Pi number is the most famous number in existence. It is also the strangest, the most obsessed over, and arguably the most important.
You probably know it as 3.14. You probably know it has something to do with circles. What you might not know is that Pi turns up where circles are nowhere in sight, that it contains every sequence of numbers that has ever existed or ever will, and that a man once memorised 3,142,958 decimal places of it purely for the love of doing so.
This is the number that broke ancient mathematicians, defeated supercomputers, and quietly runs the universe in the background while everyone ignores it.
What Pi actually is
Pi is the ratio of a circle’s circumference to its diameter. That is it. Measure any circle ever drawn, divide the distance around it by the distance across it, and you get 3.14159265358979… going on forever.
The remarkable thing is that this ratio never changes. A circle the size of a coin and a circle the size of the observable universe produce the same number. Pi is a constant woven into the geometry of space itself. It does not depend on measurement systems, cultural conventions, or the preferences of the mathematician doing the calculation. It simply is.
The Egyptians and Babylonians knew about it four thousand years ago, working with approximations close enough to build the pyramids and the hanging gardens. Archimedes in 250 BCE drew polygons inside and outside circles, doubling the sides until he had a 96-sided shape, and calculated Pi to what we now know were the correct first few decimal places. He did this by hand, with no algebra, no calculus, and no concept of the number system we use today.
He got very close. Annoyingly close, given the tools available.
The number that never ends and never repeats
Pi is irrational. This means it cannot be expressed as a fraction. No combination of whole numbers, however large, produces it exactly. Its decimal expansion runs forever without settling into a repeating pattern.
This is stranger than it sounds.
The fraction 1/7, for instance, produces 0.142857142857142857… the same six digits cycling endlessly. You can predict any digit at any point without calculating. Pi offers no such luxury. The digits appear to fall without pattern, without cycle, without end. Mathematicians have been searching for a hidden structure in Pi for centuries. They have not found one.
In 2021, researchers at the University of Applied Sciences Graubünden in Switzerland calculated Pi to 62.8 trillion decimal places. The computation took 108 days and 9 hours. If those digits were printed out, they would fill every book in the British Library ten times over.
And they still found no pattern.
How much Pi does the universe actually need
Here is where it gets interesting.
NASA’s Jet Propulsion Laboratory uses Pi to 15 decimal places for interplanetary navigation. The engineers who calculate spacecraft trajectories to Jupiter and Saturn, who have sent Voyager 1 past the edge of the solar system, need no more than 3.141592653589793 to do it accurately.
With 32 decimal places, you could calculate the circumference of the Milky Way galaxy to within the width of a single hydrogen atom.
With 39 decimal places, you have enough precision to perform any cosmological calculation that physics currently allows. Any calculation. About anything. In the known universe.
The current record stands at over 100 trillion digits.
This means humanity has calculated Pi to a precision approximately 100 trillion times more precise than the universe requires. The remaining digits serve no practical purpose. They exist because the question of what comes next is simply impossible to leave unanswered. That impulse is human, not mathematical. The universe stopped needing those digits a very long time ago. We kept going anyway.
Pi is hiding in places that have nothing to do with circles
This is the part that unsettles mathematicians the most.
Pi appears in the formula for the normal distribution, the bell curve that describes everything from human height to exam results to measurement errors in physics. It appears in the equation for calculating the probability of two randomly chosen numbers sharing no common factors. It appears in Heisenberg’s uncertainty principle in quantum mechanics. It appears in the formula describing how heat moves through a solid object over time.
None of these things are circles. None of them have any obvious geometric relationship to the ratio of a circumference to a diameter. And yet there Pi is, embedded in the equations, unavoidable and unexplained.
John Wallis discovered in 1655 that Pi divided by 4 equals a specific infinite product built from fractions of whole numbers: 1 minus 1/3 plus 1/5 minus 1/7, going on forever. That formula later reappeared, without anyone planning it, in the quantum mechanical calculations of the hydrogen atom. A seventeenth century mathematician’s work on circles showed up three centuries later in the description of an electron.
Nobody knows why. That is not a rhetorical statement. Mathematicians genuinely do not have a satisfying explanation for why Pi appears so far from home, so reliably, in so many places.
The possibility that pi contains everything
Here is the wildest claim made about Pi, and it may be true.
Mathematicians believe Pi is what they call a normal number. This means, if true, that every possible finite sequence of digits appears somewhere in its decimal expansion. Not just every sequence of digits. Every sequence, infinitely many times.
If Pi is normal, then somewhere in its infinite digits is the complete text of every book ever written. The entire works of Shakespeare, word for word, encoded in its decimal places. Every photograph ever taken, encoded as pixels. Your name, your date of birth, every thought you have ever had reduced to numbers, sitting somewhere in Pi, waiting.
The sequence of digits that represents the complete works of Homer appears infinitely often. So does every variation of every book that could ever be written. So does a version of these exact sentences.
Carl Sagan explored this idea in his 1985 novel Contact, suggesting that a message from the creator of the universe was buried in the digits of Pi. It was never proven. It cannot be, given infinite digits. But the mathematics does not rule it out.
It is worth sitting with: a number defined by the relationship between a circle’s edge and its width might contain every pattern that has ever existed or ever will.
The people who loved pi too much
Pi has produced an unusual category of human being: the person who memorises its digits.
The practice is called piphilology. World records have been set at over 70,000 digits memorised and recited aloud. Thomas W. Ferguson holds the verified record at 3,142,958 decimal places.
There are poems and songs written to help memorise Pi, where the number of letters in each word corresponds to the next digit. There is a circular room in the Palais de la Découverte in Paris where 707 digits of Pi are inscribed on the walls of the dome. Those digits contained an error, undetected from 1873 until 1946.
In the 1967 Star Trek episode Wolf in the Fold, a murderous computer is stopped by Mr Spock instructing it to calculate the last digit of Pi. An impossible task. The computer, attempting to do the impossible, is destroyed.
March 14 is Pi Day, declared officially by the US Congress in 2009. It is also the birthday of Albert Einstein. The coincidence, if it is one, would not surprise anyone who has spent time thinking about Pi.
What pi actually tells us
Every number has a personality if you look long enough. Pi’s personality is this: it is precise and unknowable simultaneously.
We know exactly what it is. It is the ratio of a circumference to a diameter. We can describe it, prove its properties, and calculate it to any desired precision. And yet we cannot write it down, cannot predict its next digit, cannot find its pattern, and cannot fully explain why it appears everywhere in nature and physics.
It is the simplest possible question about a circle, asked four thousand years ago, that has never been fully answered.
The universe is apparently very fond of circles.
Read next: Why Humans Explore: The One Drive Behind All Ambition
