Lesson Explainer: The Argument of a Complex Number Mathematics

In this explainer, we will learn how to identify the argument of a complex number and how to calculate it.

Definition: Argument of a Complex Number

The argument of a complex number 𝑧 is defined as the angle that the line from the origin to 𝑧 makes with the positive real axis (𝑥-axis), measured in a counterclockwise direction. The argument is usually denoted arg𝑧, although, in some textbooks, it might be denoted Arg𝑧.

The argument 𝜃 of a complex number is usually given in radians in the range 𝜋<𝜃𝜋. However, it is legitimate to talk about a complex number having an argument greater than 𝜋 or less than 𝜋. For this reason, if the argument is given or asked to be given in this range, it is usually referred to at the principal argument. Some sources use the range 0𝜃<2𝜋 for the principal argument, although this is less common.

Example 1: Argument of a Complex Number

Find the argument of the complex number 4+3𝑖 in radians. Give your answer correct to two decimal places.

Answer

We begin by plotting the complex number on an Argand diagram.

The argument of the complex number is the angle that it makes with the positive real axis; we have labeled this 𝜃. Using the inverse tangent function, we can calculate 𝜃 as follows: 𝜃=34=0.6435=0.64.arctan(2d.p.)

Hence, argradians(4+3𝑖)=0.64 to two decimal places.

In the previous example, we were able to calculate the argument by evaluating the inverse tangent of 𝑏𝑎. This, however, is not a general rule we can apply to any complex number. We need to be a little more careful as the next example will demonstrate.

Example 2: Argument of a Complex Number in the Second Quadrant

Given that 𝑍=12+32𝑖, find the principal argument of 𝑍.

Answer

We begin by plotting the complex number on an Argand diagram as shown below.

The argument of the complex number is the angle that it makes with the positive real axis; we have labeled this 𝜃. To calculate 𝜃, we will begin by finding 𝜙 as follows: 𝜙==3=𝜋3.arctanarctan

We can then calculate the principal argument as follows: arg(𝑧)=𝜃=𝜋𝜙=𝜋𝜋3=2𝜋3.

Had we naively tried to calculate the argument of 𝑧 by evaluating 𝛼=,arctan we would have got 𝛼=3=𝜋3.arctan

Looking at the Argand diagram, we can see that this is not the argument of the complex number. However, by adding 𝜋 to 𝛼 we would have arrived at the correct value of arg(𝑧).

The previous example demonstrates that we need to be careful when calculating the argument of a complex number when it does not lie in the first quadrant. Secondly, it highlights that there are different approaches by which we can calculate arg(𝑧).

In the two boxes below, we outline two different methods to calculate the argument of a complex number. Whichever method you chose to use, plotting the number on an Argand diagram will be extremely useful and will help you avoid common errors in calculating the argument.

How To: Finding the Argument of a Complex Number 1

To find the argument, arg𝑧, of a complex number 𝑧=𝑎+𝑏𝑖, we need to consider which quadrant it lies in.

Quadrants 1 and 4: If 𝑧 is in the first or fourth quadrant of the Argand diagram (𝑎>0), we can simply use the inverse tangent function and calculate argarctan𝑧=𝑏𝑎.

Quadrant 2: However, if the complex number is in the second quadrant (𝑎<0 and 𝑏>0), we need to add 𝜋 to the value we get using the inverse tangent function. Hence, argarctan𝑧=𝑏𝑎+𝜋.

Quadrant 3: If the complex number is in the third quadrant (𝑎<0 and 𝑏<0), we need to subtract 𝜋 from the value we get using the inverse tangent function. Hence, argarctan𝑧=𝑏𝑎𝜋.

Finally, if the complex number is purely imaginary (𝑎=0), then arg𝑧=𝜋2 if 𝑏>0, and arg𝑧=𝜋2 if 𝑏<0. If 𝑎=𝑏=0, the argument is undefined.

An alternative method for finding the argument of a complex number is explained below.

How To: Finding the Argument of a Complex Number 2

We define the angle 𝜃 to be the positive acute angle between the line linking 𝑧 to the origin and the real axis as shown in the diagram.

We can then calculate the argument of 𝑧 in different quadrants as follows.

  • Quadrant 1: arg(𝑧)=𝜃.
  • Quadrant 2: arg(𝑧)=𝜋𝜃.
  • Quadrant 3: arg(𝑧)=𝜃𝜋.
  • Quadrant 4: arg(𝑧)=𝜃.

If 𝑧=0, the argument is not defined.

Having established the basics of the definition of the argument and how to calculate it. We will now look at some of its important properties.

