1. Chapter at a Glance
The numerals 0 to 9 that we write every day are one of humanity's greatest inventions. Long before them, people counted with pebbles, knots, notches and fingers, and different civilisations invented different ways of writing numbers. This chapter tells the story of numbers: from simple tallies, through Egyptian, Roman, Babylonian and Mayan systems, to the Indian place-value system with zero that spread across the world. Along the way you will learn about bases other than ten, including the binary system used by computers.
- Early counting: one-to-one matching, tallies and grouping.
- Additive systems: Egyptian and Roman numerals.
- Place-value systems: Babylonian (base 60), Mayan (base 20) and Indian (base 10).
- The role of zero as a placeholder and as a number.
- Writing numbers in other bases, including base 2 and base 5.
- Why the Indian system became the world's system.
2. Early Counting
The earliest way to "count" was one-to-one matching: a shepherd placed one pebble in a bag for each sheep going out, and removed one for each sheep coming back. If pebbles remained, sheep were missing — no number words were needed.
- Tally marks on bones and sticks recorded counts. The Ishango bone from Africa, over 20,000 years old, has groups of notches.
- People counted on body parts — fingers, knuckles, joints.
- Grouping made large counts manageable: bundles of 5, 10 or 20. Ten fingers made 10 the most common group size.
3. Additive Systems: Egyptian and Roman
In an additive system, each symbol has a fixed value wherever it appears, and the value of a numeral is the sum of its symbols.
Egyptian numerals
The ancient Egyptians (around 3000 BCE) used separate pictures for 1, 10, 100, 1000, 10,000, 1,00,000 and 10,00,000 — a stroke, a heel bone, a coil of rope, a lotus flower, a finger, a tadpole and a kneeling figure. To write 3245 they drew 3 lotuses, 2 ropes, 4 heel bones and 5 strokes. Large numbers needed many symbols.
Roman numerals
| Symbol | I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|---|
| Value | 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
- Symbols are added when equal or decreasing: XVI = 10 + 5 + 1 = 16.
- A smaller symbol before a larger one is subtracted: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900.
- A symbol is not repeated more than three times in a row (III, XXX).
Write 1984 and 2026 in Roman numerals; read MCMXLVII.
Solution: 1984 = M + CM + LXXX + IV = MCMLXXXIV. 2026 = MMXXVI. MCMXLVII = 1000 + 900 + 40 + 7 = 1947.
Additive systems make arithmetic hard — try multiplying CXXIII by XLVII on paper! There is no zero and no place value.
4. Place-Value Systems
In a place-value system, the value of a digit depends on its position. With a base b, the places are worth 1, b, b², b³, …
Babylonian (base 60)
The Babylonians (about 2000 BCE) used base 60, with wedge-shaped marks for 1 and 10 combined to write digits up to 59. We still use their base-60 idea in 60 seconds = 1 minute, 60 minutes = 1 hour and 360° in a circle. For a long time they had no symbol for an empty place, which caused ambiguity; later they used a placeholder mark, but not a true zero that could be used in calculation.
Mayan (base 20)
The Maya of Central America used a mostly base-20 system with dots (1), bars (5) and a shell symbol for zero, written in vertical stacks.
The Indian system (base 10 with zero)
- Brāhmī numerals (from around the 3rd century BCE) gave separate signs for 1–9.
- By the early centuries CE, Indian mathematicians were writing numbers with place value and only ten symbols. The Bakhshali manuscript uses a dot for zero.
- Āryabhaṭa (499 CE) used place value in his calculations; Brahmagupta (628 CE) treated zero as a number with its own rules: a + 0 = a, a − 0 = a, a × 0 = 0.
- Through Arab scholars such as al-Khwārizmī (9th century), the system reached Europe, where it is called "Hindu–Arabic numerals".
- Only ten symbols write any number, however large.
- Zero removes ambiguity and allows clear place value.
- Arithmetic (adding, multiplying, dividing) becomes simple column work.
5. Numbers in Other Bases
Any whole number b ≥ 2 can be a base. In base b, the digits are 0 to b − 1 and the places are 1, b, b², …
Converting to base 10
Converting from base 10 — repeated division
- Divide the number by the base; write the remainder.
- Divide the quotient by the base again; write the remainder.
- Repeat until the quotient is 0.
- Read the remainders from bottom to top.
Write 45 in base 2.
Solution: 45 ÷ 2 = 22 r 1; 22 ÷ 2 = 11 r 0; 11 ÷ 2 = 5 r 1; 5 ÷ 2 = 2 r 1; 2 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1. Reading upwards: (101101)₂. Check: 32 + 8 + 4 + 1 = 45 ✔.
Binary (base 2) uses only 0 and 1 — perfect for electronic circuits that are either off or on. Every photo, song and message in a phone is stored as binary digits (bits).
6. Solved Problems (NCERT Style)
Find XLIV + LXXVI and write the answer in Roman numerals.
Solution: 44 + 76 = 120 = CXX.
Convert 99 to base 5.
Solution: 99 ÷ 5 = 19 r 4; 19 ÷ 5 = 3 r 4; 3 ÷ 5 = 0 r 3. Answer: (344)₅. Check: 75 + 20 + 4 = 99 ✔.
