|Pi Day 2022 - It's Irrational!!!|
|Written by Mike James|
|Monday, 14 March 2022|
It's Pi day again! Well don't be so surprised - it comes round every 365 days give or take a day or so. Is there anything left to say after so many meditations on this most transcendental of numbers? Forget transcendental - this is irrational!
There are so many interesting properties of Pi that it really does deserve a day to itself - and 3/14/2022 is a good day to pi...
Well OK, the 2022 bit isn't useful but the 3 and 14 are the three digits in the decimal expansion of Pi so perhaps we will just have to settle for that unless we can wind back the clock to the year 227 which would be the well known rational approximation 22/7.
What I want to concentrate on this Pi day is possibly the least glamorous aspect of Pi - its irrationality. It sounds good - how can a number be so crazy that it is irrational? Because it doesn't mean illogical - it just means that it isn't a ratio, that is it isn't ratio-nal rather ration-al. That is, you cannot find two whole numbers, integers to you and me, such that a/b is exactly Pi. No matter which a or b you choose there will be an error in the value that you are using for Pi.
For a long time I was happy with this idea. After all, there are more irrational numbers than there are rational numbers. In fact, the majority of numbers that we work with are irrational. If you pick a number out of a hat (big hat) it is almost certain to be irrational. You will never see a nice neat integer or even a useful compact ratio like 22/7. The irrational numbers swamp every other kind of number.
So Pi, being irrational isn't particularly noteworthy. Except it is because it is linked to the space we find ourselves in - Euclidean space. The universe might be curved and non-Euclidean, but our little patch of space is locally Euclidean - just as every surface is flat if you get close enough to it. This is the principle of differential geometry which is well worth finding out about.
The point is that while there may be too many irrationals to get excited about there aren't that many accurate descriptions of particular irrational numbers. To make this clearer, consider the task of writing a program to print a particular number. Each program you write is a string of binary bits and you can, if you want to, read that string of binary bits as a number. That is, every program has a number, usually called its Gödel number, and this means for every program there is an integer and for every integer a program -- but wait we already know that there are hugely more irrationals than integers and so there are an overwhelming number of irrationals that don't have programs that print them!!
Most irrational numbers can't be written down in any way at all - they don't even have symbols to distinguish them like Pi or e do. The majority of numbers are so unexceptional that they don't have any properties that pick them out as special from other numbers. They are numbers that exist, but without labels of any kind.
Talk about dark matter and dark energy --- these are dark numbers!
Now consider for a moment a circle of radius one. Its circumference is 2Pi which is irrational. So now imagine that you unroll the circumference as if it was a piece of string. Now you have a length that is irrational which means that there is no pair of whole numbers a and b for which a/b is the length. Now imagine I ask you to start at some point and walk right round the circle. To do this you have to take a step of size a and you take n steps to reach the end, but this means that n*a = 2Pi or Pi = n*a/2 but as n*a is a whole number and 2 is a whole number you cannot get back to exactly where you started by taking steps of any particular size.
Why does this happen?
You can generalize Pi to the ratio of the perimeter to the width of a range of objects. For example, Pi(Square) is 4L/(L/2)= 2 so Pi for a square is not only rational, it's an integer. If you carry on adding sides - pentagon, hexagon and so on you get a value of Pi that is always rational and a value that slowly converges in the direction of Pi for a circle - but Pi for an n-sided figure, Pi(n) only gets to be irrational when n goes to infinity.
In some non-Euclidean geometries Pi isn't irrational and in some it isn't even constant, so I think we can conclude that its irrationality is some how a consequence of living in a flat Euclidean space.
But, of course, Pi isn't just irrational, it's transendental and that leads us on to another story, perhaps next year...
|Last Updated ( Monday, 14 March 2022 )|