5 ms·

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

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?

deleted 6y ago
[deleted]
efdb 6y ago
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.
- stabbles 6y ago
  A standard technique to find explicit expressions for the `n`th number of a recursion is through generation functions; see e.g. https://en.wikipedia.org/wiki/Generating_function https://en.wikipedia.org/wiki/Generating_function. You can plugin x = 10^-1 there. Not sure if the result is always a rational number.
- magnio 6y ago
  For a linearly recursive sequence x_0, x_1, x_(n+2) = ax_(n+1) + bx_n, the general formula for the terms is 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.
  - deleted 6y ago
    [deleted]
benrbray 6y ago
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 smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.
- wizzwizz4 6y ago
  That just sounds like another formulation of the Surprise Exam paradox.[0] It falls down when you realise that “the four hundred and seventieth otherwise uninteresting number” is not a particularly interesting number, so there must be a problem with the problem statement. [0]: https://en.wikipedia.org/wiki/Unexpected_hanging_paradox https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
  - alisonkisk 6y ago
    It's a paradox related to "meta-logic", but it's different. Surprise exam is about temporal reasoning -- It's only impossible to be surprised on the last day (or else the premise of having an exam is invalidsted), and reasoning backwards in time from a contradiction is not valid. Uninteresting number is a simpler contradiction in definitions.
- DeusExMachina 6y ago
  This is not a paradox though, as your copy and paste states. It's just a theorem (as you stated) with a proof by contradiction. A paradox must be self-contradictory under all circumstances.
  - phizy 6y ago
    It is a paradox. It assumes you have a definition of uninteresting number such that you can select the least uninteresting number, and then retroactively defines that number to be interesting by brand new criteria, which contradicts that you would have ever selected it in the first place. Thus the axioms invoked are mutually contradictory: the axioms that allow you to identify the least uninteresting number, and the axioms you invoke to declare it interesting are in conflict.
bobbylarrybobby 6y ago
This is no coincidence — it’s because 89=100-10-1, and the ordinary generating function for the Fibonacci sequence is 1/(1-x-x^2) (if you go to wolfram alpha and Taylor expand that expression, you’ll see its coefficients are the Fibonacci numbers).
impendia 6y ago
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 = 0.00010102030508132134559046368320032326498... 1/998999 = 0.0000010010020030050080130210340550891442334... and so on.
- contravariant 6y ago
  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.
jerf 6y ago
Imagine the real number line is a database that you can run SELECT queries against, and you don't have to worry about giving a computational procedure that produces the result, it just magically gets produced. 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 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 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.
alisonkisk 6y ago
You might also enjoy tan(1 degree/55555555555)
- huhtenberg 6y ago
  3.1415926536212091649988555782837470430464925667776193389556...e-13 Nice
- prezjordan 6y ago
  Fun video about this one on Numberphile https://www.youtube.com/watch?v=IMY2_yzDm9I https://www.youtube.com/watch?v=IMY2_yzDm9I
  - dylan604 6y ago
    "they say there's a fine line between a numerator and a denominator" --? another one of those maths jokes that makes me smile and confuses a lot of other people of why i'm laughing
- paulpauper 6y ago
  not impressive given that more 5s are needed than correct decimals produced
JackFr 6y ago
The Penguin Book of Curious and Interesting Numbers https://www.amazon.com/Penguin-Book-Curious-Interesting-Numbers/dp/0140261494/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr= https://www.amazon.com/Penguin-Book-Curious-Interesting-Numb... The caveat here is "for some value of 'interesting'."

deleted 6y ago

[deleted]

efdb 6y ago

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.

stabbles 6y ago
A standard technique to find explicit expressions for the `n`th number of a recursion is through generation functions; see e.g. https://en.wikipedia.org/wiki/Generating_function https://en.wikipedia.org/wiki/Generating_function. You can plugin x = 10^-1 there. Not sure if the result is always a rational number.
magnio 6y ago
For a linearly recursive sequence x_0, x_1, x_(n+2) = ax_(n+1) + bx_n, the general formula for the terms is 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.
- deleted 6y ago
  [deleted]

benrbray 6y ago

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 smallest uninteresting number is itself interesting because it is the smallest uninteresting number, thus producing a contradiction.

wizzwizz4 6y ago
That just sounds like another formulation of the Surprise Exam paradox.[0] It falls down when you realise that “the four hundred and seventieth otherwise uninteresting number” is not a particularly interesting number, so there must be a problem with the problem statement. [0]: https://en.wikipedia.org/wiki/Unexpected_hanging_paradox https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
- alisonkisk 6y ago
  It's a paradox related to "meta-logic", but it's different. Surprise exam is about temporal reasoning -- It's only impossible to be surprised on the last day (or else the premise of having an exam is invalidsted), and reasoning backwards in time from a contradiction is not valid. Uninteresting number is a simpler contradiction in definitions.
DeusExMachina 6y ago
This is not a paradox though, as your copy and paste states. It's just a theorem (as you stated) with a proof by contradiction. A paradox must be self-contradictory under all circumstances.
- phizy 6y ago
  It is a paradox. It assumes you have a definition of uninteresting number such that you can select the least uninteresting number, and then retroactively defines that number to be interesting by brand new criteria, which contradicts that you would have ever selected it in the first place. Thus the axioms invoked are mutually contradictory: the axioms that allow you to identify the least uninteresting number, and the axioms you invoke to declare it interesting are in conflict.

bobbylarrybobby 6y ago

This is no coincidence — it’s because 89=100-10-1, and the ordinary generating function for the Fibonacci sequence is 1/(1-x-x^2) (if you go to wolfram alpha and Taylor expand that expression, you’ll see its coefficients are the Fibonacci numbers).

impendia 6y ago

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 = 0.00010102030508132134559046368320032326498... 1/998999 = 0.0000010010020030050080130210340550891442334... and so on.

contravariant 6y ago
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.

jerf 6y ago

Imagine the real number line is a database that you can run SELECT queries against, and you don't have to worry about giving a computational procedure that produces the result, it just magically gets produced. 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 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 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.

alisonkisk 6y ago

You might also enjoy tan(1 degree/55555555555)

huhtenberg 6y ago
3.1415926536212091649988555782837470430464925667776193389556...e-13 Nice
prezjordan 6y ago
Fun video about this one on Numberphile https://www.youtube.com/watch?v=IMY2_yzDm9I https://www.youtube.com/watch?v=IMY2_yzDm9I
- dylan604 6y ago
  "they say there's a fine line between a numerator and a denominator" --? another one of those maths jokes that makes me smile and confuses a lot of other people of why i'm laughing
paulpauper 6y ago
not impressive given that more 5s are needed than correct decimals produced

JackFr 6y ago

The Penguin Book of Curious and Interesting Numbers https://www.amazon.com/Penguin-Book-Curious-Interesting-Numbers/dp/0140261494/ref=tmm_pap_swatch_0?_encoding=UTF8&qid=&sr= https://www.amazon.com/Penguin-Book-Curious-Interesting-Numb... The caveat here is "for some value of 'interesting'."