The Last Digit of Pi
"What's the last digit of pi?" is one of those questions that sounds perfectly reasonable until you realize it quietly contains a false assumption. Let's untangle it.
The direct answer
Pi has no last digit. Its decimal expansion runs on forever — 3.14159265358979… and onward without end — because pi is an irrational number. Irrational numbers can't be written as a fraction of two whole numbers, and one consequence is that their decimals never terminate and never fall into a repeating loop. There is no final digit for the same reason there's no largest whole number: the sequence simply doesn't stop. Asking for the last digit of pi is like asking for the last step on a staircase that goes up forever.
Why people ask
The question is a natural one, and it usually comes from mixing up two different ideas: the last digit of pi and the last digit anyone has computed. Every number we meet in daily life — prices, measurements, scores — does have a last digit, so it's reasonable to expect pi to as well. And because we always see pi cut off somewhere (3.14, or 3.14159), the truncation can look like an ending. But that final digit you see is just where we stopped writing, not where pi stops. Pi has no end to reach.
What records actually measure
You'll see headlines about pi being computed to enormous lengths — well past 100 trillion digits. It's tempting to read that as "they finally got close to the end," but it means nothing of the sort. Those records measure how far computation was pushed before someone chose to stop — a milestone of hardware, algorithms, and patience, not a boundary of the number. Double the computing time and you get twice as many digits, and pi will keep obligingly handing them over forever. The "last known digit" is a fact about our machines; it says nothing about pi.
A fun twist: the nth digit is well-defined
Here's the elegant part. Even though pi has no last digit, every particular digit is perfectly well-defined — there is a definite 5th digit, a definite millionth digit, a definite trillionth digit, each one a specific value waiting to be found. Stranger still, you don't always have to compute all the digits before it to get one. The Bailey–Borwein–Plouffe (BBP) formula can pluck out a chosen hexadecimal digit of pi directly, without calculating any of the digits that come before it. So pi manages to be endless and pinned-down at the same time: no final digit exists, yet any digit you name has a definite answer.
It all comes back to irrationality
Everything here traces to a single property: pi is irrational, proven so by Johann Lambert in 1761. That one fact guarantees the decimals never terminate and never repeat, which is precisely why there's no last digit to report. If you'd like to sit with what "goes on forever" really means for pi, our companion piece on whether pi is infinite takes it a step further. The short version: pi itself is a perfectly ordinary, finite number a hair above 3 — it's only its written form that never ends.