In what quadrant does the complex conjugate of 𝑧 lie?
In this question, we’re given an Argand diagram containing the point 𝑧. And to do this, we need to determine in which quadrant would the complex conjugate of 𝑧 lie. And before we do this, there is something worth pointing out about the notation for the complex conjugate. Sometimes, you’ll see this written with a horizontal bar. Or sometimes, you’ll see this written with a star. And both of these mean exactly the same thing; 𝑧 star and 𝑧 bar both just mean the complex conjugate of our complex number 𝑧. So to answer this question, we’re first going to need to recall exactly what we mean when we say we’re taking the complex conjugate of a complex number.
To do this, let’s start by saying we have a complex number in algebraic form. So 𝑧 is equal to 𝑎 plus 𝑏𝑖 where 𝑎 and 𝑏 are real numbers. Taking the complex conjugate of this number just means we’re going to switch the sign of our coefficient of 𝑖. So instead of adding 𝑏 times 𝑖, we’re going to subtract 𝑏 times 𝑖. The complex conjugate of 𝑧 will be 𝑎 minus 𝑏𝑖. So now we know how to find the complex conjugate of 𝑧 given 𝑧 in algebraic form. But this is not the form we’re given 𝑧 in the question. Instead, we’re given the 𝑧 on an Argand diagram. So we’re going to need to recall what we mean by a point on an Argand diagram.
In an Argand diagram, each point represents a complex number. The horizontal axis tells us the real component of our complex number and the vertical axis tells us the imaginary part of our complex number. And this is particularly useful for finding the algebraic form of a complex number on our Argand diagram. If 𝑧 is given in the algebraic form 𝑎 plus 𝑏𝑖, then we say that 𝑎 is the real part of our imaginary number and our coefficient 𝑏 is the imaginary part of our complex number. And we can represent this in the following notation. The real part of 𝑧 is equal to 𝑎 and the imaginary part of 𝑧 is equal to 𝑏.
And in an Argand diagram, the real part of our complex number will be given by its horizontal coordinate and its imaginary part will be given by its vertical coordinate. So we can use our Argand diagram to find the value of 𝑧. First, we’ll want to find its horizontal coordinate. We can see from our graph this is approximately at negative 3.8. Remember, this is telling us the real part of our imaginary number 𝑧. In other words, this is giving us the value of 𝑎. So we’ll write the value of 𝑎 equal to negative 3.8. And we can do exactly the same to find the vertical coordinate of 𝑧. We can see from our diagram this is at three, and this is going to tell us the imaginary part of 𝑧 which is our value of 𝑏. So, our value of 𝑏 is just going to be equal to three.
So just by looking at our Argand diagram, we were able to show that 𝑧 is equal to negative 3.8 plus three 𝑖. But this is not what the question wants us to do. It wants to find in which quadrant the complex conjugate of 𝑧 will lie. But remember, to find the complex conjugate of 𝑧, we just need to switch the sign of our coefficient of 𝑖. So all we’re doing to find the complex conjugate of 𝑧 is instead of adding three 𝑖, we’re now subtracting three 𝑖. But we want to know in which quadrant does this lie on our Argand diagram. So we’re going to need to plot this point onto our Argand diagram.
To do this, we can see the real part of 𝑧 is negative 3.8 and the real part of our complex conjugate is also negative 3.8. So when we plot the complex conjugate of 𝑧, it’s also going to have a horizontal coordinate of negative 3.8. However, if we were to compare the imaginary part of these two numbers, we can see we’ve switched the sign. The imaginary part of the complex conjugate of 𝑧 is going to be equal to negative three. So the complex conjugate of 𝑧 is going to have vertical coordinate negative three, and we already know its horizontal coordinate is negative 3.8, and we can just plot this point onto our diagram.
Now all we need to determine is which quadrant this lies in. We label the quadrants in an Argand diagram exactly the same way we do in a Cartesian graph. When both the real and imaginary parts are positive, we call this the first quadrant. We then just number our quadrants counterclockwise. Sometimes you’ll see this written out in full. However, you’ll often see this written in Roman numerals. In either case, we can see the complex conjugate of 𝑧 is lying in the third quadrant. So we were able to show the complex conjugate of 𝑧 is going to lie in the third quadrant of our Argand diagram.
There is one thing worth pointing out here though; whenever we’re given 𝑧, we can see that the only thing we do to find the complex conjugate of 𝑧 is switch the sign of our imaginary part. So we’re never changing the real part of our complex number. This means we’re never changing its horizontal coordinate on an Argand diagram. Instead, we’re always switching the sign of its vertical coordinate. We’re switching the sign of the imaginary part of our number. This is exactly the same as just reflecting our point through the horizontal axis. So in fact, we could’ve just answered this question by reflecting the point 𝑧 in the horizontal axis and seeing that this lies in the third quadrant, and this is also a valid way to answer this question.
Therefore, we were able to show two different methods of determining in which quadrant the complex conjugate of 𝑧 lies given 𝑧 in an Argand diagram. Using both methods, we were able to show that it will lie in the third quadrant.