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1/999999999999999999999998999999999999999999999999 (futilitycloset.com)
446 points by bemmu on July 1, 2015 | hide | past | favorite | 84 comments



I feel like this would make infinitely more sense to me if not for that singular "8" hiding close to the middle. Can anyone ELI5 what's going on here?


Short answer: the denominator is (10^48 - 10^24 - 1).

Long answer follows.

It's actually easier to understand if you work backwards and arrive at the expression yourself, by asking yourself: "If I wanted the number that starts like 0.0...000 0...001 0...001 0...002 0...003 0...005 0...008 ... (with each block being 24 digits long), how would I express that number?"

Well, calling the Fibonacci numbers f_n (with f_1=0), that decimal expansion you want is sum (f_n 10^(-24n)) over n≥1. This is sum (f_n x^n) evaluated at x = 10^(-24). It is easy to work out (especially if you know the trick already) that sum (f_n x^n) = x^2 / (1 - x - x^2), which at x = 10^(-24) gives that the number we want is 1/(10^48 - 10^24 - 1), which is exactly what 1/999999999999999999999998999999999999999999999999 is.

This may be more interesting if you consider more examples:

* if we want the number 0 . 0001 0002 0004 0008 0016 0032 0064 0128..., then it is sum (2^(n-1) 10000^(-n)). We can calculate that sum(2^(n-1) x^n) is x/(1-2x), which at x = 1/10000 becomes 1/9998. And indeed 1/9998 = 0.0001000200040008001600320064012802560512102420484096...

Similarly,

* if we want the number 0 . 000 001 002 003 004 005 ..., then it is sum ((n-1) 1000^(-n)). And sum ((n-1) x^n) is x^2/(1-x)^2, and putting x = 1/1000 in it gives 1/999^2 = 1/998001, and indeed 1/998001 = 0.000001002003004005006007008009010011012013014015016...

Basically whenever sum (a_n x^n) has a nice form (aka the generating function of the sequence), you can plug in x = 1/(some power of 10) and get such pretty decimal expansions.

(Edit: Formatting, and the "short answer" at the top.)


I recommend generatingfunctionology by herbert wilf for the interested.

The beautiful idea here is to consider formal power series (encoding sequences of values in polynomials without demanding convergence) https://en.wikipedia.org/wiki/Formal_power_series


Is any of this useful for compression?


This is a great response. Thanks for taking the time to write this.


are you a master of math?


The generating function for Fibonacci numbers is

    x/(1-x-x^2) = \sum_{n\geq0} F_n x^n,
so you just let x = 1e-24. In this case it's x^2/(1-x-x^2) to make the fraction come out with numerator 1.


You can see the pattern with smaller fractions too:

1/89 = 0.0 1 1 2 3 5...

1/9899 = 0.00 01 01 02 03 05 08 13 21 34 55...

1/998999 = 0.001 002 003 005 008 013 021 034 055 089...


> You can see the pattern with smaller fractions too:

So the blog wasted a ton of space by using an unnecessarily large example! How dumb.

Maybe this submission should be converted to a "Tell HN:", featuring your comment as the content.


There it is. Clear.


I may be wrong, but I believe it's some sort of math. Or wizardry.


This explains it well.

http://www.quora.com/Which-is-the-Fibonacci-inverse-number


Wouldn't that have been a better title than that stupid number?


Well, the title piqued my curiosity and made me click on it to see what it was about.


Just like Buzzfeed.


If it were Buzzfeed the title would have been something like "You'll never believe what happened when she divided one by 999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999!!!!!1"


[flagged]


Yes, and hurray for ad hominems, right? Why discredit the "eli5", just because it's from Reddit? What if it's a useful method to convey what you mean? We just can't use it now, because it's from Reddit?


The concept you're looking for is the https://en.wikipedia.org/wiki/Genetic_fallacy, a broader form of ad hominem.

