Last night I went to the National Museum in Cardiff to attend an evening entitled “A Pythagorean Cabaret”. It was an evening of scientific entertainment, meant to make science exciting and fun. 7 different scientists gave 10-15 minute presentations on various scientific matters, including an ex-student of mine Huw James, who talked about the stability that a rotating wheel produces, and how this is used in gyroscopic devices to stabilise satellites.
However, the title got me thinking about the story I heard a long time ago about the famous Greek mathematician Pythagoras. Most people will have heard of his famous “right-angled triangle” theorem, that the square of the hypotenuse is the sum of the squares of the other two sides. Pythagoras accomplished much more in mathematics than this theory, but the story about him that has most intrigued me is the one about irrational numbers.
Many numbers are rational, in fact there are an infinite number of rational numbers. A rational number is one that can either be written as an integer (1,2,3 etc or -1, -2, -3 etc), or a fraction which can be written as one integer divided by another integer (e.g. 1/2, 2/5 etc) [Note: I am not a mathematician, so my apologies to any mathematicians who read this who want to correct my physicist’s definitions.]
The story I have heard is that Pythagoras strongly believed that all numbers were rational. However, he was wrong. The most obvious examples of irrational numbers are square roots; for example the square root of 2 is an irrational number, it cannot be written as a fraction of two integers. When a student dared to challenge Pythagoras on this matter, the story goes is that Pythagoras got another pupil to murder the troublemaker!
Whether there is any truth to this story I have no idea, its veracity is lost in the mists of time. But it does make for a nice story, and shows how passionate some people can get about maths and numbers. I have always found numbers fascinating, and when I was 11-14 I often used to look for patterns in numbers.
Within the class of irrational numbers there is a special class of numbers called transcendental numbers. The best know of these is pi, the ratio of the circumference of a circle to its diameter. Pi is, arguably, the most important mathematical constant, and certainly is important in nature. It crops up all over the place in physics.
A transcendental number is not only irrational, but also cannot be written algebraically. This means that pi, for example, cannot be written as the square root of some number (either rational or irrational) or the cubed root etc.
What fascinates me most about pi is that, if one tries to write it as a sequence of numbers, eg. 3.14149, the numbers will carry on forever and never repeat. As most people know, numbers are used to store information, be it text (using the ASCII code), or music or movies. As pi is an infinite series this means that it contains all the books, songs, movies and any other information that human beings have every created or ever will create. Even the text of this blog is there, word for word, in any language you like, buried in the infinite series of numbers that is pi.
If you do a Google search for pi, you will quite quickly stumble across links to posts where people claim to have found proof of the existence of God in pi. And, for sure, somewhere in pi the words “God exists” can be found, in any language you choose to think of. However, somewhere else in pi the words “God does not exist” also occur. Looking for meaningful messages in pi is a bit like life, if you look hard enough and long enough you will almost certainly find what you are looking for.
The wonderful Carl Sagan book “Contact” has a very interesting postscript where the book’s hero Ellie finds an interesting sequence of numbers in pi. I won’t spoil it for people who haven’t read this book, but suffice it to say that her finding is quite dramatic and very profound.
I don’t know whether I could murder someone who claimed that irrational numbers existed, but I certainly could murder a nice cappuccino right about now.
UPDATE – oops! I have just accidentally, deleted all the comments! Sorry. I thought I was just tidying up my inbox of comments, not deleting them off the post. Apologies, I am still new to WordPress.