from: don.mcdonald@welcom.gen.nz
to: jasonp@Glue.umd.edu (Jason Stratos Papadopoulos)#    #
Newsgroups: comp.lang.c,sci.math#    #
Subject: here's a program to compute pi, version 1.01#    #

Date: 28 Apr 1997 03:59:09 GMT#    #
Organization: University of Maryland, College Park#    #


# Have fun, and please send me ideas or improvements for the next version,
# which hopefully will be faster.
# 
# jasonp
# 
# PS: I'm putting together a pile of public domain pi calculators for
# uploading to the SimTel collection, so send me your favorite if you'd 
# like it included (I already have six or seven!).

    *IWORD. Don91/4ds.ILIKEPI, 12.09.89, 19.4.91, 
    17.5, 15.7, 2235  (1622 words). 
         25.4.92 (1647 words.), 22.5.92, 2415, 
    (1676 words.) 10/2/94, 1650- (1821 words.) 
     
    Calculation of Pi to 15 decimal 
    places accuracy by circle arc series. 
          By DON S McDonald, 
     
    F. 63/ 5 Hut. chi. son Rd, Wellington 2
    New Zealand. Phone 0-4- 389-6820 
     
    Abstract.  
     
     The author describes his methods for 
    calculating PI to 15 plus decimal places on 
    BBC-B home microcomputer.  In particular, the 
    infinite binomial series expansion for the 
    length of a circular arc.  Listing of Basic 2 
    program, "PI3", of 40 lines.  Calculates pi 
    using + * ^ and / of real numbers in 8 
    seconds.  Long division algorithm.  Rational 
    approximations to pi.  Mnemonics. 
     
     
    Introduction. 
     
      The decimal expansion of Pi (transcendental 
    ratio of circle circumference to diameter) 
    was "verified to 500 decimal places as early 
    as 1854, ... without inspiration of 
    computers." (Science journal, vol. 193.)  
    Would that our BBC micro- computers could 
    emulate this feat.  
      To kick off this topic I have calculated pi 
    (3.14159...) by two methods, which I did not 
    copy knowingly from any person. My methods 
    were :  
     
    1/ Area of an octant of a circle divided into 
    1024 strips by Simpson's 1/3rd rule (possibly 
    the subject of a subsequent article), and ... 
     
    2/ length of a circular arc (a better 
    method), described here. 
     
      On 19.4.91 I found out that a similar 
    series to mine, obtained by different 
    principles, was published by Newton, 
    (posthumously, 1737.)  "History books favour 
    (my) method to Newton's original work." 
     
    Circle arcs. 
     
                                    Page 1
    Archimedes' famous result for pi: 
         3 10/71  <  pi  <  3 1/7, 
    was calculated for regular polygons of 96 
    sides.  I have calculated the curved length 
    of a 30 degree unit circle arc by the 
    differential - integral calculus and set it 
    equal to the series sum, pi/6.  Hence, pi = 
    6*length of arc.  The calculation of the 
    length of a curve is known as 'Rectification 
    of the Circle'.  (Archimedes.  Contrast 
    'Squaring the circle', circle squaring, which 
    is impossible.)  
    Luke 3.5 "Straighten the curves!  Smooth out 
    the ruts!" The Living Bible. 
     
     
    Mathematics. 
     
    Let the equation for unit circle be: 
         x^2 + (y-1)^2 = 1. 
     
    Differentiate:  2x.dx + 2(y-1).dy = 0. 
      refer paragraph (80 columns) at bottom of 
    text. ............. 
     
     
    Basic machine arithmetic. 
     
    I have done multiplication of 2 positive 
    integers to 510 decimal digits, previously.  
    The algorithm is, or was, learned in primary 
    school. 
     
    Example.  1 A.U. (Astronomical Unit) is the 
    mean distance of Earth from the Sun, say 93 
    million miles.  Calculate 93E6 ^ 64. 
     
    Answer.  9.6141E509  
    <= 10 ^ (8*64) * EXP(-.07*64). 
    Compare,  Pi^6 = 961.39,  pi^3 = 31. 
     
