Question Video: Determining the Quadrant Containing a Given Complex Number in Algebraic Form Mathematics

Video Transcript

In which quadrant of the Argand diagram does the complex number three minus two 𝑖 lie?

In this question, we’re given the complex number three minus two 𝑖, and we need to determine in which quadrant of our Argand diagram does this complex number lie. To do this, let’s start by recalling what we mean by an Argand diagram. In an Argand diagram, we draw a pair of axes, a horizontal axis and a vertical axis. And we label the horizontal axis the real part of our number and the vertical axis the imaginary part of our number. So every point in our Argand diagram will have a real value and an imaginary value, based entirely on its coordinates on our diagram. And in this question, we’re going to want to plot the point three minus two 𝑖 onto our Argand diagram.

So to do this, we need to determine the real and imaginary parts of this number. We can do this directly from the number. However, there is some useful notation to write this down. First, we can represent the real part of a complex number by using the following notation. And the real part of the complex number three minus two 𝑖 is going to be equal to three. In fact, for any complex number given in algebraic form, that’s in the form 𝑎 plus 𝑏𝑖 where 𝑎 and 𝑏 are real numbers, the real part of this number is always just going to be equal to 𝑎. So what does this tell us about our Argand diagram? Well, if the real point of our complex number is equal to three, then on our Argand diagram, the horizontal coordinate of this point should be equal to three.

Next, we want to find the imaginary part of this number. We represent this in the following notation. And this time, the imaginary part of a complex number given in the algebraic form 𝑎 plus 𝑏𝑖 is just equal to 𝑏. It’s just going to be the coefficient of 𝑖. In this case, we have three minus two 𝑖, so our coefficient of 𝑖 is equal to negative two. And the imaginary part of our number being equal to negative two means that the vertical coordinate of our point is going to be equal to negative two. Now, we could just mark the point three minus two 𝑖 onto our Argand diagram. It has horizontal coordinate three and vertical coordinate negative two.

However, remember the question doesn’t want us to just plot this point. It wants us to determine which quadrant of our Argand diagram is the point in. And in an Argand diagram, we label our quadrants in exactly the same way we would for a Cartesian diagram. The first quadrant will be where the real value and the imaginary value are positive. That’s the top-right quadrant. And then we number our quadrants counterclockwise, and sometimes we might write these in Roman numerals. In either case, we can see that the number three minus two 𝑖 is in our fourth quadrant.

Therefore, we were able to show if we’re going to plot the complex number three minus two 𝑖 onto an Argand diagram, then it will lie in the fourth quadrant.