Class 8 › Mathematics › Chapter 3
A Story of Numbers
How humans learned to count and write numbers: one-to-one mapping, landmark numbers, the idea of a base, place value, and the invention of zero in the Hindu (Indian) number system.
1. Where Do Our Numerals Come From?
- Mesopotamia (present-day Iraq and neighbours) used its own number symbols about 4000 years ago.
- The structure of our spoken and written numbers began in India. The Yajurveda Samhita names numbers by powers of 10 (eka, dasha, shata, sahasra, ayuta, …) up to $10^{12}$ and beyond.
- The digits $0$ to $9$ were developed in India around 2000 years ago.
| Period | Event |
|---|---|
| c. 3rd century CE | Bakhshali manuscript: first known use of ten digits, with $0$ written as a dot |
| c. 499 CE | Aryabhata explains and uses the Indian system of 10 symbols for elaborate computations |
| c. 628 CE | Brahmagupta codifies arithmetic with $0$ as a number |
| c. 800 CE | The system reaches the Arab world |
| c. 825 / 830 | Al-Khwarizmi (On the Calculation with Hindu Numerals; "algorithm" comes from his name) and Al-Kindi (On the Use of the Hindu Numerals) |
| c. 1100 CE | Reaches Europe and parts of Africa |
| c. 1200 CE | Fibonacci argues for the Indian numerals; Roman numerals stay in use for centuries |
| By the 17th century | Widely adopted in Europe, since science could not progress without them |
Names: Europeans learned the numerals from Arabs and called them "Arabic numerals". Arab scholars called them "Hindu numerals". Preferred terms today: Hindu numerals, Indian numerals, or Hindu-Arabic numerals. "Hindu" here means the geography and people, not a religion.
2. The Mechanism of Counting
Without number names or numerals, how could a Stone Age herder check that all cows returned, compare herds with a neighbour, or find how many more cows are needed?
Method 1
Objects
Keep one stick or pebble for each cow. The collection of sticks is the number. Easy, but you need as many sticks as objects.
Method 2
Names / sounds
Match objects with letters $a, b, c, \dots, z$ in order. Convenient to say, but it stops at $26$.
Method 3
Written symbols
I, II, III, IV, V, … (Roman system). Efficient, but needs ever more symbols for larger numbers.
One-to-one mapping: matching each object with exactly one member of a standard sequence, so no two objects share the same member. Used for counting, and also for comparing two collections (Which has fewer? How many more?).
Number system: a standard sequence of objects, names or written symbols with a fixed order. A good one must be unending and easy to count with.
Numerals: the written symbols of a number system (e.g. $0, 1, 5, 36, 193$ in the Hindu system). Written numerals always went together with spoken number names.
Number system: a standard sequence of objects, names or written symbols with a fixed order. A good one must be unending and easy to count with.
Numerals: the written symbols of a number system (e.g. $0, 1, 5, 36, 193$ in the Hindu system). Written numerals always went together with spoken number names.
3. Some Early Number Systems
- Body parts: many groups used fingers and body parts as the standard sequence (e.g. people of Papua New Guinea).
- Tally marks: one notch per object, cut on bone or cave walls. Oldest examples: the Lebombo bone (South Africa, 29 notches, about 44,000 years old) and the Ishango bone (Democratic Republic of Congo, 20,000 to 35,000 years old). Tally marks work like the stick method.
- Counting in twos (Gumulgal of Australia): $1 =$ urapon, $2 =$ ukasar, $3 =$ ukasar-urapon, $4 =$ ukasar-ukasar, $5 =$ ukasar-ukasar-urapon, $6 =$ ukasar-ukasar-ukasar. Built as $3 = 2+1$, $4 = 2+2$, $5 = 2+2+1$, $6 = 2+2+2$. Anything above $6$ was called ras.
- The Bakairi (South America) and the Bushmen of South Africa developed an equivalent system with no known contact, possibly from common ancestors.
Key idea: counting in groups
Count in groups of a fixed size and use a word or symbol for the group. Common group sizes: 2, 5, 10, 20. The Roman system counts by 5s and 10s.
Why groups? Humans cannot take in the size of a collection at a glance beyond about 4. So every group of 5 tally marks gets a new symbol.
Limitation: a system with one group size is still cumbersome for large numbers (e.g. 1345 counted only by 5s).
4. Roman Numerals and Landmark Numbers
| Symbol | I | V | X | L | C | D | M |
|---|---|---|---|---|---|---|---|
| Value | 1 | 5 | 10 | 50 | 100 | 500 | 1000 |
Landmark numbers: numbers with their own basic symbol, used as reference points. A given number is grouped by landmark numbers, largest first, and written with their symbols.
