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 , the conjugate is defined as . The complex conjugate is actually a special case of this where is a negative number.

### Definition: Complex Conjugate

For a complex number , the complex conjugate, , is defined as The complex conjugate is sometimes denoted .

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

Let us begin with an example where we will find the conjugate of a given complex number.

### Example 1: Complex Conjugation

What is the conjugate of the complex number ?

### Answer

Recall the complex conjugate of is . For the complex number we have been given, and (we should be careful not to miss the minus sign). Therefore, the complex conjugate is equal to , which we can simplify to .

In the previous example, we found the conjugate of a complex number. Remember that a real number is a special case of a complex number. In the next example, we will consider the conjugate of a real number.

### Example 2: The Complex Conjugate of a Real Number

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

### Answer

The definition of the complex conjugate of is . If is a purely real number, we know that . This means that if is a real number, .

Hence, if is a real number, its complex conjugate is the same as the original number.

Similarly, we could ask this question: what is the complex conjugate of a purely imaginary number ? Using the definition of the complex conjugate, noting that we have , we find that, for a purely imaginary number,

In the next example, we will consider the sum of a complex number with its conjugate.

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

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

### Answer

The complex conjugate of is equal to . Hence, for the complex number we have been given, we have and . Hence, the complex conjugate is .

We can now add these two numbers together and get

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

We can also write this as .

We could also ask this question: what happens when we take the difference of a number with its complex conjugate? In a similar way, we can write which we can also express as .

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 that satisfies the following equations:

### Answer

We recall the identity . Using this identity, we can write the first equation as

Hence, .

Next, we recall the identity . Before we can apply this identity to the second equation, we first multiply through by , which gives

Now, we can use the identity to rewrite this as

Dividing by gives .

This gives us Hence,

In the next example, we will compute the product of a complex number with its conjugate.

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

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

### Answer

For the first part of the question: The complex conjugate of is equal to . We can write the complex number as , which tells us and . The complex conjugate of is , which is the same as .

Hence, the complex conjugate of is .

For the second part of the question: Let us consider the product of and its conjugate by expanding through the parentheses:

Since , the resulting expression is .

Hence, the product of and its conjugate is equal to 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: . Setting and gives us , and using the fact that , we get

### Property: Complex Conjugate

For a complex number ,

- ,
- ,
- ,
- is equivalent to .

In the next example, we will consider how the operation of sum or product of complex numbers interacts with the conjugate operation.

### Example 6: Complex Conjugates of Sums and Products

Consider and .

- Calculate and .
- Find and .
- Find and .

### Answer

**Part 1**

The complex conjugate of is equal to . We can write the complex number as , which tells us and , so the complex conjugate of is Similarly for , we can see that and , which leads to the conjugate

**Part 2**

Using the answers from part 1, we calculate

By gathering like terms, we can rewrite it as follows:

Now, we can calculate . Firstly, we evaluate the term inside the brackets:

To take the complex conjugate of this number, we notice that this number is in the form , where and . Since the complex conjugate takes the form , we have Noticing , we can also write this complex number as We can now see that this is the same as the number obtained from the expression .

**Part 3**

Using the answers from part 1, we can write

Distributing over the parentheses, we get

Gathering like terms and using the fact that , we can rewrite this as

We calculate by first finding and then taking the complex conjugate as follows:

Distributing over the parentheses, we find

Simplifying, we get

To take the complex conjugate of this number, we notice that this number is in the form , where and . Since the complex conjugate takes the form , we have

We can see that this is the same complex number as what we obtained from .

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

### Property: Algebraic Operations and Complex Conjugates

Given complex numbers and , we have the following identities:

- ,
- ,
- ,
- .

In our final example, we will use the properties of the complex conjugate to solve an equation using a complex variable .

### Example 7: Solving Equations Including Complex Conjugates

Solve .

### Answer

We could approach this problem in one of two ways: we could write and substitute it into the equation and then solve for and ; alternatively, we could use the properties of complex conjugates. We will demonstrate both approaches.

**Method 1**

Recall that the complex conjugate of is . Using this expression, we have

We can first distribute which simplifies to when remembering that .

Distributing over the remaining parentheses, while being careful to notice , we have

This leads to the equation Recall that the two complex numbers are equal when the real and the imaginary parts are both equal.

Equating the real parts of the complex numbers on both sides of the equation gives Equating the imaginary parts gives , which leads to . Substituting in the value for in the equation above, we get

Hence, . This gives us that could be either or , while is equal to . Since , we have two possible solutions to the equation:

**Method 2**

Let us use the properties of complex conjugates as follows. We first notice that the left-hand side of the equation has two parts:

For each of these two parts, we recall the identities

- ,
- .

To obtain the expresion (2), we can multiply both sides of the second identity to obtain Since , we can obtain . For (1), the first identity gives . Substituting these expressions into the equation above, we obtain

This leads us to the same equation as what we derived using the first method.

Using the same computations, the solutions to the given equations are

Let us finish by recapping a few important concepts from this explainer.

### Key Points

- For a complex number , its complex conjugate is defined as .
- For two complex numbers and , the following identities hold:
- ,
- ,
- ,
- ,
- .

- A complex number is equal to its conjugate if and only if it is a real number.