### Video Transcript

In this video, we’re going to learn
how to calculate the argument of a complex number. We will learn what we mean by the
terms argument and principal argument and how to calculate these. We’ll consider the properties of
the argument in relation to the complex conjugate and learn how to find the argument
of products and quotients of complex numbers.

We know that we can represent
complex numbers on a two-dimensional plane. We call this plane the Argand
diagram or Argand plane, after the amateur mathematician who discovered it. We can use it to graph a complex
number of the form 𝑧 equals 𝑎 plus 𝑏𝑖. Remember, the real part of this
complex number is 𝑎 and the imaginary part is 𝑏. We need to locate the real part 𝑎
on the real axis. That’s the horizontal axis. We then move up or down to locate
the imaginary part 𝑏 on the imaginary axis. That’s the vertical axis. 𝑎 plus 𝑏𝑖 can therefore be
represented by the point whose Cartesian coordinates are 𝑎, 𝑏 as shown.

If we had a straight line
connecting this point to the origin, we see that we can work out extra pieces of
information. We can work out the angle that this
line segment makes with the positive real axis. In fact, we call this the argument
of the complex number. We denote it as shown. And it’s important that we remember
we have to measure this from the positive real axis in a counterclockwise
direction. And it’s usually measured in
radians. And there is an extra definition
required here. The principal argument for 𝜃 in
radians is defined as 𝜃 is greater than negative 𝜋 and less than or equal to
𝜋. Though it isn’t completely
unreasonable to talk about the argument outside of this range.

Since the complex number can be
graphed in four quadrants, we also can see that the complex numbers located in the
third and fourth quadrant will have arguments measured in the wrong direction, if
you will. In this case, the principal
argument of our complex number will be negative. So how do we calculate the value of
the argument? Well, let’s say we have a complex
number of the form four plus three 𝑖. We can see that we can represent
this on the Argand diagram by the point whose Cartesian coordinates are four,
three. The argument of this complex number
is the 𝜃 that’s shown. It’s the angle measured in a
counterclockwise direction from the positive real axis.

And what we can do next is adding a
right-angled triangle which has an included angle of 𝜃. Remember, that’s our argument. The side opposite this angle is
three units in length. And the side adjacent to this angle
is four units in length. By defining the sides like this, we
can use right angle trigonometry to calculate the measure of the included angle, the
argument. We know that tan 𝜃 is equal to
opposite over adjacent. So for our right-angled triangle,
tan 𝜃 is equal to three over four.

We solve this equation for 𝜃 by
finding the inverse of tan or arctan of both sides. And we see that 𝜃 is equal to
arctan of three quarters. That gives us a value of 0.64
radians, correct to two significant figures. And therefore, the argument of this
complex number is 0.64 radians. Now, in fact, the real function
arctan of 𝑥 is a multi-valued function for real values of 𝑥. So we can’t generalize this as a
formula for every complex number. Let’s see an example of where we
might come unstuck.

Given that 𝑧 equals negative
one-half plus root three over two 𝑖, find the principal argument of 𝑧.

We have a complex number
represented in algebraic or rectangular form for which we’re trying to calculate the
principal argument. Remember, that’s the value of the
argument that is greater than negative 𝜋 and less than or equal to 𝜋. It’s always sensible to begin by
plotting this complex number on an Argand diagram. And this will help us decide which
quadrant a complex number lies in. Remember, the horizontal axis on
our Argand diagram represents the real part whereas the vertical axis represents the
imaginary part. The real part of our complex number
is negative one-half. And the imaginary part is root
three over two.

So we can represent our complex
number on the Argand diagram as a point whose Cartesian coordinates are negative
one-half, root three over two. This one lies in the second
quadrant as shown. We’ll add in a line segment joining
this point to the origin. And then we recall the definition
of the argument. It’s the angle this line segment
makes with the positive real axis measured in a counterclockwise direction. Given that angles on a straight
line sum to 𝜋 radians, it’s sensible to begin by finding the acute angle that I’ve
marked 𝛼. And, in fact, it’s actually a good
idea to try and choose the acute angle and work from there in any example.

