Version 47 of Testing Normality of Pi, Console Example

Updated 2011-05-29 18:47:30 by gold

Testing Normality of Pi, Console Example

This page is under development. Comments are welcome, but please load any comments in the comments section at the middle of the page. Thanks,gold

gold Here is an eTCL script on testing the normality of pi for the etcl console. It has not been mathematically proven that pi is a normal number, but numerical analysis with tcl can check for some aspects of normality in pi. The normality of a number means that the ten numbers (0,1,2..9) in the mantissa are equally common at infinity. Relative frequency is count of individual numbers (eg. digit 7) over the set of digits (10 in base ten) at collection size N. If pi is normal in base 10, the relative frequencies of each digit in the mantissa approach 1/10 as N approaches infinity. For example, Kanada used a set of 6.4E9 decimal digits from pi and found the relative frequency of digit 7 as 600009044/6442450000 or 0.0931336749218077. The script below was used to check the relative frequency of pi digits using the published collections of pi.

In planning any software, there is a need to develop testcases. With back of envelope calculations, we can develop a number of peg points to check output of program. For the first 20 digits of pi, one counts 2 ones so the relative frequency of one using 2/20 should be 0.1. Also for the first 20 digits of pi, one counts 3 nines so the relative frequency of nine using 3/20 should be 0.15. In the first 20 digits of pi, one finds no zeros. This means that the relative frequency of digit zero using 0/20 should be zero. In the results below, one can see the relative frequencies of 0,1,&9 digits. The sum of the relative frequencies for all decimal digits 0,1,2 ... 9 should approximate 10 times .1 or 1.0.

Other testcases could be from slices of pi digits from the big number crunchers. 

 Testcases
  frequency of "7"  for collection of N digits of pi
quantityfrequencyN digits
70.05 20 digits
7 0.08510 50 digits
70.09313 1k digits
7 0.09506 10k digits
7 0.0989 100k digits

Screenshots Section

  relative frequency of "7" in pi mantissa approaches limit of 0.1 as N>>1
  frequency versus set of PI digits  of total N (decimal log N). 
  statistics from pi314 website

http://img151.imageshack.us/img151/4917/dfdfmg.jpg

   statistics from Kanada website
  relative frequency of "7"  versus set of pi digits of total N (decimal log N).

http://img40.imageshack.us/img40/3659/ssssvs.jpg

Comments Section

Please place any comments here, Thanks.

References:

Appendix TCL programs and scripts

* Pretty Print Version

               # Pretty print version from autoindent
               # and ased editor
               # written on Windowws XP on eTCL
               # working under TCL version 8.5.6 and eTCL 1.0.1
               # gold on TCL WIKI , 25may2011
               # relative frequency of indiv. "throw" over all "throws".
               # pi mantissa used here
        package require Tk 
          console show
            proc calculation {  facen }  {
                # prob. subroutines for mimic sequence of bronze
                # prob. is throw combos of eg. "7" over all possible throws
                
                set lister [split {14159265358979323846} ""]
                
                set ee [llength  $lister ]
                set kk [ llength [ lsearch -all $lister $facen ] ]
                set prob [ expr { ($kk*1.) / $ee  } ]
                return $prob
            }
            set limit 12
            for { set i 0 } { $i <= $limit }  { incr i } {
                lappend listxxx $i
                lappend listxxx [ calculation  $i ]
                puts " $i [ calculation  $i ] "
            }
            #end
 results for first 20 numbers of pi mantissa
         
 0 0.0 
 1 0.1 
 2 0.1 
 3 0.15 
 4 0.1 
 5 0.15 
 6 0.1 
 7 0.05 
 8 0.1 
 9 0.15 
 results for small numbers of pi mantissa
 {14159265358979323846264338327950288419716939937}
 1 0.0851063829787234 
 2 0.10638297872340426 
 3 0.1702127659574468 
 4 0.0851063829787234 
 5 0.0851063829787234 
 6 0.0851063829787234 
 7 0.0851063829787234 
 8 0.10638297872340426 
 9 0.1702127659574468 
 results for 1k digits of pi 
 0 0.09117647058823529 
 1 0.11372549019607843 
 2 0.10098039215686275 
 3 0.1 
 4 0.09117647058823529 
 5 0.09509803921568627 
 6 0.09215686274509804 
 7 0.09313725490196079 
 8 0.09901960784313725 
 9 0.10392156862745099 
 results for 10k digits of pi 
 0 0.09486475891807135 
 1 0.10054880439043512 
 2 0.10005880047040376 
 3 0.09555076440611525 
 4 0.09917679341434732 
 5 0.10250882007056056 
 6 0.10005880047040376 
 7 0.09506076048608389 
 8 0.0929047432379459 
 9 0.09937279498235986 
 10 0.0 
 11 0.0 
 12 0.0 
1% 
results for 100k numbers of pi mantissa
 1 0.1000947923455181 
 2 0.09782372573414697 
 3 0.0989790074451488 
 4 0.09843592629895136 
 5 0.09899875585046508 
 6 0.09899875585046508 
 7 0.09898888164780693 
 8 0.09852479412287458 
 9 0.09777435472085629

Code scraps

        puts " timing split cmd [time { set lister [split {14159265358979323846} ""] } 1000 ]" 
        500 digits ending 1,241,100,000,000-th (1,241,099,999,501 - 1,241,100,000,000)

  3716787169 6567692125 2797286901 8503557537 6530193499
  3533850167 1616469990 5984454421 7623131551 5483436562
  7806800557 0748706663 5108659327 6579461496 7987525534
  7689068277 7037671632 7753867760 1776471900 9279382597
  6527339324 6948904759 2872702485 4618972965 3547547082
  4504016840 2350653293 6254205392 4502959326 3809170954
  8310279798 7965959470 8455199922 4435552002 5054585883
  0997016164 9607402417 5296690907 5622217705 1785600450
  0707455198 1744551596 6313820124 4825046054 2311034186
  5591198918 2262704528 2696896699 2856706487 3410311045
Category TestingCategory ToysCategory PerformanceCategory Numerical AnalysisCategory StatisticsCategory ExampleCategory Mathematics