- Machin-like formula
In
mathematics , Machin-like formulas are a class of identities involving π = 3.14159... that generalizeJohn Machin 's formula from1706 ::
which he used along with the
Taylor series expansion ofarctan to compute π to 100 decimal places.Machin-like formulas have the form
:
with and
integer s.The same method is still among the most efficient known for computing a large number of digits of π with digital
computer s.Derivation
To understand where this formula comes from, start with following basic ideas:
:: (tangent double angle identity): (tangent difference identity): (approximately): (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 can be found such that
:
Using elementary algebra, we can isolate :
:
Using the identities above, we substitute arctan(1) for π/4 and then expand the result.
:
Similarly, two applications of the double angle identity yields
:
and so
:
Other formulas may be generated using complex numbers. For example the angle of a complex number a+bI is given by and when you multiply complex numbers you add their angles. If a=b then is 45 degrees or . This means that if the real part and complex part are equal then the arctangent will equal . 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 and . If we multiply these out we will get . Therefore .
If you want to use complex numbers to show that 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 4 since the real part and imaginary part are equal
Two-term formulas
There are exactly three additional Machin-like formulas with two terms; these are Euler's
:,
Hermann's,
:,
and Hutton's
:.
More terms
The current record for digits of π, 1,241,100,000,000, by
Yasumasa Kanada ofTokyo University , was obtained in 2002. A 64-node Hitachisupercomputer with 1 terabyte of main memory, performing 2 trillion operations per second, was used to evaluate the following Machin-like formulas:::
Kikuo Takano (1982 ).: :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 ).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
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