text: # 1/109, Fibonacci, and Fractional Part
  
  Posted 2 months ago
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   I saw the following image and thought I'd check it.Turns out this only works if you use FractionalPart. Both of the following give the same output.  N[1/109, 106] N[FractionalPart[Total[Table[Fibonacci[n] 1/10^(109-n), {n,1,109}]]], 106] 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211 The part discarded from the sum is 28699473517541069311761.
   Going off the tweet linked by Shenghui in this post, here's a quick investigation of what's actually happening.As I noted previously, here is how to compute the period of a repeating decimal like $1/109$: MultiplicativeOrder[10, 109] 108 To confirm this result, let's use RealDigits[]: {{rd}, exp} = RealDigits[1/109] {{{9, 1, 7, 4, 3, 1, 1, 9, 2, 6, 6, 0, 5, 5, 0, 4, 5, 8, 7, 1, 5, 5, 9, 6, 3, 3, 0, 2, 7, 5, 2, 2, 9, 3, 5, 7, 7, 9, 8, 1, 6, 5, 1, 3, 7, 6, 1, 4, 6, 7, 8, 8, 9, 9, 0, 8, 2, 5, 6, 8, 8, 0, 7, 3, 3, 9, 4, 4, 9, 5, 4, 1, 2, 8, 4, 4, 0, 3, 6, 6, 9, 7, 2, 4, 7, 7, 0, 6, 4, 2, 2, 0, 1, 8, 3, 4, 8, 6, 2, 3, 8, 5, 3, 2, 1, 1, 0, 0}}, -2} Length[rd] 108 By convention, RealDigits starts with the first non-zero digit, so something closer to a more natural representation would be something like FromDigits[{{RotateRight[rd, 2]}, 0}] 1/109 Now let's do something similar to Ed's calculation, but now explicitly generating the Fibonacci digits in their expected places after the decimal point: tot = Sum[FromDigits[{{PadLeft[ArrayPad[IntegerDigits[Fibonacci[k]], k - 1], 108]}, 0}], {k, 109}] 8009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211/999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 Now that's a funny-looking rational. Look at the integer part: IntegerPart[tot] 8 and the fractional part: FractionalPart[tot] 9174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853219/999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 and the last one isn't quite the expected $1/109$. However, diff = % - 1/109 8/999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 which is a very tiny difference indeed: N[diff] 8. * 10^-108 RealDigits[diff] {{{8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}, -107} ...and this is one way to use Mathematica to show that the tweet was almost, but not quite, right.
explanation: The document is a detailed exploration of a mathematical observation involving the fraction 1/109 and the Fibonacci sequence. It includes code snippets and explanations to verify the relationship between the decimal expansion of 1/109 and a sum involving Fibonacci numbers. The document does not contain a specific question but rather presents a series of calculations and observations.
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distilabel_metadata.content: # 1/109, Fibonacci, and Fractional Part
  
  Posted 2 months ago
  574 Views
  |
  |
  6 Total Likes
  |
   I saw the following image and thought I'd check it.Turns out this only works if you use FractionalPart. Both of the following give the same output.  N[1/109, 106] N[FractionalPart[Total[Table[Fibonacci[n] 1/10^(109-n), {n,1,109}]]], 106] 0.009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211 The part discarded from the sum is 28699473517541069311761.
   Going off the tweet linked by Shenghui in this post, here's a quick investigation of what's actually happening.As I noted previously, here is how to compute the period of a repeating decimal like $1/109$: MultiplicativeOrder[10, 109] 108 To confirm this result, let's use RealDigits[]: {{rd}, exp} = RealDigits[1/109] {{{9, 1, 7, 4, 3, 1, 1, 9, 2, 6, 6, 0, 5, 5, 0, 4, 5, 8, 7, 1, 5, 5, 9, 6, 3, 3, 0, 2, 7, 5, 2, 2, 9, 3, 5, 7, 7, 9, 8, 1, 6, 5, 1, 3, 7, 6, 1, 4, 6, 7, 8, 8, 9, 9, 0, 8, 2, 5, 6, 8, 8, 0, 7, 3, 3, 9, 4, 4, 9, 5, 4, 1, 2, 8, 4, 4, 0, 3, 6, 6, 9, 7, 2, 4, 7, 7, 0, 6, 4, 2, 2, 0, 1, 8, 3, 4, 8, 6, 2, 3, 8, 5, 3, 2, 1, 1, 0, 0}}, -2} Length[rd] 108 By convention, RealDigits starts with the first non-zero digit, so something closer to a more natural representation would be something like FromDigits[{{RotateRight[rd, 2]}, 0}] 1/109 Now let's do something similar to Ed's calculation, but now explicitly generating the Fibonacci digits in their expected places after the decimal point: tot = Sum[FromDigits[{{PadLeft[ArrayPad[IntegerDigits[Fibonacci[k]], k - 1], 108]}, 0}], {k, 109}] 8009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211/999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 Now that's a funny-looking rational. Look at the integer part: IntegerPart[tot] 8 and the fractional part: FractionalPart[tot] 9174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853219/999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 and the last one isn't quite the expected $1/109$. However, diff = % - 1/109 8/999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 which is a very tiny difference indeed: N[diff] 8. * 10^-108 RealDigits[diff] {{{8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}, -107} ...and this is one way to use Mathematica to show that the tweet was almost, but not quite, right.
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distilabel_metadata.raw_output_q_a_detector_0: The document is a detailed exploration of a mathematical observation involving the fraction 1/109 and the Fibonacci sequence. It includes code snippets and explanations to verify the relationship between the decimal expansion of 1/109 and a sum involving Fibonacci numbers. The document does not contain a specific question but rather presents a series of calculations and observations.
  
  Label: no-question