Example 3: Relationship between the Complex Conjugate and the Argument

Consider the complex number 𝑧=45𝑖.

  1. Calculate arg(𝑧) giving your answer correct to 3 significant figures.
  2. Calculate arg𝑧 giving your answer correct to 3 significant figures.

Answer

Part 1

We begin by plotting the complex number on an Argand diagram as shown below.

As we can see, 𝑧 lies in the third quadrant. Therefore, to find the argument represented by angle 𝜃, we first calculate 𝜙 as follows: 𝜙=54=0.8960.arctan

Hence, to calculate arg(𝑧), we subtract 𝜋 from 𝜙 which gives arg(3s.f.)(𝑧)=𝜙𝜋=2.2455=2.25.

Part 2

To find the conjugate 𝑧, we switch the sign of the imaginary part. Hence, 𝑧=4+5𝑖. We now plot 𝑧 on an Argand diagram.

Given that 𝑧 is in the second quadrant, we will find the argument of 𝑧 by first calculating 𝜙: 𝜙=54=0.8960.arctan

Then, to find the argument of 𝑧, we subtract 𝜙 from 𝜋: arg(3s.f.).(𝑧)=𝜋𝜙=2.2455=2.25

The previous example demonstrates a general rule of the argument: for any complex number 𝑧, argarg(𝑧)=(𝑧).

Next, we might want to consider whether there is a simple relationship between addition and the argument of a complex number. If we consider the geometric meaning of addition to be adding vectors using the triangle or parallelogram rule, we see that knowing angles (or, specifically, the arguments) will not be sufficient to solve the triangle. This is one way we can understand why there is no simple relationship between addition and the argument.

As an alternative way to see why no such simple relationship exists, let us consider the three complex numbers 𝑧=1+𝑖, 𝑧=2+3(1+𝑖), and 𝑧=1𝑖.

We can see that argarg(𝑧)=(𝑧)=𝜋2 and arg(𝑧)=𝜋2. Furthermore, 𝑧+𝑧=2 which has an argument of 0, whereas 𝑧+𝑧=3+3+1+3𝑖 whose argument is clearly not zero. We can, in fact, calculate the exact value of the argument as follows: argarctan(𝑧+𝑧)=1+33+3.

By multiplying both the numerator and the denominator by the conjugate of the denominator, we can simplify the fraction: argarctan(𝑧+𝑧)=1+3333+333.

Expanding the brackets, we get argarctan(𝑧+𝑧)=33+33333.

Finally, we can simplify and evaluate the inverse tangent to get argarctanarctan(𝑧+𝑧)=236=33=𝜋6.

What we have demonstrated is that knowing both arg(𝑧) and arg(𝑤) is not sufficient to be able to calculate arg(𝑤+𝑧).

Example 4: Arguments of Products and Quotients

Consider the complex numbers 𝑧=1+3𝑖 and 𝑤=22𝑖.

  1. Find arg(𝑧) and arg(𝑤).
  2. Calculate arg(𝑧𝑤). How does this compare to arg(𝑧) and arg(𝑤)?
  3. Calculate arg𝑧𝑤. How does this compare to arg(𝑧) and arg(𝑤)?

Answer

Part 1

Let us start by plotting 𝑧 and 𝑤 on an Argand diagram.

Since 𝑧 and 𝑤 lie in the first and fourth quadrants, respectively, we can use the inverse tangent to find their argument as follows: argarctan(𝑧)=31=𝜋3 and argarctan(𝑤)=22=𝜋4.

Part 2

We begin by calculating 𝑧𝑤 as follows: 𝑧𝑤=1+3𝑖(22𝑖); expanding the brackets, we get 𝑧𝑤=22𝑖+232𝑖3.

Using 𝑖=1 and gathering like terms, we get 𝑧𝑤=2+23+232𝑖.

Since both the real and the imaginary parts are positive, 𝑧𝑤 lies in the first quadrant of the complex plane and we can calculate the argument by evaluating the inverse tangent as follows: argarctan(𝑧𝑤)=2322+23.

Canceling the factor of 2 from the top and the bottom, we have argarctan(𝑧𝑤)=311+3.

We can simplify the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator: argarctan(𝑧𝑤)=31131+313.

Expanding the brackets, we get argarctanarctanarctan(𝑧𝑤)=1+23313=4+232=23=𝜋12.

Comparing this with arg(𝑧) and arg(𝑤), we find that argargarg(𝑧𝑤)=(𝑧)+(𝑤).

