# Question Video: Finding the Argument of a Complex Number given the Argument of Its Reciprocal Mathematics

If π is the principal argument of a complex number π, determine the argument of 1/π.

03:10

### Video Transcript

If π is the principal argument of a complex number π, determine the argument of one divided by π.

In this question, weβre told the principal argument of a complex number π is equal to π. We need to use this to determine the argument of one divided by π. To answer this question, letβs start by recalling what we mean by the principal argument of a complex number.

The argument of a complex number π is the angle the line segment between the complex number π and the origin on an Argand diagram makes with the positive real axis. And this is a directed angle. Itβs positive if measured counterclockwise and negative if measured clockwise. And this means that thereβs many different equivalent angles for this argument. For example, we can add and subtract integer multiples of full revolutions.

For the principal argument, we restrict π to be between negative π and π if measured in radians. We need to use this to determine any argument of the complex number one divided by π. Of course, we can assume that π is not equal to zero for a few reasons. First, weβre trying to find one divided by π. This wouldnβt be defined if π was zero. And second, if π was equal to zero, it would be difficult to define an argument of π, since then the line between π and the origin on an Argand diagram would just be the origin. It would just be a single point. So we couldnβt really define an angle in this way.

To answer this question, thereβs several different methods we could take. For example, we could sketch π onto an Argand diagram with an angle of π and then try to determine the geometric relationship between one over π and π. And this could work. However, itβs quite difficult to determine this geometric relationship. Instead, weβre going to use a property of the argument. We recall for two complex numbers π€ sub one and π€ sub two, the argument of their product π€ sub one times π€ sub two is equal to the sum of the arguments, the argument of π€ sub one plus the argument of π€ sub two. And itβs worth pointing out this assumes that π€ sub one and π€ sub two are nonzero. We want to apply this to our question. So weβll set π€ sub one to be π and π€ sub two to be one divided by π. This gives us the argument of π times one over π is equal to the argument of π plus the argument of one divided by π.

We can now simplify this equation. First, π multiplied by one over π is just equal to one. So the left-hand side of this equation simplifies to give us the argument of one. Next, on the right-hand side of this equation, we have the argument of π. And weβre told in the question the principal argument of π is π. So we can also say the argument of π is π. So the right-hand side of this equation simplifies to give us π plus the argument of one divided by π.

On the left-hand side of this equation, we have the argument of a positive real number. And we can recall if π is a real number greater than zero, the argument of π is equal to zero. This is because then π on an Argand diagram lies on the positive real axis. So its angle with the positive real axis will just be zero. Therefore, zero is equal to π plus the argument of one over π. We can then subtract π from both sides of the equation, which gives us the argument of one divided by π is negative π, which is our final answer.

Therefore, we were able to show if π is the principal argument of a complex number, then the argument of one divided by π will be negative π.