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Course: Precalculus > Unit 3

Lesson 6: Polar form of complex numbers

Complex number forms review

Review the different ways in which we can represent complex numbers: rectangular, polar, and exponential forms.

What are the different complex number forms?


Rectangular		$a + b i$ ‍
Polar		$r (\cos (θ) + i \sin (θ))$ ‍
Exponential		$r \cdot e^{i θ}$ ‍

Rectangular form

a + b i

The rectangular form of a complex number is a sum of two terms: the number's

real

part and the number's

imaginary

part multiplied by

i

As such, it is really useful for adding and subtracting complex numbers.

We can also plot a complex number given in rectangular form in the complex plane. The real and imaginary parts determine the real and imaginary coordinates of the number.

Want to learn more about complex number rectangular form? Check out this video about the complex plane and this video about adding and subtracting complex numbers.

Polar form

r (\cos (θ) + i \cdot \sin (θ))

Polar form emphasizes the graphical attributes of complex numbers:

absolute value

(the distance of the number from the origin in the complex plane) and

angle

(the angle that the number forms with the positive Real axis). These are also called

modulus

and

argument

Note that if we expand the parentheses in the polar representation, we get the number's rectangular form:

r (\cos (θ) + i \cdot \sin (θ)) = \overset{a}{\overset{⏞}{r \cos (θ)}} + \overset{b}{\overset{⏞}{r \sin (θ)}} \cdot i

This form is really useful for multiplying and dividing complex numbers, because of their special behavior: the product of two numbers with absolute values

r_{1}

and

r_{2}

and angles

θ_{1}

and

θ_{2}

will have an absolute value

r_{1} r_{2}

and angle

θ_{1} + θ_{2}

Want to learn more about complex number polar form? Check out this video.

Exponential form

r \cdot e^{i θ}

Exponential form uses the same attributes as polar form,

absolute value

and

angle

. It only displays them in a different way that is more compact. For example, the multiplicative property can now be written as follows:

(r_{1} \cdot e^{i θ_{1}}) \cdot (r_{2} \cdot e^{i θ_{2}}) = r_{1} r_{2} \cdot e^{i (θ_{1} + θ_{2})}

This form stems from Euler's expansion of the exponential function

e^{z}

to any complex number

z

. The reasoning behind it is quite advanced, but its meaning is simple: for any real number

x

, we define

e^{i x}

to be

\cos (x) + i \sin (x)

Using this definition, we obtain the equivalence of exponential and polar forms:

r \cdot e^{i θ} = r (\cos (θ) + i \sin (θ))

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