Large Numbers — Place Value, Comparison & Roman Numerals
1. Place Value — Indian System
In the INDIAN system, we group digits as ONES, TENS, HUNDREDS, THOUSANDS, TEN THOUSANDS, and LAKHS.
| Period | Crores | Lakhs | Thousands | Ones |
|---|---|---|---|---|
| Places | C | TL | L | TTh |
Abbreviations:
- O = Ones (1)
- T = Tens (10)
- H = Hundreds (100)
- Th = Thousands (1,000)
- TTh = Ten Thousands (10,000)
- L = Lakhs (1,00,000)
- TL = Ten Lakhs (10,00,000)
- C = Crores (1,00,00,000)
Example:
Write 3,45,678 in words:
Step 1 — Identify each digit's place value:
- 3 → Lakhs (3,00,000)
- 4 → Ten Thousands (40,000)
- 5 → Thousands (5,000)
- 6 → Hundreds (600)
- 7 → Tens (70)
- 8 → Ones (8)
Step 2 — Read the number: Three lakh forty-five thousand six hundred seventy-eight.
Remember:
'Always start reading from the LEFT. The comma (,) helps you separate periods. After THOUSANDS comes LAKHS.'
2. Place Value — International System
The INTERNATIONAL system groups digits differently:
| Period | Millions | Thousands | Ones |
|---|---|---|---|
| Places | HM | TM | M |
Abbreviations:
- O = Ones
- T = Tens
- H = Hundreds
- Th = Thousands
- TTh = Ten Thousands
- HTh = Hundred Thousands
- M = Millions
Example:
Write 345,678 in words (International system):
Three hundred forty-five thousand six hundred seventy-eight.
Indian vs International — Key Difference
| Indian | International |
|---|---|
| 1,00,000 = One Lakh | 100,000 = One Hundred Thousand |
| 10,00,000 = Ten Lakhs | 1,000,000 = One Million |
| 1,00,00,000 = One Crore | 10,000,000 = Ten Million |
'In India, we use LAKHS and CRORES. In many other countries, they use MILLIONS.'
3. Expanded Form
Every number can be written as the SUM of its place values.
Example 1:
3,45,678 = 3,00,000 + 40,000 + 5,000 + 600 + 70 + 8
Example 2:
8,90,452 = 8,00,000 + 90,000 + 0 + 400 + 50 + 2
Practice:
Write 5,63,201 in expanded form.
Answer: 5,00,000 + 60,000 + 3,000 + 200 + 0 + 1
4. Comparing Numbers
Rules for Comparison:
- More digits = LARGER number. A 5-digit number is always larger than a 4-digit number.
- Same number of digits — Compare from the LEFTMOST digit.
Examples:
- 45,678 > 9,876 (5 digits > 4 digits)
- 67,890 > 56,789 (same 5 digits — compare 6 > 5 in Ten Thousands place)
- 78,901 < 79,012 (same 5 digits — compare 8 < 9 in Thousands place)
Key Fact:
'When comparing, always line up the numbers by their RIGHT edge (ones place) and compare from the LEFT.'
| Larger | Smaller |
|---|---|
| 1,00,000 (Lakh) | 99,999 (not a lakh) |
| 99,999 | 89,999 |
| 56,789 | 56,788 |
5. Roman Numerals
Roman numerals use LETTERS to represent numbers.
Basic Symbols:
| Roman | I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|---|
| Value | 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
Rules for Writing:
Rule 1 — Repetition:
- I, X, C, M can be repeated (up to 3 times).
- V, L, D are NEVER repeated.
- III = 3, XX = 20, CCC = 300
Rule 2 — Addition (smaller after larger):
- VI = 5 + 1 = 6
- XI = 10 + 1 = 11
- LX = 50 + 10 = 60
- DC = 500 + 100 = 600
Rule 3 — Subtraction (smaller before larger):
- IV = 5 - 1 = 4
- IX = 10 - 1 = 9
- XL = 50 - 10 = 40
- XC = 100 - 10 = 90
- CD = 500 - 100 = 400
- CM = 1000 - 100 = 900
Numbers 1 to 20:
| Hindu-Arabic | Roman | Hindu-Arabic | Roman |
|---|---|---|---|
| 1 | I | 11 | XI |
| 2 | II | 12 | XII |
| 3 | III | 13 | XIII |
| 4 | IV | 14 | XIV |
| 5 | V | 15 | XV |
| 6 | VI | 16 | XVI |
| 7 | VII | 17 | XVII |
| 8 | VIII | 18 | XVIII |
| 9 | IX | 19 | XIX |
| 10 | X | 20 | XX |
Larger Numbers:
| Number | Roman | Number | Roman |
|---|---|---|---|
| 30 | XXX | 70 | LXX |
| 40 | XL | 80 | LXXX |
| 50 | L | 90 | XC |
| 60 | LX | 100 | C |
| 500 | D | 1000 | M |
6. Common Mistakes
- Confusing Indian and International commas: '1,00,000 is One Lakh, NOT One Hundred Thousand. The commas go after every TWO digits after Thousands in Indian system.'
- Roman numeral order: 'IV is 4 (1 before 5), NOT 6. VI is 6 (5 before 1). The order MATTERS!'
- Zero in expanded form: 'If a place has 0, you can skip it in expanded form. But in the number, the 0 holds the place!'
- Comparing numbers with different digits: 'A 5-digit number is ALWAYS bigger than a 4-digit number. Don't just compare the first digit!'
7. Key Facts to Remember
- 'One Lakh = 1,00,000 = 100 Thousands.'
- 'One Crore = 100 Lakhs = 1,00,00,000.'
- 'The smallest 5-digit number is 10,000. The largest 5-digit number is 99,999.'
- 'Roman numerals have NO symbol for ZERO.'
- 'V, L, D are NEVER repeated. Only I, X, C, M can be repeated.'
8. Self-Test
Q1: Write 6,78,901 in words (Indian system).
Q2: Which is larger — 89,765 or 1,02,345? Why?
Q3: Write 2,34,567 in expanded form.
Q4: Convert to Roman numerals: (a) 14 (b) 29 (c) 45 (d) 99
Q5: Convert to Hindu-Arabic: (a) XVII (b) XLIV (c) XCIX (d) C
Q6: How many lakhs make one crore?
Q7: Write 1,50,000 in International system (words).
Answers:
A1: Six lakh seventy-eight thousand nine hundred one. A2: 1,02,345 (it has 6 digits — it is in lakhs. 89,765 has only 5 digits). A3: 2,00,000 + 30,000 + 4,000 + 500 + 60 + 7 A4: (a) XIV (b) XXIX (c) XLV (d) XCIX A5: (a) 17 (b) 44 (c) 99 (d) 100 A6: 100 lakhs = 1 crore. A7: One hundred fifty thousand.
