mathematics, Machin-like formulas are a class of identities involving π = 3.14159... that generalize John Machin's formula from 1706:
which he used along with the
Taylor seriesexpansion of arctanto compute π to 100 decimal places.
Machin-like formulas have the form
The same method is still among the most efficient known for computing a large number of digits of π with digital
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
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
There are exactly three additional Machin-like formulas with two terms; these are Euler's
The current record for digits of π, 1,241,100,000,000, by
Yasumasa Kanadaof Tokyo University, was obtained in 2002. A 64-node Hitachi supercomputerwith 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 (
The more efficient currently known Machin-like formulas for computing:
: :黃見利(Hwang Chien-Lih) (
: :黃見利(Hwang Chien-Lih) (
* [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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