= Complex plane
{title2=$\mathbb C\cong\mathbb R^2$}
{wiki}
= Argand plane
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{synonym}
= Argand diagram
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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 $\sqrt2$. More generally, <multiplication> by $a+ib$ acts through the real <matrix>
$$
\begin{pmatrix}x\\y\end{pmatrix}\longmapsto
\begin{pmatrix}a&-b\\b&a\end{pmatrix}
\begin{pmatrix}x\\y\end{pmatrix}.
$$
For $a+ib\ne0$, 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 https://fr.wikisource.org/wiki/Page:Annales_de_math%C3%A9matiques_pures_et_appliqu%C3%A9es,_1813-1814,_Tome_4.djvu/141[Argand, Annales de mathématiques, volume 4, page 133]. A https://www.hist-math.fr/textes/Argand1874_QuantitesImaginaires.pdf[BnF scan of the 1874 second edition]{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.
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