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,
Example 1: Complex Conjugation
What is the conjugate of the complex number ?
Recall the complex conjugate of is . For the complex number we have been given, and (careful not to miss the minus sign). Therefore, the complex conjugate is equal to which we can simplify to .
Example 2: The Complex Conjugate of a Real Number
If is a real number, what will its conjugate be equal to?
The definition of the complex conjugate of is . If is a purely real number, we know that . Hence, we conclude that if is a real number, .
Conversely, if we know that, for a complex number , we have gathering like terms, we get
Hence, , and we can conclude that is real.
Similarly, we could ask the 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,
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.
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 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 which satisfies the following equations:
Using the identity and the first equation, we can write
Hence, . Similarly, considering the second equation, we first multiply through by which gives
Now we can use the identity to rewrite this as
Dividing by gives . Knowing both the real and the imaginary parts of defines it uniquely. Hence,
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.
We find the complex conjugate by negating the imaginary part of a complex number. Hence, .
We can now consider the product of these two numbers:
Since , we have .
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
Properties of the Complex Conjugate
For a complex number ,
- is equivalent to .
Example 6: Complex Conjugates of Sums and Products
Consider and .
- Calculate and .
- Find and .
- Find and .
To find the conjugate of a complex number, we negate its imaginary part. Hence, and
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:
Taking complex conjugates, we have
Using the answers from part 1, we can write
Expanding the brackets, 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:
Expanding the brackets, we find
Simplifying, we get
Taking the complex conjugate of each side of the equation, we find
In the previous example, we saw that, for the complex numbers and , and . 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
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 complex conjugate identities. We will demonstrate both approaches. Starting with the first approach, we have
Expanding the brackets, we have
Equating the real and imaginary parts, we have ; that is, and . Substituting in the value for , we get
Hence, . Therefore, we have two possible solutions to the equation: and .
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:
Equating real and imaginary parts, we arrive at the same two equations we derived using the first method.
- 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.