g., -2 × (3 + 5) = -2 × 8 = -16
  • Also, (-2 × 3) + (-2 × 5) = (-6) + (-10) = -16. Holds True.
• f) Distributive Property of Multiplication over Subtraction: a × (b - c) = (a × b) - (a × c) for any integers 'a', 'b', and 'c'. Holds True.

6. Division of Integers

• Sign Rules: Similar to multiplication.
  • Positive ÷ Positive = Positive (+) ÷ (+) = + (e.g., 15 ÷ 5 = 3)
  • Positive ÷ Negative = Negative (+) ÷ (-) = - (e.g., 15 ÷ (-5) = -3)
  • Negative ÷ Positive = Negative (-) ÷ (+) = - (e.g., (-15) ÷ 5 = -3)
  • Negative ÷ Negative = Positive (-) ÷ (-) = + (e.g., (-15) ÷ (-5) = 3)
• Important Points:
  • Division by Zero: Division by zero is undefined. a ÷ 0 is not defined for any integer 'a'.
  • Zero divided by any non-zero integer is zero. 0 ÷ a = 0 (where a ≠ 0).
  • Any integer divided by 1 gives the integer itself. a ÷ 1 = a.
  • Any non-zero integer divided by itself is 1. a ÷ a = 1 (where a ≠ 0).
  • Any non-zero integer divided by -1 gives its additive inverse. a ÷ (-1) = -a.

7. Properties of Division of Integers

• a) Closure Property: Division is not closed for integers. If 'a' and 'b' (b≠0) are integers, a ÷ b may or may not be an integer. (e.g., (-10) ÷ 2 = -5 (integer), but 5 ÷ 2 = 2.5 (not an integer)). Does NOT Hold True.
• b) Commutative Property: Division is not commutative for integers. a ÷ b ≠ b ÷ a generally. (e.g., 10 ÷ 2 = 5, but 2 ÷ 10 = 0.2). Does NOT Hold True.
• c) Associative Property: Division is not associative for integers. (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) generally. (e.g., ((-16) ÷ 4) ÷ (-2) = (-4) ÷ (-2) = 2, but (-16) ÷ (4 ÷ (-2)) = (-16) ÷ (-2) = 8). Does NOT Hold True.

Key Takeaways for Exams:

• Memorize the sign rules for multiplication and division.
• Understand which properties (Closure, Commutative, Associative, Identity, Inverse, Distributive) hold true for which operations (Addition, Subtraction, Multiplication, Division). This is a common area for questions.
• Remember that division by zero is UNDEFINED.
• Practice simplifying expressions involving multiple operations on integers, remembering the order of operations (BODMAS/PEMDAS).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts covered:

1. The additive inverse of -15 is:
   a) -15
   b) 15
   c) 0
   d) 1

2. The product of (-12) × (-10) is:
   a) -120
   b) 120
   c) -22
   d) 22

3. What is the value of 8 - (-5)?
   a) 3
   b) -3
   c) 13
   d) -13

4. Which property is illustrated by the statement: (-6) + 4 = 4 + (-6)?
   a) Associative Property of Addition
   b) Commutative Property of Addition
   c) Additive Identity
   d) Closure Property of Addition

5. Calculate: (-24) ÷ 3
   a) 8
   b) -8
   c) 27
   d) -21

6. Which of the following statements is FALSE for integers?
   a) Addition is commutative.
   b) Multiplication is associative.
   c) Subtraction is commutative.
   d) Integers are closed under multiplication.

7. The integer which is its own additive inverse is:
   a) 1
   b) -1
   c) 0
   d) No such integer exists

8. Using the distributive property, simplify: (-5) × [(-2) + 7]
   a) -25
   b) 25
   c) -45
   d) 35

9. What is the result of 10 ÷ 0?
   a) 10
   b) 0
   c) 1
   d) Undefined

10. Evaluate: (-2) + (-3) × 4
    a) -20
    b) 20
    c) -14
    d) -10

Answer Key:

1. b) 15
2. b) 120
3. c) 13
4. b) Commutative Property of Addition
5. b) -8
6. c) Subtraction is commutative.
7. c) 0
8. a) -25 [(-5) × (-2)] + [(-5) × 7] = 10 + (-35) = -25 OR (-5) × [5] = -25
9. d) Undefined
10. c) -14 [Follow order of operations: Multiply first (-3) × 4 = -12, then add (-2) + (-12) = -14]

Make sure you thoroughly understand these concepts and practice solving various problems involving integers. Good luck with your preparation!