Why is 3.14 called pi? How is pi actually calculated?

 

Why is 3.14 called pi? How is pi actually calculated?

Let’s answer this in two parts:

Why is 3.14 called pi?

It’s really not. The only reason for making this faux pas is a worrying lack of basic numeracy.

How is pi actually calculated?

If π

was really 3.14

, it wouldn’t need much calculating now would it?

However, given that it is not, nowadays most calculations of π

’s decimal expansion use algorithms. Here is a venerable and simple one:

Gauss–Legendre algorithm - Wikipedia

The Gauss–Legendre algorithm is an algorithm to compute the digits of π . It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π . However, it has some drawbacks (for example, it is computer memory -intensive) and therefore all record-breaking calculations for many years have used other methods, almost always the Chudnovsky algorithm . For details, see Chronology of computation of π . The method is based on the individual work of Carl Friedrich Gauss (1777–1855) and Adrien-Marie Legendre (1752–1833) combined with modern algorithms for multiplication and square roots . It repeatedly replaces two numbers by their arithmetic and geometric mean , in order to approximate their arithmetic-geometric mean . The version presented below is also known as the Gauss–Euler , Brent–Salamin (or Salamin–Brent ) algorithm ; [1] it was independently discovered in 1975 by Richard Brent and Eugene Salamin . It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with Borwein's algorithm . Algorithm [ edit ] Initial value setting: a 0 = 1 b 0 = 1 2 t 0 = 1 4 p 0 = 1. {\displaystyle a_{0}=1\qquad b_{0}={\frac {1}{\sqrt {2}}}\qquad t_{0}={\frac {1}{4}}\qquad p_{0}=1.} Repeat the following instructions until the difference of a n {\displaystyle a_{n}} and b n {\displaystyle b_{n}} is within the desired accuracy: a n + 1 = a n + b n 2 , b n + 1 = a n b n , t n + 1 = t n − p n ( a n − a n + 1 ) 2 , p n + 1 = 2 p n . {\displaystyle {\begin{aligned}a_{n+1}&={\frac {a_{n}+b_{n}}{2}},\\\\b_{n+1}&={\sqrt {a_{n}b_{n}}},\\\\t_{n+1}&=t_{n}-p_{n}(a_{n}-a_{n+1})^{2},\\\\p_{n+1}&=2p_{n}.\\\end{aligned}}} π is then approximated as: π ≈ ( a n + 1 + b n + 1 ) 2 4 t n + 1 . {\displaystyle \pi \approx {\frac {(a_{n+1}+b_{n+1})^{2}}{4t_{n+1}}}.} The first three iterations give (approximations given up to and including the first incorrect digit): 3.140 … {\displaystyle 3.140\dots } 3.14159264 … {\displaystyle 3.14159264\dots } 3.1415926535897932382 … {\displaystyle 3.1415926535897932382\dots } The algorithm has quadratic convergence , which essentially means that the number of correct digits doubles with each iteration of the algorithm. Mathematical background [ edit ] Limits of the arithmetic–geometric mean [ edit ] The arithmetic–geometric mean of two numbers, a 0 and b 0 , is found by calculating the limit of the sequences a n + 1 = a n + b n 2 , b n + 1 = a n b n , {\displaystyle {\begin{aligned}a_{n+1}&={\frac {a_{n}+b_{n}}{2}},\\[6pt]b_{n+1}&={\sqrt {a_{n}b_{n}}},\end{aligned}}} which both converge to the same limit. If a 0 = 1 {\displaystyle a_{0}=1} and b 0 = cos ⁡ φ {\displaystyle b_{0}=\cos \varphi } then the limit is π 2 K ( sin ⁡ φ ) {\textstyle {\pi \over 2K(\sin \varphi )}} where K ( k ) {\displaystyle K(k)} is the complete elliptic integral of the first kind K ( k ) = ∫ 0 π / 2 d θ 1 − k 2 sin 2 ⁡ θ . {\displaystyle K(k)=\int _{0}^{\pi /2}{\frac {d\the

https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm

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UPDATE #1: If I read one more answer saying that π=227

or π=3.14

I think my head may explode. I know I am sounding like a grumpy old man, but what do they teach in schools nowadays?

UPDATE #2: The same goes for variants of π

is not a number or π

cannot be expressed numerically (which is kinda the same as the first misconception).

UPDATE #3: The use of π

was coined by Welsh Mathematician, William Jones, a contemporary of Newton. Various abbreviations of the word periphery had been used by others before, but Jones borrowed from its spelling in the Greek alphabet: περιφέρεια.

UPDATE #4: Thank you to readers for making this my fourth answer to get more than 300,000 views. Your support is appreciated.

UPDATE #5: I am aware that G-L is pretty inefficient by comparison to more modern algorithms. It is however easier to understand and - as I say - it’s just an example.