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## I could murder an irrational number

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.

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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.

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### 20 Responses

1. You would probably enjoy the Darren Arnofsky film “Pi: Faith in Chaos.” –LBJ

• Welcome back Mr President.

2. I am probably mathematically challenged, but why does the property that pi’s decimal digits form an infinite non-repeating sequence imply that this sequence will contain any finite sequence of digits?

• If you have an infinite sequence of non repeating numbers then any and every possible combination of finite sequences must occur in that infinite sequence.

3. I don’t think your reply qualifies as a mathematical proof π
First, I assume that non-repeating means non-periodic, i.e. that there does not exist a (finite) period T after which the sequence repeats. Then, I can construct an infinite and non-periodic sequence as follows: 0 1 2 3 4 5 6 7 8 9 x 0 1 2 3 4 5 6 7 8 9 x …, where each ‘x’ should be replaced by a random number in the range 0-9. This sequence will never contain the finite sequence ‘0 0 0’. I am not claiming that the decimals of Pi never will contain say a million zeros in a row, I am just saying that your theorem does not seem correct to me.

• Bo – I am not a mathematician, so I doubt I could give you a rigorous mathematical proof. But I will respond later – I’m in the middle of teaching at the moment π

4. Since you deleted the comments, here they are again:

Me:
I am probably mathematically challenged, but why does the property that pi’s decimal digits form an infinite non-repeating sequence imply that this sequence will contain any finite sequence of digits?

You:
If you have an infinite sequence of non repeating numbers then any and every possible combination of finite sequences must occur in that infinite sequence.

Me:
First, I assume that non-repeating means non-periodic, i.e. that there does not exist a (finite) period T after which the sequence repeats. Then, I can construct an infinite and non-periodic sequence as follows: 0 1 2 3 4 5 6 7 8 9 x 0 1 2 3 4 5 6 7 8 9 x …, where each ‘x’ should be replaced by a random number in the range 0-9. This sequence will never contain the finite sequence ‘0 0 0Β΄. I am not claiming that the decimals of Pi never will contain say a million zeros in a row, I am just saying that your theorem does not seem correct to me.

• Thanks Bo. I certainly won’t make that mistake again, although I am sure there will be others as I learn how to use WordPress.

• Thanks Bo. I certainly won’t make that mistake again, although I am sure I will make others as I learn how to use WordPress π

5. I am no mathematician, so I would probably struggle to give you a mathematically rigorous proof. I will see if I can do a proof good enough for we astrophysicists/physicists! Or, I may just ask a number theory expert, and I happen to know one! π

6. You would probably enjoy Darren Arnofsky’s debut film, “Pi: Faith in Chaos”

• Thanks Lynn, I will check it out.

7. This was posted on Facebook by Peter Laursen in response to a thread started by Bo :

Bo started the thread : “The sequence of numbers in the range 0-9 that constitute the first 10^5 decimals of pi does not contain more than 5 zeros in a row. But is there a mathematical proof that any finite sequence of numbers (say 10^100 zeros in a row) will be found in the infinite sequence constituted by the decimals of pi? (or even by any irrational number?)”

and Peter responded :

“I did a short search for discussions on this topic, but didn’t find anything useful. However, I remember a proof from my freshman years that the property of being an infinite sequence is not a sufficient condition for containing any sequence: Just think of the number 0.10100100010000100000β¦ It doesn’t repeat, yet it doesn’t contain, say, ‘123’.”

8. As I also said on Rhodri’s FB wall: I know that there is neither a proof, nor a counterproof. However, I don’t know whether or not it has been proven if a proof is possible or not. And until I know that any sequence is not necessarily contained in pi, I will keep believing that all possible truths are hidden there. π

Although as we know all possible non-truths are there as well. It’s kinda like the internet, I guess. Utterly uselessβ¦

• As I’m not able to give any kind of mathematical proof to my conjecture that pi must contain all possible information, it shows why I am not a mathematician. I am hoping to get a mathematician on here to give us a rigorous, air-tight, mathematical proof. But, in the meantime, if we take your example of the sequence

0.10100100010000100000β¦

then, as you say, it does not contain any numbers other than 0 and 1. So, in my *very naive* way of looking at this I think that, if the series is infinite and non-repeating (both of which have been proven), then at some point the sequence *must* contain other numbers apart from 0 and 1, because it carries on forever, and forever allows all possibilities, including your sequence which is devoid of anything but 0s and 1s changing to a series that has 2-9 in it.

Not a very rigorous proof I know π

• I’d say no. The sequence is defined, and behaves as I want it to, and for any decimal you mention, I can tell you which number its place holds.

The sequence doesn’t contain any number other than 0 and 1, just like the sequence 1, 10, 100, 1000, β¦, which is just 10 to the power of [all non-negative integers], which in turn is a _countably_ infinite set.

Forever does not allow all possibilities: The decimals of all rational numbers go on forever, yer repeat in a predictable way, e.g. 22/7 = 3.142856142856β¦, i.e. the sequence “142856” repeats itself, and, say, 7 will never occur.

• Yes, but we do know pi is a *non* repeating irrational number, not a repeating one. This *may* be the crucial difference, but to be honest I don’t know π Maybe the non-repeating nature of it means that all 0-9 digits *must* occur it its infinite sequence…… Dunno.