### Video Transcript

What is the argument of the complex
number ππ, where π is less than zero.

In this question, weβre asked to
find the argument of a complex number. And weβre given the form of this
complex number. Itβs in the form ππ, where our
value of π is negative. To answer this question, weβre
first going need to recall what we mean by the argument of a complex number. First, we recall the argument of a
complex number π§, written arg π§, is the angle that π§ makes with the positive real
axis on an Argand diagram. And there are a few things worth
pointing out about this definition.

First, technically, itβs not the
angle that π§ makes with the positive real axis on our Argand diagram. Itβs actually the angle that the
ray from the origin to π§ makes with this axis. However, it can be useful to think
of it in this way. Next, we usually measure this angle
in radians. However, we can also measure it in
degrees if we prefer. And when our angle is positive,
this means we measured it counterclockwise. And if we give a negative angle,
then we measured our angle clockwise.

Next, much like with any other
angle measured in this manner, thereβs going to be several different angles which
represent the same value. For example, a turn of zero, a turn
of two π, and a turn of four π will all represent the same angle. So usually we choose our argument
to be greater than negative π and less than or equal to π.

So to find the argument of the
complex number given to us in the question, weβre first going to need to sketch an
Argand diagram. Remember, in an Argand diagram, our
horizontal axis represents the real part of our complex number and the vertical axis
represents the imaginary part of our complex number. In our case, we want to plot the
number ππ onto our Argand diagram. We can see that ππ is an
imaginary number. It has no real component. So the real part of ππ is going
to be equal zero. Similarly, the imaginary part of
ππ is going to be the coefficient of π, which in this case is π. And itβs worth pointing out here we
know that our value of π is negative.

Remember in an Argand diagram, the
horizontal coordinate represents the real part of our imaginary number and the
vertical coordinate represents the imaginary part of our complex number. So for our value of ππ whose real
part is zero and imaginary part is π, its coordinates on our Argand diagram is
going to be zero, π, where of course our value of π is negative. So we need to draw this on the
negative part of our imaginary axis.

We want to find the argument of
this complex number, so it can help to add the line segment from the origin to our
value of ππ on our Argand diagram. Then the argument of π§ is going to
be the angle between this line segment and the positive real axis. And weβll measure this clockwise
because it makes the measure of our angle the smallest. Normally, with complex numbers, we
would need to use trigonometry to help us find the argument. However, itβs not necessary in this
case. We can see that this is just a
right angle.

And in radians, a right angle is
given by π by two. But remember, weβre measuring this
angle clockwise, so this needs to be negative π by two. And this gives us our final
answer. The argument of ππ is equal to
negative π by two. Now we could stop here, but weβve
actually proven a very useful result. Weβve proven if π is less than
zero, then the argument of ππ will be equal to negative π by two. So we can actually use this result
to evaluate the argument of any complex number given to us in this form. We wouldnβt need to sketch this
onto an Argand diagram and then find the resulting angle. Although, this is still a good idea
anyway.

Therefore, in this question, we
were able to prove a result about finding the argument of certain complex
numbers. We were able to show the argument
of any complex number in the form ππ, where our value of π is less than zero, is
negative π by two.