A Story of Numbers
From Egyptian pictures and Roman letters to place value: how grouping by tens changed the way we write numbers.
Original notes — ready CBSE (NCERT, English Medium) • Class 8 • Mathematics • Book: Ganita Prakash — Part I
Chapter 3: A Story of Numbers — EG's own notes, in simple words
Book: Ganita Prakash — Part I (CBSE, Class 8) | Chapter 3 | Medium: English
Diagram (labels are printed on the image): A clean number line with marked points and gold dots at the divisions. A general picture of how different kinds of numbers sit on a line, not a book scene.
Children sorting number cards into boxes on a classroom table. A general picture of grouping numbers by type, not a book scene.
What the chapter is about
Every number system is an answer to the same question: how do we record a count? The Egyptians used separate pictures for 1, 10, 100, 1000 and so on, and added them up — a system that is base 10 but has no place value, so writing a large number takes many symbols. A variation uses landmark numbers: instead of repeating a symbol many times, you use a larger landmark symbol, which shortens the writing but still needs adding.
The Roman system, built on I, V, X, L, C, D and M, counts partly by fives and uses subtraction as well as addition: IV is 4 (one before five) and IX is 9. It has no symbol for zero and no place value, so arithmetic in it is hard — try multiplying in Roman numerals. The breakthrough in the Indian system is place value with base 10: the same ten digits change their value by position, so 2367 means 2 × 1000 + 3 × 100 + 6 × 10 + 7, written as a sum of landmark numbers (2000, 300, 60, 7). Grouping by tens at every step — ten ones make a ten, ten tens make a hundred — is what makes our system short, and with zero as a place-holder it works for any size.
The chapter also plays with other bases: group objects by fives instead of tens and the same count gets a different-looking name, which shows that base 10 is a choice (a handy one, matching ten fingers), not a law of nature.
What you should learn
- Explain the Egyptian system: base 10, additive, no place value.
- Read and write Roman numerals, including subtraction pairs IV, IX, XL, XC, CD, CM.
- Break a number into landmark numbers and place values.
- Explain why place value plus zero beats the older systems.
- Show how grouping by a different base renames the same count.
Key Terms
- Base — The size of each group in a number system; our system has base 10.
- Place value — The value of a digit decided by its position; in 6523 the 5 means 500.
- Landmark number — A round number used as a building block, such as 1000, 100, 10 or 1; 2367 = 2000 + 300 + 60 + 7.
- Additive system — A system where the value is the sum of its symbols, like the Egyptian system.
- Roman numerals — I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000.
Remember
- Egyptian: base 10, many symbols, no place value, no zero.
- Roman: IV = 4, IX = 9, XL = 40, XC = 90; a smaller symbol before a larger one is subtracted.
- Our system: ten digits, place value, zero as place-holder.
- 2367 = 2000 + 300 + 60 + 7.
Exam Tip / Common Mistake
Common mistake: writing 9 as VIIII or 49 as IL. Roman subtraction uses only one smaller symbol before the next two higher ones: 9 = IX and 49 = XLIX, never IL.
Count 13 objects in groups of 5: two full groups and 3 left over. In a base-5 way of writing, that count is "23" (two fives and three ones) — the same objects, a different name.
Roman numerals have no zero, so there is no Roman way to write "0" or to hold an empty place. That alone makes large calculations painful.
Chart / Table: Roman numerals to 100
| Number | Roman | Number | Roman |
|---|---|---|---|
| 4 | IV | 40 | XL |
| 9 | IX | 90 | XC |
| 14 | XIV | 99 | XCIX |
| 39 | XXXIX | 100 | C |
Worked method: write a number in Roman numerals
- Split the number into tens and ones (for numbers up to 39, the chapter's range).
- Write the tens part: 10 = X, 20 = XX, 30 = XXX.
- Write the ones part, using subtraction where needed: 4 = IV, 9 = IX.
- Join them: 29 = XX + IX = XXIX. Check by adding back: 20 + 9 = 29.
Practice set — try it yourself
- Write 29 in Roman numerals.
Hint: Split into 20 and 9 first. - Write 47 in Roman numerals.
Hint: 40 is XL and 7 is VII. - What is the value of the digit 5 in 6523?
Hint: Count its position from the right. - Write 2367 as a sum of landmark numbers.
Hint: Start from the largest landmark, 2000. - Why is the Egyptian system harder to use for large numbers than ours?
Hint: Think about how many symbols a number like 999 would need.
Questions with Answers
- Write 14 in Roman numerals.
14 = 10 + 4 = X + IV = XIV. - Write 39 in Roman numerals.
39 = 30 + 9 = XXX + IX = XXXIX. - What does the Egyptian system add up to make a number?
It adds the values of its picture symbols: each symbol stands for 1, 10, 100, 1000 and so on, and the number is the sum of the symbols used. - Why is zero important in our number system?
Zero holds an empty place, so 205 is not confused with 25. Without it, place value cannot work. - Write 2025 as a sum of landmark numbers.
2025 = 2000 + 0 + 20 + 5, or simply 2000 + 20 + 5.
Extension questions
- Write 88 in Roman numerals.
88 = 50 + 30 + 8 = L + XXX + VIII = LXXXVIII. - Write 99 in Roman numerals and explain the subtraction steps.
99 = 90 + 9 = XC + IX = XCIX. Each 9 uses one smaller symbol before a larger one. - Why can Roman numerals not show a number like 0?
The system has no symbol for zero and no place value, so an empty count simply has no representation. - Group 23 objects by fives. How many full groups and leftovers are there?
Four groups of 5 and 3 left over (4 × 5 + 3 = 23). - In 9081, what does each digit stand for?
9 stands for 9000, 0 for 0 hundreds, 8 for 80 and 1 for 1. - Which is easier for multiplication, Roman numerals or our system? Why?
Our system, because place value lets each digit be multiplied separately and zero holds empty places, while Roman numerals must be converted and recombined.
Egyptian symbols add up with no place value; Roman letters add and sometimes subtract, with no zero; our base-10 place-value system with zero writes any number with ten digits, split into landmark numbers.
How to use this page: Read the summary, say the Key Terms in your own words, then try the practice set without looking. Use the Questions with Answers to check yourself afterwards. Ask your teacher or parent to check your answers.