A Story of Numbers

Start with a distinction that the whole chapter depends on. A number is an idea in your head: how many goats are in a field, whether you write it down or not. A numeral is the mark you make to record that idea. The number is the same for everybody; the numeral has changed enormously from one place and one century to another. This chapter is a story about numerals.

The oldest way of handling amounts needs no numerals at all. It is called one-to-one matching. A shepherd with no words for large amounts can still keep his flock safe. As each goat leaves the pen in the morning he drops one pebble into a bag. As each goat comes back in the evening he takes one pebble out. If the bag empties exactly, every goat is home. If a pebble is left over, one goat is missing. He has never counted anything, and yet he knows.

The same idea appears with fingers, with knots on a cord, with letters recited in a fixed order, and with scratches on a stick or a bone. Two very old bones are usually mentioned when this story is told. The Lebombo bone, found in southern Africa and tens of thousands of years old, is a piece of baboon leg bone carrying 29 notches. The Ishango bone, found near the headwaters of the Nile and thought to be around 20,000 years old, carries notches arranged in three separate columns. Nobody can be certain what either was for. What is certain is that somebody wanted a permanent record of an amount, and made one.

Scratches are honest but clumsy. Thirty-seven notches in a row cannot be read at a glance; you have to count them again every single time. So the next idea arrives almost immediately: put the marks in groups. Four upright strokes with a fifth struck across them makes a bundle of five, and now thirty-seven is seven bundles and two singles, which the eye takes in at once. Grouping is the first real invention in this story, and every system that follows is a different answer to the same question: what should the groups be, and how should a full group be recorded?

Some communities chose very small groups. The Gumulgal people of Australia counted in twos, using just two basic words: urapon for 1 and ukasar for 2. Bigger amounts were built by stringing these together, so ukasar-urapon is 2 + 1 = 3, ukasar-ukasar is 2 + 2 = 4, and ukasar-ukasar-urapon is 2 + 2 + 1 = 5. That is a workable system, and machines group in twos as well, although they go further and give the groups places, which these counting words never do. Its weakness is obvious the moment the amounts get big: writing 30 would take fifteen ukasars, which is no better than thirty notches.

So the story has reached its first crossroads. Grouping helps, but small groups do not help enough. Something has to be done about the fact that big numbers need short names. Everything that follows in this chapter is an attempt to solve that one problem, and the solutions get steadily cleverer.

Here is a second idea for shortening numerals. Instead of repeating one mark, choose a handful of useful amounts and give each of them a symbol of its own. Numbers picked out this way are called landmark numbers. Every other number is then written by putting landmarks together.

The Roman system takes this route. Its landmarks are seven:

I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000

The rules for putting them together are short. Write the symbols from the largest value on the left down to the smallest on the right, and add up what you see. Do not repeat any one symbol more than three times in a row, and do not repeat V, L or D at all, because two of those always make the next landmark instead: VV would simply be X, LL would be C and DD would be M. When a smaller symbol is placed immediately before a larger one, subtract instead of adding, which is how the awkward numbers just below a landmark are handled: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400 and CM = 900.

Work through some numbers slowly.

• 27 is 10 + 10 + 5 + 1 + 1, so it is written XXVII.
• 302 is 100 + 100 + 100 + 1 + 1, so it is written CCCII.
• 715 is 500 + 100 + 100 + 10 + 5, so it is written DCCXV.
• 1222 is 1000 + 100 + 100 + 10 + 10 + 1 + 1, so it is written MCCXXII.
• 2367 is 1000 + 1000 + 100 + 100 + 100 + 50 + 10 + 5 + 1 + 1, so it is written MMCCCLXVII.
• 2999 needs the subtraction rule three times over: 2000 + 900 + 90 + 9 becomes MMCMXCIX.

Reading in the other direction is just as mechanical. Take each symbol in turn; if the symbol on its right is larger, subtract it, otherwise add it. For MMCDLXIX the pieces are MM = 2000, CD = 400, L = 50, X = 10 and IX = 9, and 2000 + 400 + 50 + 10 + 9 = 2469.

Addition is not too bad. To work out LXXXVII + LXXVIII you can pool the symbols and tidy up: LXXXVII is 87 and LXXVIII is 78, so the total is 87 + 78 = 165, which is written CLXV, since 165 = 100 + 50 + 10 + 5. Even done purely with symbols the method is the same one you use in column addition: gather like symbols, and whenever you have enough of them, trade them for one symbol of the next size.

