Complex numbers Definitions

A complex number $z$ is specified by a pair of real numbers $x$ and $y$ and written as

where $i$ is the imaginary unit which has the basic property

The set of complex numbers is denoted by $\mathbb{C}$. Two elements $x+i y$ and $u+i v$ are equal $iff$ $x=u$ and $y=v$. Given $z=x+i y$ in $\mathbb{C}, x$ is the real part of $x$ and $y$ is the imaginary part,

It is convenient to represent complex numbers geometrically as points of the complex plane known as the Argand diagram shown below.

Alternatively, use polar coordinates defined through $x=r \cos \theta, y=r \sin \theta$.

Modulus & argument

The modulus $|z|$ of $z=x+i y$ is defined to be,

which is the distance from $z$ to the origin in the complex plane. The angle $\theta$ in the polar form $z=r(\cos \theta+i \sin \theta)$ is the argument of $z$, denoted as $\arg z$. Note that the angle $\theta$ is not unique: replacing $\theta$ by $\theta+2 \pi$ yields the same $z$ (this is easy to see from the figures above). For example, $z=0+i 1$ which is normally written as $i$ has modulus 1 and $\operatorname{argument} \arg (i)=\pi / 2$ as well as $\pi / 2+2 n \pi$ where $n$ is an integer.

To remove this ambiguity, we restrict the range of $\theta$ to $-\pi<\theta \leq \pi$ (or another interval of length $2 \pi$ ). Then the principal value of the argument of $z$ (denoted $\operatorname{Arg}(z)$, i.e. with upper case A) is the argument $\theta$ for which $-\pi<\theta \leq \pi$. Note that we use $-\pi<\theta \leq \pi$ and not $-\pi \leq \theta \leq \pi$ to avoid multiplicity at the point $\theta= \pm \pi$, which represents the same point on the plane. So, while $\operatorname{Arg}(z)$ is $\operatorname{single-valued,~} \arg (z)$ given by

is not. It is a multivalued function because for a given complex number $z, \arg (z)$ represents an infinite number of possibilities. Although multivalued, it becomes especially useful when we study the properties of the complex logarithm.

For example, if we wanted to calculate $\arg (z)$ and $\operatorname{Arg}(z)$ where $z=1-i \sqrt{3}$. The polar form is:

where $r=|z|=|1-\sqrt{3}|=2$. Now, comparing $z=1-i \sqrt{3}$

which gives $\tan \theta=-\sqrt{3}$.

The point (circle marker) on the Argand diagram above shows $z=1-\sqrt{3}$ and $\theta$ is the angle taken anti-clockwise between the positive $x$ axis and the line segment. Now, $\operatorname{Arg}(z)$ must be such that $-\pi<\operatorname{Arg}(z) \leq \pi$; this gives

and

for integer $n$. Note that we can show $\tan (\pi / 3)=\sqrt{3}$ using an equilateral triangle of side 2, separated into 2 right triangles as shown below. Using Pythagoras' theorem, the perpendicular has length $\sqrt{3}$ and therefore $\tan \theta=\sqrt{3}$.

Algebra in the complex plane

The rules of addition are the same as for real numbers,

Multiplication rules are also the same but with the additional condition $i^{2}=-1$.,

Note that

for any $z_{1}, z_{2}$ in $\mathbb{C}$. As for real numbers, we have the following commutative laws,

and associative laws,

for any $z_{1}, z_{2}, z_{3}$ in $\mathbb{C}$. For $z=x+i y$ in $\mathbb{C},-z$ is defined to be $(-x)+i(-y)$. Clearly, $z+(-z)=0$.

Complex conjugation

The conjugate of a complex number $z=x+i y$ in $\mathbb{C}$ is denoted by $\bar{z}$ (or $z^{*}$ ) and is the complex number $\bar{z}=x-i y$, i.e. it is the original complex number with the sign of the imaginary part reversed. In polar form, $\bar{z}=r e^{-i \theta}$ if $z=r e^{i \theta}$. In the Argand diagram, $\bar{z}$ is the reflection of $z$ about the real axis. The following hold if $z, z_{1}, z_{2}$ are complex numbers:

1. $\overline{\bar{z}}=z ;$
2. $\operatorname{Re}(z)=\frac{z+\bar{z}}{2}, \quad \operatorname{Im}(z)=\frac{z-\bar{z}}{2 i}$;
3. $\overline{z_{1}+z_{2}}=\overline{z_{1}}+\overline{z_{2}}$;
4. $\overline{z_{1} \cdot z_{2}}=\overline{z_{1}} \cdot \overline{z_{2}}$;
5. $|\bar{z}|=|z|$.

Complex reciprocal

For $z=x+i y \neq 0$, the reciprocal, $1 / z$ (also written as $z^{-1}$ ) is defined through

To derive this result, start from the reciprocal of a complex number $z=x+i y$ and multiply by $\bar{z} / \bar{z}$, where $\bar{z}$ is the conjugate:

yielding a real denominator as

from which the result follows. Like real numbers, the reciprocal has the property

Note that

To obtain the reciprocal in polar form, and starting from the equation, we have

Or simply, by replacing in the polar form $z=r \cos \theta+i r \sin \theta, r$ with $1 / r$ and $\theta$ with $-\theta$.