I would like to know if there is a mathematically correct way of describing a sequence of decimal points of a given number.
e.g. position 35-46 of π
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there is no such notation. You are free to make up anything you want. – Ittay Weiss Aug 13 '15 at 11:25
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Thats too bad. But thank you! – AKroell Aug 13 '15 at 11:31
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@awk There's no simple notation, but using the floor notation you can write a formula for it, see my answer given below. – Gregory Grant Aug 13 '15 at 11:36
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have you thought about continued fractions? – user190080 Aug 13 '15 at 11:43
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What about using "floor", the integer part of a number, indicated by $\lfloor x\rfloor$? So $\pi - \lfloor\pi\rfloor$ is the decimal part of $\pi$. So $\lfloor (\pi-\lfloor\pi\rfloor) * 10^{46} \rfloor$ would give you the first $46$ decimal places. So $\left\lfloor(30{\Large(}(\pi-\lfloor\pi\rfloor)-\lfloor30(\pi-\lfloor\pi\rfloor)\rfloor{\Large)} * 16\right\rfloor$ would give you places $35$ to $46$. I think.
Gregory Grant
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Thats what I was looking for thank you!
Would $\lfloor (\pi-\lfloor\pi\rfloor) * 10^{46} \rfloor$ - $\lfloor (\pi-\lfloor\pi\rfloor) * 10^{35} \rfloor$ be the same?
– AKroell Aug 13 '15 at 11:48 -
@awk It would be close, but it would have a bunch of trailing zeros. So divide by $10^{35}$ – Gregory Grant Aug 13 '15 at 12:40
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@awk Actually we're both wrong, you would have to multiply the second term by $10^{16}$ before subtracting, then you would still have the trailing zeros so divide the whole thing by $10^{16}$ again. – Gregory Grant Aug 13 '15 at 12:45
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@awk I had some missing parentheses as I originally stated it, but I think it's right now. – Gregory Grant Aug 13 '15 at 22:17
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In theoretical computer science, the notation $u[i,j]$ is frequently used to denote the factor $a_ia_{i+1} \dots a_j$ of an infinite word $u = a_0a_1 \dotsm$. Considering a number as an infinite word on the alphabet $\{0, \dotsm, 9\}$, you could adapt this notation and write something like $\pi[35,46]$. The obvious problem is the radix point, but you could work on the alphabet $\{0, \dotsm, 9, .\}$ instead and define the position of the radix point as $0$. With these conventions, for the real number $r = \sqrt{54321} = 233.068659412 \dotsm$, you could write $r[-2,6]= 33.068659$.
J.-E. Pin
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This might be best approached by carefully doing a small example. Suppose we'd like to capture the $926$ in $\pi=3.1415{\color\red{926}}5\ldots$. Note that $10^4\pi=31415.9265\ldots$, hence
$$10^4\pi-\lfloor10^4\pi\rfloor=0.9265\ldots$$
and thus
$$926=\lfloor10^3\cdot0.9265\ldots\rfloor=\lfloor10^7\pi-10^3\lfloor10^4\pi\rfloor\rfloor$$
For the specific example the OP mentioned, i.e. decimal positions 35-46, it should be clear the answer has the form
$$\lfloor10^n\pi-10^{12}\lfloor10^{n-12}\pi\rfloor\rfloor$$
where $n$ is either $45$ or $46$, depending on whether you count decimal positions starting at the $3$ or not. (I.e., does $926$ correspond to decimal positions 6-8 or 5-7?)
Barry Cipra
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