Explainer: Modulus of a Complex Number

In this explainer, we will learn how to use the general formula for calculating the modulus of a complex number.

Definition: Modulus of a Complex Number

The modulus of a complex number 𝑧=𝑎+𝑏𝑖 is defined as |𝑧|=𝑎+𝑏.

Equivalently, this can be written as |𝑧|=((𝑧))+((𝑧)).ReIm

If 𝑧 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 𝑧=𝑎+𝑏𝑖 to represent the vector A=𝑎,𝑏 on the Argand diagram, we see that |𝑧| represents the magnitude of the vector: ||A.

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.

Example 1: Modulus of a Complex Number

Given that 𝑍=8+4𝑖, find |𝑍|.

Answer

The definition of the modulus of a complex number 𝑤=𝑎+𝑏𝑖 is |𝑤|=𝑎+𝑏. Hence, |𝑍|=8+4=64+16=80.

Given that 80=2×5, we can rewrite this as |𝑍|=2×5=45.

Example 2: Relationship between the Complex Conjugate and the Modulus

Consider the complex number 𝑧=4+𝑖5.

  1. Calculate |𝑧|.
  2. Calculate |𝑧|.
  3. Determine 𝑧𝑧.

Answer

Part 1: Recall that for a complex number 𝑤=𝑎+𝑏𝑖 the modulus is defined as |𝑤|=𝑎+𝑏.Hence, |𝑧|=(4)+5=16+5=21.

Part 2: To find the modulus of a complex number, we change the sign of the imaginary part of the number. Hence, 𝑧=4𝑖5. Therefore, |𝑧|=(4)+5=16+5=21.

Part 3: Using the value of 𝑧 from part 2, we have 𝑧𝑧=4+𝑖54𝑖5.

Using FOIL or another method, we can expand the brackets as follows: 𝑧𝑧=(4)+4×𝑖54×𝑖5𝑖5=165𝑖.

Using the fact that 𝑖=1, we have 𝑧𝑧=21.

We could have also calculated this using the identity that, for a complex number 𝑤=𝑎+𝑏𝑖, 𝑤𝑤=𝑎+𝑏.

The previous example highlighted some of the properties of the modulus, in particular, those related to the conjugate. The box below summarizes these properties.

Properties of the Modulus of a Complex Number

For a complex number 𝑧:

  1. |𝑧|=|𝑧|,
  2. |𝑧|=𝑧𝑧.

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 𝑤=1+7𝑖 and 𝑧=53𝑖.

  1. Calculate |𝑤|+|𝑧| to two decimal places.
  2. Calculate |𝑧+𝑤| to two decimal places.
  3. Which of the following relations do 𝑤 and 𝑧 satisfy?
    1. |𝑤|+|𝑧|=|𝑧+𝑤|
    2. |𝑤|+|𝑧||𝑧+𝑤|
    3. |𝑤|+|𝑧||𝑧+𝑤|
    4. |𝑤|+|𝑧|=2|𝑧+𝑤|
    5. |𝑤|+|𝑧|=|𝑧+𝑤|

Answer

Part 1: Using the definition of the modulus of a complex number, we have |𝑤|=(1)+7=1+49=52.

Similarly, |𝑧|=5+(3)=25+9=34.

Putting these two together, we have |𝑤|+|𝑧|=52+34.

Using a calculator, we can evaluate this and round to two decimal places as follows: =12.9020=12.90(2).d.p.

Part 2: We begin by calculating 𝑧+𝑤 as follows: 𝑧+𝑤=53𝑖+(1+7𝑖)=4+4𝑖. We now calculate its modulus: |𝑧+𝑤|=4+4=42.

Evaluating this on a calculator, we get |𝑧+𝑤|=5.6568=5.66(2).d.p.

Part 3: Clearly, |𝑤|+|𝑧||𝑧+𝑤|. Hence, the answer cannot be (a). Instead, we find that |𝑤|+|𝑧||𝑧+𝑤|, which simultaneously confirms that (b) is a correct answer and that (c) is incorrect. Furthermore, by evaluating 2|𝑧+𝑤|=11.31(2)d.p. we see that option (d) is also incorrect. Finally, we check option (e) by evaluating |𝑤|+|𝑧|=3.59(2)d.p.. This confirms that option (e) is also incorrect. Hence, the only correct option is (b).

