Contents

Using Desmos: Complex Numbers

Created: 2025/3/10

This section covers how to use complex numbers in the Desmos Graphing Calculator. You can easily visualize the complex plane!

Enable Complex Numbers

By default, complex numbers are disabled. Let's start by turning on the complex number feature.

As shown in the image, there's a "Complex Mode" toggle in the settings menu. Turn it on to enable complex numbers.

What You Can Do with Complex Numbers

When you input complex numbers like the examples below, they are automatically plotted on the complex plane.

i

1+2i

Desmos supports a variety of operations with complex numbers.

(1+i)+(2-2i)\quad (=3-i)

(1+i)(2-2i)\quad (=4)

\textrm{Real part: }\mathrm{real}(2+4i)\quad (=2)

\textrm{Imaginary part: }\mathrm{imag}(2+4i)\quad (=4)

\textrm{Conjugate: }\mathrm{conj}(2+4i)\quad (=2-4i)

\textrm{Argument: }\mathrm{arg}(1+i)\quad \left(=\dfrac{\pi}{4}\right)

\textrm{Absolute value: }|3+4i|\quad (=5)

e^{i\pi}\quad (=-1)

You can also apply various functions to complex numbers.

\mathrm{round}(1.2+3.9i)\quad (=1+4i)

\mathrm{floor}(1.2+3.9i)\quad (=1+3i)

\mathrm{ceil}(1.2+3.9i)\quad (=2+4i)

\mathrm{mod}(1+3i,2i)\quad (=-1-i)

\mathrm{gcd}(4+2i,1-3i)\quad (=1+i)

\mathrm{lcm}(1.2+3.9i)\quad (=10)

\dfrac{d}{dx}\left(x+ix^2\right)\quad (=1+2ix)

Images are created with the Desmos Graphing Calculator, used with permission from Desmos Studio PBC.