Part 3

We start by calculating 𝑧𝑤 as follows: 𝑧𝑤=1+3𝑖22𝑖; multiplying both the numerator and the denominator by the conjugate of the denominator, we get 𝑧𝑤=1+3𝑖(2+2𝑖)(22𝑖)(2+2𝑖).

Expanding the brackets in the numerator and the denominator, we have 𝑧𝑤=2+2𝑖+2𝑖3+2𝑖34+4.

Using 𝑖=1 and gathering like terms, we get 𝑧𝑤=1413+141+3𝑖.

Since Re𝑧𝑤<0 and Im𝑧𝑤>0, it lies in the second quadrant and we can calculate the argument as follows: argarctan𝑧𝑤=1+313+𝜋.

Canceling the common factor of 14 from the top and the bottom, we have argarctan𝑧𝑤=1+313+𝜋.

We can rationalize the fraction by multiplying the numerator and the denominator by the conjugate of the denominator: argarctan𝑧𝑤=1+31+3131+3+𝜋.

Expanding the brackets and simplifying, we get argarctanarctan𝑧𝑤=4+232+𝜋=23+𝜋.

Evaluating the inverse tangent, we get arg𝑧𝑤=5𝜋12+𝜋=7𝜋12.

Finally, comparing this with arg(𝑧) and arg(𝑤), we find that argargarg𝑧𝑤=(𝑧)(𝑤).

Using the techniques used in the last example, it is possible to prove that, for any two complex numbers 𝑧=𝑎+𝑏𝑖 and 𝑧=𝑐+𝑑𝑖, the following identities hold: argargargargargarg(𝑧𝑧)=(𝑧)+(𝑧),𝑧𝑧=(𝑧)(𝑧).

The next example will demonstrate how we can solve problems by applying the properties of the argument.

Example 5: Effect of Multiplication on the Argument

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

Answer

We start with a complex number 𝑤; then it is multiplied by 𝑧 and 𝑧. Hence, the result is 𝑤𝑧𝑧. We are asked how the argument of the resulting complex number is related to the argument of the original complex number. Therefore, we should consider arg𝑤𝑧𝑧. Using multiplicative properties of the argument, we can rewrite this: argargargarg𝑤𝑧𝑧=(𝑤)+(𝑧)+𝑧.

We also know that the argument of a complex number equals the negative of the argument of its conjugate. Hence, argargargargarg𝑤𝑧𝑧=(𝑤)+(𝑧)(𝑧)=(𝑤).

Therefore, the argument of the resulting complex number is the same as the argument of the original complex number.

In our final example, we will consider the relationship between the argument and powers.

Example 6: Powers of Complex Numbers and the Argument

Consider the complex number 𝑧=7+7𝑖.

  1. Find the argument of 𝑧.
  2. Hence, find the argument of 𝑧.

Answer

Part 1

Since the complex number lies in the first quadrant, we can calculate its argument by simply evaluating the inverse tangent of its imaginary part over its real part as follows: argarctanarctan(𝑧)=77=1=𝜋4.

Part 2

We know that, for any two complex numbers 𝑧 and 𝑧, argargarg(𝑧𝑧)=(𝑧)+(𝑧). If the complex numbers are both equal to 𝑧, this becomes argarg𝑧=2(𝑧). Using similar logic, we can find that argarg𝑧=3(𝑧) and argarg𝑧=4(𝑧). Therefore, argarg𝑧=4(𝑧)=4×𝜋4=𝜋.

Using similar logic to that applied in the previous example, we can see that, for a complex number 𝑧, the argument of its 𝑛th power will be given by argarg(𝑧)=𝑛(𝑧).

Key Points

  • The argument of a complex number 𝑧 is defined as the angle that the line from the origin to 𝑧 makes with the positive real axis (𝑥-axis), measured in a counterclockwise direction.
  • When calculating the argument of a complex number 𝑧=𝑎+𝑏𝑖, we cannot simply calculate arctan𝑏𝑎. This is true for complex numbers in quadrants 1 and 4. However, it will not give the correct answer for complex numbers in quadrants 2 and 3: in quadrant 2, we need to add 𝜋 to the inverse tangent; in quadrant 3, we need to subtract 𝜋 from the inverse tangent.
  • The argument has the following properties:
    • argarg(𝑧)=𝑧,
    • argargarg(𝑧𝑧)=(𝑧)+(𝑧),
    • argargarg𝑧𝑧=(𝑧)(𝑧),
    • argarg(𝑧)=𝑛(𝑧).
  • There is no simple relationship between addition of complex numbers and the argument.

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