I was wondering what would happen if someone found out that Pi had a remainder of 0 down the line, some billion or trillion or whatever decimels away. What implications would this have for practical sciences? All thoughts appreciated, ty.
You're asking: what would happen if we eventually discovered that pi was a rational number? It would break a very large amount of the mathematical machinery that we have created to date. - Warren
It would be a proof by contradiction of something, most likely that the algorithm used to compute the digits is wrong.
If the universe were perfectly euclidean (all straight lines are really straight, defined in terms of parallel lines) then your question would be nonsense, but since we know that the geometry of spacetime trajectories is non-euclidean, and not perfectly understood/classified, I can for the sake of discussion consider that we inhabit a universe where every perfect measurement of a circles radius and circumfrence would have these commeasurate (that is, in a rational ratio). This would have less of an effect on macroscopic physics then quantum mechanics, and QM already has very little effect.
Thank you for the expert replies. Who proved Pi to be irrational and could a math novice such as I understand such a proof? Also, why do mathematician's bother to further refine Pi's value to millions of decimel places, are there formulas that scientists use that make more accurate predictions based on futher refinements on Pi? Perhaps, we can more accurately send a shuttle into orbit or something like this, or is this off base.
One interesting thing you get from it is to see up to what extent [itex]\pi[/itex] is random (it is assumed to be random but not proven). See for instance Is e Normal? about this but then for e.
No, there is no particular use for calculating [itex]\pi[/itex] to many decimal places, at least with regard to any practical problem. My computer can calculate [itex]\pi[/itex] to 100,000 digits in a second, and that is with 77 Firefox windows, two Maple windows, music playing, and a number of other programs running (for some reason I've never liked tabs much )! If I required it, I could have 10,000,000 digits in half an hour. Even at the 1,000 digit level there would be no practical calculation that wouldn't include other sources of uncertainty much, much larger. I'm sure there are many proofs of pi's irrationality online. Just google "proof that pi is irrational," or some such.
http://www.mathpages.com/home/kmath313.htm Here's a proof, and in the first sentence it says who proved it first. You can follow it if you know calculus. There's not really anything to understand though. It seems they arrived at a useful inequality more or less by luck.
That's not true. [itex]\pi[/itex] is a specific number that is definitely irrational. What you should say is that the could exist a portion of space in which the ratio of circumference to diamer of a circle is a rational number rather than [itex]\pi[/itex]- but that has no effect on the number [itex]\pi[/itex].
I don't know much about this. But even if we were in a different space where the ratio was rational, we probably have to use Pi itself to even solve the ratio in the first place. :uhh:
No, I see no reason for that. Certainly we can imagine a world where ratio of circumference to diameter of some circles is 3. No need to introduce pi at all. (I said some circles because while in Euclidean geometry that ratio is the same for all circles, pi, in any non-Euclidean geometry, it varies with the size of the circle.)
Simplest Definition for Pi - Ratio of the Circumference to the Diameter of a Circle IN EUCLIDEAN SPACE. If we were in a different space where the ratio of rational, we would make bigger problems for ourselves by using pi because since pi is irrational, the constant by which you multiply pi to get this rational value must also be irrational and an expression involving pi. I seemed to be quite confusing there...
I agree with you all, archimede's constant [tex] \pi [/tex] is a well defined irrational number, sorry for the confusion. Because of the following remark: I thought that OP was interested in the usual physical interpretation of [tex] \pi [/tex] --- as the empirical ratio of circumfrence to diameter in our reality. I realize this is the math forum, but the math interpretation of the OP's question is so nonsensical I treated it as a misplaced physics question.
What if pi was based in a different system. There could be a different base or altogether different way to add up numbers. Think of a quadratic number system instead of base10, base2, etc...
Since Pi is irrational, would it be logical to think that the string of digits go through every possible number of combinations? If that were true somewhere down the line of digits is this very post is encoded in binary; It may take massive amounts of calculations and perhaps more time than the age of the universe, but you would find it starting at the nth decimal place at some inconcievably large number. Just a thought, nothing more.
If pi is normal, then yes, otherwise that's not at all guaranteed. 1.01001000100001... is irrational, but the number combination "2" never appears at all (this is a decimal representation!). All evidence suggests that pi is normal, though no one's been able to prove it yet...