Pi revisited still again

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Issues of interest not visited in a recent Pi thread

http://able2know.org/topic/249161-1

…in which Con shows Pi to be irrational; but just how random is this irrationality, what conditions does its calculation impose; eg,

(1) To determine the number of digits preceding a particular sequence, that is eg, seven zeroes in a row or the sequence 0-9. We're pretty sure we'd find 'em but how much further apart (or even closer??) than in a truly random series

(2) We'd suppose the only way of finding such sequences is to run our PC for years or centuries until the desired sequence appears, then noting the number of preceding digits. But am I wrong, could there be a shortcut

(3) Owing to any deviation from randomness whether certain sequences might be impossible, maybe for instance 127 consecutive zeroes

Incidentally in Con's posting #….857 if I do eg, a highlight-command-find 53, Mac duly highlights only one such instance; not necessarily the first, as if it selected this particular rep randomly . Furthermore it wouldn't respond at all to a 3-digit sequence, eg 535

If I persist at such experimentation Mac gives up altogether, ignoring my highlight and looks for the sequence within another posting, as if the procedure were not being carried out by a pc editing routine but a 4-year-old child pecking at the keyboard

This all just doesn't make sense but only increases my amazement with and disgust at Mac's typical editing software

But maybe it's I whose expectations are based on the 4-year-old's reasoning process

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@dalehileman,

Fooling around with the search-fid routine a little bit more in Con's rendition of Pi, highlighting and searching for the number 5 results in only 10 hits, again apparently at random. I suppose I shoujld be grateful it didn't select some other digit

Incidentally illustrating Mc's editing insanity he was unable to recognize "fid" above as a misspelling of "find". But asd I said, I guess I must expect too much. Oh Mac, "asd" is a misspelling of "as"; note the proximity of the "d" to the "s". Oh, I give up, Mac, you can find that word in a dictionary….

Yeah guys I know, I'm exaggerating this incidental just a little, the problem might not be with Mac but with a2k; and yes I know if you have a masters in Software Development you know how to use "find all" in a situation like this

..while you fellas have my apologies for straying so far from the math of Pi. But then I did say, "Incidentally…." didn't I….

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@dalehileman,

Again my apologies for the OP OT (1) - (3) but if there's a math wizard in the crowd…..

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@dalehileman,

dalehileman wrote:

(1) To determine the number of digits preceding a particular sequence, that is eg, seven zeroes in a row or the sequence 0-9. We're pretty sure we'd find 'em but how much further apart (or even closer??) than in a truly random series

I suspect you will find this page interesting:

http://mathworld.wolfram.com/PiDigits.html

As far as can be determined, the distribution of digits in the decimal expansion of pi is uniform. There are some known results on how well pi can be approximated by rationals, which imply (for example) that we know a priori that the next n as-yet-uncomputed digits of pi can't all be zero. It is strongly suspected, I gather, that pi is a 'normal number' - this term has a specialised meaning: a real number (a value that represents a quantity along a continuous line) whose infinite sequence of digits in every base is distributed uniformly. A normal number can be thought of as an infinite sequence, in base 10, of spins of a roulette wheel with 10 pockets. This means that no digit, or combination of digits, occurs more frequently than any other.

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(2) We'd suppose the only way of finding such sequences is to run our PC for years or centuries until the desired sequence appears, then noting the number of preceding digits. But am I wrong, could there be a shortcut

To determine whether a sequence occurs, you have to calculate digits; to find the first or nth occurrence of any arbitrary sequence - Cousin Jemima's zip code or your Social Security number - you have to calculate the preceding digits.

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(3) Owing to any deviation from randomness whether certain sequences might be impossible, maybe for instance 127 consecutive zeroes

See above. It has not been proved that certain sequences are impossible, but if the distribution of digits is truly uniform, then no sequence is ruled out.[/quote]

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@contrex,

Thank you Con for that link. I shall peruse it

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we know a priori that the next n as-yet-uncomputed digits of pi can't all be zero.

Yes of course, even I can understand that

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suspected….. 'normal…..real number …...whose infinite sequence of digits in every base is distributed uniformly

Pretty good guess I think

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To determine whether a sequence occurs,…... you have to calculate the preceding digits.

