Radices sophisticae, racines imaginaires: the Origin of Complex Numbers in the Late Renaissance

The aim of this paper is to clarify what is meant by the ‘’invention of complex numbers’’ by the Renaissance Italian algebraists Girolamo Cardano and Rafael Bombelli. The paper demonstrates that, despite the radix sophistica found in Cardano's Ars Magna that indicates the expressions a ±b√(-1), Cardano could not arithmetically operate with them, because he was not able to determine the sign of the square root of a negative number. Viceversa, Bombelli overcame this problem by inventing not “imaginary numbers”, but rather the new signs “plus of minus” (più di meno) and “minus of minus” (meno di meno) and their rules of composition. The radices sophisticae of Cardano and Bombelli were thus entities able to give meaning to Tartaglia's solution formula for a cubic equation in the irreducible case, just as the racines imaginaires of Albert Girard and René Descartes gave meaning to the first (weak) formulations of the Fundamental Theorem of Algebra. Ultimately, I show that in the late Renaissance the radice sophisticae or racines imaginaires were something quite different from the modern “complex numbers”, essentially because they appeared only as a useful tool to solve problems, and not yet as a true mathematical object to be studied.