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Concept
Machin-like formula
Summary
In mathematics, Machin-like formulae are a popular technique for computing pi (the ratio of the circumference to the diameter of a circle) to a large number of digits. They are generalizations of John Machin's formula from 1706:
which he used to compute pi to 100 decimal places.
Machin-like formulas have the form
where is a positive integer, are signed non-zero integers, and and are positive integers such that .
These formulas are used in conjunction with Gregory's series, the Taylor series expansion for arctangent:
The angle addition formula for arctangent asserts that
if
All of the Machin-like formulas can be derived by repeated application of equation . As an example, we show the derivation of Machin's original formula one has:
and consequently
Therefore also
and so finally
An insightful way to visualize equation is to picture what happens when two complex numbers are multiplied together:
The angle associated with a complex number is given by:
Thus, in equation , the angle associated with the product is:
Note that this is the same expression as occurs in equation . Thus equation can be interpreted as saying that multiplying two complex numbers means adding their associated angles (see multiplication of complex numbers).
The expression:
is the angle associated with:
Equation can be re-written as:
Here is an arbitrary constant that accounts for the difference in magnitude between the vectors on the two sides of the equation. The magnitudes can be ignored, only the angles are significant.
Other formulas may be generated using complex numbers. For example, the angle of a complex number is given by and, when one multiplies complex numbers, one adds their angles. If then is 45 degrees or radians. 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 .
Official source
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