Two questions about Pi

'Cause you know, it’s Pi day and all…

Would anything interesting happen if some day down the line some giant computer found that Pi repeats? As in, at the 13 trillionth place it starts over at 3145…

Anything interesting to other parts of math/science (theoretical or otherwise) beyond a lot of people scratching their head and saying “well damn, first we had that brontosaurs fiasco, then there was the whole bit with Pluto, and now this.”?
Second, how did they know, before computers, that it didn’t repeat? Is it a proof or a supposition? And now that we have computers that have calculated it to a zillion places, do these computers then check to see if any of the sequences have come up before?

We know that pi is irrational,so it never repeats. In fact it is transcendental, which is a stronger condition than irrational.

The discovery that it appears to repeat would be interesting, but you could not check this directly in a finite amount of time, so no one would believe that it keeps repeating.

If, on the other hand, someone proved that it does repeat, this would mean that there is a mistake either in the new proof or the old proof.

For the proof see Proof that π is irrational

It’s been known for quite a while that pi is irrational (the proof of it being transcendental came later.) It’s also been known for quite some time that irrational numbers have non-repeating expansions. In particular, one can prove that all repeating decimals are rational, and this proof also demonstrates that if we had a non-repeating decimal, it must not be rational.

We can’t check every digit of pi to see if there are repeats, since the decimal expansion goes on forever. But we can prove that pi is irrational, and we know that irrational numbers don’t repeat. So if a repeating pattern were found (beyond mere random chance), that would cast serious doubt on some pretty fundamental proofs of pi’s irrationality.

If by repetition you mean, that at the 26 trillionth place, the 13 trillion digits repeat in order again, and also at the 39 trillionth place, and so on forever, then pi would be rational. This was proved impossible by J.H. Lambert in 1761.

In 1882, F. Lindemann proved the much stronger result that pi cannot be algebraic.

Proofs of these results are provided at the Wikipedia articles, but if you need help following the proofs please ask one of the Board’s mathematicians. I’m afraid a full pot of coffee would not be enough for me to properly follow proof of the Lindemann-Weierstrass Theorem and caffeine is contraindicated for my cardio-vascular problem.

Calculating the digits of a number and doesn’t tell you an awful lot about the number and calculating the digits of a number in sequence doesn’t tell you if a number is irrational.

If pi is a normal number as it is believed to be (though this is not proven), then there would always come a point in its digits where it starts …3145… and repeats the first nth digits of pi for a finite but arbitrarily large value of n.

Thanks. I’m going to try a bit too now that I have a better idea of what to search on.

That should of course be “…31415…”

Pi Day? Heck, we had better than that. At twenty six minutes and fifty four seconds after 9:00 AM this morning, we had Pi Seconds.


Actually, if a computer found pi was rational, I’d suspect the computer, or more accurately the program used to compute digits of pi.

We already know mathematically that pi is transcendental. A computation showing otherwise wouldn’t constitute proof, but it could very easily show user error.

The thing about irrational numbers is that brute force computation really can’t prove irrationality, though it can potentially prove rationality.

It’s not really the same thing, but in one respect the concept is similar to proving there are infinitely many primes. You don’t have to list all primes (which is impossible, as it happens) to prove there are infinitely many primes, just as you don’t have to compute all the digits of the decimal expansion of pi to prove the decimal expansion is infinite.

The heat from generations of mathematicians turning in their graves would melt the Earth’s crust down to the core…

If you mean the “practical” value of pi ( measured circumference of a true circle divided by diameter), I don’t see why it couldn’t be rational in places. Gravity does strange things…

“Rational in places?”

Given the expansion of Pi is infinite, it is absolutely certain that any and every finite sub-sequence in the expansion of Pi will be repeated in an infinite number of other locations.

In fact if a sub-sequence isn’t repeated somewhere then it would be serious cause for concern.

e.g. My Phone number occurs twice in the first 200 million digits of Pi

Only if pi is normal, which, as noted above, we don’t know for sure.

Well, it wouldn’t be the same ratio everywhere. Gravity bends space, an you know…

And it doesn’t matter if you try writing pi in some base other than 10 either.

The rationality, or irrationality, or transcendence of any number is an intrinsic property of that number itself, entirely regardless of how you write the number. A number that is rational/irrational/transcendental in base 10 is likewise in any other base.

And the rule about repeating digits applies in all bases too. A rational number, written as a “decimal” fraction but in any base other than decimal, still terminates or repeats. An irrational number doesn’t, in any base.

While pi is probably normal as mentioned above (though this has not been proven), the reasoning given here is definitely not correct.

An infinite decimal expansion is insufficient reason by itself. Here’s one example that demonstrates this fact:


This is a sequence of zeros interspersed with 1’s at predictable intervals. It’s not rational but it doesn’t contain every finite subsequence of whole numbers. In fact, it only contains a single finite subsequence - the subsequence 1 itself.

Yes, it’s a rather contrived example but it shows that an infinite decimal expansion is not sufficient on its own to demonstrate normality.

Technically, it’s equally correct to say that all rational numbers repeat, either in an infinity of zeroes or, if written another way, in an infinity of nines. So 1/1 is rational, and it can be written 1.0… or 0.9… and both forms end in an endless repetition.

Decimal fractions written that way use endless sequences of natural numbers (assuming you define the natural numbers to include zero); if you do something similar by creating a system of numbers using endless sequences of real numbers, you get the hyperreals, which can be used in nonstandard analysis to bring infinitesimals back into calculus.

That is completely untrue. Pi never repeats, but that does not mean that every finite sub-sequence can be found in its digits. If, for example, Pi did this:


And kept going, adding one more zero between each pair of ones, it would not repeat itself but would certainly not contain every possible subset of digits. There’s a substantial difference between an irrational number and an infinity of random digits. Even random digits cannot guarantee all subsets. What if a number is randomly composed of all digits except 9? Or is random in every way except that it never repeats the same digit consecutively?

Edit… Whoa, Great Antibob gave the same example a few posts up.

It’s worth noting that just calculating more and more digits could give you the observation “Oh, digits #123 through #128 are the same as digits #1 through 6” (since that would just require calculating digits through, in this example, #128), but couldn’t assure you “Oh, from hereon, everything just repeats over and over, cyclically in the same pattern, forever” (since that would require first calculating all infinitely many digits and then surveying the results). There is a difference between these two senses of the word “repeat”.

If you’re just curious about whether π ever repeats any of the digits it started with, it sure does. For example, it starts with 3, and we see later another 3 pop up [3.141592653…].

If you’re curious instead as to whether π consists of a particular pattern repeated cyclically over and over, forever [or even, like 1/6 = 0.1666…, consists of a finite bit of random junk prefacing what eventually becomes a particular pattern repeated cyclically over and over forever], well, the answer is, computation of digits doesn’t resolve this question, but mathematical proof does. We know that π is not of this form because A) being of this form is equivalent to being rational and B) π is irrational.

A) is because to say that the decimal expansion of a number x is eventually cyclic is to say that 10[sup]m[/sup]x - 10[sup]n[/sup] x is an integer for distinct integers m and n, which is to say that x can be written as a fraction with denominator equal to (or, just as well, dividing) 10[sup]m[/sup] - 10[sup]n[/sup]. Clearly, any x of this form is rational, and conversely, as every positive integer divides a difference of powers of 10 (because there are infinitely many powers of 10, but only a finite number of possible remainders modulo any particular positive integer), we see that every rational is of this form.

B) is trickier. A certain familiarity with calculus will be necessary to follow its proof. But it is proven. See Wikipedia for details.

I saw what you did there.