Large Numbers
1. Place Value — Indian System
The Indian place value system groups digits into periods: ONES, THOUSANDS, LAKHS, CRORES.
| Period | Crores | Lakhs | Thousands | Ones |
|---|---|---|---|---|
| Places | TC (Ten Crores) | C (Crores) | TL (Ten Lakhs) | L (Lakhs) |
Example: 57,34,82,916 is read as 'Fifty-seven crore thirty-four lakh eighty-two thousand nine hundred sixteen.'
'Always start from the RIGHT when grouping digits into periods.'
2. Place Value — International System
The International system groups by MILLIONS and BILLIONS.
| Period | Billions | Millions | Thousands | Ones |
|---|---|---|---|---|
| Places | TB | B | HM | TM |
Example: 573,482,916 is read as 'Five hundred seventy-three million four hundred eighty-two thousand nine hundred sixteen.'
Key Difference
| Indian System | International System |
|---|---|
| 1,00,000 = 1 Lakh | 100,000 = 1 Hundred Thousand |
| 1,00,00,000 = 1 Crore | 10,000,000 = 10 Million |
'Remember: In the Indian system, commas go after EVERY TWO digits after the thousands period. In the International system, commas go after EVERY THREE digits.'
3. Reading and Writing Large Numbers
Expanded Form
Write a number as the sum of its place values.
Example: 4,56,78,903 = 4,00,00,000 + 50,00,000 + 6,00,000 + 70,000 + 8,000 + 900 + 0 + 3
Number Names
| Digit | Place | Value |
|---|---|---|
| 4 | Crores (TC) | 40,00,00,000 |
| 5 | Ten Lakhs (TL) | 50,00,000 |
| 6 | Lakhs (L) | 6,00,000 |
| 7 | Ten Thousands (TTh) | 70,000 |
| 8 | Thousands (Th) | 8,000 |
| 9 | Hundreds (H) | 900 |
| 0 | Tens (T) | 0 |
| 3 | Ones (O) | 3 |
'When a digit is ZERO, you still write its place — but you skip it while reading the number name.'
4. Comparison and Ordering
Ascending Order — SMALLEST to LARGEST
Descending Order — LARGEST to SMALLEST
Comparing Numbers
| Rule | Example |
|---|---|
| 1. More digits = Larger number | 99,999 < 1,00,000 (5 digits vs 6 digits) |
| 2. Same digits — compare LEFT to RIGHT | 45,678 vs 45,987 — compare hundreds: 6 < 9, so 45,678 < 45,987 |
'Rule 1 is the MOST important. A 6-digit number is ALWAYS bigger than a 5-digit number, no matter what the digits are.'
Forming the Largest and Smallest Numbers
- Largest: Arrange digits in DESCENDING order. Example: Use 3, 7, 1, 9, 4 → 97431.
- Smallest: Arrange digits in ASCENDING order. Example: Use 3, 7, 1, 9, 4 → 13479.
'For the smallest number, remember that the FIRST digit cannot be ZERO. If zero is included, place it in the SECOND position.'
5. Rounding Off
Rounding to the Nearest Ten
Look at the ones digit. If 0-4 → round DOWN. If 5-9 → round UP.
47 → ones digit is 7 (≥5), so round UP to 50.
Rounding to the Nearest Hundred
Look at the tens digit. If 0-4 → round DOWN. If 5-9 → round UP.
247 → tens digit is 4 (≤4), so round DOWN to 200.
Rounding to the Nearest Thousand
Look at the hundreds digit. If 0-4 → round DOWN. If 5-9 → round UP.
3,678 → hundreds digit is 6 (≥5), so round UP to 4,000.
| To Round To | Look At | Example | Result |
|---|---|---|---|
| Ten | Ones | 374 | 370 |
| Hundred | Tens | 374 | 400 |
| Thousand | Hundreds | 6,239 | 6,000 |
'Rounding is used when you need an APPROXIMATE answer — for example, 'about 500 people attended.' It is NOT exact. Use the symbol ≈ (approximately equal to).'
