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:

$frac\{pi\}\{4\}\; =\; arctan(1)$

$tan(2arctan(a))\; =\; frac\{2\; a\}\; \{\; 1\; -\; a\; ^\; 2\}$ (tangent double angle identity)

$tan(a-arctan(b))\; =\; frac\{tan(a)-b\}\; \{\; 1\; +\; tan(a)\; b\}$ (tangent difference identity)

$frac\{pi\}\{16\}\; =\; 0.196349dots$ (approximately)

$arctanleft(frac\{1\}\{5\}right)\; =\; arctan(0.2)\; =\; 0.197395dots$ (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 $q$:

$q\; =\; tanleft(4\; arctanleft(frac\{1\}\{5\}right)\; -\; frac\{pi\}\{4\}right)$

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

$q\; =\; frac\{tanleft(4\; arctanleft(frac\{1\}\{5\}right)right)\; -\; 1\}\; \{\; 1\; +\; tanleft(4\; arctanleft(frac\{1\}\{5\}right)right)\}$

Similarly, two applications of the double angle identity yields

$tanleft(4\; arctanleft(frac\{1\}\{5\}right)right)\; =\; frac\{120\}\{119\}$

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:

$$

begin{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).

$$

begin{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

• The constant π
• Lists of Machin-type
• Machin's Merit at MathPages