What is the largest 4-digit number in base 2, and its value in base 10?
Solution: (1111)₂ = 8 + 4 + 2 + 1 = 15.
Write 5000 seconds in hours, minutes and seconds (a base-60 idea).
Solution: 5000 ÷ 60 = 83 r 20 → 83 min 20 s; 83 ÷ 60 = 1 r 23 → 1 h 23 min 20 s.
7. Master Formula Table
| Idea | Rule |
|---|---|
| Value in base b | dₙbⁿ + … + d₁b + d₀ |
| Digits in base b | 0 to b − 1 |
| Base 10 → base b | divide repeatedly; read remainders upwards |
| Roman subtraction | IV 4, IX 9, XL 40, XC 90, CD 400, CM 900 |
| Largest n-digit number in base b | bⁿ − 1 |
| Zero's rules (Brahmagupta) | a + 0 = a, a × 0 = 0 |
8. Master Summary Table
| System | Type | Base | Zero? |
|---|---|---|---|
| Tally marks | Additive | Groups of 5 | No |
| Egyptian | Additive | 10 (symbols for powers of 10) | No |
| Roman | Additive/subtractive | 5 and 10 mixed | No |
| Babylonian | Place value | 60 | Placeholder only (later) |
| Mayan | Place value | 20 (mostly) | Yes |
| Indian | Place value | 10 | Yes — as a number |
| Binary | Place value | 2 | Yes |
9. Mind Map
- Pebbles, tallies
- Body parts, grouping
- Egyptian
- Roman
- Babylonian 60
- Mayan 20
- Indian 10 with zero
- Binary
- Base 5
- Conversions
10. Tips and Tricks
- Write the place values (…, b³, b², b, 1) above the digits before converting to base 10.
- Memorise powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
- For Roman numerals, break the number into thousands, hundreds, tens and ones.
- Reading remainders top to bottom when converting bases.
- Using the digit 5 in base 5 (allowed digits are 0–4).
- Writing 49 as IL (correct: XLIX).
11. Important One-liners
- One-to-one matching was the earliest way of keeping count.
- Egyptian and Roman numerals are additive and have no zero.
- The Babylonians used base 60 — still used for time and angles.
- The Indian place-value system uses ten digits including zero.
- Brahmagupta treated zero as a number with its own rules.
- The system reached Europe through Arab scholars.
- Computers use base 2 (binary).
- In base b, place values are powers of b.
12. Exam-pattern Questions with Detailed Answers
The Roman numeral XC stands for:
- 110
- 90
- 190
- 60
Show answer
X (10) before C (100) is subtracted: 100 − 10 = 90.
Answer: (B) 90
Convert (110)₂ to base 10.
Show answer
1 × 4 + 1 × 2 + 0 = 6.
Answer: 6
Which civilisation used base 60?
Show answer
The Babylonians.
Answer: Babylonian
Write 2999 in Roman numerals.
Show answer
2000 = MM, 900 = CM, 90 = XC, 9 = IX → MMCMXCIX.
Answer: MMCMXCIX
Convert 100 to base 2.
Show answer
100 = 64 + 32 + 4 → (1100100)₂.
Answer: (1100100)₂
Which feature of the Indian system makes it better than the Roman system for calculation?
- Larger symbols
- Place value with zero
- Subtraction rule
- Use of letters
Show answer
Place value with zero lets us write any number with ten digits and calculate column by column.
Answer: (B)
Convert (2304)₅ to base 10.
Show answer
2 × 125 + 3 × 25 + 0 × 5 + 4 = 250 + 75 + 4 = 329.
Answer: 329
Why did the Babylonian system face confusion before a placeholder was used?
Show answer
Without a symbol for an empty place, numbers such as 1 × 60 + 0 and 1 could look the same; readers had to guess from context.
Answer: No symbol for an empty place → ambiguous numbers
Add (1011)₂ + (111)₂ and give the answer in base 2 and base 10.
Show answer
11 + 7 = 18 = 16 + 2 → (10010)₂.
Answer: (10010)₂ = 18
How many different numbers can be written with 8 binary digits (bits)?
Show answer
Each of the 8 places can be 0 or 1: 2⁸ = 256 numbers (0 to 255).
Answer: 256
In which base does 25 + 25 = 52? (Digits are read in that base.)
Show answer
In base b: (2b + 5) + (2b + 5) = 5b + 2 → 4b + 10 = 5b + 2 → b = 8. Check: in base 8, 25 = 21 and 52 = 42; 21 + 21 = 42 ✔.
Answer: Base 8
Using weights of 1, 2, 4, 8, 16 and 32 g (one of each), can you weigh every whole number of grams from 1 to 63? Explain.
Show answer
Yes. Every number from 1 to 63 has a unique binary form with at most 6 digits; each 1 tells you which weight to use. Example: 45 = (101101)₂ → 32 + 8 + 4 + 1.
Answer: Yes — binary representation
13. Quick Revision Summary
- Counting began with matching, tallies and grouping.
- Egyptian and Roman systems are additive (no zero, no place value).
- Babylonian base 60 and Mayan base 20 used place value.
- India gave base-10 place value with zero as a true number; it spread through the Arab world.
- Base b: digits 0 to b − 1, places are powers of b; convert by repeated division.