Formulate Thing

The formula of Thing was discovered in 1706 by John Machin and connects the number $\ pi$ to the function Arctangente:

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

This formula makes it possible to calculate an approximation of the number $\ pi$ thanks to the development in whole series of the function arctangente. John Machin used it to obtain the first 100 decimals of $\ pi$ .

Demonstration

It is possible to show the formula of Thing, by using the trigonometrical relation elementary following:

$\ tan (has + b) =$

By posing $\ theta = \ arctan {1 \ over 5}$ , one from of deduced then successively:

$\ tan 2 \ theta = =$

$\ tan 4 \ theta = = {120 \ over 119}$

$\ tan (4 \ theta - {\ pi \ over 4}) = = {1 \ over 239}$

Like $\ theta, \ theta^2, \ theta^4 \ in] 0; 2 \ pi one a: {\ pi \ over 4} = 4 \ times \ arctan {1 \ over 5} - \ arctan {1 \ over 239}$

A modern way to have the result is to connect it to the properties Complex numbers. The formula of Thing rises then from the following identity between complex numbers: ${(5+i) ^4 \ over (239+i)}=2 \ times (1+i)$

Use

The development of $\ arctan \,$ in whole series provides the following method of calculating:

${\ pi \ over 4} = 4* \ sum_ {n=0} ^ {\ infty} {(- 1) ^n \ over {2n+1}} - \ sum_ {n=0} ^ {\ infty} {(- 1) ^n \ over {2n+1}}$

Formulas of the type of Thing

Other formulas of the same type were discovered, and " is called; formulas of the type of Machin" formulas of the form:

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

where the $a_n$ and the $b_n$ are Entier S.

There only exist three other formulas of the type of Thing with two terms. They were respectively discovered by Euler, Hermann and Hutton:

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

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

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

They rise respectively from the following identities between complex numbers:

${(2+i) * (3+i) =5* (1+i)}$

{(2+i) ^2 \ over (7+i)}=2* (1+i)

{(3+i) ^2* (7+i) =50* (1+i)}

It is in fact possible to build an infinity of formulas of this type by using more the most effective terms, but only formulas historically to calculate the number $\ pi$ became famous.

\ 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).

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

Kikuo Takano (1982).

The search for effective formulas of Thing is done from now on by a systematic research using computers. The most effective formulas of the type of Thing currently known to calculate $\ pi$ are:

\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 Dog-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}

黃見利 (Hwang Dog-Lih) (2003).

There exist other formulas which converge more quickly towards the number $\ pi$ , as the formula of Ramanujan, but they are not of the type of Thing.

See too

trigonometrical Formulas
John Thing
whole Series

Random links: