Complex Number Representations

Introduction

Complex numbers have been widely used in modern physics, such as Fourier transform and Quantum mechanics. So I thought it would be important at least to become familiar with some of the basic expressions and algebra for complex numbers, even though they might have been taught in high school.

In this blog post, I would like to cover several basic expressions of complex numbers and some of the simplest algebra based on some of the expressions.

Complex Number

If we assume $\sqrt{-1}$ is valid and $i = \sqrt{-1}$, we could define the following expression as complex number

$$
\begin{align}
c = a + bi
\end{align}
$$

where $a$ and $b$ are real numbers, and $a$ is called the real part of $c$, whereas $b$ is its imaginary part.

Complex Number Polar Coordinates

We could further use Cartesian coordinates $(a, b)$ to uniquely represent any complex number.

$$
c \mapsto (a, b)
$$

It is also equivalent to represent the Cartesian coordinates $(a, b)$ using polar coordinates $(\rho, \theta)$. Here, $\rho$ are also called the modulus or magnitude and $\theta$ is also called the phase of $c$, respectively.

It is trivial to find

$$
\begin{align}
a &= \rho \cos \theta \\
b &= \rho \sin \theta
\end{align}
$$

Conversely,

$$
\begin{align}
\rho &= \sqrt{a^2 + b^2} \\
\theta &= \arctan{\frac{b}{a}}
\end{align}
$$

Therefore, we could represent a complex number using polar coordinates.

$$
\begin{align}
c &= \rho (\cos \theta + i \sin \theta)
\end{align}
$$

One of the reasons to consider using polar coordinates to represent complex numbers is it has better physical interpretations on complex number algebra.

Euler’s Formula

Euler’s Formula states that

$$
e^{i\theta} = \cos \theta + i \sin \theta
$$

This could be simply proved by expanding $e^{i\theta}$, $\cos \theta$ and $\sin \theta$ using Taylor expansions, which I would not elaborate here.

Complex Number Exponential Form

We were surprised to find that a complex number could be further simplified in its expression.

$$
\begin{align}
c &= \rho e^{i\theta}
\end{align}
$$

This expression is called the exponential form of a complex number.

Complex Number Algebra

We have two complex numbers, $c_1$ and $c_2$.

$$
\begin{align}
c_1 = a_1 + b_1 i = \rho_1 e^{i\theta_1} \\
c_2 = a_2 + b_2 i = \rho_2 e^{i\theta_2}
\end{align}
$$

We would like to compute the addition, subtraction, multiplication, and division for them.

The addition and subtraction are trivial.

$$
\begin{align}
c_1 + c_2 &= (a_1 + a_2) + (b_1 + b_2) i \\
c_1 - c_2 &= (a_1 - a_2) + (b_1 - b_2) i
\end{align}
$$

The physical meanings of addition and subtraction are just simply the 2D vector addition and subtraction.

The multiplication and division are slightly tedious.

$$
\begin{align}
c_1 c_2 &= (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 + b_1) i \\
\frac{c_1}{c_2} &= \frac{a_1 a_2 + b_1 b_2}{a_2^2 + b_2^2} + \frac{a_2 b_1 - a_1 b_2}{a_2^2 + b_2^2} i
\end{align}
$$

It is not technically hard to verify the two equations above. But it is just sometimes tedious to compute, and the physical meanings of multiplication and division are not obvious.

If we use the exponential form of complex numbers, life becomes much easier.

$$
\begin{align}
c_1 c_2 &= \rho_1 \rho_2 e^{i(\theta_1 + \theta_2)} \\
\frac{c_1}{c_2} &= \frac{\rho_1}{\rho_2}e^{i(\theta_1 - \theta_2)}
\end{align}
$$

The physical meanings of multiplication and division are polar coordinates rotation with modulus expansion or contraction. Concretely, the modulus of the two complex numbers are multiplied or divided, and the phase of the two complex numbers are added or subtracted.

Based on the multiplication and division algebra, we further have the algebra for powers.

$$
\begin{align}
c^n = \rho^n e^{in\theta}
\end{align}
$$

Imagine the polar representation of $\{c, c^2, c^3, \cdots, c^n\}$ on polar coordinates. Starting from $c$, it is just like rotating the vector of $c$, where each neighboring $c_{i+1}$ and $c_i$ has a phase difference of $\theta$ and the magnitude was $\rho$ times higher.

How about the algebra for roots. Intuitively, we have

$$
\begin{align}
c^{\frac{1}{n}} = \rho^{\frac{1}{n}} e^{i\frac{\theta}{n}}
\end{align}
$$

However, note that

$$
\begin{align}
c &= \rho e^{i(\theta + 2k\pi)}
\end{align}
$$

for $k \in \mathbb{Z}$.

So actually we have

$$
\begin{align}
c^{\frac{1}{n}} &= \rho^{\frac{1}{n}} e^{i\frac{\theta + 2k\pi}{n}} \\
&= \rho^{\frac{1}{n}} e^{i \big( \frac{\theta}{n} + \frac{k}{n} 2\pi \big)} \\
\end{align}
$$

for $k \in \mathbb{Z}$.

What is the number of unique roots could we have? We denote the root for $k$ as $\nu_n^{k}$. It is obvious that we could take $k = 0, 1, 2, \cdots, n-1$ before we see $\nu_n^{0} = \nu_n^{n}$. Therefore, the number of unique roots for complex number is $n$.

We further have a special case for unity where the unity $c = 1 = 1 + 0i = e^{i0}$. The $n$ different roots for the unity are denoted as $\omega_n^{0}, \omega_n^{1}, \cdots, \omega_n^{n-1}$, where $\omega_n^{j} = e^{i \big(\frac{j}{n} 2\pi \big)}$. They are also called the $n$th root of the unity.

For any $j, k \in [0, n-1]$,

$$
\begin{align}
\omega_n^{j} \omega_n^{k} &= e^{i \big(\frac{j}{n} 2\pi \big)} e^{i \big(\frac{k}{n} 2\pi \big)} \\
&= e^{i \big(\frac{j + k}{n} 2\pi \big)} \\
&= \omega_n^{j + k} \\
\end{align}
$$

If $k = n - j$,

$$
\begin{align}
\omega_n^{j} \omega_n^{k} &= \omega_n^{j} \omega_n^{n - j} \\
&= \omega_n^{j + n - j} \\
&= \omega_n^{n} \\
&= \omega_n^{0} \\
&= e^{i0} \\
&= 1 \\
\end{align}
$$

Therefore,

$$
\begin{align}
\omega_n^{n - j} &= {\overline{\omega}}_n^{j} \\
&= e^{ - i \big(\frac{j}{n} 2\pi \big)} \\
\end{align}
$$