Just Thinking

This is not the sort of pie you can eat, but rather the ratio of the distance around the sort of pie that you can eat compared to the distance across it. The history of the accuracy to which the value of pi has been determined is quite interesting and that is what this article is about.

When I was at school we were often told to use the approximation for pi of 22/7 which is 3.142857 ... although this is a bit of a rough value being wrong in the 3rd decimal place, a better fraction being 355/113 = 3.14159 292... but the nice thing about this very accurate fraction is when it is expressed as a division of 113)355 because it can easily be remembered from the 113355.

I have always liked that fraction of 355/113 ever since I came across it as a young man. I have even seen an explanation for this one in the bible, which is much better than the pi equals three that is also claimed to be there.

These can be compared to the true value being something like this:

pi=3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 ...

Historically pi had to first be determined by making a circle and rolling it and comparing to the diameter by measurement and early values around the world tended to be between 3 1/8 and 3 1/6 and centered around 3 1/7 which is quite close to the true value. In other words the accuracy was only a little better than 1% although careful measurement can reasonably easily achieve at least one more decimal place than that.

This graph shows several thousand years of the history of the accuracy of pi determinations and there are two main phases although they are not quite distinct. The first phase was of actual measurement and shows only a little improvement over a long period.

The second phase is based on finding ways to calculate pi based on various geometric constructions and using trigonometry to determine the value. Usually this means dividing the circle into many small identical triangles that are thin slithers and adding the circumference up as the many sided figure approximates a circle more accurately as the number of sides increases.

The accuracy of Pi as determined by measurement or calculated by humans.

The second phase of calculation really got going from around 1400 onwards when Madhava in India calculated pi to 11 decimal places. However before that, in about 480, Zu Chongzhi (or Tsu Ch'ung Chi) had found the fraction 355/113 which is accurate to 7 decimal places.

Eventually some formula for pi were discovered based on trigonometric expansions and these were easier to go to more places, but still required a lot of long calculations when done by hand. Remarkably people did hand calculations of pi to more and more digits, reaching 626 digits by Ferguson by 1946. One poor chap, Shanks, did 707 places back in 1874 but made a mistake after the 527th digit.

However from 1947 onwards computers took over from humans as the pace-setters of pi, starting with 710 places in that year and going ahead in leaps and bound to 1,240,000,000,000 places in 2005. This is pretty remarkable as disk technology today will only hold the answer on the very largest single disk drives manufactured. You need something more for the working. Note that this diagram expands the time axis and compacts the accuracy axis, so the rate of improvement is very much faster.

The accuracy of Pi as calculated by computers.

The rate of improvement in the value of pi was very slow by measurement, perhaps doubling the accuracy (by which I mean halving the percentage error) every 500 years roughly. But with formula based calculation by humans the improvement became some 7 times faster, doubling the accuracy about every 72 years as shown by the trend line in the first graph. But computer calculation meant a doubling in accuracy every 1.8 years as shown by the trend line in the second graph, a further 40 fold improvement in the rate of progress.

For comparison, the record for remembering pi is shown on the second graph. For a while human memory was gaining on computer calculation, but when computer calculation interest in pi calculation was rekindled in the 1980s, people got left in the dust. There is no chance that ever again people will be able to learn all the places of pi that computers can calculate. In fact, if a long lived person spent their whole waking life reciting pi they could only get to about 10^9 decimal places which is only 0.1% of the present determination ... but by then, at the present rate of improvement, the value of pi would be known to 10^28 digits, so they would be far further behind than when they started.

References used in preparation of this article:
* Wikipedia article about pi
* Wikipedia article about approximations to pi
* Chronology of pi determinations
* Zu Chongzhi who determined pi in 480