Pi..?

How can a number with an endless array of seemingly random decimal places with no pattern whatsoever be so useful to maths? Take Pi for example: Have you ever wondered how your calculator actually comes up with the number in its tiny computer brain? How did we even calculate it in the first place?

GEOMETRY

Some of the first methods used to calculate pi were geometrical. 

Although Pi is equal to the ratio of the diameter to the circumference of a circle, you may be thinking why don't I just measure the diameter, work out the circumference and then calculate the ratio. If only it were that easy :} Length's are continuous, which means if you measure something, your ruler might say it is 20cm, a more accurate ruler would tell you it was 20.3cm, a more accurate one would tell you 20.34 and so on.. forever, so it wouldn't measure a definite value. And anyway, if you didn't have pi to start with how would you calculate the circumference of the circle to begin with. 

In 250bc, Archimedes found upper and lower bounds of pi using an algorithm involving shapes. He drew a regular hexagon inside and outside the circle. then he used a regular polygon with 7 sides, then 8 sides and so on until he reached a 96 sided regular polygon. By finding the perimeters of these polygons, he proved that

223/71<pi< 22/7 (3.1408<pi<3.1429)

In 480 AD, Chinese mathematician Zu Chongzhi used a similar method on a 12,288 sided polygon, and he found  π ≈ 355/113, this remained the most accurate approximation to pi for the next 800 years (accurate to 6 decimal places).

INFINITE SERIES

The real revolution in calculating the digits of pi came with the invention of infinite series.

The series formula:

Is a striking representation of pi. But it's still not easy to compute the digits of pi since this is an infinite series

There are many representations of pi as an infinite series, Wallis's formula states:

But these methods can be just as tedious as the geometrical method, As Newton himself states "I am ashamed to tell you to how many figures I carried these computations" 

COMPUTERS

But with the development of computers in the mid-20th century, the hunt for pi was revolutionised. Simply by using a desk calculator, John Wrench and Levi Smith reached 1,120 digits. The mathematics and computing were inextricably linked, as in the same year, a team led by George Reitwiesner and John von Neumann used  an inverse tangent infinite series achieved 2037 digits with a calculation that took 70 hours of computer time on the ENIAC computer. 

The record, now always relying on an inverse tangent series, was broken repeatedly until 1 million digits was reached in 1973. But the current record.. by computer scientists Alexander J. Yee &Shigeru Kondo is 10 trillion digits!!

But really.. what's the use for calculating all these digits of pi? Did you know that you only need 39 digits of pi to calculate the circumference of the entire observable universe within the width of one hydrogen atom. Yep. 

To conclude. my best guess is that your calculator just stores the digits of pi to 9 places. 

Also check out this video where Matt Parker calculates Pi with Pies

Admin O.