     
      In program PI3 I required the more 
    difficult long division calculation of 
    dividing a real by a real to 20 decimal 
    places.  Thus to extend the value of the BBC 
    PI function by 6 additional decimal digits 
    accuracy, (one million times more accurate.) 
     
    I calculated within the constraint that BBC 
    Basic 2 reals have exact accuracy, similar to 
    4-byte integers (1 part in 2^31), provided 
    that they are at all times the product (less 
    than 2^31) of odd primes, multiplied by a 
    positive or negative integer power of 2, to 
    2^127. 
     
                                    Page 2
      Rationals (recurring decimals) can be 
    represented by decimal / base 100 / or base 
    1000 sequences to any number of places, (e.g. 
    3.141,592,653,...), provided that at all 
    times the numerator and denominator have 
    exact accuracy. 
     
    When the numerator and denominator grow 'too 
    big', then they may be reduced to smaller 
    integers, for a little while only, by taking 
    out any common factor, (The programmer must 
    anticipate this situation to prevent 
    overflow.) 
     
    In program PI3, loss of exact accuracy 
    occurred from the 12th term onwards, when the 
    numerator was going to be greater than 2^31 
    <> 2E9, in spite of repeatedly removing a 
    factor, 2835.  The 12th and subsequent terms 
    should be accurate to about 9 significant 
    digits. 
     
    It is feasible to make approximate allowance 
    for termination of the infinite series after 
    a finite number of terms.  As above, the 
    remainder can be bounded, above and below, by 
    the sums of 2 geometric series, a/(1-r). 
     
     
    Results. 
     
    The program output, Table 1, shows that the 
    circle arc calculation of pi converges to pi 
    (15 decimal places) after 25 terms, (using 
    the long division algorithm method.) 
     
      My series of positive terms converges too 
    slowly to allow a long decimal expansion of 
    pi, although "incomparably more quickly than 
    the Gregory-Leibniz series;  
     
    " pi/4 = 1-1/3+1/5-1/7+...", (Beckmann).   
    It is difficult to do as well as the best 
    known methods. 
     
     
     ....  (insert Table 1.) ..... 
     
     
    Rational approximations to Pi. 
     
    Some rational approximations for pi are: 
    PI=(16/9)^2,  PI^3 = 31,  PI^4 = 2143/22,  or 
    PI=355/113.  These results are obtainable by 
    the use of the  Euclidean algorithm, 
    described in BEEBLET, October 1987: 
    "Continued Fractions, a code for scrambling 
                                    Page 3
    your PIN number.", by D S McDonald. 
     
    Three of my own rational approximations are: 
    pi^34 = 8E16,  pi = 88^2/(44^2+23^2), 
     and pi = sqrt(10) = 3 + 53353/328776 
     = (79/25) + (2.7.17.29/3.101.10001.) 
     
     
    Mnemonics. 
     
    What is the true value of pi to 30 (or 402) 
    decimal places, etc.? 
     
    John Harper, Reader in Mathematics, (now 
    Professor of Applied Mathematics) Victoria 
    University of Wellington, replied 7/8/89, 
    sending a photocopy of "Circle Digits: A 
    Self-Referential Story" by Michael Keith, 
    Mathematical Intelligencer 8(3)56-7, 1986.  
     
     The full text of "Circle digits" is a 
    mnemonic for the first 402 decimals of the 
    number pi: at decimal 762 we encounter the 
    amazing sequence 999 999 8.  Two other 
    mnemonics are a French mathematical poem, 
    which gives 30 decimal places, and a sentence 
    from Everymans Encyclopedia (1978), 14 
    decimal places. 
     
     
    Other methods. 
     
    Dr Harper suggests that the best algorithm if 
    you want a few digits of pi is:   
     16ATAN(1/5) - 4ATAN(1/239);     where 
    ATAN(x) = x-(1/3)x^3 +(1/5)x^5 -(1/7)x^7 + .. 
     
    ...The best if you want a lot of digits is 
    complicated: see Scientific American, FEB 
    1988, for the details. (John Harper.)  
     
