Question Video: Using Multiplication of Complex Numbers to Describe Rotations Mathematics

Video Transcript

A complex number is multiplied by another complex number 𝑧 and then by the complex conjugate 𝑧 bar. How is the argument of the resulting complex number related to the argument of the original complex number?

In this question, we’re told that a complex number, which we will label 𝑤, is multiplied by another complex number, called 𝑧, and its complex conjugate 𝑧 bar. We need to use this to determine how the argument of the resulting complex number — that’s 𝑤 times 𝑧 times 𝑧 bar — is related to the argument of the original complex number, which is 𝑤. To answer this question, we’ll start by recalling a very useful property of the arguments of complex numbers. If we have two nonzero complex numbers 𝑤 sub one and 𝑤 sub two, then the argument of the products of these numbers is the sum of their arguments. In other words, the argument of 𝑤 sub one times 𝑤 sub two is equal to the argument of 𝑤 sub one plus the argument of 𝑤 sub two.

We want to apply this to the argument of 𝑤 times 𝑧 times 𝑧 bar so we can find an expression for the argument of the resulting complex number. And since the question wants us to find the relationship between this argument and the argument of the original complex number, we want to include the argument of 𝑤 in our expression. So, in our formula, we want to set 𝑤 sub one equal to 𝑤 and 𝑤 sub two equal to 𝑧 multiplied by 𝑧 bar. However, there is one small thing we do need to check before we do this. Remember, this result only holds true if 𝑤 sub one and 𝑤 sub two are nonzero, because if either of these two complex numbers is zero, then their arguments are not defined.

And in the question, we’re not explicitly told that 𝑤, 𝑧, or 𝑧 bar are nonzero. However, if 𝑤 is zero, then 𝑤 times 𝑧 times 𝑧 bar is also equal to zero. And this means the question is then asking us to determine the relationship between the argument of zero and the argument of zero, which is of course undefined. So, we can assume that 𝑤 is nonzero. We get a similar story if we assumed that 𝑧 was zero. If 𝑧 is zero, then once again 𝑤 times 𝑧 times 𝑧 bar is also equal to zero. Then the question wants us to determine a relationship between the argument of 𝑤 and the argument of zero. And the argument of 𝑤 can be anything because we can choose any complex number 𝑤, and the argument of zero is undefined. So, once again, we’ll assume that 𝑧 is nonzero. And this justifies our use of the argument of a product is the sum of their arguments.

Let’s now see if we can simplify this equation. We have 𝑧 multiplied by its complex conjugate 𝑧 bar. And we can recall when we multiply a complex number by its conjugate, we get the magnitude squared. This is equal to the magnitude of 𝑧 squared. And this is a useful result because the magnitude of a complex number is a nonnegative number. In fact, since 𝑧 is nonzero, this is a positive real number. And we can recall the argument of a positive real number 𝑐 must be zero. Since then, on our Argand diagram, 𝑐 will lie on the positive real axis, so its angle with the positive real axis is zero. Therefore, the right-hand side of this equation simplifies to give us the argument of 𝑤. We can see the argument of 𝑤 times 𝑧 times 𝑧 bar is equal to the argument of 𝑤. In other words, the argument of 𝑤 is unchanged when multiplied by 𝑧 and the complex conjugate of 𝑧 provided 𝑧 is nonzero.

Therefore, we were able to show if a complex number is multiplied by a nonzero complex number 𝑧 and its complex conjugate 𝑧 bar, then its argument remains unchanged.