Things About Pi You've Probably Never Heard

It is impossible to write the exact value of pi as a fraction, because pi is an irrational number. Even so, mathematicians have long searched for simple fractions that sit as close to pi as possible. The most surprising of these is 355/113. Worked out, it yields 3.1415929, and it sits so near pi's true value that the gap between them is smaller than one in a million. For this reason 355/113 is sometimes called "not pi, but an incredible simulation."

What makes this closeness so remarkable is how plain the fraction is. It uses only the digits 1, 3, and 5, each appearing twice. It is easy to remember and short to write, and yet it gives pi correctly to seven digits. Getting an approximation this good out of such small numbers is a rare thing in the world of fractions.

The fraction traces far back, to the fifth-century Chinese mathematician Zu Chongzhi. When Zu expressed pi with it, Europe would need roughly a thousand years to reach the same accuracy. Because it carries both simplicity and precision at once, 355/113 is still remembered today as the most elegant fractional approximation of pi.

Pi has two separate days of celebration each year. The first is March 14, written "3/14" in the American style. Because this date lines up exactly with pi's first three digits, 3.14, it is known as Pi Day. In many parts of the world math lovers mark it with activities built around the circle, contests to recite pi's digits, and pies and tarts whose name echoes the number itself.

The second day is July 22, or "22/7." This date comes from 22/7, a fractional approximation of pi used since ancient times. Twenty-two divided by seven gives roughly 3.142, sitting even a touch closer to pi than 3.14 does. For this reason the day is called Pi Approximation Day. The difference between the two days is that one highlights pi's decimal form while the other highlights its expression as a fraction.

That a single number has separate days for both its decimal form and its fractional form shows how special a place pi holds in the human imagination. While most numbers find no place on any calendar, pi is one of the rare few celebrated twice a year.

Every spring, MIT releases its undergraduate admission decisions on March 14 — written 3/14 in the American style, the opening digits of pi. The day is known worldwide as Pi Day, and for thousands of applicants it is also the hour when an email arrives that may change everything.

The timing is deliberate. MIT has long tied its acceptance letters to a date that already belongs to mathematics, turning a calendar coincidence into a small ritual. Pi is not merely studied there; once a year, it marks the moment good news is delivered.

Somewhere deep in the endless, patternless decimals of pi sits a small surprise. Between the 762nd and 767th decimal places there are six 9s in a row — a short, insistent run of the same digit standing out from the disorder around it. In a number whose digits are believed to fall as randomly as any, stumbling on six identical ones this early feels almost like a wink.

The stretch is known as the Feynman point. The name comes from the physicist Richard Feynman, who is said to have remarked that he would love to memorize the digits of pi all the way out to that spot, so that he could recite the first 761 of them and finish with “… nine, nine, nine, nine, nine, nine, and so on.” It is a joke that works only because pi never actually does that: the digits refuse to settle into any pattern, so “and so on” would be a beautiful lie.

What makes the run remarkable is how soon it arrives. The odds of six matching digits turning up this early are slim, and it is in fact the first place in pi where a single digit repeats six times in a row. After those six nines the digits simply resume their wandering — but for six brief places, pi seems to hold a single note.

Pi’s decimal expansion is essentially random — yet occasionally a string of digits turns up exactly where its numerical value says it should. These are sometimes called self-locating strings: a block of digits that appears at the position numbered by those same digits.

In the decimal expansion for pi, the digits 16470 appear in position 16,470. And the digits 44899 appear in position 44,899.

Neither occurrence is especially deep — with enough digits, coincidences like this are expected to turn up. But each one is a small, satisfying echo: the number telling you where to find itself.

Take pi’s opening digits, delete the decimal point, and you get an ordinary-looking integer. But this one is not ordinary at all: 31415926535897932384626433832795028841 is a prime number. It is also exactly the first 38 digits of pi — the 3 and the 37 decimals that follow.

The coincidence is tight. You cannot extend the string by even one more digit of pi and keep a prime; the run stops here. Among the many ways pi’s digits can be read, this is one of the earliest surprises: a stretch that looks like pi and behaves, briefly, like a prime.

Pi itself is transcendental and certainly not an integer. Yet its very first digits, once stitched together, land on one of the oldest ideas in mathematics. For a moment, the most famous constant in geometry wears the mask of a prime.

Isaac Asimov once offered a sentence for memorizing the opening digits of pi. It runs: "How I want a drink, alcoholic, of course, after the heavy lectures involving quantum mechanics!" Count the letters in each word — How has three, I has one, want has four — and the digits fall out in order.

The trick yields fifteen digits: 3.14159265358979. It is not pi's whole story, only its opening, but for a mnemonic that lives in ordinary English, it has remarkable staying power.

In 1906 the Literary Digest printed a small poem by A. C. Orr in praise of Archimedes — "that Immortal Syracusan," first to work out how to measure a circle. Read on its own it sounds like ordinary Victorian verse. Count the letters in each word, though, and the poem gives up thirty-one digits of pi, one word at a time.

It is the same trick Isaac Asimov would later use in a single sentence, stretched across six short stanzas instead: word length stands in for a digit. Orr's version reaches far past Asimov's fifteen digits, carrying pi out to 3.1415926535897932384626433832795 without a single numeral on the page.