What is zero, and why did it take so long to invent?

The full story

This is the episode's narration, word for word. Headings jump to that point in the video.

Nothing worth a hundred and eighty 0:00

Here's a puzzle. Write a 2 and a 5 side by side, and you've got 25. Now slip a little circle in between them, and you've got 205. You added nothing, and the number went up by 180. So how can nothing be worth a hundred and eighty? And if that little circle is so useful, why did people go so long without writing it down?

The empty column 0:27

Picture the number on a counting board, with pebbles in columns. Each column is worth ten times the one to its right. Two pebbles in the hundreds, none in the tens, five in the ones. But write it down as a row of symbols and the gap vanishes, so the 2 slides into the tens and you've written 25. The circle is a sign that says this column exists, and it's empty. That idea is place value, and the circle is a placeholder.

Two jobs, one circle 0:59

But zero has a second job. Five take away five leaves zero. Here zero isn't marking an empty column; it's an answer, a number on the number line like any other. Historians keep the two jobs apart, because a culture can have the first without the second. And once zero is a number, it needs rules. What's three times zero? What's three divided by zero?

Babylon's gap 1:29

In Babylon, by about 1700 BC, scribes were pressing numbers into clay in base sixty, not base ten. For a long time they had no sign for an empty place. A one meaning one, and a one meaning sixty, with nothing in the ones place, looked exactly the same. Much later, scribes began marking a gap: one tablet from Kish, thought to date from around 700 BC, uses three little hooks, and by the last few centuries BC two slanting wedges did the job.

A sign that isn't a number 2:05

The catch: the Babylonian mark only ever went between two digits, never at the end of a number. So a one and a sixty still looked the same. It was punctuation, not a number you could add or multiply. Greek astronomers later used a small round sign for an empty place, even at the end.

A shell in the Americas 2:27

On the other side of the world, the Maya and their neighbours came up with it again, as far as anyone can tell entirely on their own. Their Long Count calendar wrote dates as a stack of places, and its earliest surviving date, from 36 BC, comes from Chiapa de Corzo in Mexico, just outside the Maya area. The Maya counted in twenties, except that the third place counted three hundred and sixties, to fit the Long Count's 360-day year. They drew zero as a shell. By the year 665 at the latest, they were writing numbers by position with a zero sign, and their zero was older still.

Why it took so long 3:13

So why did it take so long? Three reasons. First, you only need a zero sign if your numbers work by position, and many systems, like the everyday Greek one, didn't. Second, a lot of calculating happened on boards, with rods or tokens in columns, where an empty column takes care of itself. The trouble only starts when you write the columns down. And third, treating nothing as a number means writing rules for it, and the oldest rules we have weren't written down until the 600s.

Emptiness, in Sanskrit 3:49

In India, all three pieces came together. Sanskrit had a word, shunya, meaning empty, and by the early centuries of the Common Era it was already doing a job in notation. By the fifth century, texts were writing place-value numbers in words. And by about 550, an astronomy book defined a constant as sixty minus zero, the earliest definite sign of zero being calculated with.

The oldest zero you can see 4:18

And the oldest zero you can actually see isn't in India at all. It's a dot, carved in Cambodia in 683, in a date written as year 605 of the local calendar. The French scholar George Coedès pointed it out in 1931, and in 1996 Anthony Diller, of the Australian National University, wrote that it still held the record as the oldest known zero. India's oldest firmly dated zero is from 876, in a temple at Gwalior. It records land for a flower garden, 270 hastas long.

The birch-bark puzzle 5:02

Those are the oldest zeros that survive, not necessarily the first ever written, and one manuscript started a real fight. In 1881, near Peshawar in what's now Pakistan, a man digging found a bundle of birch bark. It's called the Bakhshali manuscript, about seventy leaves of maths for traders, and it writes zero as a dot. In 2017, Oxford's Bodleian Library carbon-dated three of its leaves, and announced that one came back as early as the third or fourth century. Headlines called it the world's oldest zero.

What carbon dating dates 5:43

But radiocarbon dates the bark, not the writing. The clock starts when the birch grows it, so old bark can carry new ink. And the three leaves had come back centuries apart. Historians pointed out that two of them share the same handwriting, and a worked problem on one runs straight onto the next. If it was written out in one go, the writing can't be older than its youngest leaf. Then the lab tested more leaves, and tested that early one again. In its 2024 report, it said the first result had been wrong. Every leaf it kept now dates from the late 700s to about 1100. So the Bakhshali dot is a genuinely old zero, but not the oldest; it's younger than the one carved in Cambodia.

