Testing Normality of Pi, Console Example

This page is under development. Constructive comments are welcome, but please load any constructive comments in the comments section at the bottom of the page. Please include your wiki MONIKER and date in your comment with the same courtesy that I will give you. Aside from your courtesy, your wiki MONIKER and date as a signature and minimal good faith of any internet post are the rules of this TCL-WIKI. Its very hard to reply reasonably without some background of the correspondent on his WIKI bio page. Thanks, gold 08Jul2020

• Introduction
• Speculations on Normality of Pi
  • Summary
• Testcases
  • References:
    • Appendix TCL programs and scripts
  • * Pretty Print Version
• results for first 20 numbers of pi mantissa
• results for small numbers of pi mantissa
• results for 1k digits of pi
• results for 10k digits of pi
• results for 100k numbers of pi mantissa
• results for 1E6 numbers of pi mantissa
• slice of published Kanada sequence of pi digits.
  • Notes & Code scraps
• Hidden Comments Section

Introduction

gold Here is a TCL script on testing the normality of pi in the TCL 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. 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 a TCL 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. A slice of published Kanada sequence of pi digits can be loaded into the TCL script on the console.

Speculations on Normality of Pi

Probably numerical coincidence in the first 20 digits of the pi mantissa, but note that the even digits of 2,4,6,&8 have the average of 0.1, like the even digits would at infinity in theory.The average of the even numbers is eval </ <+ .1 .1 .1 .1 ] 4.> or 0.1, as might be expected. The average of the odd numbers is eval M/ <+ .15 .15 .05 .15] 4.] or 0.125 , slightly higher.If pi is normal at infinity, then the average frequency of all digits of 0-9 should average eval </ <+ .1 .1 .1 .1 .1 .1 .1 .1.1 ] 9.> or 0.1. Restating the original premise, the sum of the estimated frequencies for all digits in the pi mantissa should equal to one, meaning at infinity. For the first 20 digits of pi, the sum of frequencies for the 0-9 digits <+ 0. .1 .1 .15 .1 .15 .1 .05 .1 .15> equals 1.0. If not a real proof over the entire number line, at least the sum for frequencies of first 20 digits agree with the restated premise. For the first 10K digits of pi, the sum of 0-9 frequencies was 0.9801058. The “relative error” in the sum of frequencies was 2.02 percent for the first 10k digits. The difference or “absolute error” from the condition at infinity was <- 1. 0.9801058> or 0.0198941. Not sure these results for pi can be extended to other bases, but possibly for pi in base N1, the N1 digits of base N1 should have an average frequency of 1/N1.

Summary

gold 2/28/2024 Update. Alternate Gist : It is not yet mathematically proven whether pi, a well-known irrational number, is a normal number. However, numerical analysis with TCL can be employed to explore some aspects of normality in pi. A normal number is one in which the ten digits (0-9) in its mantissa are equally common when approaching infinity. To determine the normality of a number, we can calculate its relative frequency, which is the count of individual numbers (e.g., digit 7) divided by the total set of digits (10 in base ten) in a collection of size N. If pi is normal in base 10, the relative frequencies of each digit in its mantissa should approach 1/10 as N approaches infinity. To investigate this, a script was used to analyze the relative frequency of pi digits using published collections of pi.

Testcases

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

References:

• http://pi314.at/math/normal.html
• http://mathworld.wolfram.com/NormalNumber.html
• http://www.lbl.gov/Science-Articles/Archive/pi-random.html
• http://www.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
• http://www.eveandersson.com/pi/digits/
• http://www.super-computing.org/
• http://www.piworld.de/pi-statistics/
• http://www.maa.org/mathland/mathtrek_12_16_02.html
• David Richeson, "Circular reasoning:
• Who first proved that C/d (PI) is a constant?" arXiv, March 14, 2013.
• Carlos Sevcik, Fractal analysis of π (PI) normality, arXiv, July 28, 2016

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

set lister [split {14159265358979323846} {}]

results for small numbers of pi mantissa

set lister [split {14159265358979323846264338327950288419716939937} {}] 

results for 1k digits of pi

results for 10k digits of pi

results for 100k numbers of pi mantissa

results for 1E6 numbers of pi mantissa

slice of published Kanada sequence of pi digits.

500 digits ending 1,241,100,000,000-th (1,241,099,999,501 - 1,241,100,000,000)

Notes & Code scraps

Here vowels are equal probilities to consonants. Maybe vowels should be set to equal prob. in English or other language.

   # test of transmuting integers of pi into words and letters.
   # [Mapping words to integers]
   # Small program for console
   console show 
   source int2wordetc.tcl
   set bid  "14159265358979323846264338327950288419716939937510"
   puts " word  [ int2word   $bid ]"
   #output
   #jzilvrzbjfkfzpwznsqdbwjzsjzeqfxkvop

Hidden Comments Section

Please place any comments here with your wiki MONIKER and date, Thanks.gold12Dec2018.

 ----