The side opposite this acute angle
𝛼 is root three over two units. And the length of the side adjacent
to it is a half of a unit. Now, remember, we’re working with
length. So we’re interested in positive
values only. We might call those the moduli of
the real and imaginary parts. Tan of 𝛼 is equal to opposite over
adjacent. It’s root three over two divided by
one-half. Now, in fact, this simplifies
somewhat to tan of 𝛼 equals root three. We can solve this equation for 𝛼
by finding the inverse of tan. 𝛼 is equal to arctan of root three
which is 𝜋 by three radians.

We said that angles on a straight
line sum to 𝜋 radians. So we can find the value of the
argument of 𝑧 by subtracting 𝜋 over three radians from 𝜋. And, of course, another way of
thinking about 𝜋 is as three 𝜋 over three. And this will allow us to subtract
these fractions as normal. Three 𝜋 over three minus 𝜋 over
three is two 𝜋 over three radians. And if we compare this to the value
for the principal argument that’s greater than negative 𝜋 and less than or equal to
𝜋, we see that the value for our argument of 𝑧 does indeed satisfy this
criteria. It’s two 𝜋 over three radians.

Now, in fact, if we tried to
generalize our formula from the previous example and just said that argument of 𝑧
is equal to arctan of 𝑏 divided by 𝑎, which is the arctan of root three over two
divided by negative one-half, we would’ve gotten negative 𝜋 by three, which is
clearly incorrect.

Let’s see if we can find a general
rule for finding the argument of a complex number plotted in any quadrant.

Here, we have four Argand diagrams,
with a complex number plotted in each of the four quadrants. For each of these examples, we can
begin by finding the measure of the acute angle. We can find the argument for a
complex number plotted in the first quadrant by finding the arctan of 𝑏 divided by
𝑎. That’s the arctan of the imaginary
part divided by the real part. That’s actually enough for the
argument of a complex number plotted in the first quadrant.

In the second quadrant, we know
that the acute angle is found by finding the arctan of the modulus of 𝑏 divided by
the modulus of 𝑎. Remember, in the previous example,
we said we were dealing with lengths. So they need to be positive. And then we use the fact that
angles on a straight line sum to 𝜋 radians. And we can find the argument by
subtracting arctan of the modulus of 𝑏 divided by the modulus of 𝑎 from 𝜋.

Now, actually, for the complex
number plotted in the third quadrant, we could work out the size of the acute
angle. But actually, the method’s almost
identical for that we used to work out the argument of a complex number plotted in
the second quadrant. The only difference is, it’s like a
reflection in the horizontal axis. We’re essentially travelling in the
opposite direction. So we can say that it’s equal to
negative 𝜋 minus arctan of the modulus of 𝑏 divided by the modulus of 𝑎. And if we distribute these
parentheses, we see that the argument of the complex number plotted in the third
quadrant is the arctan of the modulus of 𝑏 divided by the modulus of 𝑎 minus
𝜋.

And in a similar way, we can say
that the argument of the complex number plotted in the fourth quadrant is equal to
the negative arctan of the modulus of 𝑏 divided by the modulus of 𝑎. It’s also worth being aware that if
the complex number is purely imaginary and the imaginary part is greater than zero,
then it’s argument will be 𝜋 over two radians. And if the imaginary part is less
than zero, the argument will be negative 𝜋 by two radians. If the real and imaginary parts are
zero, then the argument of 𝑧 is undefined.

There is an alternative rule that
we can remember. In the first quadrant, once again,
the argument is equal to arctan of 𝑏 divided by 𝑎. In the second quadrant, it’s the
arctan 𝑏 divided by 𝑎 plus 𝜋. And what’s quite nice here is we
can take the real and imaginary parts from the complex number. And we don’t need to worry about
making them both positive. For the complex number in the third
quadrant, it’s the arctan of 𝑏 over 𝑎 minus 𝜋. And in the fourth quadrant, it’s
just simply the arctan of 𝑏 over 𝑎.