    Pi has now been calculated to over 1 billion 
    (2E9) decimal digits by Gregory and David 
    Chudnovsky at IBM's Thomas J Watson Research 
    Centre, The Dominion, 31.8.89, page 4, 
    (exactly 1,011,196,691 digits). 
     
    James Lawry has written an interesting short 
    article about calculating PI, "As easy as 
    pi", in Microbytes (insert to Bits and Bytes 
    magazine), May 1988, page 6. 

     
                                    Page 4
    Books. 
     
    Philip Davis, "3.1416 and all that" (title 
    essay) (Birkhauser, Boston 1985) 
     
    Petr Beckmann, "A history of 'pi'", Golem 
    Press, Boulder, Colorado, 1977 (4th edn.) 
     
    David Wells, "Penguin Dictionary of Curious 
    and Interesting Numbers", 2 edn., 1987. 
     
     Van Nostrand's Scientific Encyclopedia, 6-th 
    edn 1983. 
    "Interesting background information on 
    efforts made to calculate the value of pi out 
    to hundreds of decimal places is given in 
    Science, v.193, p.836(1976)", 
     
    S.J. Smith, R.M. Cornwell, et al, 
    "Mathematics: a study in pattern", Vol. 4 
    (1980) The Jacaranda Press, New Zealand, 
    Frontispiece: A History of Pi (NZ 6 Form 
    text.) 
    --------------------------------- 
     
    Three Pi mnemonics. 
     
    QUE J'AIME A FAIRE APPRENDRE  
    FOR A TIME I STOOD PONDERING  
    HOW I WANT A DRINK,ALCOHOLIC  
    3 . 1 4    1  5     9         
     
    UN NOMBRE UTILE AUX SAGES  
    ON CIRCLE SIZES.THE LARGE  
    OF COURSE,AFTER THE HEAVY  
    2   6      5     3   5  
     
    IMMORTEL ARCHIMEDE ARTISTE INGENIEUR  
    COMPUTER MAINFRAME QUIETLY PROCESSED  
    LECTURES INVOLVING QUANTUM MECHANICS.  
    8         9          7       9  
     
    QUI DE TON JUGEMENT  
    ALL OF ITS ASSEMBLY  
    3    2  3    8       
     
    PEUT  PRISER LA VALEUR  
    CODE. INSIDE MY ENTIRE  
      4   6      2  6  
     
    POUR MOI TON PROBLEME  
    HOPE LAY FOR FIGURING  
    4    3   3   8        
     
    EUT DE PAREILS AVANTAGES.  
    OUT AN ELUSIVE EXPANSION. VALUE: PI   
    3   2   7       9           5  0  2 + 
                                    Page 5
     -------------------------------------- 
     
    rem  Spooled program listing.  "SPITB1". 
    ====== ooooooooo ========== ooooooo 
     
 
Mathematics. 
 
Let the equation for unit circle be: 
         x^2 + (y-1)^2 = 1. 
Differentiate:  2x.dx + 2(y-1).dy = 0. 
 
Then, allowing the use of "%" symbol as substitute for integration symbol:
 
arc length, s= Integral (1+(dy/dx)^2)^(1/2)dx 
             = Integral     (1-x^2)^(-1/2)dx 
               (x=0 to .5) 
 
= Integral (1 +1/2x^2  + 1/2.3/2 x^4 + 1/2.3/2.5/2 x^6 + ...)dx 
                          1 .2          1 . 2 . 3 
 
%(Infinite binomial series expansion of positive terms) = arc sin x 
<=  Integral (1 + x^2 + x^4      +  x^6   + ...)dx 
=    [(x + x^3/3 +x^5/5   +  x^7/7 + ...) ] x=0 to .5 
<=  Sum geometric series with ratio, r= x^2. 
 
Converges for 0<= x^2 < 1.  In particular, for 0 <=x<=.5. 
 
Having proved that the integral converges,
we may calculate the series for the
arc, x= 0 to .5.  After a very few lines, we obtain: 
 
PI/6 = .5*(1+1/(3*8)+3/(2*5*8^2) + 15/(6*7*8^3) +105/(24*9*8^4)+...) 
 
