Pi Time

Ben - This last week has been the British Science Festival based in the city of Birmingham. The events there covered the full range of sciences, technology, engineering, and maths, and one event even celebrated Pi time, that was 3:14 pm on the 15th of the 9th month - 3.14 15/9. It's the closest you can get - I'm sorry in America that one's just not going to work (as it would be 3.14 9/15). So Julia Graham met Robin Wilson - he's Emeritus Professor of Pure Mathematics at the Open University, to find out a bit more about Pi.

Robin -

It is a favourite project of mineA new value of Pi to assignI would fix it at 3For it's simpler, you seeThan 3.14159

That number, 3.14159, of course is not the exact value of Pi. The exact value of Pi - no one knows - it's a number you go on writing forever and ever. So what is Pi? We all learned about Pi at school. It's to do with measuring circles. First of all, it is the ratio of the circumference of a circle to its diameter. We all know the formula of circumference is 2πr where r is the radius which is the same as Pi x diameter. But we also know that Pi comes up in the area of a circle, πr². But trying to find the value of Pi to more and more degrees of accuracy is something that mathematicians have been trying to do for 4,000 years. So the earliest were the Egyptians and the Egyptians had a value which was actually 256 / 81. Now that's about 3.16 - which is just a little bit too large.

Julia - Do you know how they arrived at that number?

Robin - Yes. They regarded the area of a circle as approximately the same as an area of an octagon. They found that what they should do to work out the area of the circle, they take 8/9 of the diameter and square it, then that gives you the area of the circle, and that corresponds to about 3.16. It's a little too large, but not much.

But the real breakthrough came with the Greeks. We all know that the ancient Greeks really launched mathematics. Archimedes, who of course is well-known for many, many things - he was interested in finding the value of Pi. And the way he did it was, he drew a circle and he drew a hexagon. He put it inside the circle then he did another hexagon outside, and then he measured the perimeter of the hexagons. So the one inside gave him a value of 3 and the one outside gave a value about 3.4, so that tells you the Pi and circumference of the circle lies between those. Well that's not very accurate, so he then replaced the hexagon by dodecagon. In other words, he multiplied the number of sides from 6 to 12 and that gave him a better upper estimate, and a better lower estimate. Still not very good, so he went up to 24, 48, 96, and he stopped at 96. And his calculation showed that Pi was a little bit less than 22/7 and a little bit more than 3 and 10 71st, but basically that gave Pi to two decimal places, 3.14.

And then for the next 2,000 years, people used the same method - exactly the same method. In 1596 a German mathematician living in the Netherlands called Ludolph van Ceulen went up to 2 to the power of 62 sides, basically millions and millions of sides, and he found Pi to 35 decimal places, and he was so thrilled at this that he instructed that those Pi to 35 decimal places should be inscribed on his tomb stone.

Julia - What is it that still fascinates mathematicians today about Pi?

Robin - If you give me any pattern of numbers like 6, 4, 8, 2, 3, 5, 9 for example - I just made that up now - we believe that any pattern of numbers like that, that you'll find it somewhere in the digits of Pi. We don't know that. That has never been proved and this is why people try now to find Pi to more and more decimal places. I mean, if you want to study practical problems, you probably don't need to go for more than about 5 or 6 decimal places, but on the other hand, if you want to find out what Pi is all about, is it true that every pattern of numbers that you give me appears somewhere in that list? So that's one of the fascinating things that people don't know the answer to.

Julia - How many decimal places do we know Pi to now?

Robin - In 2002, a Japanese mathematician called Kanada reached a trillion, and then just last month, August 2010, we've reached 5 trillion. No practical use whatsoever, but why do you do it? Because it's there, just like climbing mountains.

Ben - Professor Robin Wilson there, explaining why understanding Pi is a bit like climbing a mountain. Unfortunately, that gave me images of mountains of pie, so I'm now very much looking forward to my dinner. He was talking to Julia Graham.