In this explainer, we will learn how to equate, add, and subtract complex numbers.
We begin by recalling the definition of a complex number.
Definition: Complex Numbers
A complex number is a number of the form , where both and are real numbers and is defined as the solution to the equation . The set of all complex numbers is denoted by .
For a complex number , we define the real part of to be and write
Similarly, we define the imaginary part of to be and write
Some books and articles use the notation and to refer to the real and imaginary parts of .
Before we start doing arithmetic with complex numbers, we need to understand what it means for two complex numbers to be equal. We define equality of complex numbers in a similar way to how we define equality of algebraic expressions involving variables. For example, if is a variable and , , , and are real numbers, saying the two algebraic expressions and are equal is equivalent to stating that and . We define the equality of complex numbers in a similar way.
Definition: Equality of Complex Numbers
Two complex numbers and are said to be equal if and . Conversely, if , then and . Equivalently, we can state that two complex numbers and are equal if and and the equivalent converse statement.
Oftentimes, working with the second version of this is easiest as we will see in some of the examples below.
Example 1: Equality of Complex Numbers
If the complex numbers and are equal, what are the values of and ?
Answer
Recall that two complex numbers are said to be equal if both their real and imaginary parts are equal. Beginning with equating the real parts, we have . Similarly, equating the imaginary parts, we arrive at the equation (we need to be careful not to miss the negative sign here). Hence, we have and .
Similar to how we defined the equality of complex numbers, the basic tenets of addition and subtraction of complex numbers are analogous to their equivalents within the algebra of polynomials. To add and subtract polynomials, we add and subtract the corresponding coefficients.
Addition and Subtraction of Complex Numbers
For two complex numbers and , we define
Similarly,
This is equivalent to the following statement: we add complex numbers by separately adding their real parts and imaginary parts.
Example 2: Adding and Subtracting Complex Numbers
What is ?
Answer
We can approach this either by considering both the real and the imaginary parts separately or by gathering like terms. We will demonstrate both methods here. Firstly, we will use the method of considering the real and imaginary parts. Starting with the real part, we have
So the real part of the result is . Similarly, for the imaginary part, we have
Putting these two parts together, we have that the result is .
Using the method of gathering like terms, we can write which simplifies to .
In practice, we often use the method of gathering like terms. However, occasionally, it can be useful to remember that there is an alternative method as we will see in the example below.
Example 3: Subtracting Complex Numbers
If and , find .
Answer
We begin by calculating by gathering like terms. Firstly, substituting the values of and , we have
At this point, we need to be careful with the minus signs. Multiplying each term in the bracket by gives
By gathering like terms, this simplifies to
Taking the real part, we have
This method is fine, butwe have had to do more calculations than necessary. Specifically, we have unnecessarily calculated the imaginary part of . By remembering that the real parts of the difference of two complex numbers is the difference of their real parts, we can simplify our calculation as follows:
Example 4: Solving Simple Equations Involving Complex Numbers
Determine the real numbers and that satisfy the equation .
Answer
By considering the real and imaginary parts separately, we can derive two equations which we can then solve for and . Starting with the real part, we have
Taking 2 from both sides gives
Then, by dividing by 5, we get
Taking the imaginary parts of both sides, we have
Adding 5 to both sides gives then dividing by 3 gives
We finish by looking at one slightly more advanced example.
Example 5: Solving Equations Involving Complex Numbers
Let and where , . Given that , find and .
Answer
By considering the real and imaginary parts separately, we can derive two equations which we can then solve for and . Starting with the real part, we have
Substituting in and gives us our first equation:
Similarly, by considering the imaginary parts, we have
Substituting in the values of and , we derive our second equation:
Rearranging to make the subject, we get
Substituting this into our first equation gives
At this point, we could multiply out the brackets. However, it is more efficient to divide both sides of the equation by 4 as follows:
Adding two to both sides and simplifying gives . Substituting this back into the equation for gives
Having found and , we might be tempted to stop. However, the question actually asked us for and . Hence, we still need to substitute these values back into the equations for and to finish. Starting with , we have
Similarly,
It is always a good practice to check your answer. Therefore, we subtract from and get as expected.
Key Points
- Addition, subtraction, and equality of complex numbers are defined in an analogous way to addition, subtraction, and equality of polynomial expressions.
- By applying familiar rules of algebra, we can begin to work effectively with complex numbers.
- For complex numbers and ,
- = is equivalent to saying both their real and imaginary parts are equal or and ;
- ;
- ;
- .