PI       = 2009/640          +3*( 15/(6*7*8^3) +105/(24*9*8^4)+...) ) 
         =3.1390625          +3*8.433845298E-4  (fortuitous accuracy) 
         + .0025301535894   (mult x 3, by hand)  
 
         =3.1415926535894,  (add, by hand) 
         error = -4E-13.  
 
------------------------------------------- 
 
References. 
========== 
 
Luke 3:4,5,  Prepare a road for the Lord to travel on! ... Straighten the 
curves!  Smooth out the ruts!.  (The Living Bible.) 
 
Peterson, Ivars.  Islands of truth : a mathematical
mystery cruise.  (1990) 325
pp, includes bibliographical references.  W H Freeman. 
 
Don S  McDonald, Advert, rectify 128-point circle
 (no. 1,215,306,625 is a sum
of 2 squares in 64 ways.), NZ Science Monthly,
Bulletin Board, Dec/Jan 1993/94.
 
Preston Richard, The mountains of Pi :
Profiles (David and Gregory Chudnovsky),
in The New Yorker magazine, 2 March 1992, pp. 36-67.
 (Very interesting, almost
book length.) 
                                    Page 6
 
Waters, Tom. Mathematics 1989 : Pi in the sky.
In Discover journal, January
1990, pp 76-77. 
 
If you desire further references, I recommend
'Readers Guide Abstracts' on CD
ROM (Computer) at your local library. 
 
 
Table 1. 
 
Circular arc series calculation of Pi to 15 dec. pl.  
 
N |2N-1 |NUM  |  DENOM | FACT       | DECIMAL = 3*NUM/(DENOM*FACT*8^N)  
         prod(2N-1), (2N+1) 
 
1   1   1            3   1                 12, 50, 00, 00, 00, 00  
2   3   3            5   2                 01, 40, 62, 50, 00, 00  
3   5   15           7   6                 00, 20, 92, 63, 39, 28.5714286  
4   7   105          9   24                00, 03, 56, 03, 84, 11.4583333  
5   9   945          11  120               00, 00, 65, 54, 34, 34.8366477  
6   11  10395        13  720               00, 00, 12, 70, 95, 44.2551833  
7   13  135135       15  5040              00, 00, 02, 55, 70, 39.261171  
8   15  2027025      17  40320             00, 00, 00, 52, 87, 99.4793946  
DIVIDE NUM, F, BY FACTOR 2835  
9   17  12155        19  128               00, 00, 00, 11, 17, 12.7555579  
10  19  230945       21  1280              00, 00, 00, 02, 40, 04.9433027  
11  21  4849845      23  14080             00, 00, 00, 00, 52, 30.3260869  
DIVIDE NUM, F, BY FACTOR 2835  
  loss of exact accuracy                                                ?? 
12  23  39346.18518  25  59.5978836        00, 00, 00, 00, 11, 52.8510413  
13  25  983654.6296  27  774.7724867       00, 00, 00, 00, 02, 56.5996798  
.... 
24  47  810236.4548  49  0.4215355705      00, 00, 00, 00, 00, 00.0000249  
DIVIDE NUM, F, BY FACTOR 2835  
25  49  14004.08687  51  3.717244889E-3    00, 00, 00, 00, 00, 00.0000059  
TOTAL ===============================                          89.79346156, 
                                                                      ????? 
                                                           35,  
                                                       65,  
                                                   92,  
                                               15,  
                                           14,  
                                        3  
TRUE VALUE OF PI =                       _____________________________  
                                        3. 14, 15, 92, 65, 35, 89,793238+  
1022 CENTISEC  
PROGRAM  PI3   E N D. rem 16th - 20th dec. places subject to errors. 
--------------------------------------------- 
 
                                    Page 7

From    : JASONP@GLUE.UMD.EDU                   Number    : 207,655 of 207,664
To      : DON MCDONALD                          Date      : 1/05/97 06:47
Subject : Re: PI314 15 d.p. circle arc series   Reference : NONE
Read    : 1/05/97 11:03 (0 replies)             Private   : Yes
Conf    : 0 - Internet E-Mail

Date: Wed, 30 Apr 1997 18:51:21 -0400 (EDT)
From: Jason Stratos Papadopoulos <jasonp@Glue.umd.edu>
Reply-To: Jason Stratos Papadopoulos <jasonp@Glue.umd.edu>
To: Don Mcdonald <don.mcdonald@welcom.gen.nz>
In-Reply-To: <862322218@welcom.gen.nz>
Message-ID: <Pine.SOL.3.95.970430122535.6212A-100000@franc.isr.umd.edu>

On Tue, 29 Apr 1997, Don Mcdonald wrote:

Okay, a reply in depth, with extra stuff you may enjoy.

