Pi
Pi (normally written with a Greek alphabet symbol with [1]Unicode value
U+03C0) is one of the most important and famous [2]numbers, equal to
approximately 3.14, most popularly defined as the ratio of a circle's
circumference to its diameter (but also definable in other ways). It is
one of the most fundamental mathematical constants of our universe and
appears extremely commonly in [3]mathematics, nature and, of course,
[4]programming. When written down in traditional decimal system, its
digits go on and on without end and show no repetition or simple pattern,
appearing "random" and [5]chaotic -- as of 2021 pi has been evaluated by
[6]computers to 62831853071796 digits, although approximate values have
been known from very early times (e.g. the value (16/9)^2 ~= 3.16 has been
known as early as around 1800 BC). In significance and properties pi is
similar to another famous number: [7]e. Pi day is celebrated on March 14.
{ Very nice site about pi: http://www.pi314.net. ~drummyfish }
Pi is a [8]real [9]transcendental number, i.e. simply put it cannot be
defined by a "simple" equation (it is not a root of any [10]polynomial
equation). As a transcendental number it is also an [11]irrational number,
i.e. it cannot be written as an integer [12]fraction. Mathematicians
nowadays define pi via the period of the [13]exponential function rather
than geometry of circles. If we stick to circles, it is [14]interesting
that in [15]non-Euclidean geometry the value of "pi" could be measured to
different values (if we draw a circle on an equator of a ball, its
circumference is just twice its diameter, i.e. "pi" would be measured to
be just 2, reveling the curvature of space).
Pi to 100 decimal digits is:
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...
Pi to 100 binary fractional digits is:
11.001001000011111101101010100010001000010110100011000010001101001100010011000110011000101000101110000...
Among the first 50 billion digits the most common one is 8, then 4, 2, 7,
0, 5, 9, 1, 6 and 3.
Some people memorize digits of pi for [16]fun and competition, the
official world record as of 2022 is 70030 memorized digits, however Akira
Haraguchi allegedly holds an unofficial record of 100000 digits (made in
2006). Some people make [17]mnemonics for remembering the digits of pi
(this is known as PiPhilology), for example "Now I fuck a pussy screaming
in orgasm" is a sentence that helps remember the first 8 digits (number of
letters in each word encodes the digit).
PI IS NOT INFINITE. [18]Soyence popularizators and nubs often say shit
like "OH LOOK pi is so special because it infiniiiiiite". Pi is completely
finite with an exact value that's not even greater than 4, what's infinite
is just its expansion in [19]decimal (or similar) numeral system, however
this is nothing special, even numbers such as 1/3 have infinite decimal
expansion -- yes, pi is more interesting because its decimal digits are
non-repeating and appear [20]chaotic, but that's nothing special either,
there are infinitely many numbers with the same properties and mysteries
in this sense (most famously the number [21]e but besides it an infinity
of other no-name numbers). The fact we get an infinitely many digits in
expansion of pi is given by the fact that we're simply using a system of
writing numbers that is made to handle integers and simple fractions --
once we try to write an unusual number with our system, our [22]algorithm
simply ends up stuck in an [23]infinite loop. We can create systems of
writing numbers in which pi has a finite expansion (e.g. base pi), in fact
we can already write pi with a single symbol: pi. So yes, pi digits are
interesting, but they are NOT what makes pi special among other numbers.
Additionally contrary to what's sometimes claimed it is also unproven
(though believed to be true), whether pi in its digits contains all
possible finite strings -- note that the fact that the series of digits is
infinite doesn't alone guarantee this (as e.g. the infinite series
010011000111... doesn't contain any possible combinations of 1s and 0s
either). This would hold if pi was [24]normal -- then pi's digits would
contain e.g. every book that will ever be written (see also [25]Library Of
Babel). But again, there are many other such numbers.
