3.1 Anaya’s Puzzle
On a slow Sunday Anaya was leafing through a dusty atlas when a folded sheet slid out and fluttered onto the carpet. It was a pencil rubbing taken from some old object, covered in odd little wedge-shaped dents. “What on earth is this?” she muttered, and carried it off to her grandmother.
Her grandmother peered at it and chuckled. “About four thousand years ago a remarkable people lived in a land called Mesopotamia, in the western part of Asia, most of what is now Iraq and the countries around it. One of the ways they recorded their numbers looked exactly like that!”
Anaya’s jaw dropped. Those little dents were numbers? A whole flock of questions rose up at once: When did people first start counting? What were they counting? When did anyone begin writing numbers the way we do now? How might a Mesopotamian have set down 20, or 50, or 100?
This entire chapter is one long answer to Anaya’s questions. We will not march through the systems strictly by date; instead we will trace the growth of a single idea, the idea of a clever way to write numbers, through its main stages.
Where Our Numbers Came From
Counting was already useful in the Stone Age, to keep track of food, herds, exchanges of goods, ritual offerings, and the march of the days (when is the next new moon, the next change of season?). But what those people said and wrote bore no resemblance to our numbers.
The shape of our modern spoken numbers took root thousands of years ago in India. Old texts such as the Yajurveda Samhita set down names for numbers built on powers of 10, eka (one), dasha (ten), shata (hundred), sahasra (thousand), ayuta (ten thousand), climbing all the way to 10^{12} and past it.
The way we write numbers, the ten digits 0 to 9, including a 0 first set down as a dot, also took shape in India roughly 2000 years ago. Its earliest surviving appearance is in the Bakhshali manuscript (around the 3rd century CE), and Aryabhata (around 499 CE) was the first to perform serious calculation with this ten-symbol scheme.
These digits reached the Arab world by about 800 CE, spread by scholars such as Al-Khwarizmi (whose name became our word algorithm) and Al-Kindi. From there they crossed into Europe by around 1100 CE, with the Italian mathematician Fibonacci making the case for them around the year 1200. Roman numerals were so entrenched that the newcomers needed several more centuries to win the argument, yet by the Renaissance of the 17th century, abandoning them had become unimaginable.
The great French mathematician Pierre-Simon Laplace (1749 to 1827) summed it up beautifully:
“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated.”
A Note on Names Because Europeans picked up these digits via the Arab world, they labelled them “Arabic numerals.” But the Arab scholars themselves called them “Hindu numerals,” since that is where the digits had originated. Today the more accurate names, Hindu numerals, Indian numerals, or the bridging Hindu–Arabic numerals, are reappearing in textbooks worldwide. Here “Hindu” points to a land and its people, not a religion.
How counting works
Picture yourself in the Stone Age, ten thousand years back, owning a herd of goats but with no number words and no written digits. Three perfectly natural worries crop up:
- Q1. How can I be sure every goat came home after grazing?
- Q2. Do I have fewer goats than the family next door?
- Q3. If fewer, how many more would close the gap?
How could anyone settle these without numbers? Here are the methods people genuinely used.
Method 1, Objects (pebbles or sticks). For each goat, set aside one pebble. The finished heap of pebbles stands in for the herd. If a pebble is left with no goat to match it, a goat has wandered off.
This pairing, each goat tied to exactly one pebble, with no pebble shared between two goats, is a one-to-one mapping. It sits at the centre of all counting.
The Key Idea: One-to-One Mapping To count a collection is to build a one-to-one mapping between it and a fixed, ordered standard sequence, pebbles, sounds, or symbols. To compare two herds (Q2), pair their pebbles one-to-one: whoever ends up with spare pebbles owns more goats. The count of those spares answers Q3.
Method 2, Sounds or names. Rather than objects, run through the letters of a language in order: a, b, c, \dots Counting becomes chanting letters while pointing. With the English letters a–z you can only reach 26, a glaring ceiling.
Method 3, Written symbols. Use a settled sequence of marks. The Romans, for example, wrote:
| Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Symbol | I | II | III | IV | V | VI | VII | VIII | IX | X |
These three methods reveal what every number scheme needs: a standard sequence in a fixed order. The snag is that numbers never stop, so we want a sequence that is both endless and easy to use. Pebbles are endless but unwieldy for large herds; letters are handy but expire at 26. The marks that survive into a written scheme are called numerals (for instance 0, 1, 5, 36, 193 are Hindu numerals).
Figure it Out
- Using only the pebbles method (Method 1), no number names, no Hindu numerals, describe how you would add, subtract, multiply and divide two collections of pebbles.
- The letters method (Method 2) gives out at 26. One repair is to use strings of letters, say “aa” for 27. Extend the scheme so it can name every number. (Many routes work!)
- Try inventing a number system of your very own.
- Take the two heaps of pebbles. Add: slide both heaps together; the merged heap is the sum. Subtract: lift out of the bigger heap as many pebbles as the smaller heap holds; whatever stays behind is the difference. Multiply: form several equal heaps of pebbles and pour them all into one. Divide: keep scooping out heaps of the wanted size and count how many full heaps you managed (any pebbles left over are the remainder).
- One easy route: after a, b, \dots, z (1 to 26), carry on with the doubled letters aa, bb, cc, \dots, zz (27 to 52), then aaa, bbb, \dots and so on, tacking on one more letter every time you run dry. (Any consistent rule that never repeats a string will do.)
- Open-ended, any fixed, ordered, never-ending sequence of marks or sounds is a legitimate number system.