Why does pi never end?

The full story

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

A bit more than three 0:00

Here's a puzzle you can do in the kitchen. Wrap a string once around a jar lid, then lay it straight across. It goes across three times, with a little left over. A coin or a bike wheel: three, and a bit. That bit starts point one four one five nine, and it doesn't stop. So why is it the same for every circle? And why does it never end?

The same for every circle 0:28

All circles are the same shape; a big one is just a zoomed-in copy of a small one. Zoom in, and the distance around and the distance across grow by the same factor, so their ratio can't change. A hexagon drawn inside any circle has sides exactly one radius long, so its way around is three diameters. A square outside measures four. Every circle sits between them, and the number in between is pi.

Clay and papyrus 0:59

The ratio has been known so long that nobody can say who noticed it first. A Babylonian clay tablet from about 1900 to 1600 BC seems to use three, and at least one Babylonian value was three and an eighth. In Egypt, around 1650 BC, the scribe Ahmes copied out the Rhind papyrus from a book he says was two hundred years older. His problem fifty finds the area of a round field nine units across: take away a ninth, leaving eight, and square it. That works out to pi of about three point one six.

Archimedes builds a cage 1:41

The first person we know of to calculate pi, rather than measure it, was Archimedes of Syracuse, in the two hundreds BC. His trick was a cage. He drew a polygon just inside a circle, and another just outside. Then he doubled the sides, from six to twelve, twenty-four, forty-eight, and ninety-six, and each time the cage closed in.

Trapped, not found 2:11

With no decimals and no algebra, he worked every square root out by hand. His answer: pi is more than three and ten seventy-firsts, and less than three and a seventh. And notice what he did not claim. He never said pi was twenty-two sevenths; he had trapped it, not found it.

Three thousand sides, and a lost book 2:32

In China, around the year 263, Liu Hui wrote a commentary on an older classic, the Nine Chapters. He pushed the polygon to three thousand and seventy-two sides, and got five decimal places right. Two centuries later, Zu Chongzhi pinned pi to seven decimal places, and gave three hundred and fifty-five over one hundred and thirteen. His book is lost; we know his result because a later dynastic history recorded it. Nobody is known to have done better for about nine hundred years.

Madhava adds forever 3:12

The one who did was Madhava of Sangamagrama, in Kerala, in southern India, around 1400. He took a different road. Instead he found a sum that never stops: one, take away a third, add a fifth, take away a seventh, forever, and you get a quarter of pi. His mathematical writings are lost; we know this work because later mathematicians in Kerala wrote it down. Europeans found the same series again more than two centuries later, and it's often still named after Leibniz.

How a never-ending sum lands 3:54

It's beautiful, and nearly useless. Four is too big; take away four thirds and you're too small; add four fifths and you're too big again. It zig-zags around pi so slowly that after ten thousand terms you're still out by about a ten-thousandth. So Madhava sped it up, with correction terms and a faster series, and gave pi correct to eleven decimal places. The strange part: infinitely many pieces add up to one ordinary, finite number.

Never ending isn't special 4:28

Is a decimal that goes on forever actually special? One third is nought point three three three, forever. One seventh is nought point one four two eight five seven, then those same six digits again, and again. Every fraction either stops or repeats. Divide by seven and there are only seven possible remainders, so one soon comes back and the digits cycle. The real claim about pi is that it never ends and never repeats, which is exactly what it means to not be a fraction.

Lambert's proof 5:05

Proving that took until the seventeen-sixties. Johann Lambert, at the Berlin Academy, showed that if you feed the tangent function, from trigonometry, any fraction except zero, what comes out is never a fraction. But the tangent of a quarter of pi is exactly one. So a quarter of pi can't be a fraction, and neither can pi. A number like that is called irrational. The proof is usually dated 1761, the year on the Academy's volume, but its own first page says it was read in 1767.

Irrational isn't enough 5:44

But irrational doesn't mean impossible to draw. The square root of two is irrational too, yet it's just the diagonal of a square with sides of one. That mattered for an ancient puzzle: with only a straight edge and a compass, can you draw a square with exactly the same area as a circle? Lambert's proof didn't settle it. Attempts kept pouring in, and in 1775 the Paris Academy of Sciences announced it would stop examining them.