Multiplication is where the system falls apart. There is no quick way to multiply MMCCCLXVII by XLII, because the landmarks do not fit together neatly. Look at why: 5 is a landmark and so is 10, but 5 x 5 = 25 is not a landmark at all, and neither is 50 x 50 = 2500. Sometimes a product does land on a landmark, as with 10 x 50 = 500, but you cannot rely on it. Because the landmark values jump 1, 5, 10, 50, 100, 500, 1000, the products of landmarks keep falling into the gaps. Roman traders and officials therefore did their multiplying on an abacus and wrote only the answer in numerals. A system in which you cannot calculate, only record, is doing half a job.

Two other weaknesses matter for what comes next. There is no symbol at all for nothing, so the Roman system cannot write zero. And a numeral gives no hint of size from its position: the C in CCCII means one hundred wherever it sits. Both of those gaps are about to be filled.

The Roman landmarks were chosen by convenience. The next idea is to choose them by a rule. Pick one number, call it n, and let the landmarks be the powers of n:

n to the power 0 = 1, then n, then n x n, then n x n x n, and so on

That fixed number n is called the base of the system. In base 10 the landmarks are 1, 10, 100, 1000, ... In base 2 they are 1, 2, 4, 8, 16, 32, ... In base 5 they are 1, 5, 25, 125, 625, ... In base 60 they are 1, 60, 3600, 216000, ...

Why is choosing powers so much better than choosing landmarks by taste? Because a landmark times a landmark is always another landmark. In base 10, 100 x 1000 = 100000, which is again a power of ten. That single fact is what makes multiplication systematic, and it is exactly what the Roman list could not promise, since 5 x 5 = 25 falls into a gap.

A base also fixes how many different digit symbols you need: exactly n of them, standing for 0, 1, 2, ..., up to n minus 1. Base 10 needs ten digits, 0 to 9. Base 5 needs five digits, 0 to 4. Base 2 needs only two, 0 and 1, which is why machines use it. A base-60 system would in principle need sixty different digits, and you will see shortly how Mesopotamia dodged that.

Why did most of the world settle on 10? Almost certainly because of hands. Ten fingers make ten the natural size of a full group. Other choices leave traces too. Counting in twenties, using fingers and toes, survives in the number words of several languages. Counting in sixties survives every time you look at a clock.

Writing a number in another base. The method is to strip off the largest landmark repeatedly. Take 137 into base 5. The landmarks are 1, 5, 25, 125.

• How many 125s fit into 137? One, leaving 137 - 125 = 12.
• How many 25s fit into 12? None, so that place gets 0.
• How many 5s fit into 12? Two, leaving 12 - 10 = 2.
• How many 1s are left? Two.

So 137 is written 1022 in base 5. Check it by expanding: 1 x 125 + 0 x 25 + 2 x 5 + 2 x 1 = 125 + 0 + 10 + 2 = 137. Correct.

Two more, done the same way. For 50: no 125s, two 25s giving 50, nothing left, so 50 is 200 in base 5, and 2 x 25 = 50 checks out. For 651: one 625, leaving 26; no 125s; one 25, leaving 1; no 5s; one 1. So 651 is 10101 in base 5, and 625 + 25 + 1 = 651 checks out.

Notice something important in those answers. Both of them needed a 0 to hold an empty place. Without a zero, 200 and 20 and 2 would look identical, and the whole scheme would collapse. Keep that thought; it is where the story is heading.

The Egyptians were writing numbers about five thousand years ago, and theirs is one of the earliest large systems built squarely on base 10. Each power of ten was given its own picture symbol, and the number was written by repeating those symbols as often as needed.

• 1 was a single vertical stroke.
• 10 was a shape like an arch, drawn from a heel bone.
• 100 was a coil of rope.
• 1000 was a lotus flower.
• 10000 was a bent finger.
• 100000 was a tadpole.
• 1000000 was a kneeling figure with raised arms.

To write 324 you draw three rope coils, two arches and four strokes, because 324 = 3 x 100 + 2 x 10 + 4 x 1. That is nine symbols altogether. There is no rule about the order, either: the coils could be drawn on the right and the strokes on the left and the number would still be 324, because a symbol carries its value with it. The position of a symbol tells you nothing, which is precisely the difference from our own system.

Adding is genuinely pleasant. To add two Egyptian numerals, sweep all the strokes together, all the arches together, all the coils together, and so on. Then tidy up: wherever you have collected ten identical symbols, rub them out and draw one symbol of the next size up. That trading of ten-for-one is exactly the carrying you do in column addition, and it works for the same reason, because the base is ten in both systems.

Multiplying by ten is even nicer. Every symbol simply gets promoted to the next one along: strokes become arches, arches become coils, coils become lotus flowers. One sweep of the pen and the job is done.

So what is wrong with it? Two things, and both are serious.

