1.3 Complex or Argand Plane

1.3 Complex or Argand Plane

A complex number $z = x + iy$ is exclusively determined by an ordered pair of real number $(x, y)$. The numbers $2 - 5i, 8$ and $-7i$ are equivalent to $(2, -5), (8, 0)$ and $(0, -7)$ respectively. In this manner, a complex number $z = x + iy$ can be represented by the point $(x, y)$ in the coordinate plane.

If we consider $x$-axis as real axis and $y$-axis as imaginary axis to represent a complex number, then the $xy$-plane is called complex plane or Argand plane. It is named in honour of the Swiss mathematician Jean Argand (1768-1822).

A complex number is represented not only by a point, but also by a position vector pointing from the origin to the point. The complex number, the corresponding point and the vector are all typically denoted by the same symbol, $z$. Geometrically, a complex number can be interpreted either as a point in the complex plane $C$ or as a vector in the Argand plane.

[Diagram of complex plane with point $a+ib$]
Complex number as a point.
Figure 1.8

[Diagram of complex plane with vector $a+ib$]
Complex number by a position vector pointing from origin to the point.
Figure 1.9

Here some complex numbers are plotted on the complex plane.

[Graph of complex plane with points $-3+4i$, $4+i$, $-3i$, $3-4i$]
Figure 1.10