In this explainer, we will learn how to identify the argument of a complex number and how to calculate it.
When we plot complex numbers on an Argand diagram, we can see that complex numbers share many properties with vectors. For instance, adding and subtracting complex numbers is geometrically equivalent to the corresponding operations of vectors. We know that the characteristics of a vector are its direction and magnitude, so a complex number must have equivalent characteristics. We recall that the magnitude of a complex number is called its modulus. The direction of a complex number in the Argand diagram is the argument of the complex number.
Definition: Argument of a Complex Number
The argument of a complex number is the angle, in radians, between the positive real axis in an Argand diagram and the line segment between the origin and the complex number, measured counterclockwise. The argument is denoted , or .
The argument of a complex number is, by convention, given in the range . However, we can also discuss a complex number with an argument greater than or less than . The argument of a complex number within the range is called the principal argument. Other conventions use the range for the principal argument, but this is less common.
If we are given the Cartesian form, , of a complex number, we can use right triangle trigonometry to find the argument of the complex number. For instance, consider the complex number given in the Argand diagram above. Since this complex number lies in the first quadrant, we can see that the argument of this complex number is an angle in the right triangle whose sides are the blue, green, and purple line segments. In this case, the tangent of this angle is the ratio ; hence,
We can then compute by applying the inverse tangent function to both sides of this equation:
This method can be used whenever a complex number lies in the first quadrant. In our first example, we will find the principal argument of a complex number in the first quadrant by using right triangle trigonometry.
Example 1: Finding the Argument of a Complex Number in Radians
Find the argument of the complex number in radians. Give your answer correct to two decimal places.
Answer
Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range .
We begin by plotting the complex number on an Argand diagram.
We have labeled the argument of the complex number in the Argand diagram above . We can see that the argument of this complex number is an angle in the right triangle whose sides are the blue, green, and purple line segments. Applying right triangle trigonometry, we obtain
We can then apply the inverse tangent function to both sides of this equation to find
Hence, to two decimal places.
In the previous example, we were able to calculate the argument of a complex number, , by evaluating the inverse tangent of . However, this is not the case for all complex numbers as the next example will demonstrate.
Example 2: Finding the Principal Argument of a Complex Number
Given that , find the principal argument of .
Answer
Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. Additionally, we remember that the principal argument of a complex number is the argument that lies in the range .
We begin by plotting the complex number on an Argand diagram as shown below.
We have labeled the argument of the complex number in the Argand diagram above and the supplementary angle . We can see that is an angle in the right triangle whose sides are the blue, green, and purple line segments. Applying right triangle trigonometry, we obtain
We can then apply the inverse tangent function to both sides of this equation to find
We can then calculate the argument by subtracting from :
We note that this argument lies in the range ; hence, it is the principal argument.
We conclude that the principal argument of the given complex number is .
In the previous example, we saw that the argument of a complex number, , is not always equal to the inverse tangent of . In fact, if we had naively tried to calculate the argument of by evaluating we would have ended up with
This argument represents a clockwise angle from the positive real axis of radians, which would place the complex number in the fourth quadrant. It is apparent from the Argand diagram in the previous example that this is not the argument of the complex number. However, we can arrive at the correct value of by adding to .
This effect demonstrates that we need to be careful when calculating the argument of a complex number that does not lie in the first quadrant. Also, we can see that there are different approaches to obtain .
We outline two different methods of calculating the argument of a complex number. Whichever method we choose to use, plotting the number on an Argand diagram will be extremely useful and will help us avoid common errors in calculating the argument.
How To: Finding the Argument of a Complex Number Using the Inverse Tangent Function
To find the argument, , of a complex number , we need to consider which quadrant it lies in. The argument of a complex number can be obtained using the inverse tangent function in each quadrant as follows:
- If lies in the first or the fourth quadrant,
- If lies in the second quadrant,
- If lies in the third quadrant,
If the complex number does not lie on a quadrant, then it is either purely real or purely imaginary. If it is purely imaginary , then
If purely real , then
Lastly, if , the argument is undefined.