And yes, it’s useful to have a
rule. But it’s always sensible to sketch
the Argand diagram out to ensure that you’re choosing the right values for the
argument of your complex number. Let’s practice finding the argument
of a complex number located outside of the first quadrant and then extend this into
finding a relationship between the argument of a complex number and the argument of
its conjugate.

Consider the complex number 𝑧 is
equal to negative four minus five 𝑖. 1) Calculate the argument of 𝑧
giving your answer correct to three significant figures. 2) Calculate the argument of 𝑧
star giving your answer correct to three significant figures.

To calculate the argument of the
complex number 𝑧, let’s begin by plotting it on the Argand diagram. This complex number is represented
by the point whose Cartesian coordinates are negative four, negative five. It lies in the third quadrant. Since the argument is measured in
the counterclockwise direction from the positive real axis, we’re interested in this
angle here. And we can begin by finding the
value of the acute angle. Let’s call that 𝛼. We have a right-angled triangle for
which the side opposite the included angle is five units and the side adjacent to it
is four units.

So we can find the measure of this
angle 𝛼 using the formula arctan of five divided by four. And remember, one way to think
about this is to find the arctan of the modulus of the imaginary part divided by the
modulus of the real part. That’s 0.8960 radians. Angles on a straight line sum to 𝜋
radians. So to find the measure of the angle
we’ve marked 𝜃, we subtract 0.8960 and so on from 𝜋. That’s 2.2455. Since we’re measuring in the
opposite direction, we’re measuring clockwise as opposed to counterclockwise. We can say that the argument of 𝑧
is negative 2.25 radians, correct to three significant figures.

And alternatively, we could’ve
applied a formula that says that the argument of a complex number plotted in the
third quadrant is the arctan of this imaginary part divided by its real part minus
𝜋. The imaginary part of our complex
number is negative five. And the real part is negative
four. And if we type the arctan of
negative five divided by negative four minus 𝜋 into our calculator, we get negative
2.2455 and so on, once again. Either method is just fine. It’s more about personal preference
here.

For part two, we need to calculate
the argument of the conjugate of 𝑧. Remember, we find the conjugate of
a complex number by changing the sign of its imaginary part. So the conjugate of our complex
number is negative four plus five 𝑖. Let’s plot this on the same Argand
diagram. It’s probably quite clear that
there’s a shortcut here. But let’s perform the calculations
to be sure. And this time, we’ll use a
formula. The argument for a complex number
plotted in the second quadrant is arctan of 𝑏 divided by 𝑎 plus 𝜋. For the conjugate of 𝑧, 𝑏 is five
and 𝑎 is negative four. And typing this into our
calculator, we get 2.2455 and so on. Correct to three significant
figures, that’s 2.25.

Now, actually, we can say that the
geometric interpretation of the conjugate is a reflection of the complex number 𝑧
in the horizontal axis. So it makes perfect sense that the
argument of 𝑧 is equal to the negative argument of the conjugate of 𝑧, and vice
versa.

But what about the relationship
between addition and the argument. Well, in fact, there exists no nice
relationship between the addition of two complex numbers and the argument. There does, however, exist a
relationship between the product and quotient and the argument. Let’s see what this looks like.

Consider the complex numbers 𝑧
equals one plus root three 𝑖 and 𝑤 equals two minus two 𝑖. 1) Find the argument of 𝑧 and the
argument of 𝑤. 2) Calculate the argument of
𝑧𝑤. How does this compare to the
argument of 𝑧 and the argument of 𝑤? 3) Calculate the argument of 𝑧
divided by 𝑤. How does this compare to the
argument of 𝑧 and the argument of 𝑤?

To answer this question, we’ll
begin by plotting the complex numbers 𝑧 and 𝑤 on an Argand diagram. 𝑧 is represented by the point
whose Cartesian coordinates are one, root three. And 𝑤 is represented by the point
whose Cartesian coordinates are two, negative two. Let’s use one of the rules we
discussed earlier. We said that for a complex number
plotted in the first and the fourth quadrant, we can use the formula arctan of 𝑏
divided by 𝑎 to find its argument. The imaginary part of 𝑧 is root
three and the real part is one. So the argument of 𝑧 is arctan of
root three over one. That’s 𝜋 by three radians. The imaginary part of 𝑤 is
negative two and its real part is two. So the argument of 𝑤 is the arctan
of negative two over two. And so, the argument of 𝑤 is
negative 𝜋 by four radians.

