Complex numbers are not just abstract algebra. They are used in electrical engineering, physics,
rotations, signal analysis, and computer graphics. A complex number:
\[
z=a+bi
\]
can represent a point, a vector, a phasor, an impedance, or an iteration state.
1. Electrical impedance
In AC circuits, resistance and reactance are combined into one complex number called impedance:
\[
Z=R+iX
\]
The real part \(R\) represents resistance. The imaginary part \(X\) represents reactance.
Inductors and capacitors have imaginary impedance:
\[
Z_L=i\omega L,
\qquad
Z_C=\frac{1}{i\omega C}
\]
where:
\[
\omega=2\pi f
\]
For a series circuit, impedances add:
\[
Z_{\text{series}}=Z_1+Z_2+\cdots
\]
For a parallel circuit, admittances add:
\[
\frac{1}{Z_{\text{parallel}}}
=
\frac{1}{Z_1}
+
\frac{1}{Z_2}
+\cdots
\]
2. Phasor Ohm’s law
In AC analysis, voltage and current can be represented as phasors. Ohm’s law becomes:
\[
\widetilde I=\frac{\widetilde V}{Z}
\]
The magnitude of \(Z\) affects the current size, while the angle of \(Z\) affects the phase shift.
A positive imaginary part is inductive, and a negative imaginary part is capacitive.
3. Complex rotations
Complex numbers also describe rotations in the plane. Multiplying by:
\[
e^{i\theta}=\cos\theta+i\sin\theta
\]
rotates a vector by angle \(\theta\). If \(z=x+yi\), then:
\[
z'=ze^{i\theta}
\]
This is the same as using the rotation matrix:
\[
\begin{pmatrix}
x'\\y'
\end{pmatrix}
=
\begin{pmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{pmatrix}
\begin{pmatrix}
x\\y
\end{pmatrix}
\]
The complex-number method is often shorter and clearer than matrix multiplication.
4. Julia set iteration
Complex numbers are also used in fractals. A Julia set is created by repeatedly applying:
\[
z_{n+1}=z_n^2+c
\]
Here \(c\) is a fixed complex parameter. Each starting point \(z_0\) either stays bounded for many
iterations or escapes far away. The boundary between these behaviors creates a fractal shape.
A common escape test is:
\[
|z_n|>2
\]
If this happens, the orbit is treated as escaping.
5. Formula summary
| Application |
Formula |
Meaning |
| Impedance |
\(Z=R+iX\) |
Resistance and reactance in one complex number |
| Inductor |
\(Z_L=i\omega L\) |
Positive imaginary reactance |
| Capacitor |
\(Z_C=\dfrac{1}{i\omega C}\) |
Negative imaginary reactance |
| Phasor Ohm’s law |
\(\widetilde I=\dfrac{\widetilde V}{Z}\) |
Current from voltage and impedance |
| Rotation |
\(z'=ze^{i\theta}\) |
Rotate a vector by \(\theta\) |
| Julia iteration |
\(z_{n+1}=z_n^2+c\) |
Repeated complex squaring creates fractal behavior |
6. Common mistakes
- Do not treat impedance as only a real resistance; the imaginary part controls phase.
- For capacitors, remember that \(Z_C\) has a negative imaginary part.
- For parallel circuits, add admittances \(1/Z\), not impedances directly.
- Multiplying by \(e^{i\theta}\) rotates without changing length.
- In Julia iteration, a point that has not escaped yet is not proof that it never escapes.
- The Julia preview is a visual approximation, not a proof of exact fractal membership.
