Complex Numbers#

Definition: Complex Numbers

A complex number $z$ is an expression of the form

\[ a+b \mathrm{i}, \]

where $a$ and $b$ are real numbers.

Notation

We can also notate $z$ as $a + \mathrm{i}b$. If $a = 0$, we can write just $z = \mathrm{i}b$ or $z = b\mathrm{i}$. If $b = 0$, we can write just $z = a$.

Definition: Imaginary Unit

The symbol $\mathrm{i}$ is called the imaginary unit.

Definition: Real Part

We call $a$ the real part of $a + b\mathrm{i}$.

Notation

\[ \operatorname{Re}(z) \qquad \Re(z) \]

Definition: Imaginary Part

We call $b$ the imaginary part of $a + b \mathrm{i}$.

Notation

\[ \operatorname{Im}(z) \qquad \Im(z) \]

Notation: The Set of Complex Numbers

The set of all complex numbers is denoted by $\mathbb{C}$.

\[ \mathbb{C} \overset{\text{def}}{=} \{a + b\mathrm{i} \mid a,b \in \mathbb{R}\} \]

Definition: Modulus

The modulus of a complex number $z = a + b\mathrm{i}$ is the square root of the sum of the squares of its real part and its imaginary part:

\[ |z| \overset{\text{def}}{=} \sqrt{\Re (z)^2 + \Im (z)^2} = \sqrt{a^2 + b^2} \]

Definition: Argument

The argument of a complex number $z = x + y \mathrm{i} \ne 0$ is defined using the arctan function as

\[ \arg(z) \overset{\text{def}}{=} \begin{cases}\displaystyle\hphantom{-}\arccos \left(\frac{x}{|z|}\right) \qquad \text{ if } y \ge 0 \\ \displaystyle -\arccos \left(\frac{x}{|z|}\right) \qquad \text{ if } y \lt 0\end{cases} \]

Theorem: Image of $\arg$

The range of $\arg$ is $(-\pi; \pi]$.

Proof

TODO

Forms#

Definition: Cartesian Form

Given a complex number $z = a + \mathrm{i}b$, we call $a + \mathrm{i}b$ the Cartesian form of $z$.

The Cartesian form of a complex number is just the one resulting from its definition. However, there are other, equivalent ways to specify $z$ which often make the solutions of some problems easier and more intuitive.

Theorem: Polar Form

Each complex number $z \ne 0$ can be specified using its modulus $r = |z|$ and the real trigonometric functions of its argument $\varphi = \arg z$:

\[ z = r(\cos \varphi + \mathrm{i} \sin \varphi) \]

Proof

TODO

Definition: Polar Form

We call $r(\cos \varphi + \mathrm{i} \sin \varphi)$ the polar form of $z$.

Note: Infinitely Many Polar Forms

We can also construct other, still equivalent polar forms of $z$ by adding an integer multiple of $2\pi$ to $\varphi$ because $\cos$ and $\sin$ are periodic.

There is also a third way to specify complex numbers using Euler's formula.

Theorem: Exponential Form

Each complex number $z \ne 0$ can be specified using its modulus and the complex exponential of its argument:

\[ z = |z| \mathrm{e}^{\mathrm{i} \arg(z)} \]

Proof

Using Euler's formula we obtain

\[ |z|\mathrm{e}^{\mathrm{i} \arg(z)} = |z|(\cos (\arg (z)) + \mathrm{i} \sin (\arg (z))), \]

which is the canonical polar form of $z$.

Definition: Exponential Form

We call $|z| \mathrm{e}^{\mathrm{i} \arg(z)}$ the exponential form of $z$.

Operations#

Definition: Complex Conjugation

The (complex) conjugate of a complex number $z = a + \mathrm{i}b$ is the complex number

\[ \bar{z} \overset{\text{def}}{=} a + \mathrm{i}(-b) = a - \mathrm{i}b \]

Definition: Addition

The sum of two complex numbers $z_1 = a_1 + \mathrm{i}b_1$ and $z_2 = a_2 + \mathrm{i} b_2$ is defined as the complex number

\[ z_1 + z_2 \overset{\text{def}}{=} (a_1 + a_2) + \mathrm{i}(b_1 + b_2) \]

Notation: Subtraction

We write $-z_2$ for $-a_2 - \mathrm{i}b_2$ and write $z_1 - z_2$ instead of $z_1 + (-z_2)$.

Definition: Multiplication

The product of two complex numbers $z_1 = a_1 + \mathrm{i}b_1$ and $z_2 = a_2 + \mathrm{i} b_2$ is defined as the complex number

\[ z_1 \cdot z_2 \overset{\text{def}}{=} (a_1 a_2 - b_1 b_2) + \mathrm{i}(a_1 b_2 + b_1 a_2) \]

Theorem: Multiplication in Polar Form

If the polar forms of $z_1$ and $z_2$ are

\[ \begin{aligned} &z_1 = r_1 (\cos \varphi_1 + \mathrm{i} \sin \varphi_1) \\ &z_2 = r_2 (\cos \varphi_2 + \mathrm{i} \sin \varphi_2) \end{aligned} \]

then the polar form of $z_1 \cdot z_2$ is

\[ z_1 \cdot z_2 = r_1 r_2 (\cos (\varphi_1 + \varphi_2) + \mathrm{i} \sin (\varphi_1 + \varphi_2)) \]

Proof

TODO

Theorem: Multiplication in Exponential Form

The exponential form of $z_1 \cdot z_2$ is

\[ z_1 \cdot z_2 = |z_1| \, |z_2| \mathrm{e}^{\mathrm{i}(\arg z_1 + \arg z_2)} \]

