Square Root of a Multivector of Clifford Algebras in 3D: A Game with Signs

An algorithm is presented to extract the square root from a multivector (MV) in real Clifford algebras Clp,q, where n=p+q≤3, in radicals. It is shown that in Cl3,0, Cl1,2, and Cl0,3 algebras, there are up to four isolated square roots in a case of the most general (generic) MV. The algebra Cl2,1 is an exception and, there, the MV can have up to 16 isolated roots. In addition, a continuum of roots has been found in all Clifford algebras except p+q=1. Examples which clarify computations are provided to illustrate the properties of roots in all n=3 algebras. The results may be useful in solving nonlinear equations, like for example, the Clifford–Riccati equation.