In this explainer, we will learn how to equate, add, and subtract complex numbers.
We begin by recalling the definition of a complex number and some notations for complex numbers.
Definition: Complex Numbers
A complex number is a number of the form , where both and are real numbers and is the square root of . 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.
Let us begin with an example where we can apply the equality of complex numbers to identify unknown values in given complex expressions.
Example 1: Equality of Complex Numbers
If the complex numbers and are equal, what are the values of and ?
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 .
In the previous example, we were able to identify two different unknown constants by using one complex equality. Since the equality for complex numbers equates the real and the imaginary parts separately, one complex equality generates two separate equations. This allows us to identify two different unknowns from one complex equality. Let us consider another example where we find two unknowns from an equation involving complex numbers.
Example 2: Solving Simple Equations Involving Complex Numbers
Determine the real numbers and that satisfy the equation .
Recall that two complex numbers are said to be equal if both their real and imaginary parts are equal. By considering the real and imaginary parts separately, we can derive two equations which we can then solve for and . Since we are told that and are real numbers, we know that and are, respectively, the real and imaginary parts of the complex number on the left-hand side of the given equation. Starting with the real parts, we have
Subtracting 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
Hence, 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.
Definition: Addition and Subtraction of Complex Numbers
For two complex numbers and , we define
In other words, we add or subtract complex numbers by separately adding or subtracting their real parts and imaginary parts.
Alternatively, we can add or subtract complex numbers by expanding through the parenthesis and gathering real and imaginary terms. Using this method,
We note that this leads to the same expression as above. To subtract two complex numbers: where we need to be careful to expand . This leads to which is also the same as above.
This statement can be generalized to encompass the situation when we add or subtract multiple complex numbers. In such cases, we separately add or subtract the real part and imaginary part of each complex number. In our next example, we will add and subtract multiple complex numbers.
Example 3: Adding and Subtracting Complex Numbers
What is ?
We will demonstrate two different methods here.
Method 1: Recall that we can add or subtract multiple complex numbers by adding or subtracting the real part and the imaginary part of each complex number separately. Starting with the real parts, we have
So the real part of the result is . Similarly, for the imaginary parts, we have
Putting these two parts together, we have that the result is .
Method 2: Alternatively, we can add or subtract complex numbers by expanding through the parenthesis and gathering real and imaginary terms. In particular, we should be careful when expanding the last parenthesis since there is a negative sign in front of the parenthesis. We know that . Using this method, 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 4: Subtracting Complex Numbers
If and , find .
We will present two different methods for this example, but we will observe that the second method is preferable since it is much simpler.
Method 1: 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
While this method does lead to a correct solution, it requires more calculations than necessary. Specifically, we have unnecessarily calculated the imaginary part of .
Method 2: We remember that the real part of the difference of two complex numbers is the difference of their real parts:
We can simplify our calculation as follows:
We finish by looking at one slightly more advanced example.
Example 5: Solving Equations Involving Complex Numbers
Let and , where , . Given that , find and .
We can subtract complex numbers by subtracting the real and the imaginary parts separately. Also, equality in complex numbers implies both the equality of the real parts and the equality of the imaginary parts of the complex numbers. 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
Since we are told that and are real numbers, we know that and are the real parts of and respectively. This tells us that . Substituting this expression in the equation above, we obtain
Similarly, by considering the imaginary parts, we have
Since and are the imaginary parts of and , respectively, we have . Substituting this expression to the equation above, we obtain
Rearranging to make the subject, we get
Substituting this into (1) 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 (2) 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
It is always a good practice to check your answer. Therefore, we subtract from and get as expected.
Let us recap a few important concepts from this explainer.
- 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 ;