The geometric interpretation of complex numbers.

The set of real numbers can be regarded as a subset of complex numbers where $Imz=0$.

Complex number $z=x+iy$ is represented on the coordinate plane $Oxy$ by a point with coordinates $(x;y)$. This plane is called the complex plane $C$ (see figure 1), where the $Ox$ axis is called the real axis, and the $Oy$ axis is called the imaginary axis. Thus, a real number $z=x+0i=x$ corresponds to a point on the real axis, and an imaginary number $z=0+iy=iy$ corresponds to a point on the imaginary axis.

It is also possible to represent a complex number in the form of a radius-vector $x,y$ and define it by specifying its length $r$ and angle $\phi$ between the $Ox$ axis and the vector.

The length of this vector is called the modulus of the complex number $$|z|=r=\sqrt{x^2+y^2}\ge 0,$$ and the angle $\phi$ is called the argument of the complex number, denoted by $Argz$. The argument is determined up to the term $2\pi k$ $(k=0,\pm 1,\pm 2,\pm 3,...)$ and for positive values, it is measured counterclockwise from the $Ox$ axis to the vector, while for negative values, it is measured clockwise.

The value of the argument belonging to the interval $(-\pi ,\pi ]$ is called the principal value of the argument and is denoted $argz$. The principal value of the argument for the number $x+iy$ can be computed by the formula $\phi =\arg z=\arctan \left(\frac{y}{x}\right)+k\pi$, where $k=0$ if $z$ lies in the first or fourth quadrants, $k=1$ if $z$ lies in the second quadrant, $k=-1$ if $z$ lies in the third quadrant. If $x=Rez=0$, then $\phi =\pi /2$ when $y=Imz>0$ and $\phi =-\pi /2$ when $y=Imz<0$.

Tags: complex analysis, complex numbers, operations with complex numbers