Class 6 Mathematics Notes Chapter 1 (Chapter 1) – Exemplar Problem (English) Book

Alright class, let's get started with Chapter 1, "Knowing Our Numbers," from your NCERT Exemplar book. This chapter builds the foundation for understanding numbers, how we represent them, compare them, and use them in practical situations. These concepts are crucial, not just for your regular studies, but also frequently appear in various government exams. Pay close attention!

Chapter 1: Knowing Our Numbers - Detailed Notes for Exam Preparation

1. Introduction to Numbers:

• Numbers help us count, measure, compare, and order things.
• We use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent any number.

2. Comparing Numbers:

• Rule 1: The number with more digits is always greater.
  • Example: 1024 (4 digits) > 987 (3 digits).
• Rule 2: If the numbers have the same number of digits, compare the digits starting from the leftmost place. The number with the greater digit at the first point of difference is the larger number.
  • Example: Compare 5678 and 5692.
    • Leftmost digit (Thousands place): 5 = 5
    • Next digit (Hundreds place): 6 = 6
    • Next digit (Tens place): 7 < 9. Therefore, 5678 < 5692.
• Ascending Order: Arranging numbers from smallest to largest.
• Descending Order: Arranging numbers from largest to smallest.

3. Forming Numbers:

• Greatest Number: Arrange the given digits in descending order.
  • Example: Digits 3, 8, 1, 5. Greatest number: 8531.
• Smallest Number: Arrange the given digits in ascending order.
  • Example: Digits 3, 8, 1, 5. Smallest number: 1358.
• Important Note (Zero): If one of the digits is 0, to form the smallest number, place 0 at the second position from the left (after the smallest non-zero digit). Never start a number with 0 (unless it's just the number zero itself).
  • Example: Digits 4, 0, 2, 9. Smallest number: 2049 (not 0249). Greatest number: 9420.

4. Place Value and Face Value:

• Face Value: The face value of a digit is the digit itself, regardless of its position in the number.
  • Example: In 789, the face value of 8 is 8.
• Place Value: The place value of a digit depends on its position (place) in the number (Units, Tens, Hundreds, etc.). Place Value = Face Value × Positional Value.
  • Example: In 789:
    • Place value of 9 is 9 × 1 = 9 (Units place)
    • Place value of 8 is 8 × 10 = 80 (Tens place)
    • Place value of 7 is 7 × 100 = 700 (Hundreds place)
• Expanded Form: Writing a number as the sum of the place values of its digits.
  • Example: 789 = 700 + 80 + 9 = (7 × 100) + (8 × 10) + (9 × 1).

5. Systems of Numeration:

• a) Indian System of Numeration:
  • Places: Units, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, Crores, Ten Crores.
  • Commas: Used after Hundreds (3 digits from right), then after every two digits (Ten Thousands, Ten Lakhs).
  • Example: 5,08,01,592 → Five crore eight lakh one thousand five hundred ninety-two.
• b) International System of Numeration:
  • Places: Units, Tens, Hundreds, Thousands, Ten Thousands, Hundred Thousands, Millions, Ten Millions, Hundred Millions, Billions...
  • Commas: Used after every three digits from the right.
  • Example: 50,801,592 → Fifty million eight hundred one thousand five hundred ninety-two.
• Key Relations:
  • 1 Lakh = 100 Thousand
  • 10 Lakhs = 1 Million
  • 1 Crore = 10 Million
  • 10 Crores = 100 Million

6. Large Numbers in Practice (Units and Conversion):

• Be comfortable with standard units and conversions:
  • Length:
    • 1 kilometre (km) = 1000 metres (m)
    • 1 metre (m) = 100 centimetres (cm)
    • 1 centimetre (cm) = 10 millimetres (mm)
  • Mass:
    • 1 kilogram (kg) = 1000 grams (g)
    • 1 gram (g) = 1000 milligrams (mg)
  • Capacity:
    • 1 kilolitre (kl) = 1000 litres (l)
    • 1 litre (l) = 1000 millilitres (ml)
• Exam Tip: Word problems often involve converting between units and performing operations (addition, subtraction, multiplication, division) with large numbers.

