From Tally Marks to Zero: Where Our Numbers Came From

Write down the number 2026. Four little marks, and it's instantly recognizable in every part of the world. It feels like that has always been the way to write numbers.

But it wasn't. The ten-digit system, and especially the zero, is one of the greatest inventions in human history. People counted sheep, measured fields and traded grain for thousands of years before anybody thought of that. Let us follow the long road from the first scratch on a bone to the numbers on your ruler.

Two problems every number system has to solve

Every way of writing numbers, in every civilisation that ever tried, has to answer two questions.

The first is how do you record how many? You need a mark that stands for a thing, so the count survives after the real sheep have wandered off.

The second is harder: how do you write big numbers without writing forever? If one mark means one sheep, then three hundred sheep means three hundred marks. Nobody wants to make those marks, and worse, nobody wants to check them.

Keep both questions in mind, because the whole story is really people getting better and better in answering them.

The beginning: one mark, one thing

The oldest counting we know of is really simple. Make a scratch for each thing. Archaeologists have found bones tens of thousands of years old with rows of notches cut carefully along them, and shepherds were still dropping one pebble into a pouch for each animal long into recorded history.

This is called one-to-one matching, and it is exactly what a small child does when they touch each object as they count. You do not even need to know the words for numbers. You just need the way to mark each object.

Tally marks are that idea tidied up. Four upright strokes and a fifth struck across them, so your eye can grab a group of five without recounting. That little diagonal is the first hint of the big idea to come: instead of reading every mark, read groups.

Egypt: a picture for every size

The Egyptians took a proper swing at the second problem. Rather than one symbol repeated forever, they drew a different picture for each power of ten: a stroke for 1, a bent arch for 10, a coil of rope for 100, a lotus flower for 1,000, a pointing finger for 10,000, a tadpole for 100,000, and for 1,000,000 a kneeling figure with both arms flung in the air.

To write a number you drew each picture as many times as you needed. So 243 is two coils, four arches and three strokes. Nine symbols instead of 243 scratches, which is an enormous improvement.

But notice what is still wrong. To write 9,999 you need thirty-six symbols. And every time numbers get bigger, you have to invent a brand new picture. There is no picture for ten million, so ten million cannot be written at all.

Babylon: counting in sixties, and the first place value

In Mesopotamia the scribes pressed wedge shapes into wet clay: a thin upright wedge for 1 and a sideways wedge for 10. So far, similar to Egyptian. Then they did something interesting that nobody had done before.

They let the position of a symbol change its value. A wedge in the first column means one. The same wedge in the next column along means sixty. In the column after that, sixty sixties, which is three thousand six hundred. This is place value, and it is the most powerful idea in the history of writing numbers, because now a handful of symbols can reach any size.

Yes, sixties. The Babylonians counted in base 60 while we count in tens.

There was one flaw, and it is a famous one. If a column was empty, the scribes left a gap. A gap is very hard to see on a clay tablet, so 61 and 3,601 could look almost identical, and readers had to work it out from context. Much later they added a mark to hold an empty column, but it was only ever a spacer between other digits, never a number in its own right and never at the end of a number. That last step was still missing.

The Maya: bars, dots, and a shell that means nothing

On the other side of the world the Maya built a system out of just three symbols. A dot is 1, a bar is 5, and a shell stands for zero. They wrote numbers in columns stacked upward, and each place is worth twenty times the one below, so this is base 20, which is what you get when you count fingers and toes.

The remarkable part is that shell. The Maya had a real symbol for an empty place, centuries before Europe had anything of the kind. Their astronomers tracked the movements of Venus across hundreds of years with it, which you simply cannot do while guessing at gaps.

Rome: letters that add up

Roman numerals are the ones most of us still half know. I, V, X, L, C, D and M, added together, with the shortcut that a smaller letter before a bigger one means subtract, so IV is four. That subtraction trick was not fixed in ancient Rome, where IIII was perfectly normal, and it only settled into the rule we teach today much later.

Roman numerals are excellent for carving on a monument. They are terrible for doing sums. There is no place value at all, so there are no columns to line up, and there is no zero.

India: the moment zero became a number

Here is the turn in the story. In India, mathematicians did something nobody had quite managed: they treated zero as a number. Not a gap, not a spacer, but a value you can add, subtract and calculate with like any other. Around 628 CE the mathematician Brahmagupta wrote down the rules for it, and a carved inscription at Gwalior shows the little round zero already looking much like ours.

Combine that with place value and ten digits, and suddenly everything works. Every number, however vast, is written with the same ten symbols. Columns line up. You can add, carry, borrow and multiply on paper. Writing numbers and calculating with them finally became the same activity.

The system travelled. Scholars in the Islamic world adopted and spread it, above all the Persian mathematician al-Khwarizmi, whose book on calculating with these numerals reached Europe in translation. His name, worn down by centuries of Latin, is where we get the word algorithm, and the title of another of his books gives us algebra. Around 1202 the Italian merchant's son Fibonacci argued in print that European traders should drop Roman numerals in favour of these. It still took roughly three hundred years to win the argument.

We call them Arabic numerals, and in Arabic they are called Indian numerals. Both names are right, because the name is really a record of the journey.

Binary: the newest system is the oldest idea

The most recent number system in wide use is also the most stripped down. Binary has only two digits, 0 and 1, and each place is worth double the one before instead of ten times. Gottfried Leibniz laid it out in 1703, long before there was any machine to use it.

That turned out to matter enormously, because a wire, a switch or a tiny transistor is very good at exactly two things: on and off. Every photo, song and message on your device is stored as those two digits. It is the Babylonian idea of place value again, running on the smallest possible set of symbols. If your explorer would like to see their own name in it, we have a whole activity on writing your name in binary.

See any number in all of them at once

Type a number below and watch seven civilisations write it. Keep an eye on the little count beside each system, showing how many separate symbols that number took.

The one big idea

Every system in this story is an attempt at the same puzzle: how do you write any number at all using a small handful of symbols?

Tallies use one symbol and give up on big numbers. Egypt uses a new symbol for every size and runs out of pictures. Rome uses seven letters and cannot calculate. Babylon and the Maya find place value, which is the key, and the Maya even find a zero. India puts every piece together, and that combination works so well that essentially the whole world now uses it, whatever language it speaks.

The zero is the piece everyone underestimates. It is not simply a symbol for nothing. It is the thing that lets a column sit empty and still be counted, and without it place value collapses. It took humanity a very long time to invent a way to write nothing at all.

Frequently Asked Questions

Numbers, like messages, turn out to be a story about codes: a small set of symbols, agreed in advance, that can carry any idea you like. If your explorer enjoyed this one, follow the digits further into the endless number pi, or take place value apart in a different direction with fractions.

Happy counting!