First, the numerals get long. The number of symbols you need is the sum of the digits. For 1111 you need 1 + 1 + 1 + 1 = 4 symbols, which is fine. For 1023 you need 1 + 0 + 2 + 3 = 6. But 10458 needs 1 + 0 + 4 + 5 + 8 = 18 symbols, 70707 needs 7 + 7 + 7 = 21 symbols, and the modest-looking 999 needs 9 + 9 + 9 = 27 symbols where we write just three digits. A scribe copying tax records would feel that difference every day.

Second, and worse, the system can never be finished. Each new power of ten demands a brand new picture. Egypt had symbols up to a million, and past that point you are left with two poor choices: keep drawing the million sign over and over, so that a thousand million would take a thousand of them, or invent a fresh hieroglyph. A number system ought to be able to write any whole number, however large, without needing anything new and without the numeral running off the page. The Egyptian one cannot.

Notice also that Egypt had no need of a zero. An empty place never arises, because there are no places at all. To write 1023 you draw one lotus, no coils whatsoever, two arches and three strokes, and drawing no coils causes not the slightest confusion. That sounds like an advantage, and in a small way it is, but it is the advantage of a system that has given up the very idea which turns out to matter most.

Here is the idea that changes everything. Stop giving each power its own symbol. Instead, agree on a fixed order of positions, and let the position announce which power is meant. That is place value, and with it a small set of digits can be reused in every place.

Mesopotamia. About four thousand years ago, scribes between the Tigris and the Euphrates pressed wedge shapes into clay using just two marks: one for 1 and one for 10. Those two marks built the digits 1 to 59, and the digits were then arranged in places worth 1, 60, 3600, 216000 and so on. The system is base 60, or sexagesimal.

• 63 = 1 x 60 + 3, so it is written as a group of 1 followed by a group of 3.
• 132 = 2 x 60 + 12, so a group of 2 followed by a group of 12.
• 200 = 3 x 60 + 20, so a group of 3 followed by a group of 20.

Now try 3605. Since 3605 = 1 x 3600 + 0 x 60 + 5, the middle place is empty. The scribes left a gap. A gap on damp clay is a poor sort of symbol: it can be too wide, too narrow, or lost when the tablet chips. Worse, a gap at the end of a numeral is invisible, so the single mark for 1 might mean 1, or 60, or 3600, and only the sense of the sentence told you which. Mesopotamia had place value and something like a zero in the middle, but never a full zero. Its base 60 is still with us: 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a full turn, which is 6 x 60.

China. Chinese counting rods, used for well over two thousand years, are base 10 with place value. The clever touch is that the rod shapes alternate. Digits in the ones, hundreds and ten-thousands places are drawn with upright rods, called zong; digits in the tens, thousands and hundred-thousands places are drawn with rods lying flat, called heng. Because neighbouring places look different, a blank place stands out instead of quietly merging with its neighbours. It is an ingenious patch on the empty-place problem, but it is still a patch: the blank itself is not a symbol.

The Maya. In Central America the Maya wrote numerals vertically, read from the bottom upwards, with three marks: a dot for 1, a bar for 5, and a shell shape for zero. Digits from 0 to 19 were built from dots and bars, so 17 is three bars and two dots, since 3 x 5 + 2 = 17.

Their places are almost base 20, but not quite. The landmarks run 1, 20, 360, 7200, 144000, and the third one is 360 rather than 400 because 360 mattered to their calendar. Work two examples:

• 77 = 3 x 20 + 17, so the upper place holds 3 (three dots) and the lower place holds 17 (three bars and two dots).
• 721 = 2 x 360 + 0 x 20 + 1, so from the top the places hold 2, then 0, then 1 - and that middle 0 is written with an actual shell.

That shell is the point. The Maya had a genuine written symbol for an empty place, centuries before Europe accepted one. What they did not do was treat zero as a number you could calculate with. It marked a gap; it did not join in the arithmetic.

So three separate civilisations invented place value, and all three discovered that place value forces the question of the empty place. Two answered it with a gap, one answered it with a shell. The complete answer was found somewhere else.

Everything so far has been building towards one system. In India, over a long stretch of centuries, the pieces came together: base 10, full place value, ten digit symbols, and a zero that is not merely a gap.

The digits themselves grew out of the Brahmi numerals, and their shapes drifted over the centuries into the ones you write now. There are exactly ten of them, 0 1 2 3 4 5 6 7 8 9, and every place uses the same ten. A numeral is read by multiplying each digit by the landmark of its place and adding:

375 = 3 x 100 + 7 x 10 + 5 x 1 = 300 + 70 + 5

Compare that with Egypt. There, 375 would take 3 + 7 + 5 = 15 separate symbols, and past the million sign a whole new picture would have to be invented. Here, three digits do the job, and no whole number is out of reach: for a bigger number you simply use more places, never a new symbol. Ten symbols are enough to write every whole number there has ever been. That is the claim no earlier system could make.

