Polar representation of complex numbers

In polar representation a complex number z is represented by two parameters r and Θ. Parameter r is the modulus of complex number and parameter Θ is the angle with the positive direction of x-axis.

This representation is very useful when we multiply or divide complex numbers.

Polar to Rectangular Form Conversion

Here we know r and Θ and we need to find a and b.

Example 1:

Convert the complex number to rectangular form.

Solution:

Exercise 1: Convert to rectangular form:

Level 1

Level 2

Rectangular to Polar Form Conversion

Here we know a and b and we need to find r and Θ. In this case we need to use formulas:

Example 2:

Convert the complex number to polar form.

Solution:

In this example :

The polar form is:

Exercise 2: Convert to polar form:

Level 1

Level 2

Product in polar representation

Example 3:

Let . Then:

Quotient two complex numbers in polar representation

Example 4:

Let . Then:

Exercise 3: Find product and quotient:

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Level 2

The inverse of a complex number in polar representation

Conjugate numbers in polar representation

Formula 'De Moivre'

Example 5:

Let z = 1 - i.

a) find polar representation

b) find z⁸

Solution:

a)

b)

Exercise 4: Find zⁿ:

Level 1

Level 2