Question Video: Relationship between Complex Conjugate and the Argument Mathematics

Video Transcript

Consider the complex number 𝑧 equals negative four minus five 𝑖. Calculate the argument of 𝑧, giving your answer correct to two decimal places in an interval from negative 𝜋 to 𝜋. Calculate the argument of 𝑧 bar, giving your answer correct to two decimal places in an interval from negative 𝜋 to 𝜋.

Let’s begin by reminding ourselves what we mean when we talk about the argument of a complex number. Imagine we’ve plotted our complex number as a point on an Argand diagram. If we join that point to the origin with a line segment, the argument is the angle that this line segment makes with the positive real axis measured generally in a counterclockwise direction. Now, when we think about our argument in a specific interval, here negative 𝜋 to 𝜋, we call that the principal argument of the complex number. And the principal argument can be found by adding or subtracting multiples of two 𝜋 to the value that we get.

So, let’s plot our complex number 𝑧 as a point on the Argand diagram. The real axis is the 𝑥-axis, and the 𝑦-axis is our imaginary axis. So, the point 𝑧 equals negative four minus five 𝑖 has coordinates negative four, negative five. If we join this point to the origin, we get the line segment shown.

Now, one way we can calculate the argument is to measure, as we said, in a counterclockwise direction from the positive real axis. However, since a half turn is 𝜋 radians, this will give us a value that is greater than 𝜋. It will be outside of our interval. So alternatively, we can measure in a clockwise direction. And whatever angle we get, we know it’s going to be negative.

Let’s add a right triangle to our line segment and call the included angle of this specific triangle 𝛼. We then know that the adjacent side in this triangle is five units and the opposite side is equal to four. So, let’s use the tan ratio to find the value of 𝛼. tan of 𝛼 is opposite over adjacent. So, in the case of our right triangle, tan of 𝛼 is four-fifths. So, taking the inverse tan or arctan of both sides, we get 𝛼 is the arctan of four-fifths, which is 0.6747 radians and so on.

Now, in order to measure the magnitude of the angle that this line segment makes with the positive real axis measured in a clockwise direction, we’re going to add 𝜋 by two radians, that quarter of a turn, to this value. 𝜋 by two plus 0.6747 and so on is 2.2455. So, that’s the magnitude of the angle. This means that the argument is going to be the negative of this. Rounding that to two decimal places, and we get 2.25. So, the argument of 𝑧 in the interval negative 𝜋 to 𝜋 is negative 2.25 radians.

Now let’s move on to the second part of this question. This time, we need to find the argument of 𝑧 bar. But what is 𝑧 bar? Well, 𝑧 bar is the conjugate. So, let’s imagine we have a complex number 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. 𝑧 bar is just found by changing the sign of the imaginary part, so it would be 𝑎 minus 𝑏𝑖. In the case of our complex number, if we change the sign of the imaginary part, we find 𝑧 bar is negative four plus five 𝑖. So, let’s plot this on the same complex plane.

Since the real part is the same and the imaginary part has changed signs, we can find 𝑧 bar by essentially reflecting point 𝑧 in the real axis. And this, in fact, is really useful because it means we don’t need to do a full calculation to find the value of the argument of 𝑧. Since we’re reflecting point 𝑧 in the real axis, the angle that 𝑧 bar makes with the real axis will in fact be the same as the angle that 𝑧 makes with the real axis just measured in the opposite direction. This means it will be the negative of the value for the argument of 𝑧. So, we need to multiply the argument of 𝑧 by negative one to find the argument of 𝑧 bar. That gives us 2.25. So, the argument of 𝑧 bar is 2.25.