Version 19 of Testing Normality of Pi, Console Example
Updated 2011-05-29 16:09:04 by goldTesting 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 numbers (0,1,2..9) in the mantissa are equally common at infinity. In base 10, the relative frequencies of the N numbers in the manitssa approach 1/10 as N approaches infinity. For example, Kanada used a set of 6.4E9 decimal digits and found the frequency s 600009044/6442450000 or 0.0931336749218077. One can paste statements into the etcl console.
% puts [ expr {1.*600009044/6442450000 }]
0.0931336749218077In planning any software, there is a need to develop testcases.
Testcase 1. Output of timing statements.
| quantity | number | units |
|---|---|---|
| photo size of bird.jpg | 1.6 | m |
| loading | 1595970 | microseconds per iteration |
Screenshots Section
frequency of "7" in pi mantissa approaches limit of 0.1 as N>>1 frequency versus set of numbers of total N.
Comments Section
Please place any comments here, Thanks.
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.geom.uiuc.edu/~huberty/math5337/groupe/digits.html
- http://www.super-computing.org/
- http://www.piworld.de/pi-statistics/
- citeseerx.ist.psu.edu/ viewdoc/ download?doi=10.1.1.133.7330 &rep=rep1 &type=pdf
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 first 20 numbers of pi mantissa
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
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, timing statements
puts " loading [time {pixane load $p -file $fin} ]"
puts "Scaling ($sourcewidth x $sourceheight) => ($virtualwidth x $virtualheight)"
puts " resizex [time {pixane resize $targetpix $virtualwidth $virtualheight } ]"
puts " blanking [time {pixane blank $targetpix } ]"
puts " rescale [time {pixane scale $targetpix $p } ]"
puts " saving [time {pixane save $targetpix -file $fout -format jpg } ]"
puts " proc time [time { jackpix } ]"