In the previous example, we showed that the complex 𝑧 and 𝑤 satisfy the relationship |𝑤|+|𝑧||𝑧+𝑤|. This relation not only holds for the specific numbers 𝑧 and 𝑤 in the example but, in fact, also holds for any two complex numbers. This relation is often referred to as the triangle inequality.

Triangle Inequality for Complex Numbers

For two complex numbers 𝑧 and 𝑧, the following inequality holds: |𝑧+𝑧||𝑧|+|𝑧|.

Equality holds when 𝑧+𝑐𝑧 for some real 𝑐0.

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 𝑧=34𝑖 and 𝑤=15+8𝑖.

  1. Find |𝑧| and |𝑤|.
  2. Calculate |𝑧𝑤|. How does this compare to |𝑧||𝑤|?
  3. Calculate ||𝑧𝑤||. How does this compare with |𝑧||𝑤|?

Answer

Part 1: Using the definition of the modulus, we calculate |𝑧|=3+(4)=9+16=25=5.

Similarly, we find |𝑤|=(15)+8=225+64=289=17.

Part 2: We now calculate the product 𝑧𝑤 as follows: 𝑧𝑤=(34𝑖)(15+8𝑖).

Expanding the brackets using FOIL or another method, we get 𝑧𝑤=45+24𝑖+60𝑖32𝑖.

Since 𝑖=1, we have 𝑧𝑤=13+84𝑖.

We now calculate the modulus: |𝑧𝑤|=(13)+84=169+7,056=7,225=85.

Using the answers from part 1, we have |𝑧||𝑤|=5×17=85. Hence, we find that |𝑧||𝑤|=|𝑧𝑤|.

Part 3: We begin by calculating 𝑧𝑤 as follows: 𝑧𝑤=34𝑖15+8𝑖.

Multiplying both the numerator and the denominator by the complex conjugate of the denominator, we have 𝑧𝑤=(34𝑖)(158𝑖)(15+8𝑖)(158𝑖).

Expanding the brackets in the numerator and denominator, we get 𝑧𝑤=4524𝑖+60𝑖+32𝑖225120𝑖+120𝑖64𝑖.

Using 𝑖=1, we have 𝑧𝑤=77+36𝑖289=77289+36289𝑖.

We can now calculate its modulus as follows: ||𝑧𝑤||=77289+36289.

We can rewrite this by taking the common denominator outside of the square root as follows: ||𝑧𝑤||=77+36289=5,929+1,296289=7,225289=85289=517.

Finally, we compare this with |𝑧||𝑤|. Using the answer from part 1, we see this is equal to 517. Hence, |𝑧||𝑤|=||𝑧𝑤||.

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

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 𝑍=1𝑍, where 𝑍 is a complex number, what is |𝑍|?

Answer

Starting from the equation 𝑍=1𝑍, we can take the modulus of both sides of the equation to get |𝑍|=|||1𝑍|||.

Since for any two complex numbers |||𝑍𝑍|||=|𝑍||𝑍|, we can rewrite the equation as |𝑍|=|1||𝑍|.

Furthermore, we know that |𝑍|=|𝑍| and |1|=1. Hence, |𝑍|=1|𝑍|.

Multiplying both sides of the equation by |𝑍| gives |𝑍|=1.

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, |𝑍|=1.

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 𝑧=𝑎+𝑏𝑖, what is the modulus of 𝑧?

Answer

We know that, for any two complex numbers, the modulus of their product is the product of their moduli: |𝑧𝑧|=|𝑧||𝑧|.

Therefore, in the special case where 𝑧=𝑧=𝑧, we have ||𝑧||=|𝑧||𝑧|=|𝑧|.

Using the definition of the modulus of |𝑧|, we have ||𝑧||=𝑎+𝑏=𝑎+𝑏.

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

Key Points

  1. The modulus of a complex number 𝑧=𝑎+𝑏𝑖 is defined as |𝑧|=𝑎+𝑏. Geometrically, it represents the distance of 𝑧 from the origin.
  2. The modulus has the following properties:
    1. |𝑧|=|𝑧|,
    2. |𝑧|=𝑧𝑧,
    3. |𝑧+𝑧||𝑧|+|𝑧|,
    4. |𝑧𝑧|=|𝑧||𝑧|,
    5. |||𝑧𝑧|||=|𝑧||𝑧|,
    6. |𝑧|=|𝑧|.

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