# Calculation Of Pi

Since it was first observed many centuries ago, mathematicians strove to find the exact value of pi. This was attempted in a variety of different ways, and this page will briefly detail a few of them.

## Ancient Times

All of the early attempts to find the value of pi were made through completely practical methods, such as comparing area of circles with rectangles, and as such all the early attempts found only very approximate values of pi. Still, that wasn't such a problem, as there was little call for highly accurate calculations, so it tended to be that most ancient civilisations used the approximation = 3 in most situations.

Anyway, below is a quick summary of some ancient civilisations' attempts at finding a more accurate approximation to pi:

Babylonians:
= 25/8. Found on a tablet of Babylonian clay in 1936.
Ancient Egyptians:
= 256/81. This was achieved by using the knowledge that the area of a circle with diameter 9 units was the same as a square with sides of 8 units.
Early Greeks:
= square root 10. Unknown origin of approximation.

## Archimedes' Method of Exhaustion

Archimedes (and also various other early mathematicians such as Eudoxos) is credited with taking one of the biggest steps forward towards more accurate approximations. He came up with the first theoretical method of approximating pi, all based around regular polygons and circles.

The premise is simple; draw a circle, then draw a regular polygon (for example an octagon) inside the circle, so that the vertices of the octagon touch the circle. Then draw another octagon, but this time so that the sides of the octagon touch the circle. If done correctly you should see a octagon, inside a circle, inside another octagon.

Here's what it should look like:

Archimedes discovered that as you increase the number of sides on the polygons, the area between the perimeters of the polygons and the circumference of the circle gets ever smaller. Eventually in theory the area gets so small that it becomes "exhausted", so the larger the number of sides on the polygon, the closer the polygon's perimeter becomes (in size) to the circumference of the circle. This fact can then be used to find a range of values, inside which must lie.

This was a revolution in the way of thinking at the time, and with slight variations, continued to be used for many, many years to come, by mathematicians such as Liu Hui and Ludolph van Ceulen. Van Ceulen found a value of pi correct to 32 digits with this method, a feat in patience and perservance if nothing else. (Go to my Introduction To Pi web page to see a little more about van Ceulen.)

## Modern Methods

Once it got to about the 16th Century mathematicians began to find new and better ways to approximate pi. Once such method involved using the arctan power series:

which, when z is set as 1, gives you an approximation for /4, which can easily be converted to give you an approximation for .

However, the real advance came with the invention of the computer. Able to perform calculations and remember the answers far better than any human could hope to, the computer made it all seem very easy. It's first computation, back in 1947, produced more than twice as many decimals as the nearest human; over 2000. By 1992 computer had found 10 figures worth of digits.

Now that so many digits have been found, there really is not much point continuing down the calculating road. I'd say over a billion digits is accurate enough for nearly anything you could possibly think of.