I am not a mathematician, although a few decades ago, I took some courses. I happened upon a website a couple of hours ago where they were discussing whether or not the decimal expansion of the square root of 2, had to be embedded in the decimal expansion of pi. Actually I suppose they could have been discussing the obverse of that.

So I started thinking along these lines. I probably won’t be using the correct mathematical terms. Does there exist a position N in pi, such that the first N-1 digits are then repeated? For example, if pi/10 = 3141593141598726 … , then N = 7. It seems to me since the expansion is infinite, this might be true. Then if we assign n = 314159, we could write pi/10 = nn …

If that is true, then doesn’t M exist, so that we could assign m = the first M-1 numbers, and we could write pi/10 = mmm …

Then O exists and pi/10 = oooo …

Then ultimately some (infinite) X exists such that pi/10 = xxxxxxx … unending.

That’s similar to a repeating decimal. And a repeating decimal is one integer divided by another. And the same principle would apply to any irrational number, not just pi.

So at the end of it, any Irrational Number = (some infinite thing)/(another infinite thing).

Sorry if this seems to be just so much nonsense, or is well developed and old hat.

I am not exactly sure what you are talking about, but here is an irrational number contradicts the property you are talking about.
Let . This number is irrational and the first two numbers never appears again since after that the number is composed of blocks of 's and 's. Likewise the first three numbers cannot appear again for the same reason. Thus, just because something is infinite does not mean it has to appear again.

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And he (Elisha) went up from thence unto Bethel: and as he was going up by the way, there came forth little children out of the city, and mocked him, and said unto him, "Go up, thou bald head"; "go up, thou bald head". And he turned back, and looked on them, and cursed them in the name of the Lord. And there came forth two she-bears out of the wood, and tore up forty and two children of them.
Second Kings 2: 23-24

Thanks. OK I can see you constructed an irrational number which cannot repeat. I was thinking in terms of irrationals with a seemingly random distribution of integers, like pi, or like square roots of non-square numbers. Maybe mathematicians have a special name for those, I don't know.

So, if you prefer, limit my question to the decimal expansion of pi. Does that N exist (given that the expansion is infinite, isn't it guaranteed to exist?). And if that N exists, doesn't the rest of the speculation fall out?

Almost all real number are normal to base (or any other positive integer base for that matter, and may be all bases but I don't know about that), that is every sequence of digits of length appears with its theoretical frequency in decimal expansion of the number.

is thought likely to be normal but not proven so. If it is normal then the first digits will reappear in the correct order infinitly often in the decimal expansion of .

CB

I have some empirical experience in this matter: The smallest 10-digit prime in the digit sequence of e is about at location 1.9 billion. If they are randomly distributed then we'd expect it to be somewhere in the first digits. That's about ball-park. So to expect the first n-repeating digits in pi, then we'd have to check about digits. That's a lot of numbers: The location of the smallest 100-digit prime in pi is completely beyond our capabilities at present.

. . . anybody wants to look for the eleventh one? Nobody on earth has found it yet you know. Guess that would be somewhere under 100 billion. Too much for my machine.

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"I am beset by the ironies in my life"

I am waiting for the discovery of the compelete text of the book of Genesis coded in ASCII to be found in the decimal expansion of pi. Now that will be a thing

CB

Go beyond that. Expect to find the proof that pi is normal in its own decimal expansion!

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And he (Elisha) went up from thence unto Bethel: and as he was going up by the way, there came forth little children out of the city, and mocked him, and said unto him, "Go up, thou bald head"; "go up, thou bald head". And he turned back, and looked on them, and cursed them in the name of the Lord. And there came forth two she-bears out of the wood, and tore up forty and two children of them.
Second Kings 2: 23-24

Even more amusing if pi is normal expect to find a subtly erroneous proof that it is not normal encoded in its decimal expansion!!

Or even a proof that the normality of pi is undecidable!!!

CB

Hey thanks for your comments.

So if normal, then pi can ultimately be expressed as a repeating chain (pppppppp ... ) endlessly, analogous to a repeating decimal, so all that is left is to find the integrand and divisor that generates it ... some infinities X/Y = pi Similarly with any other irrational, but with different X and Y and I'm in way too deep to understand what I'm saying ....

No, nobody said any such thing is not periodic as then it could not be normal.

CB