Also, they didn't say it's bad because it's from reddit, the statement could be interpreted as saying that it's bad, and also that it's from reddit. Imagine http://laurencetennant.com/bonds/adhominem.html applied to the genetic fallacy, and you'll see what I mean.


Some of us here do not use reddit much, so using slang specific to that site can be confusing. In this case, I don't think that saying "eli5" instead "explain" is worth risking this potential for confusion, as only 3 letters are saved.


"Eli5" is 14 characters shorter than "explain like I'm 5".


Yes, except that's not actually what people mean when they type those letters. A five year old has almost zero understanding of math. Where would you even begin?

If you want a simple explanation, just say so. There's really no need to use internet memes for basic communication.


Wow. Based on the responses, you'd think I wrote a style guide that said to use eli5 whenever possible. If you don't like it, don't use it. I don't. I was only pointing out that the abbreviation neither stands for just "explain" nor saves a mere three letters.


I always figured it was supposed to be a single word that started with "eli" followed by 5 letters, in the manner of i18n.

The only English words I could find matching that pattern are elicited, elicitor, elidible, eligible, eligibly, elisions, elitisms and elitists - none of which seem apt, although "elicited" is tantalisingly close. From context I always read it as "elucidate" (elu6?)

So why isn't it Elif?


Because that's what some languages use for else if?

Probably not the real reason...


"explain like I'm 5" is almost certain to be understood regardless of the reader. "Eli5" is unintelligible unless you are part of a particular, narrow, culture. I had no idea what it meant.

(In truth, the "like I'm 5" part is not obvious so could be misunderstood, but is easier to grasp the meaning.)


Ha, I first read the 5 and a capital s.


I like where your head is at but I'm not sure this is an ad hominem. My understanding is that an ad hominem is an attack on a persons character[1]

[1]https://en.wikipedia.org/wiki/Ad_hominem


Thanks. The ad hominem is a counter-argument; you're discrediting an argument based on a person's character. So "jack is an idiot" is not an ad hominem; "that can't be true, because jack says it, and he's an idiot" is.

I thought the guy meant: "I don't like it because it's from reddit."

But ikeboy has it right: in this case, because reddit is not a person, it's a genetic fallacy, not an ad hominem. Which I didn't even know existed. And, of course, he didn't say that it was because of reddit, so it wasn't even that :)


I'm all for not letting reddit jargon get a foothold here. Reddit culture is downright unlikeable since the entire reason that site seems to exist is to foster echo chambers and cliquish behavior. I'd prefer to keep any vestige of that out of HN. This site is a rare jewel in the sewer of the Internet and it's better that way.


I completely agree with you. We should stand together in the face of such behavior.


yeah, let's not become like those damn proles from reddit. /sarcasm


Yeah because there's no echo chambers and cliquish behavior on HN /s


> Why discredit the "eli5", just because it's from Reddit?

Redditit is childish, racist, sexist, ignorant and a hive mind.

eli5 like most things on reddit includes this. There are a lot of under currents in the term.

It shouts it's ok to be 'proud' of being ignorant on a topic.

Rather than the adult, it's ok not to know, just ask someone for a simple explanation. Adults should know you don't need a meme to hide behind to say I'm not good at this.

I'd be very surprised if the majority of people on HN don't also Reddit for our fix of what humans are at their base level but the point is HN is supposed to be better than all that.


>Redditit is childish, racist, sexist, ignorant and a hive mind.

Some people on Reddit may be that way, but that doesn't mean everyone is. Some people everywhere are that way, still doesn't mean everyone is. Try not to be so hateful.


What the hell does eli5 mean? Actually never mind, I don't want to know. I'll just downvote and move on.


Well I'll tell you that it doesn't mean "act like I'm five".


That is every bit as cliquish as whatever it is you think you're rebelling against, if not more. Probably more.


"explain like i'm five."


HN, where a11y and i10n are perfectly comprehensible but eli5 freaks people out.


