### Video Transcript

In what quadrant does the complex conjugate of ๐ง lie?

In this question, weโre given an Argand diagram containing the point ๐ง. And to do this, we need to determine in which quadrant would the complex conjugate of ๐ง lie. And before we do this, there is something worth pointing out about the notation for the complex conjugate. Sometimes, youโll see this written with a horizontal bar. Or sometimes, youโll see this written with a star. And both of these mean exactly the same thing; ๐ง star and ๐ง bar both just mean the complex conjugate of our complex number ๐ง. So to answer this question, weโre first going to need to recall exactly what we mean when we say weโre taking the complex conjugate of a complex number.

To do this, letโs start by saying we have a complex number in algebraic form. So ๐ง is equal to ๐ plus ๐๐ where ๐ and ๐ are real numbers. Taking the complex conjugate of this number just means weโre going to switch the sign of our coefficient of ๐. So instead of adding ๐ times ๐, weโre going to subtract ๐ times ๐. The complex conjugate of ๐ง will be ๐ minus ๐๐. So now we know how to find the complex conjugate of ๐ง given ๐ง in algebraic form. But this is not the form weโre given ๐ง in the question. Instead, weโre given the ๐ง on an Argand diagram. So weโre 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 ๐ง is given in the algebraic form ๐ plus ๐๐, then we say that ๐ is the real part of our imaginary number and our coefficient ๐ is the imaginary part of our complex number. And we can represent this in the following notation. The real part of ๐ง is equal to ๐ and the imaginary part of ๐ง is equal to ๐.

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 ๐ง. First, weโll 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 ๐ง. In other words, this is giving us the value of ๐. So weโll write the value of ๐ equal to negative 3.8. And we can do exactly the same to find the vertical coordinate of ๐ง. We can see from our diagram this is at three, and this is going to tell us the imaginary part of ๐ง which is our value of ๐. So, our value of ๐ is just going to be equal to three.

So just by looking at our Argand diagram, we were able to show that ๐ง is equal to negative 3.8 plus three ๐. But this is not what the question wants us to do. It wants to find in which quadrant the complex conjugate of ๐ง will lie. But remember, to find the complex conjugate of ๐ง, we just need to switch the sign of our coefficient of ๐. So all weโre doing to find the complex conjugate of ๐ง is instead of adding three ๐, weโre now subtracting three ๐. But we want to know in which quadrant does this lie on our Argand diagram. So weโre going to need to plot this point onto our Argand diagram.

To do this, we can see the real part of ๐ง 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 ๐ง, itโs 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 weโve switched the sign. The imaginary part of the complex conjugate of ๐ง is going to be equal to negative three. So the complex conjugate of ๐ง 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 youโll see this written out in full. However, youโll often see this written in Roman numerals. In either case, we can see the complex conjugate of ๐ง is lying in the third quadrant. So we were able to show the complex conjugate of ๐ง is going to lie in the third quadrant of our Argand diagram.

There is one thing worth pointing out here though; whenever weโre given ๐ง, we can see that the only thing we do to find the complex conjugate of ๐ง is switch the sign of our imaginary part. So weโre never changing the real part of our complex number. This means weโre never changing its horizontal coordinate on an Argand diagram. Instead, weโre always switching the sign of its vertical coordinate. Weโre 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 couldโve just answered this question by reflecting the point ๐ง 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 ๐ง lies given ๐ง in an Argand diagram. Using both methods, we were able to show that it will lie in the third quadrant.