What makes pi special then? Well, mostly its significance as one of the
most fundamental constants that seems to appear extremely commonly in math
and nature, it seems to stand very close to the root of description of our
universe -- not only does pi show that circles are embedded everywhere in
nature, even in very abstract ways, but we find it in [26]Euler's
identity, one of the most important equations, it is related to
[27]complex exponential and so to [28]Fourier transform, waves,
oscillation, trigonometry ([29]sin, [30]cos, ...) and angles ([31]radians
use pi), it even starts appearing in [32]number theory, e.g. the
probability of two numbers being relative primes is 6/(pi^2), and so on.
Approximations, Estimations, Measuring And Programming
Evaluating many digits of pi is mathematically [33]interesting, programs
for computing pi are sometimes used as [34]CPU [35]benchmarks. There are
programs that can search for a position of arbitrary string encoded in
pi's digits. However in practical computations we can easily get away with
pi approximated to just a few decimal digits, you will NEVER need more
than 20 decimal digits, not even for space flights (NASA said they use 15
places).
One way to judge the quality of pi approximation can be to take the number
of pi digits it accurately represents versus how many digits there are in
the approximation formula -- this says kind of the approximation's
[36]compression ratio. But other factors may be important too, e.g.
simplicity of evaluation, functions used etc.
Also remember, you can measure pi in real life by many methods: you can
draw a big circle, measure its radius and circumference and then make the
division, you can also manually perform the Monte Carlo algorithm (see
below) by drawing a circle and then throwing objects around, counting how
many fall inside and outside (just watch out to do it correctly, for
example you must have the fall spot probability as random as possible, not
biased in any way), or you can similarly make a square from wood, then cut
out its inscribed circle, weight both parts and compute pi (with the same
formula as for Monte Carlo).
{ I tried this -- I took a pizza box, cut out four squares, then used a
pencil on string to draw quarter circles on each, cut them and weighted
both groups. All the circle parts weighted 61 grams, the rest weighted 16
grams, this gives me a nice estimate value of pi of about 3.16.
~drummyfish }
An ugly engineering [37]approximation that's actually usable sometimes
(e.g. for fast rough estimates with integer-only hardware) is just
(something like this was infamously almost made the legal value of pi by
the so called Indiana bill in 1897)
pi ~= 3
A simple fractional approximation (correct to 6 decimal fractional digits,
by Tsu Chung Chih) is
pi ~= 355/113
Such a fraction can again be used even without [38]floating point -- let's
say we want to multiply number 123 by pi, then we can use the above
fraction and compute 355/113 * 123 = (355 * 123) / 113.
Srinivasa Ramanujan made a great number of pi approximations, e.g. an
improvement of the previous to (14 correct digits):
pi ~= 355/113 * (1 - 0.0003/3533)
Similarly Plouffe, e.g. (30 correct digits):
pi ~= ln(262537412640768744)/sqrt(163)
Leibnitz formula for pi is an infinite series that converges to the value
of pi, however it converges very slowly { Quickly checked, after adding
million terms it was accurate to 5 decimal fractional places. ~drummyfish
}. It goes as
pi = 4 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...
Nilakantha series converges much more quickly { After adding only 1000
terms the result was correct to 9 decimal fractional places for me.
~drummyfish }. It goes as
pi = 3 + 4/(2 * 3 * 4) + 4/(4 * 5 * 6) + 4/(6 * 7 * 8) + ...
A simple [39]algorithm for computing approximate pi value can be based on
approach used in further [40]history: approximating a circle with
many-sided regular [41]polygon and then computing the ratio of its
circumference to diameter -- as a diameter here we can take the average of
the "big" and "small" diameter of the polygon. For example if we use a
simple square as the polygon, we get pi ~= 3.31 -- this is not very
accurate but we'll get a much higher accuracy as we increase the number of
sides of the polygon. In 15th century pi was computed to 16 decimal digits
with this method. Using inscribed and circumscribed polygons we can use
this to get lower and upper bounds on the value of pi.
