How to demystify π ?
maXbox Starter 92
This tutor explains why PI could be a normal irrational number. So DRINK stands for Decimal Representation of a Irrational Number Kind.
There is nothing especially or extremely interesting in the digits of a decimal representation of irrational numbers like Pi, E or the golden ratio. Or in the digits to any whole number base I think.
*An irrational function is a function whose analytic expression has the independent variable under the root symbol, e.g.: y = sqrt(x)
First we have to create a big PI (1000 digits):
with TBigfloat.Create1(150) do begin
PIconst(1000)
writeln(toString(normal)) ;
free;
end;
The same we can get at:
https://www.wolframalpha.com/input/?i=pi+to+1000+digits
But how can we write a function which will return Pi (π) to a versatile given number of decimal places?
In calculus there is a thing called Taylor Series which provides an easy way to calculate many irrational values to arbitrary precision.
Pi/4 = 1–1/3 + 1/5–1/7 + …
But this Taylor series is probably also one of the worst ways to generate PI on a computer. You have to have huge precision on your calculations and it’ll take many billions of iterations to get past 3.14159.
Next try could be the use of atan, trouble is, the Taylor series for atan (PI/4 = atan(1) = 1–1/3 + 1/5–1/7 + 1/9… )converges very slowly for values of x near one, and it converges extremely slowly when x is equal to one.
So how we did it: with the AGM calculation of Pi. In 1799, Gauss was startled to discover that his arithmetic-geometric mean connected, the half-
circumference of a curve known as the lemniscate, with π, the half-circumference of a unit circle.
procedure TBigFloat.PiConst(const MaxSig: TMaxSig);
begin
self.CheckPiPlaces(MaxSig);
end;procedure TBigFloat.CheckPiPlaces(const MaxSig: TMaxSig);
var localPrecision: word;
begin
localPrecision:= MaxSig+2;
if localPrecision>zpi.sigdigits then
calculate_PI_AGM(localPrecision+5)
else begin
self.Assign(zpi);
self.RoundToPrec(MaxSig);
end;
end;
Then we compare the result of the code with the wolfram alpha link above to make sure we get the right 1000 places in it:
writeln(‘sha test1: ‘+sha1tohex(synsha1(tostring(normal))));
writeln(‘sha test2: ‘+sha1tohex(synsha1(PI1000)));
>>> sha test1: b7805c4fb1662666d7741fa8f915daacf706cd01
>>> sha test2: b7805c4fb1662666d7741fa8f915daacf706cd01
Got it. So why doesn’t Pi have a 0 in its first 30 digits?
3.141592653589793238462643383279502884
Let’s assume (likely but is not proven at present) that Pi is a Normal Number1. Amongst other things, this means that the frequency of occurrence of any digit in its decimal range is precisely 1/10. This assumption is in accord with statistical analysis of many trillions of decimal places of Pi.
We can use this to calculate the probability that there are no zeros in the first thirty digits of our π; In order for this to happen, zero cannot appear in the first place, a probability of 9/10 and also not in the second place, also a probability of 9/10 and so on. We get the overall probability to be:
(9/10)³⁰ = ˜ 4.24% //4.23911
So it is unlikely that no zero (or another selected number) appears in the first thirty digits.
Can we say there is even less of interest in the decimal digits as a particular transcendental number, except that so many people think there is something special about it, for example you find your birth date (as 8 digits) in it.
What can we do?
1 https://peterjamesthomas.com/maths-science/the-irrational-ratio/#normal-again
We can count the frequency of each digit of Pi to assume a uniform distribution and independence and non predictability of each digit:
Piconst(10000)
writeln(‘zero count: ‘+ itoa(StrCharCount(toString(normal), ‘0’)));
for it:= 0 to 9 do
println(itoa(it)+’count: ‘+ itoa(StrCount(toString(normal),+itoa(it)[1])));
ref: zero count: 968
0 count: 968
1 count: 1026
2 count: 1022
3 count: 976
4 count: 1012
5 count: 1047
6 count: 1023
7 count: 971
8 count: 948
9 count: 1014
But wait a second. Why does zero lag behind in the count. It is a perfectly non number digit? Just had to ask, its a rhetorical question. Lets check it with only 1000 digits:
zero count: 93
0 count: 93
1 count: 116
2 count: 103
3 count: 103
4 count: 93
5 count: 97
6 count: 94
7 count: 95
8 count: 100
9 count: 106
for it:= 0 to 9 do
FormatF(‘%d count: %d’,[it,StrCount(toString(normal),+itoa(it)[1])]);
Obviously, some digit will lag. Which digit has a reason to lag? None has a reason to lag. Zero is none. Therefore, zero lags.
— Larry Hosken
This means as said before that each number is equally likely to be the next number so each has a 1/10 chance. Therefore, the occurrence of each digit should be equal once we reach an infinite number of decimal places.
http://www.eveandersson.com/pi/precalculated-frequencies
But Pi is obviously going to look different if we calculate it in base 8 or base 12 or any base other than 10.
I was also reading a recent blog post by Evelyn Lamb where she mentioned in passing that 314159 is a prime number and that made me curious how many such primes there are.
https://www.johndcook.com/blog/2018/09/04/pi-primes/
Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas, see appendix prime-counting.
Prime-counting
def is_prime(n):
if n>1:
divs=[k for k in range(2,n) if n%k==0]
return len(divs)==0
else:
return False
def nth_prime(n):
primes=[p for p in range(n*n+2) if is_prime(p)]
return primes[n-1]
If you are lost into the source code then you could easily add parameters to your app to write output to a file instead of the console: -o out.txt, since it’s your tool doing the writing, you can build wherever you want for example to start out of the shell and get output to the shell and in the end plot an image to another file output as a graphic like below:
The script can be found:
http://www.softwareschule.ch/examples/1093_XMLUtils_Tutor92tester.txt
Release Notes maXbox 4.7.6.10 IV Jan. 2022 mX476 **************************************************
Add 25 Units + 6 Tutorials
Total of Function Calls: 35371 SHA1: A2B2B2D1596C6A5F3ACCED90D0C2246172A3DE2C
CRC32: 285ACBCB 31.3 MB (32,921,928 bytes)
