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New geometrical derivations of Machin type equation

Author:

Peled, Aaron

Category:

Research Papers

Sub-Category:

Mathematics and Applied Mathematics

Language:

English

Date Published:

June 28, 2021

File:

/Research Papers-Mathematics and Applied Mathematics/Download/8842

Downloads:

1200

Keywords:

arctangent, Machin formula, π calculations

Abstract:

Two Machin type equations were derived by summing arctangent functions giving π. New triangles constructions on a Cartesian squares lattice were used to obtain by induction these equations, as combinations of single variable functions f(x), linear in x and 1/x. Using the Wolfram Alpha mathematical engine (ME) we found that the equations hold true for most of the real numbers domain x ∊ ℝ such as the algebraic, irrational and transcendental excluding some integer numbers to avoid division by zero in denominators. A further generalization to the complex numbers domain x ∈ ∁ and under specific conditions also for hyper complex numbers such as quaternions was also tested numerically.

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