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

$$\mathrm{Re}(z)\Re (z)$$

Definition: Imaginary Part

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

NOTATION

$$\mathrm{Im}(z)\Im (z)$$

Notation: The Set of Complex Numbers

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

$$\mathbb{C}\stackrel{\text{def}}{=}\{a+b\mathrm{i}\mid a,b\in \mathbb{R}\}$$

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:

$$|z|\stackrel{\text{def}}{=}\sqrt{\Re (z{)}^2+\Im (z{)}^2}=\sqrt{a^2+b^2}$$

Definition: Argument

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

$$\arg (z)\stackrel{\text{def}}{=}\{\begin{array}{l}\arccos \left(\frac{x}{|z|}\right)\text{ if }y\ge 0\\ -\arccos \left(\frac{x}{|z|}\right)\text{ if }y<0\end{array}$$

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 $\phi =\arg z$:

$$z=r(\cos \phi +\mathrm{i}\sin \phi )$$

PROOF

TODO

Definition: Polar Form

We call $r(\cos \phi +\mathrm{i}\sin \phi )$ 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 $\phi$ 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

$$\bar{z}\stackrel{\text{def}}{=}a+\mathrm{i}(-b)=a-\mathrm{i}b$$

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

$$z_1+z_2\stackrel{\text{def}}{=}(a_1+a_2)+\mathrm{i}(b_1+b_2)$$

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

$$z_1\cdot z_2\stackrel{\text{def}}{=}(a_1{a}_2-b_1{b}_2)+\mathrm{i}(a_1{b}_2+b_1{a}_2)$$

Theorem: Multiplication in Polar Form

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

$$\begin{array}{rl}& z_1=r_1(\cos {\phi }_1+\mathrm{i}\sin {\phi }_1)\\ & z_2=r_2(\cos {\phi }_2+\mathrm{i}\sin {\phi }_2)\end{array}$$

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

$$z_1\cdot z_2=r_1{r}_2(\cos ({\phi }_1+{\phi }_2)+\mathrm{i}\sin ({\phi }_1+{\phi }_2))$$

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

$$\frac{z_1}{z_2}\stackrel{\text{def}}{=}\frac{z_1\cdot {\bar{z}}_2}{|z_2{|}^2}$$

Theorem: Division in Polar Form

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

$$\begin{array}{rl}& z_1=r_1(\cos {\phi }_1+\mathrm{i}\sin {\phi }_1)\\ & z_2=r_2(\cos {\phi }_2+\mathrm{i}\sin {\phi }_2)\end{array}$$

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

$$\frac{z_1}{z_2}=\frac{r_1}{r_2}(\cos ({\phi }_1-{\phi }_2)+\mathrm{i}\sin ({\phi }_1-{\phi }_2))$$

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 with the addition, multiplication defined on them, i.e.:

• $z_1+(z_2+z_3)=(z_1+z_2)+z_3$ for all $z_1,z_2,z_3\in \mathbb{C}$;

• $z_1\cdot (z_2\cdot z_3)=(z_1\cdot z_2)\cdot z_3$ for all $z_1,z_2,z_3\in \mathbb{C}$;

• $z+w=w+z$ for all $z,w\in \mathbb{C}$;

• $z\cdot w=w\cdot z$ for all $z,w\in \mathbb{C}$;

• $z+0=z$ for all $z\in \mathbb{C}$;

• $z\cdot 1=z$ for all $z\in \mathbb{C}$;

• $z-z=0$ for all $z\in \mathbb{C}$;

• $z\cdot \frac{1}{z}=1$ for all $z\in \mathbb{C}$;

• $z\cdot (u+v)=z\cdot u+z\cdot v$ for all $z,u,v\in \mathbb{C}$.

PROOF

TODO

Theorem: Distributivity of Complex Conjugation

Complex conjugation is distributive over addition, multiplication and division:

$$\begin{array}{rl}\overline{z_1+z_2}& ={\bar{z}}_1+{\bar{z}}_2\\ \overline{z_1\cdot z_2}& ={\bar{z}}_1\cdot {\bar{z}}_2\\ \overline{\left(\frac{z_1}{z_2}\right)}& =\frac{{\bar{z}}_1}{{\bar{z}}_2}{z}_2\ne 0\end{array}$$

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.