Argand Diagram or Complex Plane

Argand Diagram or Complex Plane

A complex number $z=x+iy$, with $x,y\in\mathbb{R}$ and $i=\sqrt{-1}$, can be represented as a point in a plane whose coordinates are the ordered pair $(x,y)$. Jean Robert Argand used exactly this one-to-one correspondence between complex numbers and points in a plane.

Let $z=x+iy$ be a complex number. Then the point $P(x,y)$ represents the complex number $z=x+iy$ (Fig. 1.2), i.e. $x+iy\equiv(x,y)$. Here $x=\mathrm{Re}(z)$ is represented on the X-axis, so the X-axis is called the real axis. Similarly, $y=\mathrm{Im}(z)$ is represented on the Y-axis, so the Y-axis is called the imaginary axis.

Examples of the correspondence:

1. $(1,2)\equiv1+2i$
2. $-4+3i\equiv(-4,3)$
3. $(0,0)\equiv0+0i$
4. $5+0i\equiv(5,0)$
5. $(0,-1)\equiv0-i$
6. $-2-2i\equiv(-2,-2)$ …