A Story of Numbers
Numbers have a long history. This Class 8 Ganita Prakash chapter traces how people first counted with tally marks and body parts, how different civilizations built number systems like the Roman and Egyptian ones, and how the Indian decimal system with the digits 0–9 and place value spread across the world to become the numerals we use today.
Learning objectives
- Describe early ways of counting and recording numbers.
- Read and write Roman numerals.
- Explain place value in the decimal (base-10) system.
- Understand how numbers can be grouped in other bases.
Key concepts
Early counting
Before written numerals, people counted by one-to-one matching — a pebble or a notch for each object — and by using body parts and tally marks. Tally marks group counts (often in fives) so large numbers can be read quickly, but they become clumsy for very big numbers.
Number systems of civilizations
Many civilizations invented symbols for numbers. The Egyptians and Romans used additive systems, writing a number by adding the values of its symbols. In Roman numerals I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000, so XII means 12; a smaller symbol before a larger one means subtraction, as in IX = 9.
The Indian decimal system and zero
The Indian number system uses just ten digits, 0 to 9, together with place value, and importantly includes zero as a number and a place holder. This idea spread through the Arab world to the rest of the world, which is why these are called Hindu–Arabic numerals. Place value makes arithmetic far easier than additive systems.
Place value and bases
In base ten, each place is ten times the one to its right, so 345 means 3 hundreds, 4 tens and 5 ones: 3×100 + 4×10 + 5×1. Numbers can also be grouped in other bases — for example grouping in twos gives base two — which shows that place value is a general idea, not tied only to ten.
Important formulas
Place value expansion
345 = 3×100 + 4×10 + 5×1
Roman numerals
I=1, V=5, X=10, L=50, C=100, D=500, M=1000
Key definitions
- Numeral
- A symbol or group of symbols used to represent a number.
- Place value
- The value of a digit according to its position in a number.
- Base
- The number of single digits a system uses before grouping; base ten uses 0–9.
- Tally marks
- Simple strokes used to count, usually grouped in fives.
Solved examples
Q1. Write 27 in Roman numerals.
Solution: 27 = 20 + 7 = XX + VII = XXVII.
Q2. Write the place-value expansion of 4072.
Solution: 4072 = 4×1000 + 0×100 + 7×10 + 2×1.
Q3. What does the Roman numeral XL represent, and why?
Solution: A smaller symbol (X = 10) before a larger one (L = 50) means subtraction, so XL = 50 − 10 = 40.
Common mistakes to avoid
- Writing Roman numerals with four identical symbols (using IIII instead of IV).
- Forgetting that zero acts as a place holder, e.g. dropping the 0 in 4072.
- Mixing up place value with face value (in 345 the 3 has value 300, not 3).
- Thinking number symbols were the same in all ancient cultures.
A Story of Numbers — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The Roman numeral for 10 is:
Practice questions
Short answer
Why was the invention of zero important?
Zero acts as a number and a place holder, making place-value notation and easy arithmetic possible.
Write 44 in Roman numerals.
XLIV (XL = 40 and IV = 4).
What is the difference between place value and face value?
Face value is the digit itself; place value is the digit multiplied by the value of its position.
Long answer
Explain how the Roman number system works, with examples and a limitation.
Roman numerals use symbols I, V, X, L, C, D and M with fixed values, combined mainly by addition: XV = 10 + 5 = 15. When a smaller symbol is placed before a larger one it is subtracted: IV = 4, IX = 9, XL = 40. A limitation is that there is no zero and no place value, so writing and calculating with very large numbers is awkward compared with the decimal system.
Describe how the decimal place-value system represents numbers and why it is powerful.
The decimal system uses ten digits (0–9) and gives each position a value ten times the one to its right: ones, tens, hundreds, thousands, and so on. A number like 4072 means 4×1000 + 0×100 + 7×10 + 2×1, where zero holds the empty hundreds place. Because only ten symbols and their positions encode any number, arithmetic operations like addition and multiplication follow simple, regular rules — which is why this system spread worldwide.
HOTS (Higher Order Thinking)
Why is 2025 easier to write in decimal than in a purely additive system?
In decimal, place value lets four digits capture the whole value; an additive system would need many repeated symbols, making it long and error-prone.
If we grouped in twos instead of tens, how would counting change?
We would use base two, where each place is twice the previous one and only the digits 0 and 1 appear; the same number would be written with more digits but the place-value idea stays the same.
Quick revision
Revision notes
- Counting began with one-to-one matching, body parts and tally marks.
- Roman/Egyptian systems are additive; Roman uses I, V, X, L, C, D, M.
- Indian decimal system: digits 0–9, place value and zero.
- Place value: each place is ten times the one to its right.
Key takeaways
- Zero and place value transformed how we write numbers.
- Additive systems are clumsy for large numbers.
- Place value is a general idea that works in any base.
Frequently asked questions
Why are our digits called Hindu–Arabic numerals?
They were developed in India and reached Europe through the Arab world.
How do you know when to subtract in Roman numerals?
When a smaller-value symbol is written before a larger one, such as IV (4) or IX (9).
What is a base in a number system?
The number of digits used before grouping; base ten groups in tens using digits 0–9.