In this explainer, we will learn how to use the general formula for calculating the modulus of a complex number.
Remember that a complex number is a complex of two things, a real part and an imaginary part . The purely imaginary number is defined as , or . Complex number is represented by the point in the Argand diagram.
We will now define the modulus of a complex number.
Definition: Modulus of a Complex Number
The modulus of a complex number is defined as
Equivalently, this can be written as
If is a real number, its modulus just corresponds to the absolute value. For this reason, the modulus is sometimes referred to as the absolute value of a complex number. Similarly, if we consider to represent the vector on the Argand diagram, we see that represents the magnitude of the vector: .
Consequently, the modulus is sometimes also referred to as the magnitude of a complex number. This also highlights the geometric interpretation of the modulus as the magnitude of the complex number or its distance from the origin.
Let us look at our first example where we have to find the modulus of a given complex number.
Example 1: Modulus of a Complex Number
Given that , find .
The definition of the modulus of a complex number is . Hence,
Given that , we can rewrite this as
We will now look, in our second example, at the relationship between the complex conjugate and the modulus of a complex number.
Example 2: Relationship between the Complex Conjugate and the Modulus
Consider the complex number .
- Calculate .
- Calculate .
- Determine .
Recall that for a complex number the modulus is defined as .Hence,
To find the modulus of a complex number, we change the sign of the imaginary part of the number. Hence, . Therefore,
Using the value of from part 2, we have
Using FOIL or another method, we can expand the brackets as follows:
Using the fact that , we have
We could have also calculated this using the identity that, for a complex number , .
The previous example highlighted some of the properties of the modulus, in particular, those related to the conjugate. The box below summarizes these properties.
Property: Properties of the Modulus of a Complex Number
For a complex number :
We will now consider the properties of the modulus in relation to other operations with complex numbers including addition, multiplication, and division. We will start by looking at addition.
Example 3: Relationship between Addition and the Modulus of a Complex Number
Consider the two complex numbers and .
- Calculate to two decimal places.
- Calculate to two decimal places.
- Which of the following relations do and satisfy?
Using the definition of the modulus of a complex number, we have
Putting these two together, we have
Using a calculator, we can evaluate this and round to two decimal places (2 d.p.) as follows:
We begin by calculating as follows: We now calculate its modulus:
Evaluating this on a calculator, we get
Clearly, . Hence, the answer cannot be (A). Instead, we find that , which simultaneously confirms that (B) is a correct answer and that (C) is incorrect. Furthermore, by evaluating we see that option (D) is also incorrect. Finally, we check option (E) by evaluating . This confirms that option (E) is also incorrect. Hence, the only correct option is (B).
In the previous example, we showed that the complex and satisfy the relationship . This relation not only holds for the specific numbers and in the example but, in fact, also holds for any two complex numbers. This relation is often referred to as the triangle inequality.
Inequality: Triangle Inequality for Complex Numbers
For two complex numbers and , the following inequality holds:
Equality holds when for some real .
Let us visualize the triangle inequality on an Argand diagram.
We see that we can form a triangle with sides of lengths , , and . The triangle inequality, as its name suggests, states that the sum of the lengths of two sides is always longer than the length of one side. If it were not the case, then the triangle could not be formed because the two sides would not join.
We have the equality when for some real because, in this case, and are aligned with the origin and are on the same side of the origin. The complex number is then on the same ray from the origin: we do not have a triangle but three segments on the same ray.
Algebraically, we see that
Since , we have
We will now explore the properties of the modulus in relation to multiplication and division.
Example 4: Modulus of Products and Quotients
Consider the complex numbers and .
- Find and .
- Calculate . How does this compare to ?
- Calculate . How does this compare with ?
Using the definition of the modulus, we calculate
Similarly, we find
We now calculate the product as follows:
Expanding the brackets using FOIL or another method, we get
Since , we have
We now calculate the modulus:
Using the answers from part 1, we have . Hence, we find that .
We begin by calculating as follows:
Multiplying both the numerator and the denominator by the complex conjugate of the denominator, we have
Expanding the brackets in the numerator and denominator, we get
Using , we have
We can now calculate its modulus as follows:
We can rewrite this by taking the common denominator outside of the square root as follows:
Finally, we compare this with . Using the answer from part 1, we see this is equal to . Hence,
Using the techniques used in the last example, it is fairly straightforward to prove that, for any two complex numbers and , we can generalize what we have shown in the last example and write the following multiplication and division identities.
Identity: Multiplication and Division Identities
For any two complex numbers and , we have
The next example will demonstrate how we can solve problems by applying the properties of the modulus.
Example 5: Solving Equations Involving the Modulus
If , where is a complex number, what is ?
Starting from the equation we can take the modulus of both sides of the equation to get
Since for any two complex numbers , we can rewrite the equation as
Furthermore, we know that and . Hence,
Multiplying both sides of the equation by gives
Finally, we can take the square root of both sides of this equation. Because the modulus is always a positive number, we only need to consider the positive square root. Hence,
In our final example, we will consider the relationship between the modulus and powers.
Example 6: Powers of Complex Numbers and the Modulus
Given the complex number , what is the modulus of ?
We know that, for any two complex numbers, the modulus of their product is the product of their moduli:
Therefore, in the special case where , we have
Using the definition of the modulus of , we have
Using the similar logic to that applied in the previous example, we can see that, for a complex number the modulus of its th power will be given by
- The modulus of a complex number is defined as Geometrically, it represents the distance of from the origin.
The modulus has the following properties: