Jean-Robert Argand was born on July 18, 1768. He was a bookkeeper and amateur mathematician, and is remembered for having introduced a geometric interpretation of the complex numbers as points in the Cartesian plane (a discovery that had been anticipated by Caspar Wessel in a paper published in 1799). Argand’s treatment of the subject appeared in a self-published anonymous monograph in 1806, which found its way to Legendre and eventually Jacques Français, who published an article about the idea and some of its ramifications, and asked for help in identifying the originator of the idea. Argand came forward, and a debate ensued in print as to the validity of working with complex numbers as geometric quantities, not merely algebraic objects (Argand and Français arguing in favor of geometry, François-Joseph Servois arguing against). This exchange served to solidify the association between Argand’s name and the geometric representation, and henceforth the name has stuck.
Curiously enough, Servois and Argand nearly share a birthday.
Mathematicians with birth or death anniversaries during the week of July 13 – 19:
July 13: Birthday of John Dee 
July 14: Birthday of Laurence C. Young  (generalized curves); deaths of Lorenzo Mascheroni  (analysis) , Augustin Fresnel  (wave theory of light), and Benjamin Gompertz  (mortality rates)
July 15: Birthdays of Adolph Pavlovich Yushkevich  (historian of mathematics) and Stephen Smale  (differential geometry, Field’s Medalist 1966)
July 16: Birthday of Siegfried Aronhold  (theory of invariants)
July 17: Birthday of Wilhelm Lexis  (time series); deaths of Henri Poincaré  (algebraic topology, polymath) and Adolph Pavlovich Yushkevich 
July 18: Birthdays of Robert Hooke  and Jean-Robert Argand ; death of Abraham Sharp  (calculator, mathematical and astronomical tables)
July 19: Birthdays of François-Joseph Servois  (operator theory) and Charles Briot  (theory of functions)