Complex Number

algebra complex-numbers

Definition

Complex Number

The set of complex numbers $\mathbb{C}$ is defined as:

$$\mathbb{C}=\{a+bi\mid a,b\in \mathbb{R},i^2=-1\},$$

where $i$ is called the imaginary unit.

The structure $⟨\mathbb{C},+,\cdot ⟩$ forms a field.

Motivation

Algebraically, we defined the set of complex numbers $\mathbb{C}$ to resolve the fact that certain polynomial equations have no solutions in the set of real numbers $\mathbb{R}$.

In $\mathbb{R}$, the equation $x^2=-1$ has no solution because the squares of any real number is non-negative. To address this, we introduce an imaginary unit $i$, defined by the property:

$$i^2=-1$$

A complex number $z$ is then defined as an expression of the form:

$$z=x+iy$$

where $x,y\in \mathbb{R}$. The value $x$ is the real part $\mathrm{Re}(z)$, and $y$ is the imaginary part $\mathrm{Im}(z)$.

Gaussian Plane

The Gaussian plane represents a complex number as a point or vector in the plane.

The horizontal axis is the real part and the vertical axis is the imaginary part.

Forms

The reason there are three different representation (Cartesian, Polar, Matrix) is that each highlights a different mathematical personality of complex numbers: arithmetic, geometry, and linear transformations.

Cartesian (Component) Form

The Cartesian (component) form is:

$$z=x+yix,y\in \mathbb{R},i^2=-1$$

It treats complex numbers as a 2D vector space over $\mathbb{R}$.

Polar Form

Based on the Euler’s formula $e^{iz}=\cos z+i\sin z$, complex numbers can be viewed as points in a plane defined by a distance from the origin ($r=|z|$) and an angle ($\phi$).

Every non-zero $z\in \mathbb{C}$ admits a polar representation:

$$z=r{e}^{i\phi },r=|z|,\phi \in \mathbb{R}.$$

where $\phi$ is called the phase factor of $z$. This identifies multiplication in $\mathbb{C}$ with a product of a scaling by $r$ and a rotation by $\theta$ in the complex plane.

Matrix Form

There’s a structural isomorphism between $\mathbb{C}$ and a specific subset of $2\times 2$ real matrices. Thus, we can represent a complex number $z$ as:

$$\hat{z}=\left(\begin{array}{cc}x& -y\\ y& x\end{array}\right)\in {\mathbb{R}}^{2\times 2}$$

We map $1$ to the identity matrix $I=I_2$ and $i$ to a rotation matrix $J$:

$$I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$

Using that mapping, we can represent $\hat{z}$ with $I$ and $J$:

$$\hat{z}=xI+yJ$$

More generally, $1$ and $i$ can be represented by any pair of matrices $I$ and $J$ satisfying

$$I^2=I,IJ=JI=J,J^2=-I$$

Arithmetic

Conjugation

For $z=a+bi\in \mathbb{C}$, the complex conjugate is

$$\overline{z}:=a-bi.$$

It is the unique non-trivial automorphism of $\mathbb{C}$ over $\mathbb{R}$.

Moreover,

$$\Re (z)=\frac{z+\overline{z}}{2},\Im (z)=\frac{z-\overline{z}}{2i}.$$

Modulus

The modulus is defined by

$$|z|:=\sqrt{z\overline{z}}=\sqrt{a^2+b^2}.$$

It satisfies

$$|zw|=|z||w|.$$

Addition

For $z_1=a+bi$ and $z_2=c+di$,

$$z_1+z_2=(a+c)+(b+d)i.$$

Addition is performed component-wise.

Additive Inverse: For $z=a+bi$,

$$-z=(-a)+(-b)i.$$

This satisfies

$$z+(-z)=0.$$

Multiplication

For $z_1=a+bi$ and $z_2=c+di$,

$$z_1{z}_2=(ac-bd)+(ad+bc)i.$$

This follows from distributivity and the relation $i^2=-1$.

Multiplicative Inverse: For $z=a+bi\ne 0$,

$$z^{-1}=\frac{a-bi}{a^2+b^2}.$$

This satisfies

$$z\cdot z^{-1}=1.$$