# 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

### 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:

We can also visualize it after expanding the formula:

Because of the scaling, we see that complex-multiplication preserves the ratio of 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)}

### 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.

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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