These points are summarized in the following diagram.
The main benefit of the method described above is that we are given a formula to follow for each situation. However, this method also requires us either to memorize each rule or to have an available reference for the rules. An alternative method for finding the argument of a complex number is to use right triangle trigonometry to first identify the positive acute angle between the real axis and the line segment between the origin and the complex number in an Argand diagram. After finding the positive acute angle, we can find the argument of the complex number geometrically.
How To: Finding the Argument of a Complex Number Using Positive Acute Angles
We define the angle to be the positive acute angle between the line linking to the origin and the real axis as shown in the diagram.
We can then calculate the argument of in different quadrants as follows:
- Quadrant 1:
- Quadrant 2:
- Quadrant 3:
- Quadrant 4:
The two different methods for obtaining the argument of a complex number will lead to the same answer. The second method, which uses the positive acute angle, is more intuitive and requires less memorization. Using this method, we first compute the positive acute angle and then use it to find the argument of the complex number, which is the counterclockwise angle from the positive real axis, lying in the range .
In the next example, we will apply this method to find the argument of a complex number lying in the third quadrant.
Example 3: The Relationship between the Complex Conjugate and the Argument
Consider the complex number .
- Calculate , giving your answer correct to two decimal places in an interval from to .
- Calculate , giving your answer correct to two decimal places in an interval from to .
Answer
Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range .
Part 1
We begin by plotting the complex number on an Argand diagram as shown below.
We have labeled the acute angle , which is related to the argument of the complex number . If we can find angle , the argument of this number can be obtained by adding to this angle. However, we can see that this argument will not lie in the range . We must then subtract the full revolution from this resulting angle, which leads to the relation
We can see that is an angle in the right triangle whose sides are the blue, green, and purple line segments. Applying right triangle trigonometry, we obtain
We can then apply the inverse tangent function to both sides of this equation to find
Hence, to calculate , we subtract from , which gives
Part 2
Recall that the conjugate is obtained by switching the sign of the imaginary part of the complex number . Hence, . We now plot on an Argand diagram.
Similar to the previous part, we will find the argument of by first calculating :
Since and are supplementary, we can obtain by subtracting from :
In the previous example, we computed the arguments of a complex number and its conjugate. We can note that the argument of the complex conjugate in this example is the negative of the argument of the original complex number. This demonstrates a general rule of the argument.
Property: Argument of the Conjugate of a Complex Number
Given any nonzero complex number and its conjugate (also denoted ),
In the next example, we will demonstrate how the multiplication and division of complex numbers is associated with the arguments of the complex numbers.
Example 4: Arguments of Products and Quotients
Consider the complex numbers and .
- Find and .
- Calculate . How does this compare to and ?
- Calculate . How does this compare to and ?
Answer
Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range .
Part 1
Let us start by plotting and on an Argand diagram.
Recall that the argument of a complex number lying in the first or the fourth quadrant is given by
Since and lie in the first and fourth quadrants, respectively, we can use the inverse tangent to find their arguments as follows: and
Part 2
We begin by calculating as follows:
Multiplying through the parenthesis, we obtain
Using and gathering real and imaginary terms, we obtain
Since both the real and the imaginary parts are positive, lies in the first quadrant of the Argand diagram and we can calculate the argument by evaluating the inverse tangent as follows:
Canceling the factor 2 from the top and the bottom, we have
We can simplify the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator:
Multiplying through the parenthesis, we obtain
Comparing this with and , we find that .
Part 3
We start by calculating as follows:
To write this complex number in the Cartesian form, , we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is :
Multiplying through the parenthesis, we have
Using and gathering real and imaginary terms,
Since and , the complex number lies in the second quadrant. Recall that if a complex number lies in the second quadrant,
Hence, we have
Canceling the common factor , we have
Evaluating the inverse tangent, we obtain
Finally, comparing this with and , we find that .
