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« Complex number arithmetic
Complex Numbers: (lesson 2 of 2)

Polar representation

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.

The polar form of a complex number is:
Complex numbers - polar form

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:

Polar to Rectangular Form Conversion


Exercise 1: Convert to rectangular form:

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4

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:

Rectangular to Polar Form Conversion

Example 2:

Convert the complex number to polar form.

Solution:

In this example Rectangular to Polar Form Conversion:

Rectangular to Polar Form Conversion

The polar form is:

Rectangular to Polar Form Conversion

Exercise 2: Convert to polar form:

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4

Product in polar representation

Product in polar representation

Example 3:

Let Product in polar representation. Then:

Product in polar representation

Quotient two complex numbers in polar representation

Quotient two complex numbers

Example 4:

Let Quotient two complex numbers. Then:

Quotient two complex numbers

Exercise 3: Find product and quotient:

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4

The inverse of a complex number in polar representation

inverse of a complex number

Conjugate numbers in polar representation

Conjugate numbers in polar representation

Formula 'De Moivre'

Formula 'De Moivre'

Example 5:

Let z = 1 - i.

a) find polar representation

b) find z8

Solution:

a) Formula 'De Moivre'

b)

Formula 'De Moivre'

Exercise 4: Find zn:

Level 1

Exercise 1 answer 1
answer 2
answer 3
answer 4

Level 2

Exercise 2 answer 1
answer 2
answer 3
answer 4