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The Remarkable Number 1/89 (2004)

www2.math.ou.edu

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Re: The Remarkable Number 1/89 (2004)

#71

Post by sabujp »

btw, there's a typo on the page for the value of 1/89, it should be :

   .01123595505617977528
but the math does work of course, it converges :

   >>> .01 + .001 + .0002 + .00003 + .000005 + .0000008 + .00000013 +  .000000021 + 
   .0000000034 + .00000000055 + .000000000089 + .0000000000144 + .00000000000233 + 
   .000000000000377
   
   0.011235955056107002

Re: The Remarkable Number 1/89 (2004)

#72

Post by jlokier »

grahamlee wrote: post #63
forinti wrote: post #42

Earlier quoted context omitted.

Aren't most transcendental numbers unremarkable?

No number is unremarkable. Let's construct the set of all unremarkable numbers. Now, let's construct the sequence of those numbers in order. The first member of that sequence has the remarkable property that it is the smallest unremarkable number. That is remarkable, so remove it from the set. By induction, the set must be empty.

> Now, let's construct the sequence of those numbers in order. The first member of that sequence

Not happening. Sets of numbers do not always have a first in order.

Consider the set of unremarkable real numbers > 0 under the regular arithmetic ordering. Which one is first?

Re: The Remarkable Number 1/89 (2004)

#74

Post by dreamcompiler »

dylan604 wrote: post #70
dreamcompiler wrote: post #68

Earlier quoted context omitted.

I had forgotten why the number 1001 mattered in video (it's been too long since I worked with NTSC circuits), so I looked it up. It has to do with avoiding dot crawl in color analog video. https://en.wikipedia.org/wiki/Frame_rate

It also had to do with allowing the additional of the color information to not break compatibility with the existing B&W TVs in existence. Had the decided to not make 1 video signal that could be broadcast to both color and B&W TVs, they could have just broadcast color at 30fps (and man would my life had been so much easier).

Indeed. NTSC is such a kludgy hack. An incredibly clever kludgy hack, but it's still kludgy.

Re: The Remarkable Number 1/89 (2004)

#75

Post by chowned »

munificent wrote: post #58
kazinator wrote: post #47

Earlier quoted context omitted.

Since there had to be a first person who discovered that all multiples of 11 have the same parity in the respective sums of their odd and even digits, that question is obviously interesting to some historians. Due to what circumstances was that person first? What hindered others before that person? How long was the lag before use of decimal and the discovery?

It's important to remember that while technically there is a chronologically first person to discover X for all X, that doesn't imply that that person is the only person to discover X. For sufficiently obvious X, there are likely to be many independent discoverers and highlighting the chronologically first one heaps praise somewhat arbitrarily on one of them.

"History isn't perfect" is not a good reason to avoid doing history. Historians are all well aware of these facts.

Re: The Remarkable Number 1/89 (2004)

#76

Post by adenadel »

I think I found a typo in the final line (not impactful for the result, however).

11/89 should be 11/8900

This can be verified by using the following Sage code (simply go to https://sagecell.sagemath.org/ to avoid installing sage)

  b = 10

  A = Matrix([[0, 1],[1/(b**2), 1/b]])
  I = matrix.identity(2)

  show((I - A).inverse() * Matrix([[1/b**2],[1/b**3]]))

Re: The Remarkable Number 1/89 (2004)

#77

Post by adenadel »

In case anyone is interested, I ran this procedure using integer bases besides 10 and obtained the following sequence

1, 5, 11, 19, 29, 41, 55, 71, 89, ...

Searching it on the OEIS (a great resource for mathematics) gave

http://oeis.org/search?q=1%2C5%2C+11%2C+19%2C+29%2C+41%2C+55...

It turns out that these are precisely the first values of the Fibonacci polynomial n^2 - n - 1

I haven't verified this fact, but it seems like it comes from an application of the generating function of the Fibonacci numbers.

I posted my code in another comment

https://news.ycombinator.com/item?id=24933085

Re: The Remarkable Number 1/89 (2004)

#78

Post by ehsankia »

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

Re: The Remarkable Number 1/89 (2004)

#79

Post by JackFr »

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 ?

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'."

Re: The Remarkable Number 1/89 (2004)

#80

Post by NohatCoder »

chrismorgan wrote: post #34

On 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…

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.
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