- Method: take as many $1000$s as possible, then $500$s, $100$s, and so on. $27 = 10 + 10 + 5 + 1 + 1 = $ XXVII.
- Subtractive form: $4 = $ IV (1 less than 5), $9 = $ IX, $40 = $ XL (10 less than 50). Not always followed: $40$ was sometimes written XXXX.
- Example: $2367 = 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1 = $ MMCCCLXVII.
- Practice: $1222 = $ MCCXXII; $\;2999 = $ MMCMXCIX; $\;302 = $ CCCII; $\;715 = $ DCCXV.
Arithmetic with Roman numerals
Addition: count the I's, X's, C's, …, then regroup from the largest. CCXXXII + CCCCXIII has 6 C's; 5 C's make a D, so the sum is DCXLV ($232 + 413 = 645$). LXXXVII + LXXVIII $= 87 + 78 = 165 = $ CLXV.
Multiplication is hard: V $\times$ L = CCL ($250$), V $\times$ D = MMD ($2500$), VII $\times$ IX = LXIII ($63$), L $\times$ D = 25 M's ($25{,}000$). Products of landmark numbers are not always landmark numbers, and regrouping sizes keep changing (5s and 2s alternate).
Abacus: Romans used it for calculations, but only specially trained people could.
Drawbacks of Roman numerals: needs ever more symbols for bigger numbers; no zero; difficult multiplication and division. It evolved from the Greek system around the 8th century BCE and spread with the Roman empire.
The chapter stresses that these systems did not develop as neat improvements over one another; the real history is complex.
5. The Idea of a Base
Egyptian system (about 3000 BCE): landmark numbers are $1, 10, 10^2, 10^3, \dots$ (each is 10 times the previous), each with its own picture symbol: 1 (a stroke), 10 (a heel bone), 100 (a coiled rope), 1000 (a lotus flower), 10,000 (a finger), and so on. $324 = 100 + 100 + 100 + 10 + 10 + 4$ uses 3 rope symbols, 2 heel-bone symbols and 4 strokes. $784 = 700 + 80 + 4$ and $1111$ uses each of the first four symbols once.
Base-$n$ number system
A number system is base-$n$ if (a) the first landmark number is $1$, and (b) every next landmark number is the previous one times the fixed number $n$.
$$\text{Landmark numbers} = n^0,\ n^1,\ n^2,\ n^3,\ \dots$$- Base 10 (decimal): Egyptian. Landmarks $1, 10, 100, 1000, \dots$
- Base 5: landmarks $1, 5, 25, 125, 625, 3125, \dots$ ($5^0$ to $5^5$). Example: $143 = 125 + 5 + 5 + 5 + 1 + 1 + 1$. Also $137 = 125 + 5 + 5 + 1 + 1$, $\;651 = 625 + 25 + 1$.
- Base 7: landmarks $1, 7, 49, 343, \dots$
- No symbol repeats $n$ or more times: in base 10, $10$ copies of a symbol are replaced by the next symbol, so no Egyptian symbol appears 10 or more times.
- Limitation: there is no zero, so the base-5 system above cannot write $0$. And the system needs a new symbol for every higher power (the Egyptian system was efficient up to about a crore, $10^7$).
6. Why a Base Helps: Arithmetic
- Addition: count the symbols of each landmark, then regroup: in base 10, 10 of a symbol make one of the next (in base 5, 5 make the next). It is the same carrying you do in the Hindu system: $47 + 56 = 103$.
- Product of landmark numbers is a landmark number: $10^a \times 10^b = 10^{a+b}$. Multiplying by $10$ moves every symbol up one landmark; by $10^2$, up two. The same is true for $5^a \times 5^b$ in base 5, and for any base.
- Multiplying general numbers: break one number into its landmark parts and use the distributive property, $(a + b + c) \times n = an + bn + cn$, then add.
- Contrast with Roman: the regrouping size changes (5s and 2s), and landmark products are not landmark numbers, hence multiplication is difficult.
Abacus (decimal): from about the 11th century, even people using Roman numerals used an abacus board with lines. Each line stands for a power of $10$ starting with $1$; counters on a line show how many times it occurs, and a counter above a line is worth $5$. To add, bring counters on each line together and replace $10$ counters by one on the next line (e.g. $2907 + 43$: $7 + 3 = 10$ ones become one ten).
