The Argand Plane: Picturing Complex Numbers and Their Operations Geometrically

Gauss popularised the idea, now standard, of plotting a complex number $z=x+iy$ as the point $(x,y)$ in a plane with perpendicular axes: the horizontal axis (the real axis) carries the real numbers, and the vertical axis (the imaginary axis) carries the purely imaginary numbers. This picture is called the Argand plane, and a specific drawing of points/vectors on it is an Argand diagram. Since $z=x+iy$ corresponds to a unique point $(x,y)$ and vice versa, every algebraic fact about complex numbers has a geometric shadow — and several operations become easy to see rather than compute.

Addition and subtraction — the parallelogram law. If $z_1$ and $z_2$ are represented by points $P_1$ and $P_2$, then $z_1+z_2$ is the fourth vertex of the parallelogram built on $OP_1$ and $OP_2$ (the familiar 'vector addition' picture — because complex addition is literally coordinate-wise addition, exactly like adding position vectors). Subtraction $z_1-z_2$ works the same way after first reflecting $z_2$ through the origin to $-z_2$.

Multiplication and division — rotate and scale. Section 1.4 showed algebraically that multiplying $z_1=r_1\,\mathrm{cis}\,\theta_1$ by $z_2=r_2\,\mathrm{cis}\,\theta_2$ gives $r_1r_2\,\mathrm{cis}(\theta_1+\theta_2)$. Geometrically this means: to multiply by $z_2$, take $z_1$'s point, rotate it about the origin by the angle $\theta_2$, and scale its distance from the origin by the factor $r_2$. This can be shown directly with similar triangles: constructing a triangle at $z_1$ similar to the triangle formed by $z_2$ and the reference point $(1,0)$ produces exactly the point $r_1r_2\,\mathrm{cis}(\theta_1+\theta_2)$. Division is the same idea with the rotation and scaling reversed. …