By Paul Goldbrum on December 21, 1998 :

I have just started teaching a group of year eight about

π

, and during our discussions one of them asked me how we actually calculate

π

. This question stopped me dead as I suddenly realised that I did not know. I had been telling them of people that memorised it to so many digits, and then I could not tell them how these people went about it. Could you please answer this for me.

Cheers, Paul

By Alex Barnard on February 11, 1999 :

Very good web page is JoyofPi.com ... it has links to all the good lots and lots of different ways of calculating $π$ .

The formula $4 - 4 / 3 + 4 / 5 - 4 / 7 - \dots$ is VERY slow and is never used to calculate $π$ don't tell your children this one! But if you don't mind slow methods then there is a way of calculating $π$ by performing experiments:

Take any sewing needle and take a large piece of blank paper. Now mark parallel lines on the paper which are exactly the length of the needle apart. Now, work out mathematically the probability of dropping a needle onto the paper and it crossing one of the lines. This turns out to be $1 / π$ ... write back if you want details of how to work this out.

So method is... get a needle for each child and a very big bit of correctly lined paper. Get all the children to drop needles and count how many times they dropped them and how many of these crossed the lines. And after a lot (several thousand at least) of drops you will be able to work out $π$ to a decimal place or two!

There is a link from the web page above to this method (called Buffon's needle).

Hope this helps....

Alex B.

By Graham Lee (P1021) on Saturday, August 7, 1999 - 09:28 pm :

Another Monte Carlo method for calculating pi is to take a square of side length 2, and draw a circle exactly within it. The area of the square is 4 units, and the area of the circle is pi units. If you drop, say, a pea into the square, then the probability that it lands within the circle is pi/4. Simply repeat say 100 times, find the ratio (number of hits in circle)/100, then multiply by 4 for an approximation of pi.

By the way, is it true that

e^{i π} - 1 = 0

, or is this apocryphal?

Graham Lee.

By Dave Sheridan (Dms22) on Saturday, August 7, 1999 - 11:00 pm :

Indeed. However, if you repeat the experiment 100 times you probably won't even get that pi is 3 and a bit. As Alex said, it takes absurdly large amounts of time to calculate pi in this way, and for each extra decimal place, you should expect to double the number of experiments. Don't try this at home...

You're almost right about the equation. It should read:

e^{i π} + 1 = 0

which is pretty much trivial once you give meaning to

f (x) = e^{i x}

. Any good book about complex numbers will explain this, and I believe it's been done in another topic, so I won't add any more here. Unless it's requested.

-Dave