Machin-like formula

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

$\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239},$

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

Machin-like formulas have the form

$\frac{\pi}{4} = \sum_{n}^N a_n \arctan\frac{1}{b_n}$

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.

1 Derivation
2 Two-term formulas
3 More terms
4 External links

[edit] Derivation

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

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

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

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

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

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

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

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

$q = \frac{\tan\left(4 \arctan\left(\frac{1}{5}\right)\right) - 1} { 1 + \tan\left(4 \arctan\left(\frac{1}{5}\right)\right)}$

Similarly, two applications of the double angle identity yields

$\tan\left(4 \arctan\left(\frac{1}{5}\right)\right) = \frac{120}{119}$

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 $\arctan\frac{b}{a}$ and when you multiply complex numbers you add their angles. If a=b then $\arctan\frac{b}{a}$ 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 $(5 + 5 i)$ . Therefore $\arctan\frac{1}{2} + \arctan\frac{1}{3} = \frac{\pi}{4}$ .

If you want to use complex numbers to show that $\frac{\pi}{4} = 4\arctan\frac{1}{5} - \arctan\frac{1}{239}$ 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 $( - 239 + i) = ( - 114244 - 114244 i)$ since the real part and imaginary part are equal $4\arctan\frac{1}{5} - \arctan\frac{1}{239} = \frac{\pi}{4}$

[edit] Two-term formulas

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

$\frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3}$ ,

Hermann's,

$\frac{\pi}{4} = 2 \arctan\frac{1}{2} - \arctan\frac{1}{7}$ ,

and Hutton's

$\frac{\pi}{4} = 2 \arctan\frac{1}{3} + \arctan\frac{1}{7}$ .

[edit] 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:

$\frac{\pi}{4} = 12 \arctan\frac{1}{49} + 32 \arctan\frac{1}{57} - 5 \arctan\frac{1}{239} + 12 \arctan\frac{1}{110443}$

Kikuo Takano (1982).

$\frac{\pi}{4} = 44 \arctan\frac{1}{57} + 7 \arctan\frac{1}{239} - 12 \arctan\frac{1}{682} + 24 \arctan\frac{1}{12943}$

F. C. W. Störmer (1896).

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

$\begin{align} \frac{\pi}{4} =& 183\arctan\frac{1}{239} + 32\arctan\frac{1}{1023} - 68\arctan\frac{1}{5832} + 12\arctan\frac{1}{110443}\\ & - 12\arctan\frac{1}{4841182} - 100\arctan\frac{1}{6826318}\\ \end{align}$

黃見利(Hwang Chien-Lih) (1997).

$\begin{align} \frac{\pi}{4} =& 183\arctan\frac{1}{239} + 32\arctan\frac{1}{1023} - 68\arctan\frac{1}{5832} + 12\arctan\frac{1}{113021}\\ & - 100\arctan\frac{1}{6826318} - 12\arctan\frac{1}{33366019650} + 12\arctan\frac{1}{43599522992503626068}\\ \end{align}$