What is a random integer? Is an integer choosen randomly between 1 and 1000000 really representative for all
integers choosen randomly? We can see that our experience is only an approximation of an ideal model. Here, we’re
going to modify the method for generating random integers... We won’t use the primitive random, we’re going to
generate random integers with the π digits sequence.
π digits have always interested mathematicians:
- The numbers 0 to 9, do some appear more often than others?
- Is there some sequence of integers that appear frequently?
In reality, it seems that the π digit sequence is a really randomly sequence. (Result not demonstrated yet). It’s not
possible to predict the following digit after the others, there’s no period.
Here is the method we’re going to use to generate integers randomly choosen:
- First, we need the first digit of π (For example, one billion)
- First way: some programs calculate the π digits. For example, PiFast in Windows environment
and SchnellPi for Linux.
- Second way: you can download this file from XLOGO website:
- To generate the integers, we’re going to read the digits sequence in packet of 7 digits:
First number Second number Third number338327950288419716939
I remove the point “. ” in 3.14 ....that will cause problem when we’re going to extract the digits
Let’s create now a new procedure called randompi and let’s modify the procedure test
to gcd :a :b
if (modulo :a :b)=0 [output :b][output gcd :b modulo :a :b]
to test :tries
# We open a flow whose identifier is 1 towards the file millionpi.txt
# Here we suppose that millionpi.txt is in the current directory
# Otherwise, fix it with changedirectory
openflow 1 "millionpi.txt
# Set the variable line to the first line of the file millionpi.text
globalmake "line first readlineflow 1
# Set the variable counter to 0
globalmake "counter 0
repeat :tries [
if 1=gcd randompi 7 randompi 7 [globalmake "counter :counter+1]
# Calculate frequency
globalmake "f :counter/:tries
# Display th pi approximation
print sentence [ pi approximation:] squareroot (6/:f)
to randompi :n
localmake "number "
repeat :n [
# If there’s no char yet on the line
if 0=count :line [globalmake "line first readlineflow 1]
# Set the variable char to the first character of the line
globalmake "char first :line
# Then remove first character from the line.
globalmake "line butfirst :line
globalmake "number word :number :char
approximation de pi: 3.4641016151377544
approximation de pi: 3.1108550841912757
approximation de pi: 3.081180112566604
approximation de pi: 3.1403714651066386
approximation de pi: 3.1361767950325627
We find a correct approximation of π with its own digits!
It’s still possible to imrove the program by indicating the time for the computation. We add on the first line of the
globalmake "begin pasttime
Then we append before closeflow:
print sentence [pasttime mis: ] pasttime - :begin