Update

pi.md

@@ -14,9 +14,9 @@ Pi to 100 binary fractional digits is:
 
 Some people memorize the digits of pi for [fun](fun.md) and competition, the world record as of 2022 is 70030 memorized digits.
 
-**PI IS NOT INFINITE**. [Soyence](soyence.md) 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 [decimal](decimal.md) (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 [chaotic](chaos.md), but that's nothing special either, there are infinitely many numbers with the same properties and mysteries in this sense (most famously the number [e](e.md) 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 [algorithm](algorithm.md) simply ends up stuck in an [infinite loop](infinite_loop.md). We can create systems of writing numbers in which pi has a finite expansion (e.g. base pi or a base with varying radix fractions), 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.
+**PI IS NOT INFINITE**. [Soyence](soyence.md) 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 [decimal](decimal.md) (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 [chaotic](chaos.md), but that's nothing special either, there are infinitely many numbers with the same properties and mysteries in this sense (most famously the number [e](e.md) 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 [algorithm](algorithm.md) simply ends up stuck in an [infinite loop](infinite_loop.md). 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... also doesn't contain any possible combination of 1s and 0s). This would hold if pi was [normal](normal_number.md), but again, there are many other such 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 [normal](normal_number.md) -- then pi's digits would contain e.g. every book that will ever be written (see also [Library Of Babel](library_of_babel.md)). 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 [Euler's identity](eulers_identity.md), one of the most important equations, it is related to [complex exponential](complex_exponential.md) and so to [Fourier transform](fourier_transform.md), waves, oscillation, trigonometry ([sin](sin.md), [cos](cos.md), ...) and angles ([radians](radian.md) use pi), it even starts appearing in [number theory](number_theory.md), e.g. the probability of two numbers being relative primes is 6/(pi^2), and so on.
 
@@ -38,11 +38,11 @@ Leibnitz formula for pi is an infinite series that converges to the value of pi,
 
 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
+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 **[algorithm](algorithm.md)** for computing approximate pi value can be based on approach used in further history: approximating a circle with many-sided regular [polygon](polygon.md) 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.
+A simple **[algorithm](algorithm.md)** for computing approximate pi value can be based on approach used in further [history](history.md): approximating a circle with many-sided regular [polygon](polygon.md) 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 [monte carlo](monte_carlo.md) 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.