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MathGroup Archive 2005

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DigitDifferences?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60677] DigitDifferences?
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Sat, 24 Sep 2005 02:55:22 -0400 (EDT)
  • Organization: The University of Western Australia
  • Sender: owner-wri-mathgroup at wolfram.com

I would like some suggestions on good implementations of the following 
idea: suppose that you have two numbers whose floating point 
representation has many common digits, say

  2`50 Sum[(-1)^(k - 1)/(2 k - 1), {k, 1, 50000}]

  1.5707863267948976192313211916397520520985833146875579625874449850433

and

  N[Pi/2, 50]

  1.5707963267948966192313216916397514420985846996875529104874722962622

Although these two numbers disagree in the 5th decimal place, they still 
have a great many common digits from that point onwards.

(By the way, if you are using OS X you can use the "Evaluate to Text" 
MathService under the Services menu to do inline Mathematica evaluations 
in your favourite newsreader, avoiding much of the cutting and pasting 
to and from Mathematica. I used that here for the numerical evaluations.)

I would like to display the differences between these numbers a la
Borwein Borwein and Dilcher (1989) in "Pi, Euler numbers and asymptotic 
expansions", Amer. Math. Monthly 96 (1989), 681-687. To get an idea, 
paste either cell below into a Notebook. These attempts were done by 
hand -- so automated code for their production would be nice -- and they 
have one unfortunate problem: the spacing between numbers is not 
uniform. Suggestions for a DigitDifferences function most welcome.

Cheers,
Paul

Here my two attempts to display the sort of formatting required -- there 
may be other better solutions:

  Cell[BoxData[
    FormBox[
      RowBox[{"1.5707", 
        UnderscriptBox["8", 
          UnderscriptBox["_", "1"]], "632679489", 
        UnderscriptBox[
          UnderscriptBox["7", "_"], 
          RowBox[{"-", "1"}]], "619231321", 
        UnderscriptBox["1", 
          UnderscriptBox["_", "5"]], "9163975", 
        UnderscriptBox[
          UnderscriptBox["205", "_"], 
          RowBox[{"-", "61"}]], "209858", 
        UnderscriptBox["3314", 
          UnderscriptBox["_", "1385"]], "6876"}], 
      TraditionalForm]], "Text"]



   Cell[BoxData[
    FormBox[
      RowBox[{"1.5707", " ", 
        UnderscriptBox["8", 
          UnderscriptBox["_", "1"]], " ", "632679489", " ", 
        OverscriptBox[
          OverscriptBox["7", "_"], "1"], " ", "619231321", " ", 
        UnderscriptBox["1", 
          UnderscriptBox["_", "5"]], " ", "9163975", " ", 
        OverscriptBox[
          OverscriptBox["205", "_"], "61"], " ", "209858", " ", 
        UnderscriptBox["3314", 
          UnderscriptBox["_", "1385"]], " ", "6876"}], 
      TraditionalForm]], "Text"]

_______________________________________________________________________
Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    
AUSTRALIA                               http://physics.uwa.edu.au/~paul


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