Proof

TODO

Definition: Division

The division of a complex number $z_1$ by a complex number $z_2$ is the complex number

\[ \frac{z_1}{z_2} \overset{\text{def}}{=} \frac{z_1\cdot \bar{z}_2}{|z_2|^2} \]

Theorem: Division in Polar Form

If the polar forms of $z_1$ and $z_2$ are

\[ \begin{aligned} &z_1 = r_1 (\cos \varphi_1 + \mathrm{i} \sin \varphi_1) \\ &z_2 = r_2 (\cos \varphi_2 + \mathrm{i} \sin \varphi_2) \end{aligned} \]

then the polar form of $\frac{z_1}{z_2}$ is

\[ \frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos (\varphi_1 - \varphi_2) + \mathrm{i} \sin (\varphi_1 - \varphi_2)) \]

Proof

TODO

Theorem: Division in Exponential Form

The exponential form of $\frac{z_1}{z_2}$ is

\[ \frac{z_1}{z_2} = \frac{|z_1|}{|z_2|} \mathrm{e}^{\mathrm{i}(\arg z_1 - \arg z_2)} \]

Proof

TODO

Theorem: The Field of Complex Numbers

The complex numbers $\mathbb{C}$ form a field $(\mathbb{C}, +, \cdot)$ with the addition, multiplication defined on them.

Proof

We need to prove the following:

(1) $z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$ for all $z_1, z_2, z_3 \in \mathbb{C}$;
(2) $z_1 \cdot (z_2 \cdot z_3) = (z_1 \cdot z_2) \cdot z_3$ for all $z_1, z_2, z_3 \in \mathbb{C}$;
(3) $z + w = w + z$ for all $z, w \in \mathbb{C}$;
(4) $z \cdot w = w \cdot z$ for all $z, w \in \mathbb{C}$;
(5) $z + 0 = z$ for all $z \in \mathbb{C}$;
(6) $z \cdot 1 = z$ for all $z \in \mathbb{C}$;
(7) $z - z = 0$ for all $z \in \mathbb{C}$;
(8) $z \cdot \frac{1}{z} = 1$ for all $z \in \mathbb{C}$;
(9) $z \cdot (u + v) = z \cdot u + z \cdot v$ for all $z, u, v \in \mathbb{C}$.

Proof of (1): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

We want to show $z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$.

\[ \begin{aligned} z_1 + (z_2 + z_3) &= (a_1 + b_1i) + ((a_2 + b_2i) + (a_3 + b_3i)) \\ &= (a_1 + b_1i) + ((a_2 + a_3) + (b_2 + b_3)i) \\ &= (a_1 + (a_2 + a_3)) + (b_1 + (b_2 + b_3))i \\ &= ((a_1 + a_2) + a_3) + ((b_1 + b_2) + b_3)i \\ &= ((a_1 + a_2) + (b_1 + b_2)i) + (a_3 + b_3i) \\ &= ((a_1 + b_1i) + (a_2 + b_2i)) + (a_3 + b_3i) \\ &= (z_1 + z_2) + z_3 \end{aligned} \]

Proof of (2): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

We want to show $z_1 \cdot (z_2 \cdot z_3) = (z_1 \cdot z_2) \cdot z_3$.

First, we compute the left-hand side:
$$
\begin{aligned}
z_2 \cdot z_3 &= (a_2a_3 - b_2b_3) + (a_2b_3 + b_2a_3)i \
z_1 \cdot (z_2 \cdot z_3) &= (a_1 + b_1i) \cdot ((a_2a_3 - b_2b_3) + (a_2b_3 + b_2a_3)i) \
&= (a_1(a_2a_3 - b_2b_3) - b_1(a_2b_3 + b_2a_3)) + (a_1(a_2b_3 + b_2a_3) + b_1(a_2a_3 - b_2b_3))i \
&= (a_1a_2a_3 - a_1b_2b_3 - b_1a_2b_3 - b_1b_2a_3) + (a_1a_2b_3 + a_1b_2a_3 + b_1a_2a_3 - b_1b_2b_3)i
\end{aligned}
$$

Proof of (3): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Proof of (4): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Proof of (5): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Proof of (6): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Proof of (7): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Proof of (8): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Proof of (9): The following proof was generated by AI and has not been human-verified. As such, it may contain mistakes.

Theorem: Distributivity of Complex Conjugation

Complex conjugation is distributive over addition, multiplication and division:

\[ \begin{aligned} \overline{z_1 + z_2} &= \bar{z}_1 + \bar{z}_2 \\ \overline{z_1 \cdot z_2} &= \bar{z}_1 \cdot \bar{z}_2 \\ \overline{\left(\frac{z_1}{z_2}\right)} &= \frac{\bar{z}_1}{\bar{z}_2} \qquad z_2 \ne 0 \end{aligned} \]

Proof

TODO

Theorem: Conjugate Multiplication and Modulus

Multiplying a complex number by its conjugate results in the square of its modulus:

\[ z\cdot\bar{z} = |z|^2 \]

Proof

TODO

Theorem: Triangle Inequality

The modulus has the following property for all $z_1, z_2 \in \mathbb{C}$:

\[ |z_1 + z_2| \le |z_1| + |z_2| \]

Proof

TODO

Theorem: Modulus Product

The modulus has the following property for all $z, w \in \mathbb{C}$:

\[ |z \cdot w| = |z|\cdot|w| \]

Proof

TODO

The Complex Plane#

Complex numbers can be plotted on a plane where the horizontal axis contains the real numbers and the vertical axis contains the imaginary numbers.

The Complex Plane