George Takei posted the following thing to Facebook recently:
It got reposted by a bunch of people and provoked a tremendous amount of discussion (for a math topic, anyway), much of which was somewhere in the continuum between merely wrong and psychedelically incoherent. It’s not a new subject – a version of the image got discussed on Stack Exchange last year – but it’s an interesting one and hey, it’s not all that often that the subtle properties of the set of real numbers get press on Facebook. Let’s do a taxonomy of the real numbers and see what we can figure out about pi and whether or not it has the properties stated in the picture.
The Natural Numbers and Integers
These are the counting numbers: 0, 1, 2, 3, 4… There’s an infinity of them, but there are gaps. If you have 5 dollars and you give half of them to your friend, you’re stuck. The number you need is not a natural number. If we want to be able to deal with ratios of natural numbers, we need more numbers so we can deal with those gaps between the natural numbers. We can include the set of natural numbers with negative signs in from of them, and we have what’s called the integers: …-3, -2, -1, 0, 1, 2, 3… Later on I won’t worry about explicitly discussing negative numbers, but of course all of the subsequent sets include negative numbers.
The Rational Numbers
These are the ratios of integers, or fractions. Divide 1 by 4 and you get the rational number 1/4. We can write it in decimal notation as 0.25. Divide 1 by 3 and you have the rational number 1/3 = 0.333… All rational numbers have a decimal representation that either terminates or repeats infinitely. In fact, it’s better to say that all rational numbers have a decimal representation that repeats infinitely: 1/4 = 0.25000000… and we just happen to have a notation that suppresses trailing zeros. Sometimes you have to go out quite a ways before the repeat happens, but it always does. 115/151= 0.761589403973509933774834437086092715231788079470198675496688741721854304635761589… All rationals have repeating decimal representations, and all repeating decimals represent rational numbers.
The rational numbers are dense. Between any two rational numbers, there is another rational number. Which immediately implies that between any two rational numbers, there are an infinite number of rational numbers. Pick any point on the number line, and you’re guaranteed that you can find a rational number as close as you want to it. But alas, you’re not guaranteed that every point on the number line is a rational number. Some of them aren’t.
The Irrational Numbers Part 1: The Algebraic Numbers
The square root of 2 is the most famous example of an irrational number. It’s the number which, when squared, gives exactly 2. It’s equal to 1.41421356237…, but the decimal representation never repeats. This is because there are no two integers A and B such that (A/B)2 = 2. You can get as close as you want: 7/5 = 1.4 is kind of close, and 3363/2378 is much closer still, but you’ll never find a rational number whose square is exactly 2. This can be rigorously proven and means that the square root of 2 is irrational, and never repeats.
The square root of two is the solution to the equation . This is an example of a polynomial with integer coefficients. Another random example is , which happens to have the irrational number x = 1.84302… as one of its solutions. Numbers which are solutions to these kinds of polynomials are the algebraic numbers.
Does all this mean the decimal expansion of the square root of 2 includes any and every combination of digits? Maybe. Maybe not.
The Irrational Numbers Part 2: The Transcendental Numbers
Not all irrational numbers can be written in terms of the solutions of polynomials with integer cofficients. The ones that can’t are called transcendental numbers. Pi is one of them. So is Euler’s number e = 2.71828… Transcendental numbers are all irrational.
In a precise but somewhat technical mathematical sense, “almost all” real numbers are irrational. Throw a dart at the real number line and you will hit an irrational number with probability 1. This makes some intuitive sense. If you just start mashing random digits after a decimal point, it seems reasonable that you won’t just happen to make an infinitely repeating sequence. It turns out that the same thing is true of the transcendental numbers. “Almost all” real numbers are transcendental. But at the present time, even with hundreds of years of brilliant mathematicians pouring unfathomable effort into the problem, our toolkit for dealing with transcendental numbers is pretty sparse. It’s very difficult to prove that specific numbers are transcendental, even if they pretty obviously seem to be. Is transcendental? Almost certainly, but nobody has proved it.
Here’s a number called Liouville’s constant which is proven to be transcendental: 0.110001000000000000000001000000… (It has 1s at positions corresponding to factorials, 0s elsewhere.) It was among the first numbers known to be transcendental and was in fact explicitly constructed as an example of a transcendental number. It’s irrational, of course. It is an “infinite, nonrepeating decimal”, as the Facebook picture puts it. But is my DNA in it? Heck no, my phone number’s not even in it. Infinite and nonrepeating is not synonymous with “contains everything”.
The Normal Numbers
A normal number is one whose decimal representation contains every string of digits on average as often as you’d expect them to occur by chance. So the digit 4 occurs 1/10th of the time, the digit string 39 occurs 1/100th of the time, the digit string 721 occurs 1/1000th of the time, and so on. All normal numbers are irrational. Normal numbers satisfy Takei’s criteria. Any finite string of digits occurs in the decimal representation of a normal number with probability 1.
Is pi a normal number? Nobody knows. If our toolkit is sparse for proving things about transcendental numbers, it’s almost completely empty for proving anything about normal numbers. There are a few contrived examples. The number 0.123456789101112131415… is normal in base 10 at least, and in fact it contains every finite string of digits, because it was constructed so that it would. It also satisfies the properties which Takei’s image ascribes to pi, though it also shows that these criteria aren’t especially profound. A string that contains all numbers turns out to contain all numbers, which is true but not all that impressive.
But is this specific number normal in other bases? Nobody knows. Are there numbers that are normal in every base? Yes – again, “almost all” of them. Can I actually write out the first few digits of one? Nope. As far as I can tell, while examples of absolutely normal numbers have been given at in terms of algorithms, there’s not yet been anyone who’s been able to start generating the digits of a provably absolutely normal number. [Edit: I think in the comments we've found in the literature an example of the first few digits of a provably absolutely normal number.]
Mathematicians love proof. I’m a physicist. I love proof too, but I’m a lot more willing to work with intuition and experiment. Do the billions of digits of pi that we’ve calculated act as though they’re distributed in the “random” way that the digits of an absolutely normal number ought to be distributed? Yes. Just about everyone suspects pi is absolutely normal. Same for e and the square root of 2 and the rest of the famous irrationals of math other than the ones that are obviously not normal. Numerical evidence is not dispositive though, and has misled mathematicians before.
If pi is absolutely normal, than Takei’s image is true. If you can prove this conjecture, you will have boldly gone where no one has gone before.