I was afraid that might be the case

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if the distribution of digits is truly uniform, then no sequence is ruled out

I wonder about that. The uniform distribution of digits doesn't sound exactly the same as absolute randomness, so makes one wonder whether certain combos or lengths of digits might be impossible. Pretty deep kind of math but still leads me to speculate how long the best computer might take to find that string of, eg, 33 zeroes or even whether it would be possible to estimate without actually carrying it out

I think it would. Instead of actually carrying out the calculation of Pi until 33 zeroes happens, simply observe the frequency of reps in shorter sequences of zeroes; so as their size increases an estimated time (or number of intermediate digits) for rep of even larger such sequences might be calculated

I hope I'm making sense as a2k won't let me perform any further editing

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@dalehileman,

dalehileman wrote:

The uniform distribution of digits doesn't sound exactly the same as absolute randomness,

Why not?

Quote:

Instead of actually carrying out the calculation of Pi until 33 zeroes happens, simply observe the frequency of reps in shorter sequences of zeroes; so as their size increases an estimated time (or number of intermediate digits) for rep of even larger such sequences might be calculated

It doesn't work like that. You cannot predict future digits or sequences of digits based on what has already been calculated. This is also true of lotteries and roulette wheels. There is no pattern. To suppose otherwise is to succumb to the "gambler's fallacy".
http://en.wikipedia.org/wiki/Gambler%27s_fallacy

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@contrex,

dalehileman wrote:
The uniform distribution of digits doesn't sound exactly the same as absolute randomness,

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Why not?

Dunno, I'm no mathematician but if the figure in question (in this case Pi) is determined by a formula, intuition says the likelihood of certain sequences might be greater or lesser than if every digit is absolutely random

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Instead of actually carrying out the calculation of Pi until 33 zeroes happens, simply observe the frequency of reps…. [so] even larger such sequences might be calculated

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It doesn't work like that. You cannot predict future digits or sequences of digits based on what has already been calculated,,,,,, [don't] succumb to the "gambler's fallacy".

Thank you Con for that link, I shall peruse it at leisure. However my own intuition rebels at the notion that whenever the series isn't truly random, the frequency certain early patterns don't provide a means for estimating that of later patterns

Of course with roulette the perfect machine would yield a truly random result every time. In the case of Pi however, the next digit is determined by a formula, even though irrationality assures its total increasing at the same rate as any other digit, owing to requirements of the formula one might expect certain patterns to appear at fairly regular intervals, however far separated

Like I said though I'm no mathematician, as you may have guessed

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@dalehileman,

dalehileman wrote:

Of course with roulette the perfect machine would yield a truly random result every time. In the case of Pi however, the next digit is determined by a formula, even though irrationality assures its total increasing at the same rate as any other digit

Drop that "however". Pi isn't calculated by a "formula" in the sense that I think you mean. To explain what I mean I have to delve a little bit into number theory. In mathematics, a real number is a value that represents a quantity along a continuous line (the "number line"). The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356…, the square root of two, an irrational algebraic number) and π (3.14159265…, a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation.

Pi has a value that can only be approximated to. What is "calculated" when digits of pi are produced, is an approximation. One (slow) formula for the approximation is to successively add and subtract the odd fractions like this:

pi /4 = 1 - 1/3 + 1/5 -1/7 + 1/9 - 1/11 ... (continued indefinitely)

Can you see how the sign flips from plus to minus after every new fraction? Can you see how that means the approximation will flip from just below pi to just above it, the gap getting narrower and narrower? Can you appreciate that it will never end (because there is an infinite supply of integers for the denominators of the fractions?)

Pi is not so much calculated as discovered. If there was a pattern then it would have been found sometime in the last 400 years.

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@contrex,

Golly Con thank you for that rundown. I have to take your word for much of that since my take is almost purely intuitional

Quote:

Can you see how the sign flips from plus to minus...how that means the approximation will flip from just below pi to just above it, the gap getting narrower.that it will never end…...?)

Yes of course I see all that, it's intuitional. However to the left of the last digit Intuition insists there ought to be found patterns

Incidentally it seems to me that yes, in the case of Pi the patterns would be few and subtle but still they'd be

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Pi is not so much calculated as discovered. If there was a pattern then it would have been found sometime in the last 400 years.

Isn't there another mathematician in the crowd who can testify either way, or isn't the subject of much general interest

Wonder if anyone hereabout also participates in a math site where we might post the q

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@contrex,

Con, I don't blame you for having given up on me, I'm just now becoming vaguely conscious about the official notion of "pattern." To me a pattern might for instance reveal the approximate frequency and distribution of certain sequences in a number, say, Pi.; whereas your interpretation is the ability to write a formula that might determine the next digit

http://able2know.org/topic/249623-1

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