7. Estimation (Rounding Off):

• Estimation gives a rough idea or an approximate value.
• Rounding to the nearest Tens: Look at the units digit. If it's 0-4, round down (keep tens digit same, units becomes 0). If it's 5-9, round up (increase tens digit by 1, units becomes 0).
  • Example: 53 ≈ 50; 58 ≈ 60.
• Rounding to the nearest Hundreds: Look at the tens digit. If it's 0-4 (00-49), round down. If it's 5-9 (50-99), round up.
  • Example: 734 ≈ 700; 781 ≈ 800.
• Rounding to the nearest Thousands: Look at the hundreds digit. If it's 0-4 (000-499), round down. If it's 5-9 (500-999), round up.
  • Example: 4275 ≈ 4000; 4875 ≈ 5000.
• Estimating Sums, Differences, Products: Round off the numbers involved to a suitable place value (depending on the context or instructions) and then perform the operation.
  • Example: Estimate 5290 + 17986. Round to thousands: 5000 + 18000 = 23000.

8. Using Brackets:

• Brackets ( ) are used to group parts of an expression. The operation inside the brackets is performed first.
  • Example: 6 × (5 + 3) = 6 × 8 = 48. (Whereas 6 × 5 + 3 = 30 + 3 = 33).

9. Roman Numerals:

• System using letters to represent numbers.
• Basic Symbols:
  • I = 1
  • V = 5
  • X = 10
  • L = 50
  • C = 100
  • D = 500
  • M = 1000
• Rules:
  1. Repetition: I, X, C, M can be repeated up to 3 times (e.g., III = 3, XX = 20, CCC = 300). V, L, D are never repeated.
  2. Addition: A smaller value symbol written after a larger value symbol is added (e.g., VI = 5 + 1 = 6; LX = 50 + 10 = 60; MC = 1000 + 100 = 1100).
  3. Subtraction: A smaller value symbol written before a larger value symbol is subtracted.
     • I can be subtracted from V and X only (IV = 4, IX = 9).
     • X can be subtracted from L and C only (XL = 40, XC = 90).
     • C can be subtracted from D and M only (CD = 400, CM = 900).
     • V, L, D are never subtracted.
  4. Only subtract one symbol from a larger one (e.g., 8 is VIII, not IIX).
• Exam Tip: Questions often involve converting between Hindu-Arabic and Roman numerals or identifying invalid Roman numerals.

Practice MCQs:

Here are 10 multiple-choice questions based on this chapter for your practice:

1. The smallest 5-digit number using the digits 5, 1, 0, 8, 2 exactly once is:
   (a) 01258
   (b) 10258
   (c) 12580
   (d) 02158

2. The difference between the place value and face value of the digit 7 in the number 97435 is:
   (a) 7000
   (b) 7
   (c) 6993
   (d) 0

3. How is the number 67,502,148 written according to the Indian System of Numeration?
   (a) 67,50,21,48
   (b) 6,75,02,148
   (c) 675,02,148
   (d) 6,750,2148

4. Which of the following Roman numerals is meaningless?
   (a) LXIV
   (b) XCIX
   (c) VVII
   (d) CDXL

5. Estimate the product 598 × 31 by rounding off each number to the nearest tens.
   (a) 18000
   (b) 17700
   (c) 18600
   (d) 1800

6. A box contains 50 packets of biscuits, each weighing 120g. What is the total weight of biscuits in 25 such boxes in kilograms?
   (a) 150 kg
   (b) 1500 kg
   (c) 6 kg
   (d) 15 kg

7. One million is equal to:
   (a) 1 Lakh
   (b) 10 Lakh
   (c) 1 Crore
   (d) 10 Crore

8. The number 947 in Roman numerals is:
   (a) CMXLVII
   (b) CMLXVII
   (c) CMXLIII
   (d) CMXXXVII

9. Rounding off 8456 to the nearest hundreds gives:
   (a) 8000
   (b) 8500
   (c) 8400
   (d) 9000

10. The greatest number which on rounding off to the nearest thousands gives 5000, is:
    (a) 5001
    (b) 5559
    (c) 5999
    (d) 5499

Answers to MCQs:

1. (b) 10258
2. (c) 6993 (Place Value = 7000, Face Value = 7. Difference = 7000 - 7 = 6993)
3. (b) 6,75,02,148
4. (c) VVII (V is never repeated)
5. (a) 18000 (600 × 30 = 18000)
6. (a) 150 kg (Weight per box = 50 × 120g = 6000g = 6kg. Total weight = 25 × 6kg = 150kg)
7. (b) 10 Lakh
8. (a) CMXLVII (CM = 900, XL = 40, VII = 7. 900 + 40 + 7 = 947)
9. (b) 8500 (Hundreds digit is 4, look at tens digit 5. Round up.)
10. (d) 5499 (Numbers from 4500 to 5499 round off to 5000. The greatest among these is 5499).

Revise these notes thoroughly. Understanding place value, number systems, estimation, and Roman numerals is essential. Good luck with your preparation!