Another simple approach is [42]monte carlo estimation of the area of a
unit circle -- by generating random (or even regularly spaced) 2D points
(samples) with coordinates in the range from -1 to 1 and seeing what
portion of them falls inside the circle we can estimate the value of pi as
pi = 4 * x/N where x is the number of points that fall in the circle and N
the total number of generated points.
Digits of pi also emerge when trying to measure some distances inside
[43]Mandelbrot set (see David Bolle, 1991) -- this can perhaps also be
exploited.
[44]Spigot algorithm can be used for computing digits of pi one by one,
without [45]floating point. Bailey-Borwein-Plouffe formula (discovered in
1995) interestingly allows computing Nth hexadecimal (or binary) digit of
pi, WITHOUT having to compute previous digits (and in a time faster than
such computation would take). In 2022 Plouffe discovered a similar formula
for computing Nth decimal digit.
The following is a [46]C implementation of the Spigot algorithm for
calculating digits of pi one by one that doesn't need [47]floating point
or special arbitrary length data types, adapted from the original 1995
paper. It works on the principle of converting pi to the decimal base from
a special mixed radix base 1/3, 2/5, 3/7, 4/9, ... in which pi is
expressed just as 2.22222... { For copyright clarity, this is NOT a web
copy paste, it's been written by me according to the paper. ~drummyfish }
#include <stdio.h>
#define DIGITS 1000
#define ARRAY_LEN ((10 * DIGITS) / 3)
unsigned int pi[ARRAY_LEN];
void writeDigit(unsigned int digit)
{
putchar('0' + digit);
}
int main(void)
{
unsigned int carry, digit = 0, queue = 0;
for (unsigned int i = 0; i < ARRAY_LEN; ++i)
pi[i] = 2; // initially pi in this base is just 2s
for (unsigned int i = 0; i < DIGITS; ++i)
{
carry = 0;
for (int j = ARRAY_LEN - 1; j >= 0; --j)
{ // convert to base 10 and multiply by 10 (shift one to left)
unsigned int divisor = (j + 1) * 2 - 1; // mixed radix denom.
pi[j] = 10 * pi[j] + (j + 1) * carry;
carry = pi[j] / divisor;
pi[j] %= divisor;
}
pi[0] = carry % 10;
carry /= 10;
switch (carry)
{ // latter digits may influence earlier digits, hence these buffers
case 9: // remember consecutive 9s
queue++;
break;
case 10: // ..X 99999.. becomes ..X+1 00000...
writeDigit(digit + 1);
for (unsigned int k = 1; k <= queue; ++k)
writeDigit(0);
queue = 0;
digit = 0;
break;
default: // normal digit, just print
if (i != 0) // skip the first 0
writeDigit(digit);
if (i == 1) // write the decimal point after 1st digit
putchar('.');
digit = carry;
for (unsigned int k = 1; k <= queue; ++k)
writeDigit(9);
queue = 0;
break;
}
}
writeDigit(digit); // write the last one
return 0;
}
See Also
* [48]e
* [49]phi, or golden ratio
Links:
1. unicode.md
2. number.md
3. math.md
4. programming.md
5. chaos.md
6. computer.md
7. e.md
8. real_number.md
9. transcendental.md
10. polynomial.md
11. irrational.md
12. fraction.md
13. exp.md
14. interesting.md
15. non_euclidean.md
16. fun.md
17. mnemonic.md
18. soyence.md
19. decimal.md
20. chaos.md
21. e.md
22. algorithm.md
23. infinite_loop.md
24. normal_number.md
25. library_of_babel.md
26. eulers_identity.md
27. complex_exponential.md
28. fourier_transform.md
29. sin.md
30. cos.md
31. radian.md
32. number_theory.md
33. interesting.md
34. cpu.md
35. benchmark.md
36. compression.md
37. approximation.md
38. float.md
39. algorithm.md
40. history.md
41. polygon.md
42. monte_carlo.md
43. mandelbrot_set.md
44. spigot.md
45. float.md
46. c.md
47. float.md
48. e.md
49. phi.md