7. Place Value Representation
Place value (positional) system: a number system with a base in which the position of a symbol decides which landmark number it belongs to. It solves the "endless symbols" problem: a finite set of symbols can write every number.
Mesopotamian (Babylonian) system: base 60
- Base $60$ (sexagesimal), with a wedge symbol for $1$ and another for $10$; numbers $1$ to $59$ are made from these. Its influence survives: $1$ hour $= 60$ minutes, $1$ minute $= 60$ seconds.
- Why 60? Several theories: lunar month of 30 days, the Sun's apparent yearly cycle, ease of fractions, and their older landmarks $1, 10, 60, 600, 3600, \dots$ shrinking to powers of $60$.
- Landmark numbers: $1,\ 60,\ 60^2 = 3600,\ 60^3 = 216000, \dots$. Rightmost group = number of $1$s, next = number of $60$s, next = number of $3600$s, and so on.
- $640 = (10) \times 60 + 40$ and $7530 = (2) \times 3600 + (5) \times 60 + 30$. Examples: $63 = (1) \times 60 + 3$; $\;132 = (2) \times 60 + 12$; $\;200 = (3) \times 60 + 20$; $\;3605 = (1) \times 3600 + 0 \times 60 + 5$.
- No place may hold $60$ or more: $(1) \times 3600 + (70) \times 60 + 2$ regroups to $(2) \times 60^2 + (10) \times 60 + 2$.
- Defect: blanks were written by spacing, and spacing was inconsistent, so the same numeral could be read several ways. Later a placeholder symbol (like our $0$) was introduced, but it was mostly used in the middle of numbers, not at the end, so ambiguity remained.
Mayan system (3rd to 10th century CE, Central America)
- Developed place value independently of Asia. Dot $= 1$, bar $= 5$; a seashell-like symbol for zero (a placeholder). These write $0$ to $19$ in each place.
- Digits are stacked vertically: bottom = $1$s, above it $20$s, then $360$s ($= 20 \times 18$, not $400$, possibly calendar-related).
- Examples: $77 = 3 \times 20 + 17$; $\;100 = 5 \times 20 + 0$; $\;361 = 1 \times 360 + 0 \times 20 + 1$; $\;721 = 2 \times 360 + 0 \times 20 + 1$.
- Not a true base-20 system (the third landmark is $360$), so it lacks the computing advantages of a pure base. Base-20 survives in the number names of some European languages.
Mayan numeral for 77: the upper level counts 20s (3 dots), the lower level counts 1s (3 bars + 2 dots = 17).
Chinese rod numerals (by the 3rd century CE to 17th century)
- Base 10 (decimal) and a place value system, used for computing with rods.
- Symbols for $1$ to $9$, written alternately as vertical (zong: units, hundreds, ten-thousands) and horizontal (heng: tens, thousands) rods, so adjacent places are not confused.
- Example: heng $2$, zong $6$, heng $3$, zong $4$ $= 2 \times 10^3 + 6 \times 10^2 + 3 \times 10 + 4 = 2634$.
- A blank space showed a skipped place. Because the symbols were more uniform, blanks were easier to spot than in the Mesopotamian system. With a symbol for zero, it would be a fully developed place value system.
- If only zong rods were used, $41$ could be misread as $23$, $32$ or $122$.
8. The Hindu Number System and Zero
Hindu number system
Base 10, ten symbols $0, 1, 2, \dots, 9$, place value, one digit per position.
$$375 = (3) \times 10^2 + (7) \times 10 + (5) \times 1$$- Why it won the world: $0$ is a digit, and only one digit sits in each place, so there is no ambiguity; any number can be written with finitely many symbols; and computation is easy.
- Zero: a symbol for $0$ existed in India at least as early as 200 BCE. It was used both as a placeholder and as a number in its own right.
- Aryabhata (499 CE) used the properties of $0$ (e.g. $0 +$ any number is that number; $0 \times$ any number is $0$) in his computations.
- Brahmagupta (628 CE) codified arithmetic with zero in the Brahmasphutasiddhanta. With $0$ and negative numbers he created what is called a ring: a set closed under addition, subtraction and multiplication.