6. Roman Numerals
Symbols
| Symbol | I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|---|
| Value | 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
Rules
| Rule | Example | Explanation |
|---|---|---|
| Smaller AFTER larger — ADD | VI = 5 + 1 = 6 | V(5) then I(1) |
| Smaller BEFORE larger — SUBTRACT | IV = 5 − 1 = 4 | I(1) before V(5) |
| You can repeat I, X, C, M up to THREE times | XXX = 30 | X repeated 3 times |
| V, L, D are NEVER repeated | VV is NOT allowed | 10 is written as X |
Common Roman Numerals
| Hindu-Arabic | Roman | Hindu-Arabic | Roman |
|---|---|---|---|
| 1 | I | 50 | L |
| 4 | IV | 90 | XC |
| 5 | V | 100 | C |
| 9 | IX | 400 | CD |
| 10 | X | 500 | D |
| 19 | XIX | 900 | CM |
| 40 | XL | 1000 | M |
'You never write the same symbol more than THREE times in a row. 4 is IV (5-1), not IIII.'
Large Numbers in Roman Numerals
To write numbers up to 1000, combine the symbols:
- 1987 = M (1000) + CM (900) + LXXX (80) + VII (7) = MCMLXXXVII
- 2025 = MM (2000) + XX (20) + V (5) = MMXXV
Key Facts to Remember
- The Indian place value system uses periods: Ones, Thousands, Lakhs, Crores.
- The International system uses: Ones, Thousands, Millions, Billions.
- A 10 crore number has 9 digits in the Indian system and 8 digits in the International system.
- 'Zero is a PLACEHOLDER — without it, we could not tell 205 from 25.'
- Roman numerals do NOT have a symbol for zero.
- M = 1000 is the largest basic Roman numeral symbol.
Common Mistakes
| Mistake | Why It Is Wrong | Correct Way |
|---|---|---|
| Writing 49 as IL | I(1) before L(50) = 49, but this rule only works with specific pairs | 49 = XLIX (40 + 9) |
| Writing 99 as IC | Same issue — I before C only works for 99 in limited contexts | 99 = XCIX |
| Placing comma after 3 digits in Indian system | The Indian system groups differently after thousands | 5,67,890 not 567,890 |
| Writing 'crore' for 10 million | Crore = 10 million = 1,00,00,000 | 1 crore = 100 lakhs |
Exam Focus (ICSE Class 5)
| Topic | Marks (Typical) | Question Type |
|---|---|---|
| Write number names | 2-3 marks | Fill in the blanks |
| Place value and expanded form | 2-3 marks | Direct questions |
| Comparing numbers | 2 marks | True/False or arrange in order |
| Rounding | 2 marks | Round off to nearest 10/100/1000 |
| Roman numerals (up to 100) | 3-4 marks | Convert Hindu-Arabic to Roman and vice versa |
'In ICSE exams, Roman numeral questions appear in ALMOST every paper. Practice converting numbers between 1 and 100 thoroughly.'
Self-Test: 5 Questions
Q1. Write the number name for 7,89,45,231 in the Indian system.
Q2. Arrange in descending order: 67,890; 67,980; 76,089; 67,099.
Q3. Round 7,849 to the nearest hundred.
Q4. Convert to Roman numerals: (a) 49 (b) 94 (c) 2025
Q5. Write the expanded form of 30,08,07,006.
Answers
A1. Seven crore eighty-nine lakh forty-five thousand two hundred thirty-one.
A2. 76,089 > 67,980 > 67,890 > 67,099.
A3. 7,800 (tens digit is 4, so round down).
A4. (a) XLIX (b) XCIV (c) MMXXV
A5. 30,00,00,000 + 0 + 0 + 8,00,000 + 0 + 7,000 + 0 + 0 + 6.