Why zero has to be a symbol. Suppose you had place value but no zero. Then 12, 102 and 1020 would all have to be written as some arrangement of a 1 and a 2 with blank space in between, and a reader could not tell which was meant. Zero removes the guesswork. It says, out loud, that this place is empty. That is the placeholder job, and Mesopotamia, China and the Maya had all felt the need for it.

Why zero also has to be a number. This is the step that was taken in India. If zero is only a placeholder, it is a piece of punctuation and you cannot calculate with it. If zero is a number, it takes part in arithmetic like any other. An early written zero survives as a dot in the Bakhshali manuscript. Aryabhata, writing around 499, set out the working of the place-value system. Then in 628 Brahmagupta wrote down rules that treat zero as a number in its own right, alongside rules for negative amounts, which he described in terms of fortunes and debts. His rules for zero are the ones you still use:

• a + 0 = a, so adding nothing changes nothing.
• a - 0 = a, for the same reason.
• a x 0 = 0, whatever a happens to be.

Division by zero was the one case that resisted. Brahmagupta's treatment of it was not right, and it took later mathematicians to settle that dividing by zero has no answer at all. The reason is short: if 5 divided by 0 were some number, multiplying that number by 0 would have to give 5, and multiplying anything by 0 gives 0. So there is no such number. Even 0 divided by 0 fails, though for the opposite reason: every number would fit, so no single answer could be picked out. Division by zero is left undefined. Notice this is a statement about zero as a number, which is a question the Mayan shell could not even have asked.

Put the two jobs together and you have a system with four properties that no earlier one had all of: a fixed base, full place value, a finite set of digits, and a zero that both holds a place and joins in the arithmetic. Because of the last two, the same written procedures for adding, subtracting, multiplying and dividing work for every number, big or small, and can be taught to anyone. Calculation stopped being a specialist craft done on an abacus and became something an ordinary person could do on paper.

A good idea does not spread by itself; somebody has to carry it. The Indian numerals and the method of calculating with them were studied by scholars in the Arab world, and around 825 al-Khwarizmi wrote an influential account of calculation with them. His name, worn down by centuries of use in Europe, gives us the word algorithm, meaning a step-by-step method; the title of another of his works gives us the word algebra.

From there the numerals moved into Europe, slowly, over several centuries. Fibonacci, an Italian merchant's son who had learnt the system in North Africa, published a book about it in 1202 arguing that traders should abandon Roman numerals. Change still took a long time: merchants distrusted a symbol for nothing, and some cities banned the new digits in account books because a 0 was thought easy to alter into a 6 or a 9. By the time printing had spread the same shapes everywhere and science needed serious calculation, the argument was over.

One small point of naming. In Europe these numerals were long called Arabic, because Europeans met them through Arab scholars. Those scholars themselves called them Indian numerals, and knew perfectly well where they came from. Indian numerals, or Hindu-Arabic numerals, are the accurate names.

What the whole story teaches. Line the systems up and you can see the ideas arriving one at a time.

1. Match one to one, then keep the marks in groups. Tallies do this, and so do the Gumulgal counting words.
2. Give landmark numbers their own symbols. The Roman system does this, and numerals become much shorter than tallies.
3. Choose the landmarks to be powers of one base. Egypt does this with base 10, and now arithmetic becomes systematic.
4. Let position stand for the power. Mesopotamia, China and the Maya do this, and now a handful of digits is enough.
5. Make zero a symbol and a number. India does this, and the system is finally complete.

From that list you can read off what a number system needs to be good. It should use finitely many symbols. It should be able to write any whole number at all. Every numeral should have exactly one reading, with no ambiguity. And the written form should make calculating easy rather than sending you to an abacus. The Indian system is the first to score full marks on all four, which is why it is now used all over the world.

The older systems have not vanished, though. Sixty is alive every time you read a clock or measure an angle, and 360 degrees in a full turn is a direct inheritance from Mesopotamia. Roman numerals still appear on clock faces, on monuments, in the numbering of the opening pages of books and in the names of kings and queens. Tally marks are still a quick and reliable way to count votes at a school election, precisely because adding one to a tally needs no rubbing out. And base 2 runs every computer and phone in the world, because a wire can easily be on or off but cannot easily be one of ten different states, so counting in twos, where this chapter started, is the grouping the machines came back to.

The story is not really about old symbols at all. It is about the fact that how you write something changes what you can do with it. The column method you use for long multiplication cannot even be written down in Roman numerals, because every step of it leans on place value. Once the notation was right, the mathematics that followed became possible for everybody.