- Evolution of digit shapes: the written forms of $0$ to $9$ changed gradually over centuries.
| Feature | Hindu system | Roman system |
|---|---|---|
| Value of a symbol | Depends on position (place value) | Fixed value |
| Zero | Yes (digit and number) | No symbol for zero |
| Symbols needed | Only 10 for every number | New symbols for bigger numbers |
| Calculation ($+, -, \times, \div$) | Easy | Extremely difficult |
9. Writing Numbers in Other Bases
Base-$n$ representation
$$N = a_k n^k + a_{k-1} n^{k-1} + \dots + a_1 n + a_0, \qquad 0 \leq a_i \leq n-1,\ a_k \neq 0$$
The representation is written $a_k a_{k-1} \dots a_1 a_0$. A base-$n$ system needs $n$ digits, $0$ to $n-1$.
Worked: write 25 in other bases
Base 8: $25 = (3) \times 8 + (1) \times 1$, so $25 = 31$ in base 8.
Base 5: $25 = (1) \times 5^2 + (0) \times 5 + (0) \times 1$, so $25 = 100$ in base 5.
Base 2: $25 = 16 + 8 + 1 = (1) 2^4 + (1) 2^3 + (0) 2^2 + (0) 2 + (1)$, so $25 = 11001$ in base 2.
Method: take the largest power of the base that fits, record how many times it fits (the digit), subtract, and repeat for the smaller powers. Put $0$ where a power is not needed.
- Base 5 positional example: $143 = 125 + 3 \times 5 + 3 = 1 \times 5^3 + 0 \times 5^2 + 3 \times 5 + 3$, so $143 = 1033$ in base 5.
- Base 2 with Gumulgal words: take urapon $= 0$ and ukasar $= 1$. Then $2 = $ ukasar urapon, $3 = $ ukasar ukasar, $4 = $ ukasar urapon urapon. Gumulgal itself has only two landmark numbers and repeats words, while the place value base-2 system writes any number using positions.
- Why base 10? Probably because we have 10 fingers. With 8 fingers we would likely use base 8.
10. Summary and Quick Revision
Evolution of ideas in number representation.
| System | Base | Place value? | Zero? | Landmarks / notes |
|---|---|---|---|---|
| Gumulgal | Counts in 2s | No | No | Names for 1 and 2 only; above 6 is "ras" |
| Roman | None (landmarks) | No | No | 1, 5, 10, 50, 100, 500, 1000; hard to multiply |
| Egyptian | 10 | No | No | Powers of 10, a new symbol for each |
| Mesopotamian | 60 | Yes | Placeholder (later, mostly in the middle) | Ambiguity from spacing |
| Mayan | Nearly 20 (1, 20, 360, …) | Yes (vertical) | Yes (shell) | Not a true base 20 |
| Chinese rod | 10 | Yes | Blank space | Alternating zong / heng rods |
| Hindu | 10 | Yes | Yes, a digit and a number | Used worldwide |
Key definitions:
• Number system: a standard ordered sequence used to count. Numerals: its written symbols.
• Landmark numbers: easily recognised reference numbers with their own symbols.
• Base-$n$ system: landmark numbers are $1, n, n^2, n^3, \dots$ (powers of $n$).
• Place value system: a base system in which the position of a symbol shows its landmark number.
• Number system: a standard ordered sequence used to count. Numerals: its written symbols.
• Landmark numbers: easily recognised reference numbers with their own symbols.
• Base-$n$ system: landmark numbers are $1, n, n^2, n^3, \dots$ (powers of $n$).
• Place value system: a base system in which the position of a symbol shows its landmark number.
Common mistakes:
• Writing 2367 as MMCCCLXII. It is MMCCCLXVII ($50 + 10 + 5 + 2 = 67$).
• Thinking each system improved on the one before. They arose in different places and times, and the real history is complex.
• Saying the Romans or Egyptians had place value. Their symbols have fixed values.
• Calling the Mayan system a true base-20 system. Its third landmark is $360$, not $400$.
• Forgetting zeros while writing in a base: $25$ in base 5 is $100$, not $1$.
• Using a digit equal to or larger than the base (base 5 digits go only up to $4$).
• Calling the Indian numerals "Arabic" without noting they came from India through the Arab world.
• Writing 2367 as MMCCCLXII. It is MMCCCLXVII ($50 + 10 + 5 + 2 = 67$).
• Thinking each system improved on the one before. They arose in different places and times, and the real history is complex.
• Saying the Romans or Egyptians had place value. Their symbols have fixed values.
• Calling the Mayan system a true base-20 system. Its third landmark is $360$, not $400$.
• Forgetting zeros while writing in a base: $25$ in base 5 is $100$, not $1$.
• Using a digit equal to or larger than the base (base 5 digits go only up to $4$).
• Calling the Indian numerals "Arabic" without noting they came from India through the Arab world.