Complex plane

The complex plane identifies the complex number $x+iy$ with the point $(x,y)$ in Cartesian coordinates. The horizontal coordinate is its real part, the vertical coordinate its imaginary part. Arrows from zero identify the same points with vectors. Adding $1+i$ moves $3+4i$ to $4+5i$: every point undergoes the same translation.

Multiplication has a richer geometry. Multiplying by the imaginary unit sends $x+iy$ to $-y+ix$, a counterclockwise quarter-turn rotation. In particular $i(3+4i)=-4+3i$. Multiplying by $1+i$ rotates through $\pi /4$ and scales every Euclidean distance by $\sqrt{2}$. More generally, multiplication by $a+ib$ acts through the real matrixFor $a+ib\ne 0$, this preserves oriented angles between nonzero displacement vectors: every direction turns equally. Its determinant is $a^2+b^2>0$, while its common scale is $\sqrt{a^2+b^2}$. This is the elementary geometric reason complex numbers are so useful for conformal maps.

The name Argand plane recalls Jean-Robert Argand's geometric interpretation. In his 1813 article he explicitly reports publishing his Essai in 1806; the original article is available in Argand, Annales de mathématiques, volume 4, page 133. A ^[ref] preserves the essay and its geometric drawings. Reading these alongside a modern complex plane reveals how directions and lengths give arithmetic a visible meaning.