p - Sequences and Series

Take a circle, measure its diameter and circumference, then divide the circumference by the diameter. You get an answer, 3.141593ish. Try it with any number of circles, you will always get the same answer - pi.

Most people associate pi with circles, however pi crops up in all sorts of places. For example find the sum of this series:

it's pi, found by simply adding natural numbers. There are variations on this, for example:

Calculating this produces 9.87.... which is pi squared.

So why do these simple sequences produce pi? Archimedes (Greek mathematician, doh) was investigating pi. He was investigating the relationship between the area and radius of various circles. Archimedes was inscribing and circumscribing circles with regular polygons, splitting the polygons in to triangles and then working out the area. He concluded (using 96 sided polygons, the name of which escapes me) that the value of pi must lie between 3 and ten seventieths (3 + 10/70 = 22/7) and 3 and ten seventy-firsts (3 + 10/71) - not bad for mental arithmetic!

What else can you find in pi? Pi contains every single digit, statistically with an even distribution, so theoretically any combination of digits will come up eventually. For example the digits 999999 turn up at position 762, and the digits 314159 at position 176,451. In his book Contact, Carl Sagan suggested that hidden in the depths of the transcendental numbers (pi, e root2 etc) a message from the creator would be hidden.

Heavy huh? Maybe so, however the digits of pi aren't random (chaotic perhaps). In 1995 a group of mathematicians discovered a formula which allows you to find the value of a digit at any given position in pi without having to discover all of the preceding digits first. This disturbed many other mathematicians who were under the opinion that pi was random. There is a catch (there always is) the formula only works if pi is expressed in hexadecimal (base-16).

Pi, chaotic but not random? weird. The record number of decimal places to which pi has been computed stands at nearly six and a half billion digits (6,500,000,000 digits), by a Japanese corporation. See further down this article for the first 1001 digits of pi.

Links

Archimedes' Constant a page on the brilliant mathsoft site devoted to pi, and is far too complicated to understand, but looks good anyway.

Program for finding the digits of pi (hexadecimal) yet another incomprehensible item on the mathsoft site.

Listen to Pi ok, I'm sorry to put you through this, but this was voted the worlds second worst web-site, by a internet magazine. 'Elias' has taken the first 65,536 digits of pi, converted then into an audio file format, and claims that they contain a message from god. They sound very like white noise, but if you want to hear for yourself then try this address.

http://www.yahoo.com/science/mathematics/numbers/specific_numbers/pi that explains it really. (directory of other sites)

Calculator Programs

Mike Docker has written a page, which has been programmed with both the sequences mentioned above, one from the mathsoft site and one I made up, which allows you to run them and compare the results from the first 15 stages, so that you can investigate which sequence converges to pi first.Click here to see.

This first program, calculates the first series mentioned in the article, the program runs quite slowly, and will take about 10 seconds to run, producing a value for pi accurate to 3 significant figures. By increasing the number of time the program goes through the for...next loop (increase the number which is currently 999) you can improve the accuracy at the cost of the amount of time the program take to run.

This program, calculates the second series mentioned in the article, the program runs more slowly, and will take about 15 seconds to run, producing a value for pi accurate to 3 significant figures. As before you could increase the accuracy and time and waste some batteries by increasing the loop cycles in the same manner as before.

Here's the first 1000 digits of pi! If you thought that you'd got away with an article on p without some serious digits then you were very, very wrong.

If you're not impressed by the first 1000 digits of pi, then you can always look at the first 10000 digits!!

Credits: researched and written by Luke Wakeling