"Divide the number 999,999,999,999,999,999,999,998,999,999,999,999,999,999,999,999 into 1"

Isn't this poor wording? The author is dividing 1 by the large number above, not the other way around.


"Divide b into a" means the same as "divide a by b".


Which I why I hate that wording, because that means "divide b into a" is equivalent to "divide a into b parts". And that's just confusing.


It feels to me as though far too much of any given field uses redundant (the same concept expressed lots of different ways) or misleading (something made to sound far more complicated than it actually is) terminology and causes unnecessary amounts of confusion among anyone trying to learn it.


But it loses its English meaning for a=1. Whereas, "divide ten into two" would make sense.


I'm with you on the wording. The word "into" is ambiguous on its own and even more ambiguous used with "divide." There's a discussion of that here: http://mathforum.org/library/drmath/view/52296.html

As an example, a person can easily read "divide 24 into 6" as "divide 24 into 6 [parts]," i.e. 24/x = 6, x = 4 — as opposed to 6/24 = x, x = 0.25. This use is especially reinforced by everyday experiences like dividing a cake into eight slices.


I've seen it both ways- "Divide four by two" is equivalent to "Divide two into four" in my experience.


Really? I've always read "divide by" and "divide into" as meaning the exact same thing:

"Divide four by two" = 4/2 = 2

"Divide two into four" = 2/4 = 1/2

Could this be a US/UK English difference?


I think this phrasing is less confusing when it refers to labels rather than actual numerals. For instance 'divide the price into the exchange rate', which would mean the same as 'divide the exchange rate by the price'.

Why would you use this phrasing? I suppose it can flow more easily in some cases, for instance 'first work out X, then work out Y, then divide them both into Z and see which one is larger' is less of a mouthful than the alternative.


It comes from "how many times does 2 go into 4", which is "4/2".


If you go to a pizza restaurant and ask them to "Divide two (large pizzas) into four (portions)" they would indeed give each person 1/2 pizza. So for colloquial use you are correct. Which makes the math definition confusing.


Perhaps it is some kind of regional difference. I've always lived on the US West Coast, and I've heard "divide four by two" expressed as "divide two into four" many times - and never the other way around.

I never use the "into" phrasing myself because it is confusing to me. But it does make some kind of sense if you think about it this way: "How many twos can you put into four?"


To me, "divide A into B" sounds like a some sort of anachronism that you might find in a 17th century one-room schoolhouse, while "divide B by A" sounds perfectly natural.

I went to school on the US East Coast, so you might be onto something here.


"Divide two into four" to me means that I am dividing two into four parts, which would equal one half.


"Divide the cake into two pieces" does not mean I'm giving you another cake.


x into y is y/x


This would be fun response to the usual interview white boarding question.


> This would be fun response to the usual interview white boarding question.

i don't really understand what you mean here. may you please elaborate ? thanks !


A common warm up question in a coding interview is to write (on the whiteboard) a program that prints the Fibonacci numbers in sequence. Arguably, you can write this fraction and claim this answers the question, though I don't know any programming language where this would actually work without a bunch of more code around it.

Edit: other comments have examples in Python, Clojure and bc.


There's no way I'd get a job if they asked me to express the Fibonacci sequence in a series of 24 digit blah blah blah as a fraction.


Python can perform a similar calculation using large integers:

  10 ** (24 * 116) / 999999999999999999999998999999999999999999999999
Very cool math.


Interesting. Something I didn't realize: for n > 4, F(n) = F(n-2) * 3 - F(n-4)

fibonacci sequence is defined as: F(n) = F(n-1) + F(n-2)

substitute F(n-1) with F(n-2) + F(n-3): F(n) = F(n-2) + F(n-3) + F(n-2)

substitute F(n-3) with F(n-4) + F(n-5): F(n) = F(n-2) + F(n-5) + F(n-4) + F(n-2)

substitute F(n-4) with F(n-2) - F(n-3): F(n) = F(n-2) + F(n-5) + F(n-2) - F(n-3) + F(n-2)

simplify: F(n) = 3 * F(n-2) + F(n-5) - F(n-3)

because F(n-3) = F(n-4) + F(n-5), -F(n-4) = F(n-5)) - F(n-3):

F(n) = 3 * F(n-2) - F(n-4)


IT WORKS!

