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Machin-like formula

In mathematics, Machin-like formulas are a class of identities involving &pi; = 3.14159... that generalize John Machin's formula from 1706:

:$frac\left\{pi\right\}\left\{4\right\} = 4 arctanfrac\left\{1\right\}\left\{5\right\} - arctanfrac\left\{1\right\}\left\{239\right\},$

which he used along with the Taylor series expansion of arctan to compute π to 100 decimal places.

Machin-like formulas have the form

:$frac\left\{pi\right\}\left\{4\right\} = sum_\left\{n\right\}^N a_n arctanfrac\left\{1\right\}\left\{b_n\right\}$

with $a_n$ and $b_n$ integers.

The same method is still among the most efficient known for computing a large number of digits of π with digital computers.

Derivation

To understand where this formula comes from, start with following basic ideas:

:$frac\left\{pi\right\}\left\{4\right\} = arctan\left(1\right)$:$an\left(2arctan\left(a\right)\right) = frac\left\{2 a\right\} \left\{ 1 - a ^ 2\right\}$ (tangent double angle identity):$an\left(a-arctan\left(b\right)\right) = frac\left\{ an\left(a\right)-b\right\} \left\{ 1 + an\left(a\right) b\right\}$ (tangent difference identity):$frac\left\{pi\right\}\left\{16\right\} = 0.196349dots$ (approximately):$arctanleft\left(frac\left\{1\right\}\left\{5\right\} ight\right) = arctan\left(0.2\right) = 0.197395dots$ (approximately)

In other words, for small numbers, arctangent is to a good approximation just the identity function. This leads to the possibility that a number $q$ can be found such that

:$frac\left\{pi\right\}\left\{16\right\} = arctan\left(frac\left\{1\right\}\left\{5\right\}\right) - frac\left\{1\right\}\left\{4\right\} arctan\left(q\right).$

Using elementary algebra, we can isolate $q$:

:$q = anleft\left(4 arctanleft\left(frac\left\{1\right\}\left\{5\right\} ight\right) - frac\left\{pi\right\}\left\{4\right\} ight\right)$

Using the identities above, we substitute arctan(1) for π/4 and then expand the result.

:$q = frac\left\{ anleft\left(4 arctanleft\left(frac\left\{1\right\}\left\{5\right\} ight\right) ight\right) - 1\right\} \left\{ 1 + anleft\left(4 arctanleft\left(frac\left\{1\right\}\left\{5\right\} ight\right) ight\right)\right\}$

Similarly, two applications of the double angle identity yields

:$anleft\left(4 arctanleft\left(frac\left\{1\right\}\left\{5\right\} ight\right) ight\right) = frac\left\{120\right\}\left\{119\right\}$

and so

:$q = frac\left\{frac\left\{120\right\}\left\{119\right\} - 1\right\}\left\{1 +frac\left\{120\right\}\left\{119 = frac\left\{1\right\}\left\{239\right\}.$

Other formulas may be generated using complex numbers. For example the angle of a complex number a+bI is given by $arctanfrac\left\{b\right\}\left\{a\right\}$ and when you multiply complex numbers you add their angles. If a=b then $arctanfrac\left\{b\right\}\left\{a\right\}$ is 45 degrees or $frac\left\{pi\right\}\left\{4\right\}$. This means that if the real part and complex part are equal then the arctangent will equal $frac\left\{pi\right\}\left\{4\right\}$. Since the arctangent of one has a very slow convergence rate if we find two complex numbers that when multiplied will result in the same real and imaginary part we will have a Machin-like formula. An example is $\left(2 + i\right)$ and $\left(3 + i\right)$. If we multiply these out we will get $\left(5 + 5i\right)$. Therefore $arctanfrac\left\{1\right\}\left\{2\right\} + arctanfrac\left\{1\right\}\left\{3\right\} = frac\left\{pi\right\}\left\{4\right\}$.

If you want to use complex numbers to show that $frac\left\{pi\right\}\left\{4\right\} = 4arctanfrac\left\{1\right\}\left\{5\right\} - arctanfrac\left\{1\right\}\left\{239\right\}$ you first must know that when multiplying angles you put the complex number to the power of the number that you are multiplying by. So $\left(5 + i\right)$4$\left(-239+i\right) = \left(-114244 - 114244i\right)$ since the real part and imaginary part are equal $4arctanfrac\left\{1\right\}\left\{5\right\} - arctanfrac\left\{1\right\}\left\{239\right\} = frac\left\{pi\right\}\left\{4\right\}$

Two-term formulas

There are exactly three additional Machin-like formulas with two terms; these are Euler's

:$frac\left\{pi\right\}\left\{4\right\} = arctanfrac\left\{1\right\}\left\{2\right\} + arctanfrac\left\{1\right\}\left\{3\right\}$,

Hermann's,

:$frac\left\{pi\right\}\left\{4\right\} = 2 arctanfrac\left\{1\right\}\left\{2\right\} - arctanfrac\left\{1\right\}\left\{7\right\}$,

and Hutton's

:$frac\left\{pi\right\}\left\{4\right\} = 2 arctanfrac\left\{1\right\}\left\{3\right\} + arctanfrac\left\{1\right\}\left\{7\right\}$.

More terms

The current record for digits of π, 1,241,100,000,000, by Yasumasa Kanada of Tokyo University, was obtained in 2002. A 64-node Hitachi supercomputer with 1 terabyte of main memory, performing 2 trillion operations per second, was used to evaluate the following Machin-like formulas:

:$frac\left\{pi\right\}\left\{4\right\} = 12 arctanfrac\left\{1\right\}\left\{49\right\} + 32 arctanfrac\left\{1\right\}\left\{57\right\} - 5 arctanfrac\left\{1\right\}\left\{239\right\} + 12 arctanfrac\left\{1\right\}\left\{110443\right\}$: Kikuo Takano (1982).

: $frac\left\{pi\right\}\left\{4\right\} = 44 arctanfrac\left\{1\right\}\left\{57\right\} + 7 arctanfrac\left\{1\right\}\left\{239\right\} - 12 arctanfrac\left\{1\right\}\left\{682\right\} + 24 arctanfrac\left\{1\right\}\left\{12943\right\}$:F. C. W. Störmer (1896).

The more efficient currently known Machin-like formulas for computing:

: :黃見利(Hwang Chien-Lih) (1997).

: :黃見利(Hwang Chien-Lih) (2003).

*
* [http://numbers.computation.free.fr/Constants/Pi/piclassic.html The constant π]
* [http://www.mathpages.com/home/kmath373.htm Machin's Merit] at MathPages

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