# 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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Machin — may refer to: * Alfred Machin, French film director * Alfred Machin (writer), British writer on social evolution * Arnold Machin, British designer ** Machin series of British stamps using his design ** Full list of machin stamps * Cerro Machín,… …   Wikipedia

• John Machin — Infobox Scientist name = John Machin box width = 300px |100px image width = 100px caption = John Machin birth date = 1680 birth place = England death date = 9 June 1751 death place = London, England residence = flag|England citizenship =… …   Wikipedia

• Computing π — Similarly, the more complex approximations of π given below involve repeated calculations of some sort, yielding closer and closer approximations with increasing numbers of calculations.Continued fractionsBesides its simple continued fraction… …   Wikipedia

• Numerical approximations of π — This page is about the history of numerical approximations of the mathematical constant pi;. There is a summarizing table at chronology of computation of pi;. See also history of pi; for other aspects of the evolution of our knowledge about… …   Wikipedia

• Пи (число) — У этого термина существуют и другие значения, см. Пи (значения). Иррациональные числа γ ζ(3)  √2  √3  √5  φ  α  e  π  δ Система счисления Оценка числа …   Википедия

• List of formulae involving π — The following is a list of significant formulae involving the mathematical constant π. The list contains only formulae whose significance is established either in the article on the formula itself, or in the articles on π or Computing π.Classical …   Wikipedia

• Carl Størmer — Infobox Scientist name = PAGENAME box width = image width =150px caption = PAGENAME birth date = September 3, 1874 birth place = Skien death date = August 13, 1957 death place = residence = citizenship = nationality = Norwegian ethnicity = field …   Wikipedia

• List of mathematics articles (M) — NOTOC M M estimator M group M matrix M separation M set M. C. Escher s legacy M. Riesz extension theorem M/M/1 model Maass wave form Mac Lane s planarity criterion Macaulay brackets Macbeath surface MacCormack method Macdonald polynomial Machin… …   Wikipedia

• Zacharias Dase — Johann Dase Johann Martin Zacharias Dase (June 23, 1824, Hamburg September 11, 1861, Hamburg) was a German mental calculator. He attended schools in Hamburg from a very early age, but later admitted that his instruction had little influence on… …   Wikipedia