Polar form is one of the most useful ways to work with complex numbers.
Instead of writing a complex number as \(a+bi\), we write it using a magnitude and an angle:
\[
z=r\angle\theta
\]
This means the same thing as:
\[
z=r(\cos\theta+i\sin\theta)=re^{i\theta}
\]
The number \(r\) is the distance from the origin, and \(\theta\) is the direction of the vector.
1. Why polar form is efficient
Rectangular form is convenient for addition and subtraction, but polar form is especially efficient for
multiplication, division, and powers. The reason is simple:
- Magnitudes multiply or divide.
- Angles add, subtract, or multiply.
This avoids long algebraic expansion when many complex numbers are involved.
2. Multiplication in polar form
If:
\[
z_1=r_1\angle\theta_1,
\qquad
z_2=r_2\angle\theta_2
\]
then:
\[
z_1z_2=r_1r_2\angle(\theta_1+\theta_2)
\]
So to multiply polar complex numbers, multiply the magnitudes and add the angles.
3. Example: \(5\angle30^\circ\times2\angle45^\circ\)
Multiply the magnitudes:
\[
5\cdot2=10
\]
Add the angles:
\[
30^\circ+45^\circ=75^\circ
\]
Therefore:
\[
5\angle30^\circ\times2\angle45^\circ=10\angle75^\circ
\]
4. Division in polar form
Division works in the opposite way:
\[
\frac{r_1\angle\theta_1}{r_2\angle\theta_2}
=
\frac{r_1}{r_2}\angle(\theta_1-\theta_2)
\]
So to divide, divide magnitudes and subtract angles.
Division by a complex number with magnitude \(0\) is undefined.
5. Powers in polar form
Powers are handled with De Moivre’s theorem:
\[
(r\angle\theta)^m=r^m\angle(m\theta)
\]
This means that raising a complex number to a power repeatedly scales the magnitude and rotates the angle.
6. Example: \((2\angle30^\circ)^3\)
Raise the magnitude:
\[
2^3=8
\]
Multiply the angle:
\[
3\cdot30^\circ=90^\circ
\]
Therefore:
\[
(2\angle30^\circ)^3=8\angle90^\circ
\]
In rectangular form, this is:
\[
8i
\]
7. Angle conventions
The same direction can be written using many equivalent angles. For example:
\[
75^\circ,\qquad 435^\circ,\qquad -285^\circ
\]
all point in the same direction. This calculator can show accumulated angles, principal angles, or
positive angles from \(0^\circ\) to \(360^\circ\).
8. Formula summary
| Operation |
Polar rule |
Meaning |
| Polar form |
\(z=r\angle\theta\) |
Magnitude and direction |
| Multiplication |
\((r_1\angle\theta_1)(r_2\angle\theta_2)=r_1r_2\angle(\theta_1+\theta_2)\) |
Multiply magnitudes, add angles |
| Division |
\(\dfrac{r_1\angle\theta_1}{r_2\angle\theta_2}=\dfrac{r_1}{r_2}\angle(\theta_1-\theta_2)\) |
Divide magnitudes, subtract angles |
| Power |
\((r\angle\theta)^m=r^m\angle(m\theta)\) |
Raise magnitude, multiply angle |
| Rectangular conversion |
\(r\angle\theta=r(\cos\theta+i\sin\theta)\) |
Convert to \(a+bi\) |
9. Common mistakes
- Do not add magnitudes when multiplying; magnitudes multiply.
- Do not multiply angles when multiplying two different complex numbers; angles add.
- For division, subtract the denominator angle.
- For powers, multiply the angle by the exponent.
- Use the same angle unit consistently unless the input line clearly says degrees or radians.
- Remember that angles differing by \(360^\circ\) represent the same direction.
