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The Remarkable Number 1/89 (2004)

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Re: The Remarkable Number 1/89 (2004)

#81

Post by danbolt »

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).

Kludgy hacks are interesting when their value highly outweighs the lack of careful design or effort put into it. Or, I work in the video games industry and quick work can end up being charming or valuable to your audiences, even if they're a difficult thing to continue developing or maintain. The wide compatibility of NTSC likely has been very valuable to the public, but this public is also unaware of the difficult work it implies.

That said, games traditionally have a point where development stops and doesn't resume (not counting from more live-ops-style games today), so the calculus of that sort of thing changes to management.

Re: The Remarkable Number 1/89 (2004)

#82

Post by justinlloyd »

I went to college in '88 or '89 and one of my teachers showed me the 1/89 trick, along with a few others, e.g. 1/7, and so forth. My teacher claimed to have been shown the various tricks by one of his teachers back in the '60s.

In the same period I "discovered" an error detection technique which is commonly known as Hamming codes. I would never dare to claim I invented them or discovered them.

In '92 and '93, I was heavily in to the BBS scene, some people had 56K modems, others had 14.4K. With verifiable evidence of written notes and digital artifacts (a BBS door and protocol for AmiBBS and Citadel and a couple of others) , I created a technique whereby multiple peers with low-bandwidth connections could transfer small fragments of a larger dataset to a peer with a lot of bandwidth.

If you were heavily in to the warez scene and/or part of Fate, it is probably you made use of this protocol to transfer pirated software between FTP sites. A warehousing server would tell each peer who had what part of a piece of data, and any peer could make requests for any piece of the data from any other peer who happened to have a copy of that data. Today, a very similar protocl is commonly known as BitTorrent.

I would not say I "discovered" or "invented" the protocol as my work was based on the various X-, Y- & Z- modem protocols. There was a TCP/IP packet to -Modem packet translator so that a BBS talking over that new fangled internet thing could take advantage of a T1 (1.5Mbps) connection for instance, which really helped with spreading the warez around the various FTP sites by the couriers.

I doubt the veracity of the claim by the author to have "discovered it as original" in 1994 before anybody else.

Re: The Remarkable Number 1/89 (2004)

#83

Post by gnulinux »

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.

That doesn't work because real numbers are not enumerable, so you cannot induce over them. That joke "proof" only works for natural numbers and goes like this:

Theorem: all natural numbers are interesting

* Base case: 0 is interesting because it is the smallest natural number, as well as the identity element of + operation.

* Inductive case: Assume the theorem holds for all m, mBy induction, we conclude all natural numbers are interesting. QED.

Re: The Remarkable Number 1/89 (2004)

#84

Post by LolWolf »

anonu wrote: post #65

This is cool. The proof is a bit over my head though. How does someone go about learning or dissecting the syntax of this proof...?

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!

Re: The Remarkable Number 1/89 (2004)

#85

Post by samatman »

jerf wrote: post #19
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 ?

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 t…

> Most of them are, in fact, utterly pointless; SELECT statements that return no values, SELECT statements that return all values (equally pointless)

There are numerous examples of such queries which are far from pointless, such as "SELECT number FROM reals WHERE number = sqrt(-1)" for the former, and "SELECT number FROM reals WHERE number = number * 1".

I would call those results interesting, insofar as they're absolutely required to do any interesting number theory.

Not to detract from your interesting post! I enjoy the conceit, and could see myself using it in conversation.

Re: The Remarkable Number 1/89 (2004)

#86

Post by marton78 »

noncoml wrote: post #69
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…

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

No. A trivial counterexample would be base 7.

Re: The Remarkable Number 1/89 (2004)

#87

Post by dan-robertson »

cleansingfire wrote: post #41
doodpants wrote: post #4

"The successive ratios of the terms, i.e. 1/1, 2/1, 3/2, 5/3 ... tend to a number called the Golden Ratio by the Greeks." Fun fact: take any two numbers (e.g. chosen randomly), and use them as the seeds for a Fibonacci-like sequence by summing the last two terms to generate the next term. The ratio of any two consecutive terms in that series will tend towards the golden ratio.

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

Well it isn’t really calculus as there’s no differentiation (I guess you could consider the last step where you take a limit to be calculus), but it’s a bit like differential equations. You can write down the recurrence relation:

  a_(n+2) = a_(n+1) + a_n
Observe that there is a linear solution space (I.e. if you add solutions point wise or multiply each value by the same scalar, you get solutions), and the values a_0 and a_1 are sufficient to determine the sequence. Now guess that a_n = k^n is a solution:

  k^2 = k + 1
  (k - 1/2)^2 = 5/4
  k = (1 +/- sqrt(5))/2
  k = φ or -1/φ, where φ is the golden ratio
Due to linearity, there are a family of solutions a_n = Rφ^n + S(-1/φ)^n for any values of R and S. Because this family provides a solution for any choices of a_0 and a_1, it contains all the solutions.

Because |1/φ|<1, we find that asymptotically a_n ~ Rφ^n as n grows. Therefore the ratio of terms tends to φ in the limit.

Re: The Remarkable Number 1/89 (2004)

#89

Post by dylan604 »

danbolt wrote: post #81
dylan604 wrote: post #70

Earlier quoted context omitted.

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).

Kludgy hacks are interesting when their value highly outweighs the lack of careful design or effort put into it. Or, I work in the video games industry and quick work can end up being charming or valuable to your audiences, even if they're a difficult thing to continue developing or maintain. The wide compatibility of NTSC likely has been very valuable to the public, but this public is also unaware of the difficult w…

This is especially true to the people building the ROM emulators. There are so many tricks/hacks/kludges that were at the heart of some games even being usable. Timing for interlacing that was needed for the game to work in NTSC has to be accounted for on today's faster hardware and progressive scanning. Reproducing color accurately from NTSC seems to also be another thing I've seen. I'm sure there are plenty more that have been posted here before, but they are always a fun reminder that porting code can be a nightmare.

Re: The Remarkable Number 1/89 (2004)

#90

Post by Stupulous »

zekrioca wrote: post #50
alisonkisk wrote: post #23

Earlier quoted context omitted.

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.

Then it is also silly to (erroneously) mention who discovered such a property, don't you think?

The proof follows from this instance of the discovery, with an Oklahoma U student indepently discovering and an OU professor indepently proving. I think that's a perfectly reasonable way to track what happened.

I am unable to open the pdf or follow any of the links from the wiki talk page, so the proof may predate this instance, which would invalidate my point.

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