Representing Complex Numbers With Vectors

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will explore complex numbers and vectors, and we will look at how these two concepts, though seemingly unrelated, work together by representing complex numbers with vectors.

Vectors and Complex Numbers

You may be surprised to find out that there is a relationship between complex numbers and vectors. After all, consider their definitions.

  • A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary number √(-1). We call a the real part of the complex number, and we call b the imaginary part of the complex number.
  • Abstractly speaking, a vector is something that has both a direction and a length (or magnitude). To imagine this, consider a plane's path from point A to point B. The path has a direction, and it has a length (distance), so it is considered to be a vector. A more visual representation of a vector is a directed line segment, or a line segment that has one endpoint, called the head, and a ray pointing in some direction from that endpoint to another point, called the tail. The directed line segment has a length and it is pointing in some direction, so it has a direction as well. Hence, it is a vector.


Hmmm…these seemingly have nothing to do with each other. However, it turns out that we can actually use vectors to represent complex numbers in the complex plane. Let's take a look at how to do this!

Representing Complex Numbers With Vectors

In the complex plane, we call the x-axis the real axis, and we call the y-axis the imaginary axis. When viewing the complex plane this way, we can graph complex numbers of the form a + bi by locating a on the real axis, and then moving up or down to b on the imaginary axis. Simply put, we plot the point (a, b) on the complex plane.

Complex Plane

This is where vectors come into play! You see, once we've plotted the point (a, b), if we draw a directed line segment from the origin of the complex plane to the point (a, b), then we've created a vector representing the complex number a + bi on the complex plane.

a + bi as a vector

In other words, to represent a complex number, a + bi, using a vector, we use the following steps:

  1. Plot the point (a, b) on the complex plane.
  2. Draw a directed line segment from the origin of the plane to the point (a, b). This is the vector representing a + bi.

Ah-ha! There's the connection! That really is quite neat! Let's consider some examples of using vectors to represent complex numbers.


Consider the following complex numbers:

  • 3 - 4i
  • 2 + 8i
  • -5 + i
  • -7 - 7i

Let's use our steps to represent each of these complex numbers with a vector. Starting with the complex number 3 - 4i, we have that a = 3, and b = -4. Therefore the first step is to plot the point (3, -4) on the complex plane. Once we've done this, we then draw a vector with its head at the origin, and its tail at the point (3,-4). This is our vector representing the complex number 3 - 4i.


Now, let's consider the other three complex numbers by first identifying a and b in each of the numbers and finding the point that we need to plot for the first step for each of them.

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