Brahmagupta's ledger 6:35

The oldest surviving rulebook for zero as a number was written in 628, by the astronomer Brahmagupta, working in Bhillamala, in India. In a huge book on astronomy, he set out rules for positive numbers, negative numbers and zero. And he used a merchant's language: positive numbers are fortunes, negative numbers are debts. A fortune plus an equal debt leaves zero.

Rules that still work 7:08

Add zero to a number, or take zero away, and nothing changes. Take a debt away from zero and you get a fortune. Any number times zero is zero. Zero is the number that sits exactly between the fortunes and the debts, and adding it does nothing at all. Then he got to division.

Three answers, all wrong 7:30

Zero divided by zero, said Brahmagupta, is zero. A number divided by zero he simply left as a fraction with zero underneath. Two centuries later, Mahavira said a number divided by zero stays unchanged. In the 1100s, Bhaskara said it becomes an infinite quantity. Three answers, and modern maths accepts none of them. Some historians argue they were following a different convention rather than blundering.

The division puzzle 8:06

So try it yourself. Twelve divided by three is four, and here's how we know: three times four is twelve. Division is multiplication run backwards. So twelve divided by zero has to be a number that, times zero, gives twelve. Try one, ten, a thousand. Zero times anything is zero, so you never reach twelve. There's simply no answer to find.

Too many answers 8:36

Zero divided by zero fails the opposite way. Now we need a number that, times zero, gives zero. One works. Seven works. Minus five works. Every number works. So Brahmagupta's answer, zero, is just one of infinitely many numbers that fit, and a question with every answer has no single answer. That's why mathematicians call both of these undefined.

Why not infinity? 9:10

But Bhaskara's idea is tempting, isn't it? Divide twelve by a tenth and you get a hundred and twenty. By a hundredth, twelve hundred; by a thousandth, twelve thousand. Now come at zero from the other side: twelve divided by minus a tenth is minus a hundred and twenty, plunging the other way. One side says huge, the other says hugely negative, so no single value fits. And if twelve over zero and thirteen over zero were both infinity, zero times infinity would have to be twelve and thirteen at once.

How it travelled 9:49

The Indian numerals travelled west through the Islamic world. Around 770, an Indian astronomer brought a book of astronomy to the caliph's court; one historian, Georges Ifrah, argues it was probably Brahmagupta's. In Baghdad's House of Wisdom, al-Khwarizmi wrote on the Indian way of reckoning. The Arabic original is lost, but a much-changed Latin version, Algoritmi on the numbers of the Indians, gave us the word algorithm. Arabic translated shunya as sifr.

Nine figures and a sign 10:31

Leonardo of Pisa, now known as Fibonacci, learnt these numerals in North Africa. In 1202 he wrote the Liber abaci, a maths book for merchants. Its first chapter begins: the nine figures of the Indians are these, nine down to one. With these nine figures, and with this sign, zero, which in Arabic is called zephirum, any number at all is written. Notice: nine figures, and a sign. Zephirum became our word zero, but even then, zero wasn't quite one of the numbers.

So what is zero? 11:16

So what is zero? It's a sign that keeps the columns honest. It's the number between the fortunes and the debts, the one that changes nothing when you add it. And it's the one number you can never divide by. It took so long because it was really three ideas, worked out over thousands of years, in different places. But Bhaskara left a question hanging. If dividing by zero doesn't give you infinity, then what is infinity? And can one infinity be bigger than another?

Sources

Every factual claim in the episode is tied to one of these. Spotted an error? Tell us.

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Image credits

• Gwalior inscription, 876 CE: the numeral 270 · photo Ms Sarah Welch · CC0 · via Wikimedia Commons · cropped · licence: https://creativecommons.org/publicdomain/zero/1.0/
• Chaturbhuj temple, Gwalior Fort · photo Varun Shiv Kapur · CC BY 2.0 · via Wikimedia Commons · cropped (slow zoom) · licence: https://creativecommons.org/licenses/by/2.0/
• A leaf of the Bakhshali manuscript · Bodleian Library, Oxford, MS. Sansk. d. 14 · reproduction via National Geographic / Wikimedia Commons · public domain (PD-Art)

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