Introduction to the Complex Plane

Imaginary numbers had no use when it was invented except for intellectual exploration until it was linked to coordinate geometry.  The complex number $a + bi$ can be plotted as the ordered pair $(a,b)$ where $a$ is the real part and $b$ is the imaginary part.  Therefore, the complex numbers $3 + 2i$, $3- 2i$, $-2 + 0i$,  and $0 - i$ can be plotted as $(3,2)$, $(3,-2)$, $(-2,0)$, and $(0,-1)$ in a “coordinate plane” respectively as shown below.  In the figure, we can see that the origin is $0 + 0i$ and that $3 - 2i$, the complex conjugate of $3 + 2i$, is its reflection along the x-axis (Can this be generalized?).

The plane where the complex numbers are plotted above is called the complex plane.

In the complex plane, we can observe that all numbers of the form $(a,0)$ are real numbers and these numbers are represented by the horizontal axis. The numbers of the form $(0,bi)$ are all imaginary numbers and it is represented by the vertical axis. Thus, we can call them  the real axis and the imaginary axis respectively.

In addition, we can conclude that all real numbers are complex numbers since  for all  real numbers  $a$, $a = a + 0i$. So, the set of real numbers is a subset of the set of complex numbers.