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1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

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Re: 1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

#12

Post by ronaldx »

neals wrote: post #3

Ok, I hope somebody has a really good logical explanation of this, or maybe even some other awesome examples?

Explanation: 0.0001+0.00000002+0.00000000004 etc = 2^0/10^4 + 2/10^8 + 2^2/10^12 etc

which is a geometric sequence with common ratio 2/10000 and first term 1/10000

So it has an infinite sum of (1/10000)/(9998/10000) = 1/9998

Same for powers of 3: 1/9997

Actually 1/8 = 0.125 is an example of this; it just breaks down very early because 4+0.8+0.16+0.032+0.0064+... = 5

Re: 1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

#16

Post by archgoon »

pja wrote: post #4
neals wrote: post #3

Ok, I hope somebody has a really good logical explanation of this, or maybe even some other awesome examples?

The fact that 10000 - 2 = 9998 probably has something to do with it. Compare and contrast the result for 1 / 99998.

Pretty much.

1/9998 = 1/(10000-2) = 1/(10000)*1/(1-2/(10000)

Since 2/10000 is very small, it is well approximated by the taylor expansion for 1/(1-x), which is simply

Sum(x^n)

Since x is 2/10000, we get powers of two, which keep getting shifted to the right. Like a bit pattern, they don't overlap when added, so we get the sequence above.

Re: 1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

#17

Post by huhtenberg » 

  1/9998 is
  1/(10000-2) is
  (1/10000) / (1 - 2/10000)
which is an infinite sum of geometric progression with an initial value of 1/10000 and ratio of 2/10000. In other words,

  x1 = 1/10000;            //  0.0001
  x2 = x1 + x1 * 2/10000;  //  0.0001 0002
  x3 = x2 + x2 * 2/10000;  //  0.0001 0002 0004 0008
  ...
Magic

  O_O
[0] http://en.wikipedia.org/wiki/Geometric_progression

Re: 1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

#19

Post by nanofortnight »

neals wrote: post #3

Ok, I hope somebody has a really good logical explanation of this, or maybe even some other awesome examples?

Simple high-school maths.

    S = 0.00010002000400080016...

    S = 0.0001 + 0.0000 0002 + 0.0000 0000 0004 + 0.0000 0000 0000 0008 + ...

    S = 2^0 / 10000^0 + 2^1 / 10000^1 + 2^2 / 10000^2 + 2^3 / 10000^3 + ...

    S = sum to infinity of (2/10000)^i
You might have noticed this is a geometric series with ratio 2/10000 = 0.0002.

    S = 0.0001 / (1 - 0.0002) = 0.0001 / 0.9998 = 1/9998

Re: 1/9998 = 0.0001 0002 0004 0008 0016 0032 0064 0128 0256..

#20

Post by rsiqueira »

crntaylor wrote: post #10

The pattern will break down once you get past 8192, which is 2^13. That means that the pattern continues for an impressive 52 significant figures (well, it actually breaks down on the 52nd digit, which will be a 3 instead of a 2). The reason it works is that 9998 = 10^4 - 2. You can expand as 1 / (10^n - 2) = 1/10^n * 1/(1 - 2/10^n) = 1/10^n * (1 + 2/10^n + 2^2 /10^2n + 2^3 /10^3n + ...) which gives the observed patt…

The pattern is not really breaking. What happens is that 16384 doesn't fit in a 4 digit space so it's first digit "1" jumps to 8192 and it becomes 8193. Then the next number (32768) add it's first digit "3" to 16384 and it becomes 16387 and so on, so the sequence appears strange after 4096: ...409681936387...
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