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In current French mathematical tradition, when introducing complex numbers, it is common to hear about "complex plane of Argand-Cauchy". What is particular in French treatment, it is the differentiation between the geometric objects of a "complex plane" and complex numbers: complex numbers are complex coordinates of vectors and points of a complex plane. To each vector and each point of a complex plane, the French associate a complex number, called their affixe (probably this would read affix in English).

I tried to find the origin of the term "affixe" and that of the French treatment of the geometry of complex numbers in the literature but failed so far. I've looked through the essays of Caspar Wessel and of Jean-Robert Argand, but they do not use "affixes", and they could not, because for them complex numbers were vectors.

Who, when, and how used the term affixe/affix for the first time? Same question about differentiating between complex numbers and their geometric realisations.

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    $\begingroup$ I searched "Google Books" and found a book "Mathematics of the 19th Century: Geometry, Analytic Function Theory, Volume 2 edited by Andrei N. Kolmogorov, Adolf-Andrei P. Yushkevic". The authors attribute the term affix to Cauchy. See pg 124. $\endgroup$
    – AChem
    Jul 13, 2022 at 23:04
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    $\begingroup$ @AChem - good point! See AL Cauchy, Mémoir sur la quantité géométrique $i=1_{\frac {\pi}{2}}$ (1847?) [see page 245 of Tome 26 of Oeuvres Completes]: "[Soit $r,p$ les coordonnés polares d'un point $A$ ...] on pose $z=r(\cos p + i \sin p)$, la quantité géométrique $z$ serace que nous appellerons l' affixe de ce point." $\endgroup$
    – Mauro ALLEGRANZA
    Jul 14, 2022 at 7:53

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According to Eduard Study and Élie Cartan [1], it was Cauchy who introduced the term affixe in Sur la quantité géométrique $\mathrm{i} = 1_{\frac{\pi}{2}}$, et sur la réduction d'une quantité géométrique quelconque à la forme $x + y\mathrm{i}$, in Exercices d'analyse et de physique mathématique, Tome 4 (1847).

Cauchy was only talking about affixes of points, while vectors were basically identified with complex numbers, which he called "geometric quantities."

[1]: E. Study and É. Cartan, Les nombres complexes (1908), in Encyclopédie des sciences mathématiques pures et appliquées, Tome 1, Volume 1, edited by Jules Molk.

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