Polar Form of Complex Numbers

Every complex number \(z = a + ib\) corresponds to the point \((a, b)\) on the Argand plane. To express this point in a different way, we use:

• r — the distance from the origin (same as modulus)
• θ — the angle with the positive real axis (same as argument)

These two values describe the same point using polar coordinates.

The formulas are:

• \(r = \sqrt{a^2 + b^2}\)
• \(\theta = \arg(z)\)

Using modulus and argument, any complex number can be written in polar form.

Definition:

For \(z = a + ib\), the polar form is:

\(z = r(\cos \theta + i \sin \theta)\)

where:

• \(r = |z| = \sqrt{a^2 + b^2}\)
• \(\theta = \arg(z)\)

This form is very useful when multiplying, dividing, or raising complex numbers to powers.

To convert \(z = a + ib\) into polar form, follow these steps:

1. Find modulus: \(r = \sqrt{a^2 + b^2}\)
2. Find argument: \(\theta = \tan^{-1}(\frac{b}{a})\), adjusted according to the correct quadrant

Example: Convert \(3 + 3i\) to polar form.

• \(r = \sqrt{3^2 + 3^2} = 3\sqrt{2}\)
• \(\theta = \frac{\pi}{4}\)

So, the polar form is:

\(3\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})\).

To convert \(z = r(\cos \theta + i \sin \theta)\) back to Cartesian form, simply expand:

\(z = r\cos \theta + i(r\sin \theta)\)

So:

• Real part: \(a = r\cos\theta\)
• Imaginary part: \(b = r\sin\theta\)

Example: Convert \(4(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\).

• \(a = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\)
• \(b = 4 \cdot \frac{1}{2} = 2\)

So the number is \(2\sqrt{3} + 2i\).

The polar form shows the complex number as a rotation and stretching from the origin.

In \(z = r(\cos \theta + i \sin \theta)\):

• \(r\) tells how far the point is from the origin
• \(\theta\) tells the direction

So you can think of polar form as: “Start at the origin, rotate by \(\theta\), then move forward by length \(r\).”

This is useful in physics, electrical engineering, and trigonometric simplifications.

One major advantage of polar form is that multiplication and division become simple.