> here's a prog to calc pi 1million places?
> 
> My prog is in BBC BASIC V.
> The printout at end of this article can be checked manually.
> 
> Involves rational fractions only.
> 
> Do you want to see the BBC code?
>            


Sure, send it! I think, though, that as far as calculating pi
is concerned the real satisfaction comes from doing it quickly;
hence there are tips involved in large number calculations.

> # PS: I'm putting together a pile of public domain pi calculators for 
> # uploading to the SimTel collection, so send me your favorite if you'd  
> # like it included (I already have six or seven!). 

I'd be happy to throw it into the pile.

>     1/ Area of an octant of a circle divided into  
>     1024 strips by Simpson's 1/3rd rule (possibly  
>     the subject of a subsequent article), and ...  

If you're going to calculate pi by numerical integration, try
instead to find the area under 1/(1+x^2) from x=0 to x=1. This
is supposed to be pi/4, and you don't need to use square roots 
to get it.

>       
>     2/ length of a circular arc (a better  
>     method), described here.  
>       
>       On 19.4.91 I found out that a similar  
>     series to mine, obtained by different  
>     principles, was published by Newton,  
>     (posthumously, 1737.)  "History books favour  
>     (my) method to Newton's original work."  

Newton's formula was essentially an inverse sine series. The
error drops by a factor of about 4 for every extra term added.

>     sides.  I have calculated the curved length  
>     of a 30 degree unit circle arc by the  
>     differential - integral calculus and set it  
>     equal to the series sum, pi/6.  Hence, pi =  
>     6*length of arc.  The calculation of the  
>     length of a curve is known as 'Rectification  
>     of the Circle'.  (Archimedes.  Contrast  
>     'Squaring the circle', circle squaring, which  
>     is impossible.)   

Arcsine calculations (what you're doing) have a limited
appeal, because it's hard to go from one arcsine to several
arcsines of smaller numbers; your series needs 1.6 terms to
get a correct digit of pi because your arcsine is being eval-
uated at 1/2. If you could evaluate an arcsine at 1/10, you could
get a digit of pi in half a term. This is why it's easier to think in
terms of arctangents, because reduction formulas exist for them.

See DH Lehmer, "On Arccotangent Relations for Pi", Ammerican
Mathematical Monthly 1935 (I think) or Dario Castellanos, "The
Ubiquitous Pi" in Mathematics Magazine of April 1988 for details.


>     Luke 3.5 "Straighten the curves!  Smooth out  
>     the ruts!" The Living Bible.  

It's easy to mix math and religion. Both can inspire
ferocious devotion!

>     Basic machine arithmetic.  
>       
>     I have done multiplication of 2 positive  
>     integers to 510 decimal digits, previously.   
>     The algorithm is, or was, learned in primary  
>     school.  

The primary school algorithm is extremely slow; you should only
use it for numbers of 100 digits or less. The reason is that
forming all those intermediate products to add together means
multiplying n digit numbers requires time proportional to n^2.
Thus multiplying numbers twice as big requires quadruple the time.
There are two alternatives: first, use computations that don't 
require multiplying large numbers (this is why infinite series
are so nice, because you only need to be able to multiply a big
number by a number one word long) or to use more sophisticated
multiplication (this is what serious digit hunters are forced
to use, and programming a high speed FFT-boosted multiplication
is a real pain).

>       In program PI3 I required the more  
>     difficult long division calculation of  
>     dividing a real by a real to 20 decimal  
>     places.  Thus to extend the value of the BBC  
>     PI function by 6 additional decimal digits  
>     accuracy, (one million times more accurate.)  

