Question Video:  Representing Complex Numbers on an Argand Diagram Mathematics

Video Transcript

Seven complex numbers 𝑧 one, 𝑧 two, 𝑧 three, 𝑧 four, 𝑧 five, 𝑧 six, and 𝑧 seven are represented on the Argand diagram. This question has four parts. Part one, which of the complex numbers is negative three plus two 𝑖? Part two, what complex number is represented by 𝑧 sub four? Part three, which complex number has equal real and imaginary parts? And part four, which two complex numbers are a conjugate pair and what is their geometric relationship?

In this question, we’re given seven complex numbers, and we’re given them represented on an Argand diagram. So before we start answering this question, let’s start by recalling what it means to represent a complex number on an Argand diagram. In an Argand diagram, we label the horizontal axis as the real axis and the vertical axis as the imaginary axis so that the horizontal coordinate represents the real part of our complex number and the vertical coordinate represents the imaginary part of the complex number. So for a complex number written in the form 𝑎 plus 𝑏𝑖 where 𝑎 and 𝑏 are real constants, in an Argand diagram, this will have the coordinates 𝑎, 𝑏.

We can use this to answer the first part of our question. We need to determine which of the complex numbers is the point negative three plus two 𝑖. For the complex number negative three plus two 𝑖, the real part of this complex number, that’s the value of 𝑎, is going to be negative three. Similarly, the imaginary part of our complex number, that’s the coefficient of 𝑖 or 𝑏, is going to be equal to two. Therefore, in our Argand diagram, the coordinates of the point negative three plus two 𝑖 will be negative three, two. We can then mark this point on the Argand diagram given to us; the horizontal coordinate is negative three, and the vertical coordinate is two. We can see this is represented by the point 𝑧 three. Therefore, we’ve shown that 𝑧 sub three represent the complex number negative three plus two 𝑖.

We can follow the same process in reverse to answer the second part of this question. Which complex number is represented by 𝑧 sub four? Remember the coordinates of a complex number on the Argand diagram will tell us the real part of our complex number and the imaginary part of our complex number. We can see the point 𝑧 sub four in our Argand diagram has horizontal coordinate negative four and vertical coordinate negative one. Therefore, since in our Argand diagram the point with coordinates 𝑎, 𝑏 represents the complex number 𝑎 plus 𝑏𝑖, we know the point negative four, negative one will represent the complex number negative four minus 𝑖. This means we’ve shown the complex number represented by 𝑧 sub four in our Argand diagram is the complex number negative four minus 𝑖.

The third part of this question wants us to determine which of the complex numbers will have equal real and imaginary parts. If the real and imaginary parts of our complex number are equal, then our complex number is going to be of the form 𝑎 plus 𝑎𝑖 for some real constant 𝑎. But remember, in an Argand diagram the complex number 𝑎 plus 𝑏𝑖 will have coordinates 𝑎, 𝑏. Therefore, if the real and imaginary parts of our complex number are equal, then the horizontal coordinate and the vertical coordinate of our complex number in the Argand diagram must be also be equal. The complex number 𝑎 plus 𝑎𝑖 will have coordinates 𝑎, 𝑎. This means we need to determine which of our seven complex numbers has the same horizontal and vertical coordinate.

There’s a few different ways of doing this. We could check each of our complex numbers separately. However, we can also mark all of the points on the Argand diagram with equal horizontal and vertical coordinate. If we call the horizontal axis 𝑥 and the vertical axis 𝑦, this is the line 𝑦 is equal to 𝑥. We can see that only one of our complex numbers lies on this line: 𝑧 sub five. Its horizontal coordinate is negative two, and its vertical coordinate is also equal to negative two. This means that 𝑧 five represents the complex number negative two minus two 𝑖, which means it must have real and imaginary parts.

The final part of this question wants us to determine which of our two complex numbers is a conjugate pair, and it also wants us to determine the geometric relationship that these two points will have in our Argand diagram. To answer this part of the question, we’ll start by recalling what it means for two complex numbers to be a conjugate pair. We recall we say that two complex numbers are a conjugate pair if they have equal real part; however, their imaginary parts have opposite signs. In other words, for real constants 𝑎 and 𝑏, 𝑎 plus 𝑏𝑖 and 𝑎 minus 𝑏𝑖 will be a conjugate pair. We could also write this in terms of their coordinates on an Argand diagram. In an Argand diagram, 𝑎 plus 𝑏𝑖 will have coordinates 𝑎, 𝑏, and 𝑎 minus 𝑏𝑖 will have coordinates 𝑎, negative 𝑏.

So we’re looking for two points with an equal horizontal coordinate; however, their vertical coordinates need to have opposite sign. We’ll do this in numerical order starting with 𝑧 one. First, we’ll determine if there’s any complex number with the same horizontal coordinate as 𝑧 sub one. We can see that there is; the point 𝑧 sub six will also have horizontal coordinate too. For these two numbers to be complex conjugates, they need to have opposite signs of their imaginary parts. The imaginary part of 𝑧 sub one is three, and the imaginary part of 𝑧 sub six is equal to negative three. Therefore, 𝑧 sub one and 𝑧 sub six have the same real parts. However, their imaginary parts have opposite signs. This means they are a conjugate pair.

We could stop here; however, we can also check the rest of our complex numbers. Let’s check 𝑧 sub two. We can see its real part is equal to negative two. Our only other complex number with a real part of negative two is 𝑧 sub five. However, if we check their imaginary parts, we could see these don’t have opposite signs. The imaginary part of 𝑧 sub two is equal to three; however, the imaginary part of 𝑧 sub five is negative two. Now, there’s no complex number given to us with the same real part as 𝑧 sub three or 𝑧 sub four or 𝑧 sub seven, so the only conjugate pair given to us is 𝑧 sub one and 𝑧 sub six.

The last thing we need to do is determine the geometric relationship between a conjugate pair on an Argand diagram. And we can do this by considering the coordinates of a conjugate pair. Their horizontal coordinates must be equal; however, their vertical coordinates must have opposite signs. And switching the sign of the vertical coordinate will be a reflection in the horizontal axis. This means for any conjugate pair of complex numbers, in an Argand diagram, they will be related by a reflection in the real axis or 𝑥-axis.

Therefore, we were able to show from a diagram of seven given complex numbers 𝑧 sub three is the complex number negative three plus two 𝑖, 𝑧 sub four represents the complex number negative four minus 𝑖, only the complex number 𝑧 sub five has equal real and imaginary parts, and finally the only conjugate pair is the pair of complex numbers 𝑧 sub one and 𝑧 sub six, and these are related by a reflection in the real axis.