Polar form of a complex number

Eleven Standard >> Polar form of a complex number

Polar Form of a Complex Number

A complex number can be represented not only in rectangular form \( z = a + bi \), where \( a \) is the real part and \( b \) represents the imaginary component; alternatively, the number can be expressed in its polar representation. The polar form is particularly useful in multiplication, division, and finding powers and roots of complex numbers.

1. Polar Coordinates

Any complex number \( z = a + bi \) can be represented as a point in the complex plane. Alternatively, it may be written using:

\( r \): represents the absolute value or length of the complex number
\( \theta \): the argument or angle made with the positive real axis

The value of the modulus \( r \) can be calculated as:

\[ r = |z| = \sqrt{a^2 + b^2} \] The argument \( \theta \) (in radians) is:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] (Note: The angle \( \theta \) must be adjusted based on the quadrant in which the complex number lies.)

2. Polar Form Representation

After determining \( r \) and \( \theta \), the complex number \( z = a + bi \) can be represented as:

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

This is known as the polar form of the complex number. It is sometimes abbreviated as:

\[ z = r \] cis \[\theta \] where cis \(\theta = \cos \theta + i \sin \theta \).

3. Example

Convert the complex number \( z = 1 + i \) to polar form.

Real part \( a = 1 \), Imaginary part \( b = 1 \)
Modulus: \( r = \sqrt{1^2 + 1^2} = \sqrt{2} \)
Argument \( \theta \) is calculated as \( \tan^{-1}(1) = \frac{\pi}{4} \) radians.

So, the polar form of \( z \) is:

\[ z = \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) \] or \( z = \sqrt{2} \) cis \(\frac{\pi}{4} \)

4. Why Use Polar Form?

Makes multiplication and division more straightforward: \( z_1 z_2 = r_1 r_2 \) cis \((\theta_1 + \theta_2) \)
Useful in De Moivre’s Theorem: \( z^n = r^n \) cis \((n\theta) \)
Helpful in finding roots of complex numbers

Summary

The polar form is an alternative way of representing complex numbers using trigonometry. It represents both the size and angle of a complex number in the complex plane. This form is especially useful for complex number operations like multiplication, division, and exponentiation.