Explainer: Complex Number Conjugates

In this explainer, we will learn how to use the properties of conjugate numbers to evaluate an expression.

One important concept related to complex numbers is the idea of the complex conjugate. The idea of a conjugate might not be a new idea. When learning how to manipulate and simplify radicals, you might have been introduced to the idea of a conjugate. For a radical expression 𝑎+𝑏𝑐, the conjugate is defined as 𝑎𝑏𝑐. The complex conjugate is actually a special case of this where 𝑐 is a negative number.

Definition: Complex Conjugate

For a complex number 𝑧=𝑎+𝑏𝑖, the complex conjugate, 𝑧, is defined as 𝑧=𝑎𝑏𝑖. The complex conjugate is sometimes denoted 𝑧.

By simple consideration of the definition, we can see that, for any complex number 𝑧, the conjugate of the conjugate is equal to 𝑧; that is, (𝑧)=𝑧.

Example 1: Complex Conjugation

What is the conjugate of the complex number 27𝑖?

Answer

Recall the complex conjugate of 𝑧=𝑎+𝑏𝑖 is 𝑧=𝑎𝑏𝑖. For the complex number we have been given, 𝑎=2 and 𝑏=7 (careful not to miss the minus sign). Therefore, the complex conjugate is equal to 2(7)𝑖 which we can simplify to 2+7𝑖.

Example 2: The Complex Conjugate of a Real Number

If 𝑧 is a real number, what will its conjugate be equal to?

Answer

The definition of the complex conjugate of 𝑧=𝑎+𝑏𝑖 is 𝑧=𝑎𝑏𝑖. If 𝑧 is a purely real number, we know that 𝑏=0. Hence, we conclude that if 𝑧 is a real number, 𝑧=𝑧.

Conversely, if we know that, for a complex number 𝑧=𝑧, we have 𝑎+𝑏𝑖=𝑎𝑏𝑖; gathering like terms, we get 2𝑏𝑖=0.

Hence, 𝑏=0, and we can conclude that 𝑧 is real.

Similarly, we could ask the question what is the complex conjugate of a purely imaginary number 𝑧. Using the definition of the complex conjugate noting that we have 𝑎=0, we find that, for a purely imaginary number, 𝑧=𝑧.

Example 3: The Sum of a Number and Its Complex Conjugate

Find the complex conjugate of 7𝑖 and the sum of this number with its complex conjugate.

Answer

The complex conjugate of 𝑧=𝑎+𝑏𝑖 is equal to 𝑧=𝑎𝑏𝑖. Hence, for the complex number we have been given, we have 𝑎=7 and 𝑏=1. Hence, the complex conjugate is 7+𝑖.

We can now add these two numbers together and get 7𝑖+7+𝑖=14.

Notice that the result of adding this number to its complex conjugate was a real number. This is no coincidence: for any complex number 𝑧=𝑎+𝑏𝑖, we have 𝑧+𝑧=𝑎+𝑏𝑖+𝑎𝑏𝑖=2𝑎.

We can also write this as 𝑧+𝑧=2(𝑧)Re.

We could also ask what happens when we take the difference of a number with its complex conjugate. In a similar way, we can write 𝑧𝑧=𝑎+𝑏𝑖(𝑎𝑏𝑖)=2𝑏𝑖, which we can also express as 𝑧𝑧=2𝑖(𝑧)Im.

We will now look at an example in which we consider the difference of a number with its complex conjugate.

Example 4: Solving Equations Involving Complex Conjugates

Find the complex number 𝑧 which satisfies the following equations: 𝑧+𝑧=5,𝑧𝑧=3𝑖.

Answer

Using the identity 𝑧+𝑧=2(𝑧)Re and the first equation, we can write 2(𝑧)=5.Re

Hence, Re(𝑧)=52. Similarly, considering the second equation, we first multiply through by 1 which gives 𝑧𝑧=3𝑖.

Now we can use the identity 𝑧𝑧=2𝑖(𝑧)Im to rewrite this as 2𝑖(𝑧)=3𝑖.Im

Dividing by 2𝑖 gives Im(𝑧)=3𝑖2𝑖=32. Knowing both the real and the imaginary parts of 𝑧 defines it uniquely. Hence, 𝑧=5232𝑖.

Example 5: The Product of a Complex Number with Its Complex Conjugate

Find the complex conjugate of 1+𝑖 and the product of this number with its complex conjugate.

Answer

We find the complex conjugate by negating the imaginary part of a complex number. Hence, (1+𝑖)=1𝑖.

We can now consider the product of these two numbers: (1+𝑖)(1𝑖)=1+𝑖𝑖+𝑖=1𝑖.

Since 𝑖=1, we have (1+𝑖)(1𝑖)=2.

