The Complex Intermediate Route Algorithm (CIRA): Coordinate Symmetry in the Computation of Square Roots of Complex Numbers

Standard methods for computing square roots of complex numbers impose aconventional restriction requiring intermediate variables to be real. While computationally efficient, this restriction is rarely justified conceptually. In this paper,we introduce the Complex Intermediate Route Algorithm (CIRA), a symmetrypreserving framework that allows intermediate variables to take complex valueswithout altering the resulting roots. Using the language of field extensions andGalois theory, we show that such intermediate complex solutions correspond to alternative coordinate representations of the same element in the quadratic splittingfield. The so-called y · i operation is identified as a canonical coordinate rectification rather than an algebraic correction or automorphism. CIRA clarifies thatrestricting intermediates to real values is a convention rather than a necessity, whilepreserving all underlying algebraic symmetries