As everyone knows, Pi is the ratio of the circumference of a circle to its diameter. That is, Pi = C / D or C= 2 x Pi x r. Pi is an irrational number – one that can never be expressed as a fraction and one whose pattern of digits never repeats. It is infinite and patternless. Many people believe that Pi = 22 / 7 but this is only a very rough approximation and only accurate to 2 decimal places. Even by 480 AD Pi had already been calculated by Zu Chongzhito to 7 decimal places, which is very impressive when you consider that there was not yet trigonometry, calculus or any concept of infinite series. Today, Pi is known accurately to over a trillion digits (using computers of course).

As everyone knows, [pmath size=14]pi[/pmath] is the ratio of the circumference of a circle (C) to its diameter (D)

[pmath size=14]pi = C/D[/pmath] or [pmath size=14]C = 2 pi r[/pmath] (where r is the radius)

[pmath size=14]pi[/pmath] is an irrational number – one that can never be expressed as a fraction and one whose pattern of digits never repeats. It is infinite and patternless

Many people believe that [pmath size=14]pi = 22/7[/pmath] but this is only a very rough approximation and only accurate to 2 decimal places.

Even by 480 AD [pmath size=14]pi[/pmath] had already been calculated by Zu Chongzhito to 7 decimal places, which is very impressive when you consider that there was not yet trigonometry, calculus or any concept of infinite series. Today, [pmath size=14]pi[/pmath] is known accurately to over a trillion digits (using computers of course).

To 50 digits, [pmath size=14]pi = 3.14159265358979323846264338327950288419716939937510 …[/pmath]

No surprises at all so so far, but did you know that you can actually calculate the value of [pmath size=14]pi[/pmath] quite simply?

[pmath size=14]pi = (1  –  1/3  +  1/5  –  1/7  +  1/9  –  1/11  +  1/13  –  1/15  …) * 4[/pmath]

Wow, isn’t that a surprise? I think the simplicity is just beautiful. However you would have to crunch away at this infinite sum for quite some time to get an accurate answer, as it is quite slow to converge.

There are many other visually intriguing variations which are more useful to calculate accurate values, including-

[pmath size=14]pi = { } 3 + 4 / {2 * 3 * 4}   –   4 / {4 * 5 * 6} + 4 / {6 * 7 * 8}   –   4 / ( 7 * 8 * 9} … [/pmath]

and

[pmath size=14]2/pi={sqrt{2}/2} * {sqrt{2+sqrt{2}}/2}*{sqrt{2+sqrt{2+sqrt{2}}}/2} …[/pmath]

By the way, [pmath size=14]pi[/pmath] is related to two other important numbers in science and engineering, [pmath size=14]e[/pmath] (the base of the natural logarithm) and [pmath size=14]i[/pmath] (the imaginary square root of -1). Amazingly, what is called ‘Eulers Identity’ ties these three numbers together:

[pmath size=14]e^{i pi} + 1 = 0[/pmath]

(Even though that one is way beyond high school maths, it appeared once in an episode of The Simpsons. This was because several of their writers were ex-mathematicians and occasionally inserted little maths jokes into the show).