Reema was flipping through an old book when a paper with strange symbols fell out. Her father explained โ these were numbers written by the Mesopotamian civilisation ~4000 years ago, in the region of present-day Iraq!
This sparked the big question: How did numbers evolve to what we use today?
The Yajurveda Samhita mentioned number names based on powers of 10 โ eka (1), dasha (10), shata (100), sahasra (1000), ฤyuta (10,000)โฆ all the way to 10ยนยฒ!
The way we write numbers today (digits 0โ9) was developed in India. The first known instance using all 10 digits including 0 appears in the Bakhshali manuscript (c. 3rd century CE).
First mathematician to fully explain and do elaborate scientific computations with the Indian system of 10 symbols.
Indian numbers transmitted to the Arab world. Popularised by Al-Khwฤrizmฤซ (who the word algorithm is named after!) through his book On the Calculation with Hindu Numerals.
Hindu numerals reached Europe via the Arab world. Fibonacci (~1200) championed adopting Indian numerals in Europe. By the 17th century, their use became unavoidable for scientific progress!
Imagine you're in the Stone Age with a herd of cows. Without our modern numbers, how would you:
- Make sure all cows returned after grazing?
- Know if you have fewer cows than your neighbour?
- Know how many more cows you need?
Many groups worldwide used hands and body parts for counting. The Papua New Guinea people used body parts as their standard sequence/number system โ counting up to 27 using fingers, wrists, elbows, shoulders, ears, eyes and the nose!
One of the oldest methods โ making notches (cuts) on bones or cave walls. Each mark = one object counted.
The Lebombo Bone with 29 notches is one of the oldest known mathematical artefacts โ possibly a lunar calendar!
The Gumulgal (indigenous people of Australia) counted in 2s:
5 = ukasar-ukasar-urapon | 6 = ukasar-ukasar-ukasar
Numbers greater than 6 were all called ras. The pattern: 3=2+1, 4=2+2, 5=2+2+1, 6=2+2+2
Evolved from the ancient Greek system, ~8th century BCE in Rome. Spread across Europe with the Roman Empire.
Example: 27 = 10 + 10 + 5 + 1 + 1 = XXVII
Example: 2367 = 1000+1000+100+100+100+50+10+5+1+1 = MMCCCLXII
These are called Landmark Numbers โ special numbers with their own unique symbol.
The Egyptians made landmark numbers that are powers of 10 โ each one is 10 ร the previous one.
Each has a different hieroglyph symbol!
Example: 324 = 100+100+100+10+10+4 โ written using 3 hundred-symbols + 2 ten-symbols + 4 one-symbols
A number system where:
(a) First landmark number is 1
(b) Every next landmark = current landmark ร fixed number n
Egyptian = Base-10 (also called Decimal System)
Base-5 example: landmark numbers are 1, 5, 25, 125, 625, 3125โฆ
The Egyptian system's drawback: to write very large numbers, you need an unending sequence of new symbols. The solution? Use the position of symbols to tell us which power they represent!
A number system with a base that uses the position of each symbol to determine which landmark number it is associated with.
This is the highest point in number system evolution! It lets us write ALL numbers using only a finite number of different symbols!
- Base-60 (Sexagesimal System)
- Symbols: โผ for 1, โ for 10
- First to use place value idea!
- Used blank space for zero (later: placeholder symbol)
- Influence today: 60 min = 1 hr, 60 sec = 1 min
- Base-10 system
- Different hieroglyph for each power of 10
- Efficient up to 10โท (crore)
- NOT a place value system
- Drawback: needs new symbols for bigger numbers
- Almost base-20 system
- Landmarks: 1, 20, 360, 7200โฆ
- Dot (โข) for 1, bar (โ) for 5
- Symbols placed vertically
- Used seashell symbol for zero!
- Place value system โ independent invention!
- Base-10 (Decimal)
- Two sets: Zong (vertical) & Heng (horizontal)
- Alternated to avoid ambiguity between positions
- Blank space = zero (similar to Mesopotamian)
- Very similar to Hindu system!
By ~11th century CE, even people using Roman numerals started using the abacus โ a calculating device built on the decimal system. Each line represents a power of 10. A counter above a line added a value of 5.
Numbers were grouped into powers of 10, and counters placed accordingly โ making addition and subtraction visual and intuitive!
- Base-10 (Decimal)
- 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- Full place value system
- 0 as a real digit AND a number!
- No ambiguity in reading or writing
- Originated in India ~2000 years ago
- Now used throughout the entire world!
In the Hindu system, zero was not just a placeholder โ it became a real number in its own right!
Brahmagupta's work created a ring โ a set of numbers closed under addition, subtraction, and multiplication โ laying foundations for modern algebra and analysis!
The shapes of 0โ9 we use today evolved over centuries:
| Term | Meaning |
|---|---|
| Number System | Standard sequence of objects/names/symbols in fixed order for counting |
| Numerals | Symbols used in a written number system (0,1,5,36,193โฆ) |
| Landmark Numbers | Special reference numbers in a system (like I,V,X,L,C,D,M in Roman) |
| One-to-One Mapping | Each object paired with exactly one counting element |
| Base-n System | System where each landmark = previous landmark ร n |
| Decimal System | Base-10 number system |
| Place Value System | Position of a digit determines its value |
| Sexagesimal | Base-60 (Mesopotamian/Babylonian) system |
| Rod Numerals | Chinese computing numerals using vertical/horizontal rods |
| Tally Marks | Notches/marks cut on surfaces to count objects |
It forms the basis of modern science, technology, computing, accounting, surveying โ and is present in every corner of our daily lives. The discovery of 0 was truly revolutionary!