Argand Plane

A complex number $z=x+iy$ is uniquely determined by the ordered pair of real numbers $(x,y)$: for instance $3-6i,\ 8,$ and $-4i$ correspond to $(3,-6),\ (8,0)$ and $(0,-4)$ respectively. This lets us associate a complex number $z=x+iy$ with a point $(x,y)$ in a coordinate plane. Taking the $x$-axis as the real axis and the $y$-axis as the imaginary axis, this plane is called the complex plane or Argand plane, named after the Swiss mathematician Jean Argand (1768–1822).

A complex number is represented not just by a point, but also by the position vector pointing from the origin $O$ to that point — the number, the point and the vector are all denoted by the same letter $z$. (As with ordinary vectors, we identify vectors that differ only by a parallel displacement.) Geometrically, then, a complex number can be viewed either as a point in $\mathbb R^2$ or as a vector in the Argand plane.

Example. Complex numbers such as $2+i,\ -1-2i,\ 3-2i,\ -\dfrac32-\dfrac32 i,\ -2-3i$, and $\cos\dfrac\pi6+i\sin\dfrac\pi6$ can all be plotted as points (or drawn as arrows from the origin) in one Argand diagram. …