In reference to one of Cecil’s columns, I was reading something about pi on wikipedia. It noted that people really do look for patterns in pi (I always thought that that was just a plot device in Aronofsky’s eponymous film). I always thought that irrational numbers had no patterns because this would imply that it was a ratio of two integers and therefore rational. So what gives? Are there any known irrational numbers which exhibit patterns? Is the radix important?
Thanks,
Rob
Sure. But just not repeating patterns of fixed length. The classic example is
0.12345678910111213141516171819202122232425262728293031…
IIRC this number is not only irrational, it is also transcendental.
People looking for patterns in pi long predates that film. Look at Sagan’s contact, for instance.
I believe there are plenty of numbers with patterns that aren’t simple ratios – but they aren’t simple patterns of repeating digits. Like 1/10 + 1/10^3+1/10^5+1/10^7+1/10^11+ etc, with all the exponents prime numbers.
The decimal expansion of π has an infinite number of digits, so if you look long enough you’ll probably find any pattern you care to search for. 
I think the confusion revolves around what constitutes a “pattern.”
Does something like 0.1010010001000010000010000001… have a pattern? I’d say it does: once you see what’s going on, you can keep writing as many digits of that pattern as you want to. It’s certainly not random. But it is an irrational number, because it’s not just the same digit or sequence of digits repeating over and over.
What’s the function for constructing this number? (Some kind of infinite summation, I imagine; I’d be interested in seeing it.)
Yeah, that’s the expectation. It’s thought that pi is “normal”, which means that every finite string of digits occurs in the expansion of pi with the frequency expected from a uniform distribution of digits. It’s not known that pi is normal, however (not to my knowledge, anyway).
Of course “normal” depends on the base. A number that’s normal in base 10 might not be normal in binary.
A number that’s normal in any base is called “absolutely normal”. It’s known that almost all of the real numbers are absolutely normal (“almost all” meaning the set of numbers that are NOT absolutely normal has Lebesgue measure zero). In spite of this, I don’t believe an absolutely normal number has been discovered (though many believe pi is one).
A normal number will contain every piece of information that can be finitely encoded into digits. Given any such method of encryption, if you look far enough into a normal number, you’ll find all the works of Shakespeare in chronological order, Beethoven’s symphonies, and that picture of your grandmother taken when she was 73.
Who needs monkeys with typewriters when you’ve got pi? 
0.12345678910111213141516171819202122232425262728293031…
Well, here’s one formula (probably not the simplest):
[SIZE=6]Σ[sub]{d=1…Infinity}[/sub]Σ[sub]{n=10[sup]d-1[/sup]…10[sup]d[/sup]-1}[/sub]n*10[sup]-d(n+1)+(10[sup]d[/sup]-1)/9[/sup][/size]
To see how this works, consider one d at a time.
d=1:
[SIZE=6]Σ[sub]{n=1…9}[/sub]n*10[sup]-n-1+1[/sup] = 0.123456789[/size]
d=2:
[SIZE=6]Σ[sub]{n=10…99}[/sub]n10[sup]-2n-2+11[/sup] = 0.1011121314…97989910[sup]-2*9[/sup]10[sup]9[/sup] = 0.1011121314…97989910[sup]-9[/sup][/size]
d=3:
[SIZE=6]Σ[sub]{n=100…999}[/sub]n10[sup]-3n-3+111[/sup] = 0.100101102103104…99799899910[sup]-399[/sup]10[sup]9+99[/sup] = 0.100101102103104…99799899910[sup]-9-290[/sup][/size]
etc.
This is getting on the right track. It’s tough to say anything about irrational numbers in general, because the only thing they have in common is that they’re not rational. There are larger classes of real numbers than the rationals that are worth talking about; here are the ones I know of:
[ul][li]Constructible numbers: These are the numbers on the real line that you can get starting with a straightedge, a compass, and the numbers 0 and 1.[/li][li]Algebraic numbers: A number a is algebraic if there’s some polynomial P(x) with integer coefficients such that P(a) = 0. For instance, 2[sup]1/3[/sup] is algebraic because if you plug it into P(x) = x[sup]3[/sup] - 2, you get 0.[/li][li]Computable numbers: If you can write a computer program that takes no input and outputs a number r, r is said to be computable.[/li][li]Definable numbers: x is a definable number if there’s some well-formed formula of first order logic that exactly specifies the value of x.[/ul][/li]
Each category contains the one above it (and every rational is constructible), but you’re adding new numbers at each step. For instance, 2[sup]1/2[/sup] is constructible but irrational; 2[sup]1/3[/sup] is algebraic but not constructible; [symbol]p[/symbol] is computable but not algebraic; and Chaitin’s constant is definable but not computable. It’s a little tough to give examples of non-definable numbers in a finite amount of space, but they’re out there. In fact, the set of definable real numbers has Lebesgue measure 0 (as it’s a countable set), so in a very real sense almost every real number is not definable.
You, sir, have a knack for understatement.
On the other hand, you’re referring there only to mathematical definitions, so there might be some examples. It’s conceivable, for instance, that alpha (the fine structure constant) is undefinable (in the mathematical sense), or any other dimensionless constant in physics (any of the mass ratios of particles, or charge-to-mass ratios, etc.). But I don’t think that we could ever prove it to be so.
So you’re trying to tell us that you can’t define a non-definable number?
Lazy mathematicians.
I have no idea what I’m talking about, but if this is true couldn’t one simply create a line accross two other lines (and so create an isocelees triangle), call that one, trisect it (creating 1/3), and so trisect an angle, which I know to be impossible?
In terms of constructible numbers, 1 is a length, not an angle. And it is possible to trisect a line segment. Or, indeed, to n-sect one, where n is any integer.
If you’re going to go with the circles and lines approach to constructible numbers, think of the numbers as points in the complex plane. We’re given the points (0, 0) and (1, 0) to start with, and then we can draw circles and lines based on those. Anywhere that two lines intersect, or two circles, or a circle and a line is also a constructible number. The constructible reals are just the constructible numbers restricted to the real line.
Of course, the real definition has nothing to do with circles and lines, but it works well.
This thread reminds me of an essay by Isaac Asimov I read in high school. It was call “A Piece of Pi”, and near the end, he had the following footnote (bolding mine):
I’ve been annoyed by that last little comment ever since. How can he be so cruel as to taunt the reader with a comment like that, but then not say what any of these interesting points are?
