For people who are generally interested in these kind of identities relating a fraction to a recursion, please check out the generatingfunctionology book [1]. The basic idea is to define f(x) = sum_n f_n x^n where f_n satisfies some recursion equation. Very often you can find f(x) as p(x) / q(x) where p and q are polynomials in x. Now you can simply evaluate f(0.1) or more generally f(b^-l) where b is a base and l a positive integer, which gives you on the left-hand side your rational number as a fraction, and on the right-hand side the decimal expansion.
The Remarkable Number 1/89 (2004)
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Re: The Remarkable Number 1/89 (2004)
#32Presumably in a different number base, the number would be different. Would it still be in the aeries?
The fraction, the decimals, and the series, are in the same base.
Re: The Remarkable Number 1/89 (2004)
#33"The successive ratios of the terms, i.e. 1/1, 2/1, 3/2, 5/3 ... tend to a number called the Golden Ratio by the Greeks." Fun fact: take any two numbers (e.g. chosen randomly), and use them as the seeds for a Fibonacci-like sequence by summing the last two terms to generate the next term. The ratio of any two consecutive terms in that series will tend towards the golden ratio.
If you read this and you’re curious, there’s a proof that is fairly easy to follow if you understand eigenvectors. Write the operation that takes the two last elements of the sequence and produces the following two, notice it’s linear, then analyze the eigenvalues of the associated matrix and relate the original operation to the power method.
Re: The Remarkable Number 1/89 (2004)
#34On the decimal expansion part, 1⁄7 has always fascinated me, having something very similar going on. Doubling from 7, you get 14, 28, 56; and 1⁄7 is 0.1̅4̅2̅8̅5̅7̅, 2⁄7 is 0.2̅8̅5̅7̅1̅4̅, 3⁄7 is 0.4̅2̅8̅5̅7̅1̅, &c. (just changing which digit you start the recurring sequence with). https://en.wikipedia.org/wiki/142,857 talks about it a bit more; the doubling sequence thing is covered in the section 1⁄7 as an infinite sequence (including the reason the recurring decimal has 57 instead of 56—that 56 doubled is 112, so the hundred there overlaps with the six, much as the ten of the 13 adds to the 8 in the 1⁄89 expansion of this article).
Re: The Remarkable Number 1/89 (2004)
#35Okay, as a non-mathematician, I see something like this and I think... “neat coincidence?” But the world of numbers seems to be full of these neat coincidences. So do any of the math folks here have a theory or explanation of why ?
Yes. One has the identity: 1/(1 - x - x^2) = 1 + x + 2x^2 + 3x^3 + ... where the Fibonacci numbers are the coefficients on the right. Try writing it out! The basic idea is that because F_n = F_{n - 1} + F_{n - 2}, everything will neatly cancel out. This is an example of a "generating function". Anyway, plug in x = 0.1, and then divide by 100 to see the behavior described in the post. One also gets 1/9899 = 0.00010102…
More to the point this function trivially statisfies f(x)-1-x = X (f(x)-1) + X^2 f(X). You should be able to convince yourself that this is equivalent to the fact that its power series satisfies the Fibonacci recurrence, and that this power series starts with 1 + x.
You can also use this to easily find generating functions that satisfy other starting conditions.
Re: The Remarkable Number 1/89 (2004)
#36Okay, as a non-mathematician, I see something like this and I think... “neat coincidence?” But the world of numbers seems to be full of these neat coincidences. So do any of the math folks here have a theory or explanation of why ?
You might also enjoy tan(1 degree/55555555555)
3.1415926536212091649988555782837470430464925667776193389556...e-13
Nice
Re: The Remarkable Number 1/89 (2004)
#37Ah makes sense if you think of it as 1 / (100 - 10 - 1)
89 being a fib itself is quite a coincidence though?
Re: The Remarkable Number 1/89 (2004)
#38On the decimal expansion part, 1⁄7 has always fascinated me, having something very similar going on. Doubling from 7, you get 14, 28, 56; and 1⁄7 is 0.1̅4̅2̅8̅5̅7̅, 2⁄7 is 0.2̅8̅5̅7̅1̅4̅, 3⁄7 is 0.4̅2̅8̅5̅7̅1̅, &c. (just changing which digit you start the recurring sequence with). https://en.wikipedia.org/wiki/142,857 talks about it a bit more; the doubling sequence thing is covered in the section 1⁄7 as an infinite…
That really is rather fascinating.
Re: The Remarkable Number 1/89 (2004)
#39On the decimal expansion part, 1⁄7 has always fascinated me, having something very similar going on. Doubling from 7, you get 14, 28, 56; and 1⁄7 is 0.1̅4̅2̅8̅5̅7̅, 2⁄7 is 0.2̅8̅5̅7̅1̅4̅, 3⁄7 is 0.4̅2̅8̅5̅7̅1̅, &c. (just changing which digit you start the recurring sequence with). https://en.wikipedia.org/wiki/142,857 talks about it a bit more; the doubling sequence thing is covered in the section 1⁄7 as an infinite…
I've always liked 1/7 as well, and I never realized that thing about doubling from 7 giving 14, 28, 56. People are always impressed when I can rattle off the digits of x/7.
By the way, I appreciate your use of U+0305 combining overline. Did you enter those manually or do you have some neat way of doing it?
Re: The Remarkable Number 1/89 (2004)
#40So strange. Published another post about the same topic an hour before you