Mathematics and Applied Mathematics
June 28, 2021
arctangent, Machin formula, π calculations
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.