11–20 of 116 posts
"The linked page misleadingly suggests that a certain Cody Birsner discovered the relationship between the series and the fraction, whereas it had been known for a considerable time before".
Günter Köhler, 1983 (published in the The Fibonacci Quarterly, 1985; who cites earlier papers from 1977 and 1981): https://www.fq.math.ca/Scanned/23-1/kohler.pdf
[0] https://en.wikipedia.org/wiki/Talk%3AFibonacci_number%2FArch...
Okay, 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 ?
There's a "theorem" about this: > The interesting number paradox is a semi-humorous paradox which arises from the attempt to classify every natural number as either "interesting" or "uninteresting". The paradox states that every natural number is interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting natural numbers, there would be a smallest uninteresting number – but the smal…
[0]: https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
Any number is remarkable if you're smart enough to find why it's remarkable.
Any number is remarkable if you're smart enough to find why it's remarkable.
Okay, 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 ?
Okay, 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 ?
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 = 0.00010102030508132134559046368320032326498...
1/998999 = 0.0000010010020030050080130210340550891442334...
and so on.
Okay, 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 ?
Now, imagine you write a query, like, say, "SELECT number WHERE number = .01 * FIB[1] + .001 * FIB[2] + .0001 * FIB[3]" and so on until you get what the article discusses.
It isn't necessarily that surprising that you might find something with an uncountably infinite numbers to pick from.
Now, consider all possible "interesting" queries you could run, along with all their results.
There result is an inconceivably large sea of queries. Most of them are, in fact, utterly pointless; SELECT statements that return no values, SELECT statements that return all values (equally pointless), SELECT statements that return complicated sets of values but have essentially no mathematical interest because there is no practical way to represent them as anything smaller or more interesting, etc.
In this massive sea of results, you should expect a lot of interesting things to exist. Finding them is tricky; in percentage terms they make up 0% of the results, but we have mechanisms for finding some of them.
Basically, there are so infinitely many mathematical statements that there can't help but be a large supply of "interesting" statements like this.
For an interesting view on that, see https://en.wikipedia.org/wiki/Mathematical_coincidence . These are true statements or almost true statements (near equalities) about a wide variety of numbers that are essentially meaningless... it's just there's so many ways of putting things together that there are inevitably large numbers of these things (the wiki page is just a sampling).
You can even generate these mathematical coincidences yourself. Create a program that will systematically iterate over abstract syntax trees of mathematical expressions involving whatever combination of mathematical operators (+-×/, sqrt, log, sin, whatever) and numbers you like (the first ten integers, e, i, pi, whatever else you like), store up a table of results and emit any two expressions that are, say, within .01% of each other. You will rapidly find a ton of results, because it turns out that even with modest numbers of operators, there are far more mathematical expressions than there are small numbers for them to result in separated by more than .01%. If you think about it, this program can't help but emit a lot of results. Some of them will be humanly "interesting". A few of them will even be mathematically interesting (e.g., this procedure will generate the famous Euler identity relatively quickly if you included the relevant operators and numbers).
On a larger scale, this is also known as the Strong Law of Small Numbers: https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers The previous paragraph is a very bite-sized example of why this holds that you can code up yourself if you are interested.
Okay, 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 ?
In addition to this question I would like to know if you can in general say/proof that for every sequence which has some relation between the successive numbers there is a rational number whose decimal expansion is the same as the sequence.
x_n = cα^n + dβ^n,
where α, β are the roots of the quadratic x²−ax−b; c, d are solutions to the system
c + d = x_0 cα + dβ = x_1.
If the series Σ x_n⋅10^n converges then its value is
10c/(10−α) + 10d/(10−β) = ((100−10a)x_0 + 10x_1)/(100 − 10a − b).
If a, b, x_0, x_1 are all rational then the above series converges to a rational number, too. This is the case for the Fibonacci sequence, with a=b=1, x_0=0 and x_1=1.