Long division is even worse than long multiplication! DH Lehmer
has also written papers on how to do it but it's more painful
than multiplying. Your best bet is to create a program that 
doesn't have to divide multi-digit numbers by multi-digit numbers.
How? Your computer can come to the rescue, since it knows auto-
matically how divide 10-digit integers by 10-digit integers
(the "long integer" data type can do this). So, you can choose
a series that doesn't require you to divide by big numbers, 
and then break up your sum into little chunks of digits to 
divide by. Thus, dividing 3657282648533 by 4 is simple (just
go digit by digit), but your computer can handle more digits
at once. It could, for example, divide "365 728 264 853 300" by
"173" just like dividing by 4 above, even though 173 is several
digits long, because your computer can do several digits at once.
If you want, I can explain further.

This, incidentally, is another reason why arctangent formulas are so
popular: because creating a term requires just two divisions. One is
always by the same number, and the other increases in size very slowly
so you can add up hundreds of thousands of terms without worrying about
overflow.

>     Rational approximations to Pi.  
>       
>     Some rational approximations for pi are:  
>     PI=(16/9)^2,  PI^3 = 31,  PI^4 = 2143/22,  or  
      ^^^^^^^^^^^^
      ancient Egyptian value?

>     Three of my own rational approximations are:  
>     pi^34 = 8E16,  pi = 88^2/(44^2+23^2),  
>      and pi = sqrt(10) = 3 + 53353/328776  
>      = (79/25) + (2.7.17.29/3.101.10001.)  

Also, pi/2 = 6^(1/4) = (630/17) ^ (1/8)

>     Mnemonics.  
>       

Somebody else has an enormous collection of mnemonics for pi in several
different languages; if you don't have such a work and want it, tell me.

>     Other methods.  
>       
>     Dr Harper suggests that the best algorithm if  
>     you want a few digits of pi is:    
>      16ATAN(1/5) - 4ATAN(1/239);     where  
>     ATAN(x) = x-(1/3)x^3 +(1/5)x^5 -(1/7)x^7 + ..  

Without doubt. Forming the terms is fast and easy, you don't divide
by large numbers, and you don't have to divide by 25 to full precision
(the decimal value terminates after a while).

>     Pi has now been calculated to over 1 billion  
>     (2E9) decimal digits by Gregory and David  
>     Chudnovsky at IBM's Thomas J Watson Research  
>     Centre, The Dominion, 31.8.89, page 4,  
>     (exactly 1,011,196,691 digits).  

This is somewhat outdated; the record now is 6.4 billion digits,
by Y. Kanada at the University of Tokyo.

>     Petr Beckmann, "A history of 'pi'", Golem  
>     Press, Boulder, Colorado, 1977 (4th edn.)  

Excellent book (a little nutty though)

> Preston Richard, The mountains of Pi : Profiles (David and Gregory Chudnovsky),  
> in The New Yorker magazine, 2 March 1992, pp. 36-67. (Very interesting, almost  
> book length.)  

I just got a look at this article, and though it didn't answer the 
questions I had it was a great read nevertheless

> Waters, Tom. Mathematics 1989 : Pi in the sky.  In Discover journal, January  
> 1990, pp 76-77.  

There's a book of the same name having nothing to do with pi.

> Circular arc series calculation of Pi to 15 dec. pl.   
>   
> N |2N-1 |NUM  |  DENOM | FACT       | DECIMAL = 3*NUM/(DENOM*FACT*8^N)   
>          prod(2N-1), (2N+1)  
>   
> 1   1   1            3   1                 12, 50, 00, 00, 00, 00   
> 2   3   3            5   2                 01, 40, 62, 50, 00, 00   
> 3   5   15           7   6                 00, 20, 92, 63, 39, 28.5714286   
> 4   7   105          9   24                00, 03, 56, 03, 84, 11.4583333   
> 5   9   945          11  120               00, 00, 65, 54, 34, 34.8366477   
> 6   11  10395        13  720               00, 00, 12, 70, 95, 44.2551833   
> 7   13  135135       15  5040              00, 00, 02, 55, 70, 39.261171   
> 8   15  2027025      17  40320             00, 00, 00, 52, 87, 99.4793946   
> DIVIDE NUM, F, BY FACTOR 2835   
> 9   17  12155        19  128               00, 00, 00, 11, 17, 12.7555579   
> 10  19  230945       21  1280              00, 00, 00, 02, 40, 04.9433027   
> 11  21  4849845      23  14080             00, 00, 00, 00, 52, 30.3260869   
> DIVIDE NUM, F, BY FACTOR 2835   
>   loss of exact accuracy                                                ??  

