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

If πœƒ is the principal argument of a complex number 𝑍, determine the argument of 1/𝑍.

03:10

Video Transcript

If πœƒ is the principal argument of a complex number 𝑍, determine the argument of one divided by 𝑍.

In this question, we’re told the principal argument of a complex number 𝑍 is equal to πœƒ. We need to use this to determine the argument of one divided by 𝑍. To answer this question, let’s start by recalling what we mean by the principal argument of a complex number.

The argument of a complex number 𝑍 is the angle the line segment between the complex number 𝑍 and the origin on an Argand diagram makes with the positive real axis. And this is a directed angle. It’s positive if measured counterclockwise and negative if measured clockwise. And this means that there’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œƒ to be between negative πœ‹ and πœ‹ if measured in radians. We need to use this to determine any argument of the complex number one divided by 𝑍. Of course, we can assume that 𝑍 is not equal to zero for a few reasons. First, we’re trying to find one divided by 𝑍. This wouldn’t be defined if 𝑍 was zero. And second, if 𝑍 was equal to zero, it would be difficult to define an argument of 𝑍, since then the line between 𝑍 and the origin on an Argand diagram would just be the origin. It would just be a single point. So we couldn’t really define an angle in this way.

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

We can now simplify this equation. First, 𝑍 multiplied by one over 𝑍 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 𝑍. And we’re told in the question the principal argument of 𝑍 is πœƒ. So we can also say the argument of 𝑍 is πœƒ. So the right-hand side of this equation simplifies to give us πœƒ plus the argument of one divided by 𝑍.

On the left-hand side of this equation, we have the argument of a positive real number. And we can recall if 𝑐 is a real number greater than zero, the argument of 𝑍 is equal to zero. This is because then 𝑐 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œƒ plus the argument of one over 𝑍. We can then subtract πœƒ from both sides of the equation, which gives us the argument of one divided by 𝑍 is negative πœƒ, which is our final answer.

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

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