So, I got in to work this morning, and saw a tweet with the following image:
Pi is an infinite, non-repeating decimal – meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that string of digits is the name of every person you will ever love, the date, time, and manner of your death, and the answers to all the great questions of the universe. Converted into a bitmap, somewhere in that infinite string of digits is a pixel-perfect representation of the first thing you saw on this earth, the last thing you will see before your life leaves you, and all the moments, momentous and mundane, that will occur between those points.
All information that has ever existed or will ever exist, the DNA of every being in the universe.
EVERYTHING: all contaned in the ratio of a circumference and a diameter.
Things like this, that abuse misunderstandings of math in the service of pseudo-mystical rubbish, annoy the crap out of me.
Before I go into detail, let me start with one simple fact: No one knows whether or not π contains every possible finite-length sequence of digits. There is no proof that it does, and there is no proof that it does not. We don’t know. At the moment, no one does. If someone tells you for certain that it does, they’re bullshitting. If someone tell you for certain that it doesn’t,
they’re also bullshitting.
But that’s not really important. What bothers me about this is that it abuses a common misunderstanding of infinity. π is an irrational number. So is e. So are the square roots of most integers. In fact, so are most integral roots of most integers – cube roots, fourth roots, fifth roots, etc. All of these numbers are irrational.
What it means to be irrational is simple, and it can be stated in two different ways:
- An irrational number is a number that cannot be written as a ratio (fraction) of two finite integers.
- An irrational number is a number whose precise representation in decimal notation is an infinitely long non-repeating sequence of digits.
There are many infinitely long sequences of digits. Some will eventually include every finite sequence of digits; some will not.
For a simple example of a sequence that will, eventually, contain every possible sequence of digits: 0.010203040506070809010011012013014015016…. That is, take the sequence of natural numbers, and write them down after the decimal point with a 0 between them. This will, eventually, contain every possible natural number as a substring – and every finite string of digits is the representation of a natural number.
For a simple example of a sequence that will not contain every possible sequence of digits, consider 0.01011011101111011111… That is, the sequence of natural numbers written in unary form, separated by 0s. This will never include the number combination “2”. It will never contain the number sequence “4”. It will never even contain the digit sequence for four written in binary, because it will never contain a “1” followed by two “0”s. But it never repeats itself. It goes on and on forever, but it never starts repeating – it keeps adding new combinations that never existed before, in the form of longer and longer sequences of “1”s.
Infinite and non-repeating doesn’t mean without pattern, nor does it mean without structure. All that it means is non-repeating. Both of the infinite sequences I described above are infinitely long and non-repeating, but both are also highly structured and predictable. One of those has the property that the original quote talked about; one doesn’t.
That’s the worst thing about the original quotation: it’s taking a common misunderstanding of infinity, and turning it into an implication that’s incorrect. The fact that something is infinitely long and non-repeating isn’t special: most numbers are infinitely long and non-repeating. It doesn’t imply that the number contains all information, because that’s not true. </p.
Hell, it isn’t even close to true. Here’s a simple piece of information that isn’t contained anywhere in π: the decimal representation of e. e is, like π, represented in decimal form as an infinitely long sequence of non-repeating digits. e and π are, in fact, deeply related, via Euler’s equation: . But the digits of e never occur in π, because they can’t: in decimal form, they’re both different infinitely long sequences of digits, so one cannot be contained in the other.
Numbers like π and e are important, and absolutely fascinating. If you take the time to actually study them and understand them, they’re amazing. I’ve writted about both of them: π here and e here. People have spent their entire lives studying them and their properties, and they’re both interesting and important enough to deserve that degree of attention. We don’t need to make up unproven nonsense to make them interesting. We especially don’t need to make up nonsense that teaches people incorrect “fact” about how infinity works.