  Copyright 1991-1994, 1997, 1998, 2000, 2004, 2006 Free Software Foundation, Inc.
  This is free software with ABSOLUTELY NO WARRANTY.
  For details type `warranty'. 
  scale=24*20
   
  1/999999999999999999999998999999999999999999999999
  .0000000000000000000000000000000000000000000000010000000000000000000\
  00001000000000000000000000002000000000000000000000003000000000000000\
  00000000500000000000000000000000800000000000000000000001300000000000\
  00000000000210000000000000000000000340000000000000000000000550000000\
  00000000000000089000000000000000000000144000000000000000000000233000\
  00000000000000000037700000000000000000000061000000000000000000000098\
  70000000000000000000015970000000000000000000025840000000000000000000\
  04181
Don't know how.


> echo 'scale=500-1;10/(10^20-10^10-1)' | BC_LINE_LENGTH=12 bc

will give you the ten digit ones, with the last digit rounded up on the very end.

I havent quite got an exact formula for scale before the rounding is wrong, e.g. for 100 digits:

> echo 'scale=50000-2000-1;10/(10^200-10^100-1)' | BC_LINE_LENGTH=102 bc


Why not just use

     scale=499
instead of

     scale=500-1
? I'm curious.


Originally in my terminal I was experimenting to find a cutoff for the point where the numbers become inaccurate. All of those formulas were in terms of digits and had a -1 on the end.


Very cool. https://gist.github.com/clifton/97f8a862a590c15e1785


not nearly as cool, but in 8th grade I enjoyed:

  987654312 / 123456789 = 8


> 987654321 / 123456789 = 8

FTFY


No, the original division gives 8 exactly, while yours gives 8.000000072900001. (There's no way an odd number can be a multiple of 8.)


The sequence popping out of 'nowehere' like this reminds me of a upenn cis194 Haskell homework*

Fib sequence can be expressed as:

F(x) = x / (1 − x − x^2)

where x is an infinite stream of 0,1,0,0,0....

and with sane definitions of divide, add, multiply, subtract for these streams.

* http://www.seas.upenn.edu/~cis194/spring13/hw/06-laziness.pd.... Ex 6


Maybe a dumb question...so I wrote out the derivation for the generating function and am of course getting x/1-x-x^2. Is the idea just that multiplying through by x just shifts the generating function so you end up with the generating function for f(n)x^(n+1)? Which still gives the resulting decimal expansion?


Tried it in Windows cmd:

> C:\Users\X>set /a result=1/2147483647

> 0

> C:\Users\X>set /a result=1/2147483648

> Invalid number. Numbers are limited to 32-bits of precision.


mostly works in wolfram: http://www.wolframalpha.com/input/?i=1%2F9999999999999999999...


mathematica: http://pastebin.com/GfNG8bLU


It seems like day by day the stories being posted on HN more and more resembles something like reddit, and by that I mean stories like this are just so minimal in value and more geared towards self promotion


Did you just come from Reddit to tell us that?


I feel like if I posted this to facebook, I would lose at least 1 friend.


a friend not worth having!


Awesome!


What is going on with the quality of the comments in this thread?


Reddit is extremely unpopular here; mentioning it is a good way to start a flamewar.

Can't say I've seen many flamewars about HN on reddit though.


SIGFPE: floating-point exception

:-)


while not city specific we have the Silver Comet trail here in Georgia, follows old rail road tracks

http://www.silvercometga.com/


You've got the wrong thread, I'm afraid.




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