Hermite does the hard part 6:16

The answer came in 1882. Nine years earlier, the French mathematician Charles Hermite had proved something powerful about another famous number, called e. Ferdinand von Lindemann, a professor at Freiburg, had visited Hermite in Paris and learnt his methods, and he found the step that applied them to pi. Many historians feel Hermite did most of the hard work, and Lindemann spotted the finishing trick. Pi, he proved, is transcendental.

Why the compass can't do it 6:52

The square root of two answers a tidy question: what number, times itself, makes two? Numbers that solve equations like that, built from whole numbers, are called algebraic. Transcendental means pi solves none of them. A straight edge and compass can only add, subtract, multiply, divide and take square roots, so every length they make is algebraic. A circle of radius one has area pi, so the matching square needs sides of root pi. Draw that, and pi would be algebraic. It isn't, so the circle can't be squared.

The Indiana pi bill 7:33

Fifteen years later, an Indiana doctor, Edwin Goodwin, believed he'd squared the circle anyway. He offered his method to the state's schools free of royalties, if the legislature made it official. His local member introduced House Bill 246 in January 1897, and on the fifth of February the House passed it, sixty-seven votes to none. By luck, a Purdue maths professor, Clarence Waldo, was in the building lobbying for funding. Offered an introduction to Goodwin, he later recalled saying he knew as many crazy people as he cared to.

What the bill actually said 8:14

One line of the bill gives the ratio of diameter to circumference as five-quarters to four. Divide it out, and pi equals three point two. Other parts contradict that, and even make root two exactly ten sevenths. Waldo briefed senators, and on the twelfth of February the Senate mocked the bill for half an hour and postponed it indefinitely. It never became law, and it never tried to make pi equal three.

Trillions of digits 8:44

Meanwhile the digit hunt went on. In 1873 William Shanks published seven hundred and seven digits, by hand. In the mid-1940s, someone checking his work found a slip at the five hundred and twenty-eighth, which wrecked every digit after it. In 1949 the ENIAC computer reached two thousand and thirty-seven digits. The current record, finished in November 2025 by the tech site StorageReview, is three hundred and fourteen trillion digits, from one server running for a hundred and ten days.

Is pi random? 9:22

So does pi look random? Statistically, so far, yes: each digit turns up about as often as you'd expect, and the digits pass the tests for randomness. But they aren't random; they're fixed, and anyone who calculates them gets exactly the same ones. And never repeating doesn't mean patternless. Write one, nought, one, nought, nought, one, with one more nought each time. It never repeats, so it's irrational, yet it never contains a two.

Is your phone number in pi? 9:55

You may have heard that pi contains every number somewhere, including your phone number and your birthday. Maybe it does. That would follow if pi is what mathematicians call normal, meaning every string of digits turns up as often as chance predicts. But nobody has proved pi is normal. No one has even proved every digit keeps turning up forever. So is your birthday in pi? Almost certainly; it's a short string, and you can look it up in the digits we already have. But whether every string of digits, of every length, turns up somewhere, nobody knows.

Thirty-eight digits for the universe 10:37

So how many digits do we need? Marc Rayman, an engineer at NASA's Jet Propulsion Laboratory, answered that. For interplanetary navigation, JPL uses fifteen decimal places. A circle whose radius is as far as Voyager One will be from Earth in the late 2030s then comes out wrong by about one and a half centimetres. For the whole visible universe, to within the width of a hydrogen atom, thirty-eight digits is enough. We have three hundred and fourteen trillion.

So why does it never end? 11:12

So why does pi never end? Because it isn't a fraction, so its decimal can't stop or settle into a loop. It isn't even the answer to any whole-number equation, which is why no compass can square the circle. Whether its digits hide everything, nobody knows. But remember how Madhava reached it: infinitely many pieces, landing on one finite number. An ancient Greek thinker, Zeno, built a famous puzzle out of exactly that idea. So how did Achilles ever catch the tortoise?

Sources

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

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Image credits

• Rhind Mathematical Papyrus (detail), copied by the scribe Ahmes · British Museum EA 10057 · photo via Wikimedia Commons · public domain (PD-Art)
• J. H. Lambert, Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques · Mém. Acad. Berlin, vol. for 1761, printed 1768, p. 265 · scan via Internet Archive · public domain

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