This is the chief problem with using multiplications of floating point
numbers. It would be better to use double precision, or even better to
use an array of integers.

> 12  23  39346.18518  25  59.5978836        00, 00, 00, 00, 11, 52.8510413   
> 13  25  983654.6296  27  774.7724867       00, 00, 00, 00, 02, 56.5996798   
> ....  
> 24  47  810236.4548  49  0.4215355705      00, 00, 00, 00, 00, 00.0000249   
> DIVIDE NUM, F, BY FACTOR 2835   
> 25  49  14004.08687  51  3.717244889E-3    00, 00, 00, 00, 00, 00.0000059   
> TOTAL ===============================                          89.79346156,  
>                                                                       ?????  
>                                                            35,   
>                                                        65,   
>                                                    92,   
>                                                15,   
>                                            14,   
>                                         3   
> TRUE VALUE OF PI =                       _____________________________   
>                                         3. 14, 15, 92, 65, 35, 89,793238+   
> 1022 CENTISEC   
> PROGRAM  PI3   E N D. rem 16th - 20th dec. places subject to errors.  
> ---------------------------------------------  

Well, I certainly learned a few things from your kind letter; I can
only hope my response provides the same.

jasonp

Conference 325 - sci.math
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* 6  Msg:42,682Fm:HJSMITH@IX.NETC To:ALL     Sb:Re: Pi ?               



From    : HJSMITH@IX.NETCOM.COM                 Number    : 42,575 of 42,687
To      : ALL                                   Date      : 3/05/97 11:58
Subject : Re: Pi ?                              Reference : NONE
Read    : No (0 replies)                        Private   : No
Conf    : 325 - sci.math

From: HJSmith@ix.netcom.com (Harry J. Smith)
Newsgroups: sci.math
Subject: Re: Pi ?
Date: Fri, 02 May 1997 23:58:51 GMT
Organization: Litton PRC, Systems Integ., Santa Clara, CA
Lines: 41
Message-ID: <336a7c84.453609484@nntp.ix.netcom.com>

mau055@thunder (D.M.Rigby) wrote:

>Have they ever calculated pi in anything other than base 10? 
>
>--
>                                    from
>                                    Dawn    
> =========================================================================
>| Scientists have calculated that the chance of anything so patently      |
>| absurd actually existing are millions to one.                           |
>| But magicians have calculated that million-to-one chances crop up nine  |
>| times out of ten.                                                       |
>| (Terry Pratchett - Mort)                                                |
> =========================================================================

Yes, what base would you like?

Base 36 is fun! In base 36 Pi =
                          ascii47 few4d (don.)

