Knowing Our Numbers

1. Place Value Systems

Every digit in a number has a place value based on its position. For large numbers, we use two main systems.

Indian System (used in India)

Abbreviations: TC = Ten Crores, C = Crores, TL = Ten Lakhs, L = Lakhs, TTh = Ten Thousands, Th = Thousands, H = Hundreds, T = Tens, O = Ones.

Example: Write 58436217 in the Indian place value chart.

Solution: 5,84,36,217 — Five crore eighty-four lakh thirty-six thousand two hundred seventeen.

International System (used globally)

Abbreviations: HM = Hundred Millions, TM = Ten Millions, M = Millions, HTh = Hundred Thousands, TTh = Ten Thousands, Th = Thousands.

Example: Write 58436217 in the International place value chart.

Solution: 58,436,217 — Fifty-eight million four hundred thirty-six thousand two hundred seventeen.

Comparison: Indian vs International

Common Mistake: Students often confuse 'lakh' and 'million.' One lakh = 100,000, while one million = 10,00,000 (ten lakhs).

2. Comparing and Ordering Numbers

Rule 1: The number with more digits is greater.
Rule 2: If the digit count is the same, compare digits from the leftmost place.

Worked Example: Arrange 23785, 5678, 25463, 23799 in ascending order.

Step 1: 5678 has 4 digits; the rest have 5. So 5678 is the smallest.
Step 2: Among 23785, 25463, 23799 — compare ten-thousands: all 2. Compare thousands: 3 vs 5 vs 3. So 25463 is largest.
Step 3: Compare 23785 and 23799: hundreds both 7, tens both 8, ones: 5 vs 9. So 23785 < 23799.
Ascending order: 5678 < 23785 < 23799 < 25463.

Common Mistake: Assuming 23799 > 25463 because '99 > 46'. Always compare from the leftmost digit first.

3. Estimation

Estimation helps us get a 'close enough' answer quickly.

Rounding Rules

• To nearest 10: look at ones digit (0-4 round down, 5-9 round up).
• To nearest 100: look at tens digit.
• To nearest 1000: look at hundreds digit.

Worked Example: Estimate 4783 + 2196 by rounding to nearest 1000.

4783 rounds to 5000. 2196 rounds to 2000. Estimated sum = 7000. Exact sum = 6979. The estimate is close!

Exam Focus (2 marks): 'Round 67492 to the nearest 10, 100, and 1000.'

Nearest 10: 67490. Nearest 100: 67500. Nearest 1000: 67000.

4. Roman Numerals

Roman numerals use letters to represent numbers.

Rules for Reading and Writing

• Rule 1: If a smaller numeral follows a larger, add: VI = 5 + 1 = 6.
• Rule 2: If a smaller numeral precedes a larger, subtract: IV = 5 - 1 = 4.
• Rule 3: Only I, X, C can be used for subtraction.
• Rule 4: A symbol cannot repeat more than 3 times.
• Rule 5: V, L, D are never repeated or subtracted.

Worked Example: Write 94 in Roman numerals.

94 = 90 + 4 = (100 - 10) + (5 - 1) = XC + IV = XCIV.

Common Mistake: Writing 99 as IC (100 - 1). This is wrong! Write 99 = 90 + 9 = XC + IX = XCIX.

Exam Focus (3 marks): 'Convert XLVII to Hindu-Arabic.'

XLVII = (50 - 10) + 5 + 1 + 1 = 40 + 7 = 47.

5. Comparison Table: Indian vs International Periods

6. Self-Test

1. Write 60345281 in the Indian place value chart and in words.
2. Arrange in descending order: 45903, 45093, 49503, 40953.
3. Estimate 7924 - 3158 by rounding to nearest 100.
4. Convert to Roman numerals: (a) 49 (b) 88 (c) 275.
5. Convert to Hindu-Arabic: (a) LXIX (b) CDXL (c) MCMXLV.
6. Round 38465 to the nearest 10, 100, and 1000.
7. Compare: Write 'greater than' or 'less than': 5,43,210 ___ 5,43,201.
8. How many lakhs make one crore? How many millions make one crore?

7. Answers to Self-Test

1. 6,03,45,281 — Six crore three lakh forty-five thousand two hundred eighty-one.
2. 49503 > 45903 > 45093 > 40953.
3. 7900 - 3200 = 4700 (exact: 4766).
4. (a) XLIX (b) LXXXVIII (c) CCLXXV.
5. (a) 69 (b) 440 (c) 1945.
6. 38470, 38500, 38000.
7. 5,43,210 > 5,43,201.
8. 100 lakhs = 1 crore. 10 million = 1 crore.