Class 6 Mathematics Notes Chapter 6 (Chapter 6) – Exemplar Problem (English) Book

Alright class, let's get started with Chapter 6, which is all about Integers. This is a crucial topic, not just for your current class but also as a foundation for higher mathematics and competitive exams. So, pay close attention!

Chapter 6: Integers - Detailed Notes for Exam Preparation

1. What are Integers and Why Do We Need Them?

• Beyond Whole Numbers: You already know about Whole Numbers (0, 1, 2, 3,...). But what about situations like:
  • Temperature below 0°C (like -5°C)
  • Depth below sea level (like -100 meters)
  • A loss in business (like -₹500)
  • Moving backwards or downwards
  • These situations require numbers less than zero.
• Definition: Integers are the collection of all whole numbers (0, 1, 2, 3, ...) AND the negative numbers (-1, -2, -3, ...).
  • The set of integers looks like this: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}
• Types:
  • Positive Integers: 1, 2, 3, 4, ... (These are the same as Natural Numbers)
  • Negative Integers: -1, -2, -3, -4, ...
  • Zero (0): It is an integer, but it is neither positive nor negative.

2. Representing Integers on a Number Line

• A number line is a visual way to understand integers.

• Steps:

  1. Draw a straight horizontal line.
  2. Mark a point in the middle and label it '0' (Origin).
  3. Mark points at equal distances to the right of 0 and label them 1, 2, 3, ... (Positive Integers).
  4. Mark points at equal distances to the left of 0 and label them -1, -2, -3, ... (Negative Integers).

  <--|---|---|---|---|---|---|---|---|---|---|-->
  ...-5 -4 -3 -2 -1 0 1 2 3 4 5...

• Key Idea: Numbers increase as you move to the right and decrease as you move to the left.

3. Ordering and Comparing Integers

• Rule: On the number line, any number to the right is always greater than any number to its left.
• Examples:
  • 5 > 2 (5 is to the right of 2)
  • 0 > -3 (0 is to the right of -3)
  • -2 > -5 (-2 is to the right of -5)
• Important Comparisons:
  • Every positive integer is greater than 0 and every negative integer.
  • 0 is greater than every negative integer.
  • Among two negative integers, the one that looks smaller (ignoring the sign) is actually greater (because it's closer to 0 on the number line). Example: -1 is greater than -100.

4. Operations on Integers

(a) Addition:

• Using Number Line:
  • To add a positive integer, move right on the number line.
  • To add a negative integer, move left on the number line.
  • Example: (+3) + (-5): Start at 3, move 5 steps left -> Reach -2. So, (+3) + (-5) = -2.
  • Example: (-2) + (+4): Start at -2, move 4 steps right -> Reach +2. So, (-2) + (+4) = 2.
• Rules without Number Line:
  • Same Signs: Add the numbers (ignoring signs) and keep the common sign.
    • (+5) + (+3) = +(5+3) = +8
    • (-5) + (-3) = -(5+3) = -8
  • Different Signs: Subtract the smaller number (ignoring signs) from the larger number (ignoring signs). Keep the sign of the number which was larger (ignoring signs).
    • (+7) + (-3): Larger is 7 (positive). Subtract 7-3=4. Result: +4.
    • (-7) + (+3): Larger is 7 (negative). Subtract 7-3=4. Result: -4.

(b) Additive Inverse:

• For any integer 'a', its additive inverse is '-a'.
• The sum of an integer and its additive inverse is always 0.
  • a + (-a) = 0
• Examples:
  • Additive inverse of 5 is -5 (because 5 + (-5) = 0)
  • Additive inverse of -8 is +8 (or 8) (because -8 + 8 = 0)
  • Additive inverse of 0 is 0.

(c) Subtraction:

• Key Rule: Subtracting an integer is the same as adding its additive inverse.
  • a - b = a + (-b)
  • a - (-b) = a + b
• Examples:
  • (+8) - (+3) = (+8) + (-3) = +5 (Using addition rule for different signs)
  • (+8) - (-3) = (+8) + (+3) = +11 (Using addition rule for same signs)
  • (-8) - (+3) = (-8) + (-3) = -11 (Using addition rule for same signs)
  • (-8) - (-3) = (-8) + (+3) = -5 (Using addition rule for different signs)
• Using Number Line:
  • Subtracting a positive integer means moving left.
  • Subtracting a negative integer means moving right.

5. Key Takeaways for Exams:

• Be very clear about the difference between positive and negative integers and the role of zero.
• Master the number line representation – it helps visualize comparisons and operations.
• Practice addition and subtraction rules until they become second nature. Remember the "subtracting is adding the inverse" rule.
• Understand the concept of additive inverse.
• Be prepared for word problems involving temperature changes, profit/loss, sea levels, scores, etc. Translate the words into integer operations.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 6 (Integers) for your practice:

1. Which of the following is the smallest integer?
   (a) -1
   (b) 0
   (c) -100
   (d) 1

2. The additive inverse of -12 is:
   (a) -12
   (b) 0
   (c) 12
   (d) 1/12

3. Which expression results in a negative integer?
   (a) (-5) + (+8)
   (b) (-3) - (-7)
   (c) (+4) + (+6)
   (d) (+2) - (+9)

4. On the number line, the integer 5 is located:
   (a) To the left of 0
   (b) To the right of 0
   (c) To the left of -5
   (d) To the left of 2

5. What must be subtracted from -3 to get -9?
   (a) -6
   (b) 6
   (c) -12
   (d) 12

6. The sum of two integers is +15. If one of them is -7, the other is:
   (a) +8
   (b) -8
   (c) +22
   (d) -22

7. Which of the following statements is TRUE?
   (a) 0 is the smallest integer.
   (b) -1 is greater than -10.
   (c) The additive inverse of a positive integer is positive.
   (d) Every integer is a whole number.

8. Calculate: (-10) + (+6) - (-3)
   (a) -7
   (b) -1
   (c) +1
   (d) -13

9. A submarine is at a depth of 500 m below sea level. It ascends (goes up) by 200 m. What is its new position?
   (a) 700 m below sea level (-700 m)
   (b) 300 m below sea level (-300 m)
   (c) 300 m above sea level (+300 m)
   (d) 700 m above sea level (+700 m)

10. The predecessor of the integer -99 is:
    (a) -98
    (b) -100
    (c) 98
    (d) 100

Answer Key for MCQs:

1. (c) -100 (Among the given options, -100 is furthest to the left on the number line)
2. (c) 12
3. (d) (+2) - (+9) = (+2) + (-9) = -7
4. (b) To the right of 0
5. (b) 6 (Let the number be x. -3 - x = -9 => -x = -9 + 3 => -x = -6 => x = 6)
6. (c) +22 (Let the other integer be y. (-7) + y = 15 => y = 15 - (-7) => y = 15 + 7 = 22)
7. (b) -1 is greater than -10 (On the number line, -1 is to the right of -10)
8. (b) -1 ( (-10) + 6 + 3 = -4 + 3 = -1 )
9. (b) 300 m below sea level (-300 m) (Represent initial position as -500 m. Ascending means adding positive. -500 + 200 = -300 m)
10. (b) -100 (Predecessor means the integer just before it, or one less. -99 - 1 = -100)

Study these notes carefully and practice the sums from your Exemplar book. Understanding integers well is key to success! Let me know if any part is unclear.