Version 29 of Testing Normality of Pi, Console Example

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.

 Testcases
  frequency of "7"  for collection of N digits of pi

quantity	frequency	N	digits
7	0.05	20	digits
7	0.08510	50	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.

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

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
               #rank of indiv. "throws" 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 
 10 0.0 
 11 0.0 
 12 0.0 
{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 
 10 0.0 
 11 0.0 
 12 0.0 

results for small numbers of pi mantissa
 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 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 " loading [time {pixane load $p -file $fin} ]"

Category Testing

Category Toys

Category Performance

Category Numerical Analysis

Category Statistics

Category Example

Category Mathematics