Argand's "Reflexions" of 1815 -- An English Translation

TL;DR

This paper provides an English translation of Argand's 1815 work, highlighting his early proof of the fundamental theorem of algebra using complex plane representations, which influenced later mathematical proofs.

Contribution

It presents the first English translation of Argand's work, emphasizing his novel approach to complex numbers and his pioneering proof of the fundamental theorem of algebra.

Findings

Argand's proof is simple and direct.

He introduced complex plane representation in 1806.

His proof influenced 19th-century mathematics.

Abstract

This is a translation from French into English of Argand's "Reflexions sur la nouvelle th\'eorie des imaginaires, suivies d'une application \`a la d\'emonstration d'un th\'eor\`eme d'analise", published in 1815. Argand reprises the method of representing complex numbers as points in the plane, which he first introduced in 1806. He takes up complex addition, multiplication, division, root taking, and absolute value. He gives an early and perhaps the first valid proof of the fundamental theorem of algebra, assuming only that the absolute value of a polynomial with complex coefficients assumes an absolute minimum in the complex plane. Argand's proof is simple and direct, variants of it being reproduced by Cauchy and textbook writers throughout the nineteenth century.