Skip to Main Content
It looks like you're using Internet Explorer 11 or older. This website works best with modern browsers such as the latest versions of Chrome, Firefox, Safari, and Edge. If you continue with this browser, you may see unexpected results.

Complex Numbers

Polar Form Terminology

Recall that in addition to rectangular form, \(a+bi\), we can also represent complex numbers in polar form:


where \(r\) is called the modulus and \(\theta\) is called the argument of the complex number.

Sometimes, you might see \(z=r(cos\theta+isin\theta)\) abbreviated as \(z=rcis\theta\).


To add/subtract complex numbers in polar form, follow these steps:

1. Convert all of the complex numbers from polar form to rectangular form (see the Rectangular/Polar Form Conversion page).

2. Perform addition/subtraction on the complex numbers in rectangular form (see the Operations in Rectangular Form page).

3. After evaluating the sum/difference, convert it back into polar form for your answer (see the Rectangular/Polar Form Conversion page).

The following example will demonstrate how to apply these steps to find the sum/difference of complex numbers.

Example: Evaluate \(4\sqrt{2}(cos(\frac{3\pi}{4})+isin(\frac{3\pi}{4})) + 3(cos(\frac{\pi}{2})+isin(\frac{\pi}{2}))\) in polar form (in radians), rounding to two decimal places.

See this video for the solution:



To multiply/divide complex numbers in polar form, multiply/divide the two moduli and add/subtract the arguments. More specifically, for any two complex numbers, \(z_1=r_1(cos(\theta_1)+isin(\theta_1))\) and \(z_2=r_2(cos(\theta_2)+isin(\theta_2))\), we have:

\(z_1 z_2=r_1r_2[cos(\theta_1+\theta_2)+isin(\theta_1+\theta_2)]\)

\(\frac{z_1}{z_2}=\frac{r_1}{r_2}[cos(\theta_1-\theta_2)+isin(\theta_1-\theta_2)], z_2 \ne 0\)

Example: Determine the product and quotient of \(z_1=5cis(\frac{5\pi}{2})\) and \(z_2=7cis(\frac{3\pi}{4})\).


1. Identify the moduli and arguments. In this example, we have:


2. Substitute these values into the above produce and quotient formulas:

\(z_1 z_2=(5)(7)[cos(\frac{5\pi}{2}+\frac{3\pi}{4})+isin(\frac{5\pi}{2}+\frac{3\pi}{4})]\)


3. Evaluate the product and quotient by simplifying both expressions:

\(z_1 z_2=35[cos(\frac{13\pi}{4})+isin(\frac{13\pi}{4})]\)




chat loading...