Video: Argument of a Complex Number

In this video, we will learn how to find and interpret the argument of a complex number and understand some of its key properties.

17:20

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.

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