I ARE SQUARED

Before I commence with my mathematical meanderings, let me firstly confess to a most dismal loss in soccer last week. I think the score was 14-2 or something along those lines. And while I'm about it, Jesamine's Gems lost on Saturday, as did Suzy's Gems on Thursday.

Now, on to my personal challenge.

It seems from my admittedly cursory research that there is no intuitive mathematical formula from which to calculate Pi. By intuitive, I mean a formula of which, with a basic grasp of logic, you can understand the why-for. For example, the formula for the area of a square, length by width, can be understood by picturing a grid of 1-unit by 1-unit squares filling the area. I had anticipated some similarly logical reasoning would lie behind a formula for calculating Pi - and I was sadly mistaken.

'Twould appear that for some time Pi was only known by measurement. If you draw a large enough circle and measure the circumference and the diameter accurately enough, you can determine Pi to, well, around 3-4 decimal places. Not much really is it - but for most people using pencil and paper a 1 meter diameter circle and a 1 millimetre is about the practical limit, so that's a 3 decimal place limit.

Wikipedia tells me that with a circle the size of the observable universe, and a measurement precision of 1 hydrogen atom - you can still only calculate Pi to 39 decimal places. And in case you're not aware, the observable universe is very large, and a hydrogen atom considerably less so. If you imagine a teapot, you couldn't fit the observable universe in it - whereas you could easily fit two hydrogen atoms within a teacup - and still have room for milk and sugar.

I just did a quick calculation, and by my reconing, if a hydrogen atom is around 25 picametres, and the obervable universe is around 1000 billion trillion kms, then I can't see how you'd get beyond 38 decimals.


Anyway, so it seems that calculating Pi was all about drawing circles until people started stumbling across mathematical series that approach Pi. Still, these are non-intuitive, but they do work.

For example, the series (4/1)-(4/3)+(4/5)-(4/7)+(4/9) .... where the denominator increase by 2 in each term, converges towards Pi ... very slowly. You get up to (4/13) before it even gets to "3", and you're up to (4/49) before you've found Pi to one decimal place. At (4/1017), which is all that I could do using all the columns in Excel with two terms per column, I still hadn't resolved Pi to 2 decimal places.

Whe I first read about this, I thought, if that's the way it works, how do people know how many places they've calculated Pi to? You've got a raft of decimals after each result, but how do you know which ones are right?

Then I thought to myself, obviously you keep adding terms until a decimal stops changing. But they don't change with each calculation. For the first six terms you'd be convinced that Pi = 2 point something. For the next 18 terms you'd think the first decimal is 0.

So, at the moment I remain a little confused. But by gosh by golly hasn't this been educational. I shall continue my investigations a little further.

Toodle-oo