3.53I5A,B8P5F,SA5JH,K72I8,ASC47,WWZLA,CLJJ9,ZN98L,
TXM61,VYMS1,FRYTC,I4U2Q,FRA2
V,JAW70,CH6J1,53P3Z,9ZL55,UKZL0,KAPWJ,YGJOU,067IY,
9WNZD,Z9N4J,LTEDT,IW2B6,5ACRP,
IL9LJ,26ST5,C8FX8,S5LPS,F7PA6,J5BOH,CKP3H,8EYX4,
GK7MZ,AOYHS,I9N5K,6MDBD,7HAZ4,91
LP2,6Z8AK,AIN7Q,FZ6B4,9WZO4,6LNU8,G2408,0V68L,F6MAA,
9I2YU,Z0T2O,ZP27J,J9L82,EQA1
W,W00PI,P67P0,9U7PQ,P7Z39,CDIA2,2LEVW,QR7ZX,N66ZS,
WG6BU,YWYF4,RD48G,AQIYG,86TRS,
VDB6D,8W3A4,CYB8J,QG51J,2QQ11,N5SJL,FEW4D,SWWSC,OG6ON,
LXB4W,GPICV,FULRS,1XCLC,K3
4JI,43CKC,2JPZL,LHFA9,647ZG,S0CJE,X5QYI,DCJQ9,V2Y78,ICM4P,94LKF,DFTDB,WDNOY,07BE
7,6VO1X,V29B3,8UKOB,3DO5G,O0VLH,7Z6I3,NNPEY,GNZAW,WF8KZ,J8M5N,GSJYN,ICG6D,NM5Q8,
KH2LC,II50S,DRA55,I3H5A,NAI83,5B00W,BL7RF,3RP5R,2MD74,WVTY4,B6AWZ,ASGR5,ZPSKM,XM
4BF,MY92C,Y4ENL,G0Z2O,RT84F,X1BLG,NFDBI,16EY2,L4TXB,NDBJY,FIYI8,IUCOL,ZSRG9,WZ2F
M,0N1MG,3OPXU,EBYXD,ZYQW1,0XOZX,PWWSM,4H6L0,1D0A9,3KNGD,G84OC,HGA3H,GEBTN,DHB64,
927O7,UHCLZ,TJJ8V,GL3EM,K7VFL,XRHV6,UIATT,GY0F...

-Harry

-- 
| Harry J. Smith, 19628 Via Monte Dr., Saratoga, CA 95070-4522, USA
| Home Phone: 1 408 741-0406, Work Phone: 1 408 235-5088 (Voice Mail)
| E-mail: HJSmith@ix.netcom.com, Fax: 1 408 235-2019
| Web site: http://www.netcom.com/~hjsmith
--



From    : HJSMITH@IX.NETCOM.COM                 Number    : 42,682 of 42,689
To      : ALL                                   Date      : 3/05/97 11:43
Subject : Re: Pi ?                              Reference : NONE
Read    : No (0 replies)                        Private   : No
Conf    : 325 - sci.math

From: HJSmith@ix.netcom.com (Harry J. Smith)
Newsgroups: sci.math
Subject: Re: Pi ?
Date: Fri, 02 May 1997 23:43:53 GMT
Organization: Litton PRC, Systems Integ., Santa Clara, CA
Lines: 36
Message-ID: <336a7a8c.453105227@nntp.ix.netcom.com>
References: <5j2v9i$60e@buggy.news.easynet.net> <335BF2C3.1F5E@ix.netcom.com> <335E70A1.1734@mail.mia.bellsouth.net> <5k5d3u$v8m@hydraulix.bangor.ac.uk> <5k63p6$gtf@sun001.spd.dsccc.com>

jmccarty@sun1307.spd.dsccc.com (Mike McCarty) wrote:

>In article <5k5d3u$v8m@hydraulix.bangor.ac.uk>,
>D.M.Rigby <mau055@thunder> wrote:
>)Have they ever calculated pi in anything other than base 10? 
>)--
>)                                    from
>)                                    Dawn    
>
>Almost every computation of PI has been in binary, not decimal. Sorry,
>but you have a misconception, Dawn.
>
>Mike
>-- 
>----
>char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
>This message made from 100% recycled bits.
>I don't speak for DSC.         <- They make me say that.

# We are now computing Pi in decimal, and have for quite some time.
# 
# The reason that the current record is
# about 6,442,450,944 (=3*2^31) decimal
# digits of pi is that each super digit
# was stores// as base 1000 and stored in
# arrays of length 2^31.
# 
don.  hint 1999 pi record = 3*2^36 digits = 206E9.

# -Harry
# --
# | Harry J. Smith, 19628 Via Monte Dr., Saratoga, CA 95070-4522, USA
# | Home Phone: 1 408 741-0406, Work Phone: 1 408 235-5088 (Voice Mail)
# | E-mail: HJSmith@ix.netcom.com, Fax: 1 408 235-2019
# | Web site: http://www.netcom.com/~hjsmith