In the previous example, we observed the relationship between the multiplication/division of complex numbers and their arguments. This relationship shown in the example holds for general complex numbers.
Property: Arguments and Multiplication/Division of Complex Numbers
Given any nonzero complex numbers and ,
The next example will demonstrate how we can solve problems by applying the properties of the argument.
Example 5: Using Multiplication of Complex Numbers to Determine an Argument
A complex number is multiplied by another complex number and then by the complex conjugate . How is the argument of the resulting complex number related to the argument of the original complex number?
Answer
Recall that the argument of the product of a pair of complex numbers is equal to the sum of the arguments of the two complex numbers.
We start with a complex number ; then, it is multiplied by and . Hence, the result is . 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 . Using multiplicative properties of the argument, we can rewrite this as follows:
We also know that the argument of a complex number equals the negative of the argument of its conjugate. Hence, we can replace above with to write
Therefore, the argument of the complex number after it is multiplied by another complex number and then by the complex conjugate is unchanged.
In our final example, we will consider the relationship between the argument and powers.
Example 6: Finding the Argument of Powers of Complex Numbers in Algebraic Form
Consider the complex number .
- Find the argument of .
- Hence, find the argument of .
Answer
Recall that the argument of a complex number is the angle, in radians, between the positive real axis of an Argand diagram and the line between the origin and the complex number, measured counterclockwise. We also remember that the argument of a complex number is, by convention, given in the range .
Part 1
Recall that the argument of a complex number lying in the first or the fourth quadrant is given by
Since the complex number lies in the first quadrant, we can calculate its argument by evaluating the inverse tangent of its imaginary part over its real part as follows:
Part 2
We recall that for any two nonzero complex numbers and ,
If the complex numbers are both equal to , this means
Using similar logic, we can find that
Therefore,
In the previous example, we saw the relationship between the power of a complex number and its argument. Using similar logic as in that example, we can see that this relationship holds for general complex numbers for any positive integer power.
Property: Argument of the Power of a Complex Number
Given any nonzero complex number and a positive integer power , the argument of is given by
In this explainer, we have considered how the argument of a complex number relates to the conjugates, multiplication, and division of complex numbers. However, we have intentionally left out the addition and subtraction of complex numbers since there is no simple relationship between these operations and the arguments of complex numbers. We will finish this explainer by illustrating in two different ways why we do not expect to find a simple relationship between addition/subtraction and the arguments of complex numbers.
Firstly, we recall that the addition and subtraction of complex numbers are geometrically equivalent to the corresponding vector operations and hence follow the triangle or parallelogram rules. In doing so, we can see that knowing only the arguments (angles) of the complex numbers will not be sufficient to find the argument of the resulting complex number. This is one way that we can see why there is no simple relationship between these operations and the arguments of complex numbers.
As an alternative way to see why no such simple relationship exists, let us consider the three complex numbers , , and plotted on the Argand diagram below.
We can see that and that . Furthermore, , which has an argument of 0, whereas whose argument is clearly not zero. We can, in fact, calculate the exact value of the argument as follows:
By multiplying both the numerator and the denominator by the conjugate of the denominator, we can simplify the fraction:
Multiplying through the parenthesis, we obtain
Finally, we can simplify and evaluate the inverse tangent to obtain
To summarize what we have calculated here, recall that the complex numbers and have the same argument, . If a simple relationship between the arguments of complex numbers and the sum existed, the arguments of and would be the same. However, we have obtained
This demonstrates that knowing the arguments of two complex numbers is not sufficient to be able to calculate the argument of their sum.
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The argument of a complex number is defined as the angle, in radians, between the positive real axis in an Argand diagram and the line segment from the origin to the complex number, measured counterclockwise.
- The argument of a complex number can be obtained using the inverse tangent function in each quadrant as follows:
- If lies in the first or the fourth quadrant,
- If lies in the second quadrant,
- If lies in the third quadrant,
- The argument has the following properties:
- ,
- ,
- ,
- .
- There is no simple relationship between the addition of complex numbers and their arguments.
