Machin-like formula

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

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

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{pi}{4} = sum_{n}^N a_n arctanfrac{1}{b_n}

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:

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

Using elementary algebra, we can isolate $$q:

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

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

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

Similarly, two applications of the double angle identity yields

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

and so

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

Other formulas may be generated using complex numbers. For example the angle of a complex number a+bI is given by $$arctanfrac{b}{a} and when you multiply complex numbers you add their angles. If a=b then $$arctanfrac{b}{a} is 45 degrees or $$frac{pi}{4}. This means that if the real part and complex part are equal then the arctangent will equal $$frac{pi}{4}. 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 $$(2 + i) and $$(3 + i). If we multiply these out we will get $$(5 + 5i). Therefore $$arctanfrac{1}{2} + arctanfrac{1}{3} = frac{pi}{4}.

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

Two-term formulas

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

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

Hermann's,

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

and Hutton's

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

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

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

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

: $$egin{align}frac{pi}{4} =& 183arctanfrac{1}{239} + 32arctanfrac{1}{1023} - 68arctanfrac{1}{5832} + 12arctanfrac{1}{110443}\& - 12arctanfrac{1}{4841182} - 100arctanfrac{1}{6826318}\end{align}:黃見利(Hwang Chien-Lih) (1997).

: $$egin{align}frac{pi}{4} =& 183arctanfrac{1}{239} + 32arctanfrac{1}{1023} - 68arctanfrac{1}{5832} + 12arctanfrac{1}{113021}\& - 100arctanfrac{1}{6826318} - 12arctanfrac{1}{33366019650} + 12arctanfrac{1}{43599522992503626068}\end{align}:黃見利(Hwang Chien-Lih) (2003).

External links

*
* [http://numbers.computation.free.fr/Constants/Pi/piclassic.html The constant π]
* [http://machination.mysite.wanadoo-members.co.uk/ Lists of Machin-type]
* [http://www.mathpages.com/home/kmath373.htm Machin's Merit] at MathPages