Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

The Remarkable Number 1/89 (2004)

www2.math.ou.edu

Post reply on HN
Print view View original

21–30 of 116 posts

Re: The Remarkable Number 1/89 (2004)

#21

Post by alisonkisk »

wizzwizz4 wrote: post #12
benrbray wrote: post #9

Earlier quoted context omitted.

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…

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

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.

Re: The Remarkable Number 1/89 (2004)

#23

Post by alisonkisk »

wumms wrote: post #11

From an archived talk (2011) on wikipedia [0]: "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.wikip…

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.

Re: The Remarkable Number 1/89 (2004)

#24

Post by DeusExMachina »

benrbray wrote: post #9
burlesona wrote: post #2

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…

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.

Re: The Remarkable Number 1/89 (2004)

#25

Post by [deleted] »

magnio wrote: post #20
efdb wrote: post #7

Earlier quoted context omitted.

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.

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 conver…

[deleted]

Re: The Remarkable Number 1/89 (2004)

#26

Post by alisonkisk »

burlesona wrote: post #2

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 ?

You might also enjoy

tan(1 degree/55555555555)

Re: The Remarkable Number 1/89 (2004)

#28

Post by stabbles »

ckemere wrote: post #18

Presumably in a different number base (base 12, eg) it would be a different number that had this reciprocal property. Would it still be in the fibonacci series in that base?

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).

Re: The Remarkable Number 1/89 (2004)

#29

Post by Cullinet »

ckemere wrote: post #17

Presumably in a different number base, the number would be different. Would it still be in the aeries?

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.

Post reply on HN