For part two, we’re going to need
to begin by calculating the complex number 𝑧𝑤. That’s the product of one plus root
three 𝑖 and two minus two 𝑖. Let’s distribute these
parentheses. Multiplying the first term in each
bracket, we get two. Multiplying the outer terms, we get
negative two 𝑖. Multiplying the inner terms, we get
two root three 𝑖. And multiplying the last terms, we
get negative two root three 𝑖 squared. But, of course, 𝑖 squared is equal
to negative one. So this last term becomes a
positive two root three. We collect together the real
parts. That’s two and two root three. And we gather together the
imaginary parts. And we can see that 𝑧𝑤 is equal
to two plus two root three plus two root three minus two 𝑖.

All that’s left is to calculate the
argument of this complex number. Now, both the real and imaginary
parts of this complex number are actually greater than zero. So 𝑧𝑤 will lie in the first
quadrant. So the argument is the arctan of
the imaginary part divided by the real part. And we can evaluate these using the
rules for division of complex numbers. We’d need to multiply both the
numerator and the denominator by the conjugate of two plus two root three. And doing so, we’d see that the
argument of 𝑧𝑤 is arctan of two minus root three which is 𝜋 over 12. And if we compare this to the
argument of 𝑧 and the argument of 𝑤, we can see that the argument of their
products is equal to the sum of their arguments.

Let’s have a look at part
three. We need to work out 𝑧 divided by
𝑤. That’s one plus root three 𝑖
divided by two minus two 𝑖. And as before, we would need to
evaluate this by multiplying both the numerator and the denominator by the conjugate
of two minus two 𝑖. That’s two plus two 𝑖. And when we do, we see that 𝑧
divided by 𝑤 is equal to a quarter of one minus root three, that’s its real part,
plus a quarter of one plus root three, that’s its imaginary part, 𝑖. This time the real part of 𝑧
divided by 𝑤 is less than zero. But its imaginary part is greater
than zero. It lies in the second quadrant.

So we can use the formula arctan of
𝑏 divided by 𝑎 plus 𝜋 to find its argument. That gives us seven 𝜋 over 12. And, in fact, this time we can see
that the argument of 𝑧 divided by 𝑤 is equal to the argument of 𝑧 minus the
argument of 𝑤. And these are general rules that
hold for any two complex numbers. The argument of their products is
equal to the sum of their arguments. And the argument of their quotient
is equal to the difference of their arguments. And we can use these fact to solve
problems involving the properties of their argument. And we can use these facts to solve
problems involving the properties of the arguments.

Consider the complex number 𝑧
equals seven plus seven 𝑖. 1) Find the argument of 𝑧. 2) Hence, find the argument of 𝑧
to the power of four.

Here, we have a complex number
whose real and imaginary parts are positive. This means we would plot this
complex number in the first quadrant on the Argand diagram. And we can therefore find the
argument by using the formula arctan of 𝑏 divided by 𝑎, where 𝑏 is the imaginary
part and 𝑎 is the real part. In our case, that’s the arctan of
seven divided by seven. And that’s 𝜋 by four radians.

So, how do we find the argument of
𝑧 to the power of four? Well, what we’re not going to do is
evaluate the complex number 𝑧 to the power of four. Instead, we’re going to recall the
fact that the argument of the product of two complex numbers is equal to the sum of
their arguments. We’re going to extend this and say,
well, if we have 𝑧 times 𝑧 times 𝑧 times 𝑧, that’s going to be equal to the
argument of 𝑧 plus the argument of 𝑧 plus the argument of 𝑧 plus the argument of
𝑧. But actually, that’s equal to four
lots of the argument of 𝑧. And in our example, that’s equal to
four lots of 𝜋 by four, which is simply 𝜋 radians. And we can generalize this idea and
say that the argument of 𝑧 to the power of 𝑛 is equal to 𝑛 multiplied by the
argument of 𝑧.