The Remarkable Number 1/89 (2004) (opens in new tab)

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.

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).

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.

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.

The Penguin Book of Curious and Interesting Numbers

https://www.amazon.com/Penguin-Book-Curious-Interesting-Numb...

The caveat here is "for some value of 'interesting'."

You might also enjoy

tan(1 degree/55555555555)

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 . 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.

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?

Cool thing is, this is not a special property of 1/7. 100 / 7 is 14, with a remainder of 2, therefore the series starts with 14, multiplies by 2, and divides with 100 in each iteration. For instance 10 / 7 is 1, with a remainder of 3, therefore 1/7 is also equal to 0.1+0.03+0.009 etc. And 1/8 is 0.1+0.02+0.004 etc.

Coming from video, I've always been a fan of 1/1001. 30000/1001 = 29.970029797002997 and 24000/1001 = 23.97600239760023976. There's something about it's clean repeating that I liked. I hear people confusing frame rates by saying something like 29.976. I also don't like 23.98 as that rounding is going to cause problems later.

However, you have to be a special math something to have any of these kind of number "oddities" be anything meaningful. I wear mine like a badge of honour

I discovered this in my late teens and thought it was super cool and made me want to understand more about repeating decimals. After playing around for a bit I realized you could form arbitrary repeating decimals by dividing by 9, 99, 999 etc. So for example, 1/7 = 142857/999999. Or written another way, 999999/7=142857.

Which also makes it easier to find the patterns in its multiple(or just divide by the extra multiple) say 1/14 starts .07142857142857. Multiples with 3 don't give us the common repeating 1428 but still repeats in its own way.. but 1/49 is pretty cool. 1/49 looks to do what 1/89 is doing but with the powers of 2. Nice!

edit: Dont know the format for proofs but heres a try.

1/49 = i=1 towards inf

sum 2^i*100^-i

I learned about 1/7 back in my youth, and it's just been one of those things that I enjoyed knowing as I went on in life.

Imagine my amusement when I ran across a Project Euler problem where those digits were the answer. I recall just looking at it and thinking I __know__ this one, there's no need to code anything. An easy point, but I didn't feel like I cheated on it.

Does this work or other bases too? Other than 10?

That really is rather fascinating.

Wow, it keeps going also with the further digits, e.g.

=14285712

+0000000224

=1428571424

+000000000448

=142857142848

+00000000000896

=14285714285696

+0000000000001792

=142857142857....

It tickles me that there was a "Fibonacci Quarterly" where people shared their favorite new Fibonacci facts on a quarterly basis.

It's a bit silly to chase down original authorship of an idea that is a minor detail visible to many people who work in a field.

It's like asking who was the first person to discover that all multiples of 11 have the same parity in the respective sums of their odd and even digits.

This seems like a pretty basic calculus problem. Constants become irrelevant at the limit, so that the relationship (ratio) is all that's left.

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.

Without any basis whatsoever, and I really must one day put this idea out of its misery with some studying, I long suspected that quantum mechanics involved parallel universes where different bases more aptly fit with that other reality.

My thought is surely crackpot but I'll explain how my idea arose :

A fraction eg 1/3 describes a decimal number to infinite accuracy but creates a challenge for base 10 calculations.

I thought about the precision necessary for learning the math of the quantum particles and the experimental way we're trying to figure out how they behave to infer their properties. (Higgs was the other way around but I am optimistic that we'll predict much more in future instead of this convoluted observations rigmarole) and I started wondering if you could approximate extremely high precision decimals to fractions in non decimal bases and from there simplify calculations with far greater resolution.

How much more capable would Nyquist - Shannon sampling, if we could clock with the precision of infinite decimal fp digits but handle only a short simple "one third" input or do visor?

My silly mind wandered off to imagine particles jumping between different base based universes just as a sequential progression through the precision of their infinitessimal steps through space and time.

The fraction, the decimals, and the series, are in the same base.

Hardy and Srinivasa Ramanujan about interesting and uninteresting numbers, Hardy remarked that the number 1729 of the taxicab he had ridden seemed "rather a dull one", and Ramanujan immediately answered that it is interesting, being the smallest number that is the sum of two cubes in two different ways.

https://en.wikipedia.org/wiki/Interesting_number_paradox

And if it's not remarkable, then that is in itself remarkable.

This was my thought too. If 1/89 in base 10 works, and 89 in the 10th unique term of the series

> 1,1,2,3,5,8,13,21,34,55,89,144,...

then does 1/144 work in base 11, and 1/233 in base 12?

Assuming we of course translate the base-10 number 144 to the appropriate base-11 number (121) first. I'm too bad at math to do this anymore, maybe someone else can tell us ;)

Compute the generating function f(x) = sum_n f_n x^n where f_n satisfies your favorite recursion. Compute f(b^-1) where b is your base. For fibonacci f(x) = x / (1 - x - x^2).

Here's a simple way of looking at it (and we'll build up to the matrix notation).

The nth and (n+1)st fibonacci numbers can be written as a system of two equations in the nth and (n-1)st numbers:

  F(n) = F(n)
  F(n+1) = F(n) + F(n-1)

where the base cases are F(0) = 0 and F(1) = 1.

The question is then to show that (note that the indices under F are actually wrong in the OP...):

  1/89 = .01F(2) + .001F(3) + .0001F(4) + ... + 10^(-n)F(n) + ...

One beautiful (and mathematically simple) way of analyzing this system is by the use of linear algebra. The idea is that, because F(n+1) depends linearly on F(n) and F(n-1) (see the above definition), then we can write the previous system with the following (linear-algebraic) notation

  [ F(n)   ] = [ 0 1 ] [ F(n-1) ]
  [ F(n+1) ] = [ 1 1 ] [ F(n)   ].

If we write x(n+1) as the vector (F(n), F(n+1)), i.e., first entry is F(n) and the second entry is F(n+1), and the matrix as A, then x(n+1) = Ax(n). (Note that A is the matrix whose entries are exactly the coefficients of the linear equation we gave above!)

In other words, we've reduced the problem down to the question of investigating the properties of the matrix A ! Now, the sum we were looking at, originally, can be written (in terms of x(n)) as the first entry of (note that x is a vector as we've defined it!)

  x(2) + .1x(3) + .01x(4) + ... = x(2) + .1Ax(2) + .01 A Ax(3) + ... = x(2) + (.1A)x(2) + (.1A)^2 x(3) + ...

There's a slick proof (see [0]) that, if this sequence converges, then its result is given by the first entry of

  (I - .1A)^(-1)x(2),

which is exactly what the result gives. (I have changed the normalization a little bit for convenience, but it is the same proof :)

-----

[0] If

  y + By + B^2y + ... = z converges, then we can multiply both sides by B to get

  By + B^2y + ... = Bz

But, here's the magic! Let's subtract the first equation from the second to get

  y = z - Bz = (I - B)z,

so, multiplying by the inverse of B on both sides, we get:

  y + By + B^2y + ... = z = (I - B)^(-1)y,

as required!

As a self-studyer, I'm a big fan of this book https://www.people.vcu.edu/~rhammack/BookOfProof/ .

I think it would be built into the curve or sequence that the new births replace the dying ones.

Obviously though they would have limited resources and couldn't keep growing indefinitely.

8th Fibbonacci century. (0 1 1 2 3 5 8 13)