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About π (Pi)
Simple explanation (for everyone)
π (pi) is the number you get when you divide the distance around any circle by the distance across it. If you measure around a pizza and across it, the ratio is always about 3.14 — that's π! It works for every circle, big or small.
What kind of number is π?
π is irrational: its decimal digits never end and never repeat. You cannot write it as a simple fraction. It's also transcendental, meaning it's not the solution to any ordinary polynomial equation with whole-number coefficients. Unlike numbers like ½ (0.5) or ⅓ (0.333…), π goes on forever with no pattern.
Calling 3.14 or 22/7 an approximation quietly assumes there is an exact value sitting out there that no fraction can ever reach. That idea is the frame for the history below. The scribes who first wrote down a circle-number had no such concept.
How many digits have been computed?
As of 2024, over 100 trillion decimal digits of π have been calculated (by 2022, Emma Haruka Iwao computed 100 trillion digits using Google Cloud). This app uses the first 10 million digits. Mathematicians have proven π's digits continue infinitely with no repeating pattern.
π in a circle
For any circle: C = π × d (circumference = π times diameter) and A = π × r² (area = π times radius squared). π appears in waves, physics, probability, and many areas of science and engineering.
Fun facts
- March 14 (3/14) is Pi Day.
- The sequence 123456 does not appear in the first million digits.
- Every possible finite sequence of digits is believed to appear somewhere in π (π is thought to be "normal").
- Ancient Babylonians used both 3 and 3⅛ (3.125). How they got those numbers.
- Archimedes later proved π sits between 3.1408 and 3.1429; 22/7 is the everyday fraction we inherited from that world.
How π was first measured
Two early values — 3 and 3⅛ (3.125), against the true 3.14159… — come from completely different places. One is about 4.5% low; the other about 0.5% low. Neither was a guess.
Why 3 isn’t a lazy rounding
Draw a regular hexagon inside a circle. Something remarkable happens: each side of that hexagon is exactly the radius of the circle. Not approximately — exactly. You can check it with a compass: swing the radius and step it around the rim six times, and you close the figure. That is how you lay out a hexagon in the field.
The hexagon’s perimeter is 6r. The circle’s circumference is 2πr. Treat the circle as if it were the hexagon and you get 2πr = 6r, so π = 3. It is not sloppiness — it is a snapped chalk line. You get π = 3 for free the moment you can swing a compass. Every ancient culture that used 3 arrived there the same way.
Where 3⅛ came from
A clay tablet from Susa (in present-day Iran) gives the ratio of a regular hexagon’s perimeter to the circumference of the circle around it as 0;57,36 in base 60 — that is 57/60 + 36/3600 = 0.96.
Read that as a correction factor: the hexagon comes up 4% short of the circle. So the circle’s true circumference is 6r ÷ 0.96. Dividing 3 by 0.96 gives exactly 3.125, which is 3⅛.
Somebody measured the bulge. The number did not come from a proof; it came from comparing a stepped-off hexagon against the actual rim and finding the shortfall. An empirical result, written down as a working coefficient.
Why an eighth — and why we inherited 22/7
In base 60, eighths are clean. One eighth of 60 is 7.5, so 3⅛ writes as 3;7,30 — three whole units, seven sixtieths, thirty thirty-six-hundredths. Tidy.
We inherited 22/7 (≈ 3.1429) instead, which is a better approximation, but only because sevenths are workable in a system that thinks in tenths and sevenths. Neither culture picked their value purely for accuracy; both picked the most accurate value that was easy to write down in their notation. Same instinct as reaching for 3/8″ on a tape measure instead of 0.377″.
What the error costs you
Say a round cistern, 10 ft across and 6 ft deep (volumes in US gallons):
| Value used | Capacity | Off by |
|---|---|---|
| π = 3 | 3,366 gal | 159 gal short (~4.5%) |
| π = 3.125 | 3,506 gal | 19 gal short (~0.5%) |
| true π | 3,525 gal | — |
For a scribe reckoning grain in a silo or bricks in a wall, 4.5% is real money and 0.5% is noise. That is presumably why both values stayed in circulation — the coarse one for quick estimates, the better one when it mattered.
A look-up number, not a chase
We say “3.125 is an approximation to π,” which assumes there is an exact value no fraction can ever reach. The Babylonians had no such concept. To them 3;7,30 was not a stand-in for something unreachable; it was simply the coefficient for circles, sitting in a table right alongside other polygon constants (about 1;40 for pentagons and 2;37,30 for hexagons in the same tradition). A number you look up, not a number you chase.
That is also why proof was entirely absent. If you are building a table of working coefficients, you do not need to prove anything — you need the number close enough that the wall stands up.
What came next
Archimedes, working in Syracuse in the 3rd century BCE, boxed the circle between 96-sided polygons and proved that π sits between 223/71 and 22/7 (about 3.1408 and 3.1429). The method is the hexagon idea pushed further: more sides, a tighter squeeze, and now a proof that the true value lies in the gap.
This page searches the first ten million digits of that same number. The Babylonian coefficient and the digit stream above are two relationships to π: a number you look up so the cistern holds, and a number you chase because it never ends.