Once again, we notice that the product of this complex number with its complex conjugate is a real number. This is an example of a general property of the complex conjugate. In particular, we find that the product of a complex number and its conjugate is actually a special case of the difference of two squares: (𝑥𝑦)(𝑥+𝑦)=𝑥𝑦. Setting 𝑥=𝑎 and 𝑦=𝑏𝑖 gives us (𝑎+𝑏𝑖)(𝑎𝑏𝑖)=𝑎(𝑏𝑖), and using the fact that 𝑖=1, we get 𝑧𝑧=𝑎+𝑏.

Properties of the Complex Conjugate

For a complex number 𝑧=𝑎+𝑏𝑖,

  1. 𝑧+𝑧=2(𝑧)Re,
  2. 𝑧𝑧=2𝑖(𝑧)Im,
  3. 𝑧𝑧=𝑎+𝑏,
  4. 𝑧=𝑧 is equivalent to 𝑧.

Example 6: Complex Conjugates of Sums and Products

Consider 𝑧=5𝑖3 and 𝑤=2+𝑖5.

  1. Calculate 𝑧 and 𝑤.
  2. Find 𝑧+𝑤 and (𝑧+𝑤).
  3. Find 𝑧𝑤 and (𝑧𝑤).

Answer

Part 1

To find the conjugate of a complex number, we negate its imaginary part. Hence, 𝑧=5+𝑖3 and 𝑤=2𝑖5.

Part 2

Using the answers from part 1, we calculate 𝑧+𝑤=5+𝑖3+2𝑖5.

By gathering like terms, we can rewrite it as follows: 𝑧+𝑤=5+2+35𝑖.

Now we can calculate (𝑧+𝑤). Firstly, we evaluate the term inside the brackets: 𝑧+𝑤=5𝑖3+2+𝑖5=5+2+53𝑖.

Taking complex conjugates, we have (𝑧+𝑤)=5+253𝑖=5+2+35𝑖.

Part 3

Using the answers from part 1, we can write 𝑧𝑤=5+𝑖32𝑖5.

Expanding the brackets, we get 𝑧𝑤=525𝑖5+𝑖32𝑖35.

Gathering like terms and using the fact that 𝑖=1, we can rewrite this as 𝑧𝑤=52+15556𝑖.

We calculate (𝑧𝑤) by first finding 𝑧𝑤 and then taking the complex conjugate as follows: 𝑧𝑤=5𝑖32+𝑖5.

Expanding the brackets, we find 𝑧𝑤=52+5𝑖5𝑖32𝑖35.

Simplifying, we get 𝑧𝑤=52+15+556𝑖.

Taking the complex conjugate of each side of the equation, we find (𝑧𝑤)=52+15556𝑖.

In the previous example, we saw that, for the complex numbers 𝑧 and 𝑤, 𝑧+𝑤=(𝑧+𝑤) and 𝑧𝑤=(𝑧𝑤). This is, in fact, a general rule which holds for any pair of complex numbers, the derivation of which uses exactly the same techniques as used in the previous example.

Example 7: Solving Equations Including Complex Conjugates

Solve 𝑧𝑧+𝑧𝑧=4+2𝑖.

Answer

We could approach this problem in one of two ways: we could write 𝑧=𝑎+𝑏𝑖 and substitute it into the equation and then solve for 𝑎 and 𝑏; alternatively, we could use the complex conjugate identities. We will demonstrate both approaches. Starting with the first approach, we have 4+2𝑖=𝑧𝑧+𝑧𝑧=(𝑎+𝑏𝑖)(𝑎𝑏𝑖)+𝑎𝑏𝑖(𝑎+𝑏𝑖).

Expanding the brackets, we have 4+2𝑖=𝑎𝑎𝑏𝑖+𝑎𝑏𝑖𝑖𝑏2𝑏𝑖=𝑎+𝑏2𝑏𝑖.

Equating the real and imaginary parts, we have 2=2𝑏; that is, 𝑏=1 and 𝑎+𝑏=4. Substituting in the value for 𝑏, we get 𝑎+(1)=4𝑎=3.

Hence, 𝑎=±3. Therefore, we have two possible solutions to the equation: 𝑧=3𝑖 and 𝑧=3𝑖.

Alternatively, we can use complex conjugate identities as follows. We first notice that the left-hand side of the equation has two parts:

For each of these two parts, we have an identity:

  1. 𝑧𝑧=𝑎+𝑏,
  2. 𝑧𝑧=2𝑖(𝑧)Im.

Equating real and imaginary parts, we arrive at the same two equations we derived using the first method.

Key Points

  1. For a complex number 𝑧=𝑎+𝑏𝑖, its complex conjugate is defined as 𝑧=𝑎𝑏𝑖.
  2. For two complex numbers 𝑧=𝑎+𝑏𝑖 and 𝑧=𝑐+𝑑𝑖, the following identities hold:
    1. (𝑧±𝑧)=𝑧±𝑧,
    2. (𝑧𝑧)=𝑧𝑧,
    3. 𝑧+𝑧=2(𝑧)Re,
    4. 𝑧𝑧=2𝑖(𝑧)Im,
    5. 𝑧𝑧=𝑎+𝑏.
  3. A complex number is equal to its conjugate if and only if it is a real number.

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