Question Video: Graphing Complex Numbers Mathematics

Video Transcript

Let us consider a complex number 𝑧 with nonzero real and imaginary parts. If the real and imaginary parts of 𝑧 are the same sign, in which quadrant or quadrants of the Argand diagram could 𝑧 appear? And if the real and imaginary parts of 𝑧 are of opposite signs, in which quadrant or quadrants of the Argand diagram could 𝑧 appear?

Let’s begin by recalling the general form of a complex number 𝑧 and how to plot these on an Argand diagram. A complex number is of the form 𝑎 plus 𝑏𝑖. We say that 𝑎, which would be a real number, is the real part of 𝑧. Similarly, the coefficient of 𝑖, which again would be a real number, is the imaginary part of 𝑧.

In the first part of this question, we’re told that the real and imaginary parts of 𝑧 are of the same sign. This means both 𝑎 and 𝑏 could be positive, but also 𝑎 and 𝑏 could both be negative. And so let’s imagine what that might look like on an Argand diagram. Remember, an Argand diagram is a lot like the coordinate plane, except the horizontal axis represents the real component and the vertical axis represents the imaginary component of our complex number.

If we start at the origin and move directly right, we’re moving in the positive direction. If we move up, we’re also moving in the positive imaginary direction. The other two directions are considered to be negative. If we travel left, we’re traveling in the negative real direction. And if we travel down, we’re traveling in the negative imaginary direction.

So let’s imagine we have a complex number whose real and imaginary parts are positive. Note that since both the real and imaginary parts are nonzero, our complex numbers cannot lie on the axes. We’re going to need to start at the origin, move right, and then move up. And so our complex number will lie somewhere in this quadrant.

However, we could also have a complex number whose real and imaginary parts are both negative. This time, we start at the origin, but we’d move left, and then we’d move down. And so our complex number would be in this quadrant.

Then we label the quadrant as shown. We start in the top right and then move in a counterclockwise direction one through four. And we see that when the real and imaginary parts of 𝑧 are both positive, our complex number lies in the first quadrant. But when those parts are both negative, the complex number lies in the third quadrant. And so if the real and imaginary parts of 𝑧 are the same sign, we know that the quadrants in which 𝑧 could lie would be the first or the third.

The second part of this question is really similar, but this time the real and imaginary parts of 𝑧 are opposite signs. In other words, one will be positive and one will be negative. Let’s start with a complex number whose real part is positive and whose imaginary part is negative. We’ll need to start at the origin. And since the real part is positive, we’re going to move horizontally to the right. Then since the imaginary part is negative, we’re going to travel down. And so our complex number could lie somewhere in this quadrant.

But what would happen if the real part was negative and the imaginary part was positive? Once again, we start at the origin, but now we move left and then we move up. And so our complex number would be somewhere in this quadrant. As before, we label our quadrants as shown. And so if the real and imaginary parts of 𝑧 are of opposite signs, 𝑧 could appear in either the second or fourth quadrants of the Argand diagram.