Complex plane ($\mathbb{C}\cong {\mathbb{R}}^2$)

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.

The region between two rays with a common vertex, with no upper bound on the modulus. In the displayed convention the vertex and boundary rays are excluded. A holomorphic logarithm with $\alpha <\arg z<\beta$ defines powers $z^p=e^{p\mathrm{Log}z}$ there. For real $p>0$ with $p(\beta -\alpha )\le 2\pi$, the power is a conformal bijection onto the sector $p\alpha <\arg w<p\beta$: it multiplies arguments by $p$ and sends moduli bijectively by $r\mapsto r^p$.

Given a compact planar set $K$ and an open neighborhood $V$, a finite square-grid construction gives a compact polygonal neighborhood inside $V$ with finitely many complementary components. Remove thin polygonal corridors disjoint from $K$ joining holes that lie in the same component of $\mathbb{C}\setminus K$, and joining holes in its unbounded component to the exterior. Corridors can also include a prescribed point from each of finitely many specified bounded components. There remains a neighborhood whose bounded holes lie in distinct bounded components of $\mathbb{C}\setminus K$, each containing its chosen point. Standard small polygonal rounding gives a finite union of planar surfaces with boundary. The index map from its first integral homology group to one integer per bounded hole is an isomorphism. Triangulate a surrounding disc: a one-cycle bounds a two-chain whose coefficients are constant in each complementary component. Zero hole coefficients give a filling supported inside the neighborhood; the face chain of each hole supplies a dual cycle. The cycles can be sums of boundary curves in different components. This uses finite triangulation, not the general Jordan curve theorem.

Represent Cartesian coordinates $(x,y)$ in the plane by the complex number $z=x+iy$, whose real part and imaginary part recover them. Multiplication by $e^{i\theta }$ rotates the coordinates through angle $\theta$, so a rotating-frame calculation becomes a complex exponential change of variable.

For $c\ne 0$ and real $r$, this equation defines a straight line. Writing $z=x+iy$ and $c=u+iv$ gives $2ux+2vy+r=0$, so $c$ supplies a normal direction. Conversely every real affine line $Ax+By+C=0$ has this representation with $c=(A+iB)/2$ and $r=C$. The condition $c\ne 0$ excludes an empty set or the entire complex plane.

The imaginary axis is the set of complex numbers $iy$ with $y\in \mathbb{R}$. It is the vertical coordinate axis in the complex plane.

The complex unit circle is the subgroup $\{z\in \mathbb{C}:|z|=1\}$ under multiplication.

The real axis consists of the points of the complex plane whose imaginary part is zero. These points represent the real numbers.