A complex number has the form:
\[
z=a+bi
\]
On the complex plane, the real part \(a\) is plotted on the horizontal axis and the imaginary part \(b\)
is plotted on the vertical axis. So \(z=a+bi\) corresponds to the point:
\[
(a,b)
\]
The same complex number can also be drawn as a vector from the origin to the point \((a,b)\).
This is why complex numbers have both algebraic and geometric meaning.
1. Modulus and argument
The modulus of \(z=a+bi\) is the length of the vector:
\[
|z|=\sqrt{a^2+b^2}
\]
The argument is the angle the vector makes with the positive real axis:
\[
\arg(z)=\operatorname{atan2}(b,a)
\]
These two values describe the same complex number in polar form:
\[
z=r(\cos\theta+i\sin\theta)
\]
where \(r=|z|\) and \(\theta=\arg(z)\).
2. Vector addition
If:
\[
z_1=a+bi,\qquad z_2=c+di
\]
then:
\[
z_1+z_2=(a+c)+(b+d)i
\]
Geometrically, this is ordinary vector addition. You can add by placing the second vector at the tip
of the first, or by using the parallelogram rule.
3. Vector subtraction
Subtraction is also component-based:
\[
z_1-z_2=(a-c)+(b-d)i
\]
Geometrically, \(z_1-z_2\) is the vector from \(z_2\) to \(z_1\).
4. Complex multiplication
Algebraically:
\[
(a+bi)(c+di)=(ac-bd)+(ad+bc)i
\]
But geometrically, multiplication is easier to understand in polar form. If:
\[
z_1=r_1e^{i\theta_1},
\qquad
z_2=r_2e^{i\theta_2}
\]
then:
\[
z_1z_2=r_1r_2e^{i(\theta_1+\theta_2)}
\]
So complex multiplication multiplies lengths and adds angles.
5. Complex division
Division can be done by multiplying by the conjugate:
\[
\frac{a+bi}{c+di}
=
\frac{(a+bi)(c-di)}{c^2+d^2}
\]
In polar form:
\[
\frac{z_1}{z_2}
=
\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}
\]
So division divides lengths and subtracts angles.
6. Powers
If:
\[
z=re^{i\theta}
\]
then:
\[
z^m=r^m e^{im\theta}
\]
This means raising a complex number to a power scales the modulus by \(r^m\) and multiplies the angle
by \(m\). A power trail shows the sequence:
\[
1,z,z^2,z^3,\ldots,z^m
\]
7. Example: vector addition
Let:
\[
z_1=3+4i,\qquad z_2=2-5i
\]
Then:
\[
z_1+z_2=(3+2)+(4-5)i=5-i
\]
On the graph, the result vector points to \((5,-1)\).
8. Example: multiplication
Let:
\[
z_1=2+i,\qquad z_2=1+3i
\]
Then:
\[
z_1z_2=(2+i)(1+3i)
\]
\[
z_1z_2=2+6i+i+3i^2
\]
\[
z_1z_2=-1+7i
\]
The product vector is not obtained by simply adding arrows. Multiplication changes both length and direction.
9. Formula summary
| Concept |
Formula |
Meaning |
| Complex point |
\(z=a+bi\leftrightarrow(a,b)\) |
Real part is horizontal, imaginary part is vertical |
| Modulus |
\(|z|=\sqrt{a^2+b^2}\) |
Vector length |
| Argument |
\(\arg(z)=\operatorname{atan2}(b,a)\) |
Vector direction |
| Addition |
\((a+bi)+(c+di)=(a+c)+(b+d)i\) |
Add components |
| Subtraction |
\((a+bi)-(c+di)=(a-c)+(b-d)i\) |
Subtract components |
| Multiplication |
\((a+bi)(c+di)=(ac-bd)+(ad+bc)i\) |
Multiply lengths, add angles |
| Power |
\((re^{i\theta})^m=r^me^{im\theta}\) |
Scale and rotate repeatedly |
10. Common mistakes
- Do not put the imaginary part on the horizontal axis.
- Do not add moduli when adding complex numbers; add real and imaginary components.
- Do not treat multiplication like vector addition.
- For division, make sure the denominator is not zero.
- The argument of \(0\) is undefined because the zero vector has no direction.
- Dragging a point changes the input complex number, so recalculate after moving it.
