Complex_Number

As you know, the imaginary number is defined as: where $a \in \mathbb{R}$

$ai = a\sqrt{-1}$

And therefore complex number $c \in \mathbb{C}$ is imaginary number plus a real number.

Imaginary number relates to rotation. For detailed intuitive understanding of imaginary number, see this article

Different forms of complex numbers:

$a + bi$
$r(\cos \theta + i \sin \theta)$
$re^{i \theta}$

Euler's identity: $e^{i\pi} = -1$ is a result of power representation of complex numbers.

There is no natural total order of complex number as $\mathbb{C}$ is not an ordered field and there is no ordering, but one can artificially construct a total order (but not well-order):

first compare its argument and then modulus
first compare its modulus and then argument

Similar to $\mathbb{R}$ , the well-order of $\mathbb{C}$ might be possible to construct by making use of Axiom of Choice, but that's hard.

However, this is not a well-ordering

Addition

Complex number addition is the same as vector addition.

For detailed intuitive understanding of complex number, see this article

Multiplication

Notice complex number multiplication is different from vector multiplication:

$\begin{align*} &(a + bi)(c + di)\\ =& (ac - bd) + (bc + ad)i\\ \end{align*}$

whereas in vector multiplication for $\vec{i}, \vec{j}$ unit vector:

$\begin{align*} &(a\vec{i} + b\vec{j})(c\vec{i} + d\vec{j})\\ =& ac\vec{i}^2 + bd\vec{j}^2 + ad\vec{i}\vec{j} + bc\vec{i}\vec{j}\\ =& ac + bd\\ \end{align*}$

To visualize complex multiplication, we view it as vector scale, rotation, and addition:

Visualizing klzzwxh:0010 — Visualizing $(3 + 4i)(2 + 3i)$

We can also visualize it after expanding the formula:

Visualizing klzzwxh:0012 after break down — Visualizing $(3 + 4i)(2 + 3i)$ after break down

Because of the scaling, we see that complex-multiplication preserves the ratio of angle:

Visualizing klzzwxh:0014 adds the angle — Visualizing $(3 + 4i)(2 + 3i)$ adds the angle

We can also see multiplication as multiplying modulus and adding the argument using exponential form:

$z_1z_2 = r_1r_2e^{i(\theta_1 + \theta_2)}$

For detailed intuitive understanding of complex multiplication, see this article

De Moivre's Theorem

$(r(\cos \theta + i \sin \theta))^n = (re^{i \theta})^n = r^n (\cos (n\theta) + i \sin (n\theta))$

Complex Conjugation

Complex Conjugate Pair is defined as following:

$z = a + bi \implies z^* = a - bi$

When complex number and its conjugate are multiplied together, they becomes a real number:

$zz^* = (a + bi)(a - bi) = a^2 + abi - abi - b^2i^2 = a^2 + b^2 \in \mathbb{R}$

When complex number and its conjugate are added together, they becomes a real number:

$z + z^* \in \mathbb{R}$

Note that only a complex number's conjugate have both two properties above.

Roots of Quadratic Equation

When the solution to quadratic equation involves complex root $c \in \mathbb{C}$ , then $c^* \in \mathbb{C}$ is also a root.

Argand Diagram

Modulus and Argument

For $z = a + bi \in \mathbb{C}$ , we define modulus and argument.

Modulus: the magnitude of the complex vector

$r = \sqrt{a^2 + b^2}$

Argument: the angle of the complex vector

$\theta = \arctan(\frac{b}{a})$

We can also write the complex number in trignometry expression (modulus-argument form) after knowing $a = r\cos \theta, b = r\sin \theta$ :

$z = r(\cos \theta + i \sin \theta)$

However, for a point $P$ in Argand Diagram, we have infinite many $z = r(\cos \theta + i \sin \theta)$ correspond to $P$ because there are infinite many $\theta$ to choose from. So we define principal argument as a convention of choosing the angle:

$-\pi < \theta \leq \pi$

We get arithmetic property for $z_1, z_2 \in \mathbb{C}$ :

$|z_1z_2| = |z_1| |z_2|$
$|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$
$\arg(z_1z_2) = \arg(z_1) + \arg(z_2)$
$\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)$

Loci

The loci is just a set of 1D points generated by complex-functions in Argand Diagram.

We observe complex-functions:

$|z - z_1| = r$

because complex addition is vector addition, the above formula generates a circle of radius $r$ around $(x_1, y_1)$ for $z_1 = x_1, iy_1$ .

$|z - z_1| = |z - z_2|$

because complex addition is vector addition, the above formula generates a perpendicular bisector of line segment between $z_1, z_2$ .

$\arg(z - z_1) = \theta$

This generates a half-line starting from $z_1$ and shoot at angle $\theta$ to infinity.

Region

The region is just a set of 2D points generated by complex-functions in Argand Diagram.

Interesting Facts

We have relationship to sinusoids:

$\begin{align*} z =& \cos \theta + i \sin \theta\\ z^{-1} =& \cos(-\theta) + i\sin(-\theta) = \cos \theta - i \sin \theta\\ z^n + z^{-n} =& 2 \cos (n\theta)\\ z^n - z^{-n} =& 2i \sin (n\theta)\\ \cos \theta =& \frac{1}{2}(e^{i\theta} + e^{-i\theta})\\ \sin \theta =& \frac{1}{2i}(e^{i\theta} - e^{-i\theta})\\ \end{align*}$

We have properties of complex series, for $w, z \in \mathbb{C}$ :

$\begin{align*} \sum_{r = 0}^{n - 1} wz^r =& \frac{w(z^n - 1)}{z - 1}\\ \sum_{r = 0}^\infty wz^r =& \frac{w}{1 - z} \tag{for $|z| < 1$}\\ \end{align*}$

Rotational Property:

$e^{i\pi} = e^{i(\pi + 2k\pi)}$

Roots of Unity (de Moivre number): any complex number that yields $1$ when raised to some positive integer power $n \in \mathbb{Z}$

The n-th root of unity if $n \in \mathbb{Z}^+$ , $\forall k \in \{0, 1, ..., n - 1\}$ is:

$\exp\left(\frac{2k\pi i}{n}\right) = \cos \frac{2k\pi}{n} + i \sin \frac{2k\pi}{n}$

The above set of roots, if plotted in Argand Diagram, will yield vertices of a regular $n$ -gon.

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