Complex Numbers in Algebra

Abstract

Over the last few chapters it has often been claimed that certain mysteries— de Moivre’s formula for sin nθ (Section 5.6), the factorization of polynomials (Section 5.7), the classification of cubic curves (Section 7.4), branch points (Section 9.5), genus (Section 10.3), and the behavior of elliptic functions (Sections 10.6 and 11.6)—are clarified by the introduction of complex numbers. That complex numbers do all this and more is one of the miracles of mathematics. At the beginning of their history, complex numbers $a + b\sqrt { - 1}$ were considered to be “impossible numbers,” tolerated only in a limited algebraic domain because they seemed useful in the solution of cubic equations. But their significance turned out to be geometric and ultimately led to the unification of algebraic functions with conformai mapping, potential theory, and another “impossible” field, noneuclidean geometry. This resolution of the paradox of $\sqrt { - 1}$ was so powerful, unexpected, and beautiful that only the word “miracle” seems adequate to describe it.