Almost two centuries of fog

«enveloped in mystery and surrounded by darkness» — C. F. Gauss, Theoria residuorum biquadraticorum, Commentatio secunda (1831)

Gauss blamed not the mathematics but the words: had $+1$, $-1$ and $\sqrt{-1}$ been called direct, inverse and lateral from the very beginning, the fog would never have set in.

Full paragraph (translation of the abstract)

In 1831, reflecting on almost two centuries of confusion surrounding the square root of minus one, Gauss remarked that the subject was «enveloped in mystery and surrounded by darkness» — and blamed this not on the mathematics but on the words. Had $+1$, $-1$ and $\sqrt{-1}$ been called not positive, negative and imaginary, but direct, inverse and lateral, then, in his opinion, the darkness would never have formed.

The term imaginary itself came via Descartes and was cemented by Euler in Introductio in analysin infinitorum (1748), whence it was inherited by modern textbooks. Gauss's proposed correction was not adopted.

Three pieces of technical debt we will be paying off

№	What is broken	Who in history already started fixing it
1	Axes drawn "sideways" (Argand vs. Wessel)	Caspar Wessel, 1799
2	Hermiticity is an algebraic statement, but could have been a mirror	(the talk's know-how)
3	$\mathbb{Z}[i]$ is not the optimal lattice; the right one is $\mathbb{Z}[\omega]$	Gotthold Eisenstein, 1844

References

• Gauss, C. F. (1831) Theoria residuorum biquadraticorum, Commentatio secunda. Göttingische gelehrte Anzeigen, 64, 169–178. sophiararebooks.com
• Euler, L. (1748) Introductio in analysin infinitorum. Lausanne: Marcum-Michaelem Bousquet. scholarlycommons.pacific.edu