Complex Number Representations
Complex numbers as $2\times 2$ Matrices
For the complex number $z=a+ib$:
\[\mathbf M_z=\begin{pmatrix}a & -b \\ b & a\end{pmatrix}\]The following operations will then apply:
- Addition and Multiplication
- As normal.
- Conjugate
- $\mathbf M_{\bar z}=\mathbf M_z^\top$ - This is the transpose of $\mathbf M_z$.
- Modulus
- $\vert z\vert=\text{det }\mathbf M_z$
Argand Diagram
In this representation then you treat them as vectors.
- Conjugate is reflection in the real axis.
- The modulus is the size of the vector.
Multiplication and division don’t have a natural geometric analogy.
Polar Coordinates
For $z=a_ib$ with $\vert z\vert=r$. The number $z$ is described in polar form as:
\[z=(r,\theta)\]The angle (phase) of the complex number is denoted as $\text{arg}\ z$. It can be shown as:
\[\begin{aligned} \text{arg}\ z &= \cos^{-1}\left(\frac{\text{Re}(z)}{\vert z\vert}\right)\\ &= \sin^{-1}\left(\frac{\text{Im}(z)}{\vert z\vert}\right) \end{aligned}\]
Euler Form
For $z=a_ib$ with $\vert z\vert=r$ and $\text{arg}\ z=\theta$. The number $z$ is described in Euler form as:
\[z=r\times e^{i\theta}\]This gives the following identity:
\[z=re^{i\theta}=r(\cos\theta+i\sin\theta)\]This relation is known as Euler’s Formula and leads to ($\forall\alpha\in\Bbb R$):
\[(\cos\theta+i\sin\theta)^\alpha=\cos(\alpha\theta)+i\sin(\alpha\theta)\]Consequences of Euler Form
For $\vert u \vert = s$ and $\text{arg} u=\sigma$, $\vert v\vert=t$ and $\text{arg}v=\tau$:
\[uv = se^{i\sigma}te^{i\tau}=(st)e^{i(\sigma+\tau)}\]Looking at Argand diagram: the phase resulting from the product of two complex numbers is the result of adding the phase of each.
There are infinitely many representations of any single complex number. To avoid this we use the principle value of $\theta$.
Principle values can be written as $\varphi_m$.
Primitive Roots of Unity
For any $k\in\Bbb N$ there are exactly $k$ primitive roots of unity.
Example
Identify a primitive fourth root of unity modulo 8.
For each value:
\[p\in\{0,1,2,3,4,5,6,7,8\}\]Compute $p^4(\mod 8)$:
\[\{0,1,0,1,0,1,0,1\}\]If it divides wholely then we don’t count it.
The four primitive roots are:
\[\{1,3,5,7\}\]