I was in an “Introduction to Logic” lecture the other evening (back at college at night… sheeh) when the Maths Prof. stated that pi represented in decimal form is infinite and after a certain point, random.
I asked him if he meant that we just assume it’s random, 'cause we can’t find a pattern yet. I further ventured that we couldn’t prove that pi was random, since a pattern may emerge at some n-th decimal place.
The teacher just laughed at me and said - “You don’t understand.” and carried on with the lecture.
Well first up, of course I don’t understand - that’s why I’m here in the first place, but more importantly,
Depends what you mean by pattern. If you mean that it might start repeating, that’s definitely out. You can only get repeats if a number is “rational” (that is, it can be expressed as a whole-number fraction like 355/113) - but pi has been proved to be irrational.
pi is an example of an irrational number, which means that, when you write it out in decimal form, the digits after the decimal point neither terminate nor do they repeat the same pattern over and over again. But it’s not obvious that pi is irrational; it wasn’t proved until 1761. (By comparison, the ancient Greeks knew that the square root of two is irrational, for example; and e was shown to be irrational in 1737.)
As for other sorts of patterns or randomness in the digits of pi, none has been found, as far as I know (and by know we have bazillions of digits); but it’d be tough to prove there’s no pattern anywhere, partly because you’d have to specify precisely what you mean by a “pattern.”
Thudlow Boink - thanks for the understanding. So am I right in saying that irrational numbers never repeat, because they are irrational numbers. i.e. if they repeat at any point, then they’re not irrational.
So (hopefully) I understand that much. But how can we prove that maybe all irrational numbers start to repeat after the gazillionth decimal place ?
The digits of [symbol]p[/symbol] do have a pattern, although it’s not as simple as a repeating decimal. In short, there’s an algorithm that will let you calculate the nth hexadecimal digit of [symbol]p[/symbol] without knowing any of the previous ones. Given the first n digits of [symbol]p[/symbol], we can compute the nth decimal digit of [symbol]p[/symbol], although it’s a slow process.
Now, there are real numbers whose decimal representations do not follow any pattern–in fact, most real numbers are like that–but it is not possible to show that any specific real number has this property.
I think you’d use proof by contradiction, you state pi = x/y where x and y are integers, and show that leads to a a false statement. Just wait, some maths whiz will show us I bet…
You want to be careful thinking about infinite stuff it gets very counter intuitive. Same with randomness. Slippery subjects.
Why didn’t the prof justify his statments? Sounds a bit off hand to me.
It’s a matter of fact that the sequence 12345 repeats “infinitely often” if you write out all the digits of PI. Not just a billion times or a million times.
As a matter of fact, that’s true of any finite sequence, even a sequence a million digits long.
FWIW, PI goes beyond irrational. It’s transcendental. It can not be be the solution to a polynomial with integer coefficients. So even Sqrt(2), the solution to the equation X*X - 2 = 0 is not transcendental, but it is irrational.
If they repeat, we can use that to show that they’re rational, by the same process you used in grade school to convert repeating decimals to fractions.
By the way, Achilles, I agree with Thudlow Boink that it definitely wasn’t a stupid question. Probably the guy was just in mid-lecture and didn’t want to get into a lengthy discussion. But he should have suggested discussing it afterward.
The deal with pi is that there are heavier methods to show that it’s irrational, even you don’t know the digits. It comes down to a 1761 proof by one Lambert, who showed by means way beyond me that if a number x is rational, then both e^x and tan(x) are irrational. Suppose x = pi/4? You find tan(pi/4) = 1 (i.e. it’s rational), so pi/4 - and hence pi - must be irrational.
Certainly it’s true for any infinite string of numbers with each successive digit chosen independently (straightforward application of the “borel-cantelli lemma”).
I was being sloppy assuming that you might have the necessary conditions in the digits of PI since it appears so random.
Achilles, I think your prof stuck his foot in his mouth - you asked a very good question, and he backed out.
I think it’s a bad idea to try to talk about randomness of the digits of pi. For one thing, as ultrafilter pointed out, nobody has proven anything about the probability distribution of the digits of pi, which would be the only way I can see of linking probability to the explanation. And, saying that an irrational number has some kind of “random” character is not a good way to explain irrational numbers.
The term for this property of a number is normal. It’s not currently known whether pi, or just about any interesting number, is normal, but (as ultrafilter says) I think most mathematicians suspect that it’s true.
