Calculating π-approximation

13.4 Calculating π-approximation

In fact, a famous result in numbers theory says that the probability for two randomly chosen integers to be coprime is 6-- π2 ≈ 0,6079. To exhibit this result, we’re going to:

Choose randomly two integers between 0 and 1 000 000.
Calculate their GCD.
If the GCD value is 1, increment the counter variable.
Repeat this experience 1000 times.
The frequence for the couple of coprime integers can be calculated dividing the variable counter by 1000 (tries number).

 
 to test
 # We set the variable counter to 0
 globalmake "counter 0
 repeat 1000 [
   if (gcd random 1000000 random 1000000)=1 [globalmake "counter :counter+1]
 ]
 print [frequence:]
 print :counter/1000
 end

In a similar way as the precevious note, notice the parenthesis around gcd random 1000000 random 1000000. Otherwise, the interpreter will try to evaluate 1 000 000 = 1. You can write in other way: if 1=gcd random 1000000 random 1000000

We execute the program test.

 test
 0.609
 test
 0.626
 test
 0.597

We obtain some values close to the theorical probability: 0,6097. This frequency is an approximation of -6- π2 .

Thus, if I note f the frequency, we have: f ≈ -6- π2

Hence, π² ≈ 6- f and π ≈ ∘ -- 6- f .

I append to my program a line that gives this π approximation in procedure test:

 to test
 # We set the variable counter to 0
 globalmake "counter 0
 repeat 1000 [
   if (gcd random 1000000 random 1000000)=1 [globalmake "counter :counter+1]
 ]
 # We calculate te frequency
 make "f :counter/1000
 # we dispaly the pi approximation
 print sentence  [ pi approximation:] sqrt (6/:f)
 end
 
 test
 pi approximation: 3.1033560252704917
 test
 pi approximation: 3.1835726998350666
 test
 pi approximation: 3.146583877637763

Now, we modify the program because I want to set the number of tries. I want to try with 10 000 and perhaps more tries.

 to test :tries
 # We set the variable counter to 0
 globalmake "counter 0
 repeat :tries [
   if (gcd random 1000000 random 1000000)=1 [globalmake "counter :counter+1]
 ]
 # We calculate te frequency
 make "f :counter/:tries
 # we dispaly the pi approximation
 print sentence  [ pi approximation:] sqrt (6/:f)
 end
 
 test 10000
 pi approximation: 3.1426968052735447
 test 10000
 pi approximation: 3.1478827771265787
 test 10000
 pi approximation: 3.146583877637763
                                                                                                  
                                                                                                  
 test 10000

Quite interesting, isn’t it?