Explainer: Equating, Adding, and Subtracting Complex Numbers

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 𝑎+𝑏𝑖, where both 𝑎 and 𝑏 are real numbers and 𝑖 is defined as the solution to the equation 𝑥2=1. The set of all complex numbers is denoted by .

For a complex number 𝑧=𝑎+𝑏𝑖, we define the real part of 𝑧 to be 𝑎 and write Re(𝑧)=𝑎.

Similarly, we define the imaginary part of 𝑧 to be 𝑏 and write Im(𝑧)=𝑏.

Some books and articles use the notation (𝑧) and (𝑧) to refer to the real and imaginary parts of 𝑧.

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 𝑥 is a variable and 𝑎, 𝑏, 𝑐, and 𝑑 are real numbers, saying the two algebraic expressions 𝑎+𝑏𝑥 and 𝑐+𝑑𝑥 are equal is equivalent to stating that 𝑎=𝑐 and 𝑏=𝑑. We define the equality of complex numbers in a similar way.

Definition: Equality of Complex Numbers

Two complex numbers 𝑧1=𝑎+𝑏𝑖 and 𝑧2=𝑐+𝑑𝑖 are said to be equal if 𝑎=𝑐 and 𝑏=𝑑. Conversely, if 𝑧1=𝑧2, then 𝑎=𝑐 and 𝑏=𝑑. Equivalently, we can state that two complex numbers 𝑧1 and 𝑧2 are equal if Re(𝑧1)=Re(𝑧2) and Im(𝑧1)=Im(𝑧2) 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 7+𝑎𝑖 and 𝑏3𝑖 are equal, what are the values of 𝑎 and 𝑏?

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 7=𝑏. Similarly, equating the imaginary parts, we arrive at the equation 𝑎=3 (we need to be careful not to miss the negative sign here). Hence, we have 𝑎=3 and 𝑏=7.

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 𝑧1=𝑎+𝑏𝑖 and 𝑧2=𝑐+𝑑𝑖, we define 𝑧1+𝑧2=(𝑎+𝑐)+(𝑏+𝑑)𝑖.

Similarly,𝑧1𝑧2=(𝑎𝑐)+(𝑏𝑑)𝑖.

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 9+(7+4𝑖)+(44𝑖)(1+3𝑖)?

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 9+7+(4)1=7.

So the real part of the result is 7. Similarly, for the imaginary part, we have 4+(4)3=3.

Putting these two parts together, we have that the result is 73𝑖.

Using the method of gathering like terms, we can write 9+(7+4𝑖)+(44𝑖)(1+3𝑖)=(9+7+(4)1)+(4+(4)3)𝑖 which simplifies to 73𝑖.

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 𝑟=5+2𝑖 and 𝑠=9𝑖, find Re(𝑟𝑠).

Answer

We begin by calculating 𝑟𝑠 by gathering like terms. Firstly, substituting the values of 𝑟 and 𝑠, we have 𝑟𝑠=5+2𝑖(9𝑖).

At this point, we need to be careful with the minus signs. Multiplying each term in the bracket by 1 gives 𝑟𝑠=5+2𝑖9+𝑖.

By gathering like terms, this simplifies to 𝑟𝑠=4+3𝑖.

Taking the real part, we have Re(𝑟𝑠)=4.

This method is fine, butwe have had to do more calculations than necessary. Specifically, we have unnecessarily calculated the imaginary part of 𝑟𝑠. By remembering that the real parts of the difference of two complex numbers is the difference of their real parts, Re(𝑟𝑠)=Re(𝑟)Re(𝑠), we can simplify our calculation as follows: Re(𝑟𝑠)=Re(5+2𝑖)Re(9𝑖)=59=4.

Example 4: Solving Simple Equations Involving Complex Numbers

Determine the real numbers 𝑥 and 𝑦 that satisfy the equation 5𝑥+2+(3𝑦5)𝑖=3+4𝑖.

Answer

By considering the real and imaginary parts separately, we can derive two equations which we can then solve for 𝑥 and 𝑦. Starting with the real part, we have 5𝑥+2=3.

Taking 2 from both sides gives 5𝑥=5.

Then, by dividing by 5, we get 𝑥=1.

Taking the imaginary parts of both sides, we have 3𝑦5=4.

Adding 5 to both sides gives 3𝑦=9; then dividing by 3 gives 𝑦=3.

We finish by looking at one slightly more advanced example.

Example 5: Solving Equations Involving Complex Numbers

Let 𝑧1=4𝑥+2𝑦𝑖 and 𝑧2=4𝑦+𝑥𝑖 where 𝑥, 𝑦. Given that 𝑧1𝑧2=5+2𝑖, find 𝑧1 and 𝑧2.

Answer

By considering the real and imaginary parts separately, we can derive two equations which we can then solve for 𝑥 and 𝑦. Starting with the real part, we have Re(𝑧1)Re(𝑧2)=Re(5+2𝑖).

Substituting in 𝑧1 and 𝑧2 gives us our first equation: 4𝑥4𝑦=5.

Similarly, by considering the imaginary parts, we have Im(𝑧1)Im(𝑧2)=Im(5+2𝑖).

Substituting in the values of 𝑧1 and 𝑧2, we derive our second equation: 2𝑦𝑥=2.

Rearranging to make 𝑥 the subject, we get 𝑥=2𝑦2.

Substituting this into our first equation gives 4(2𝑦2)4𝑦=5.

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: 2𝑦2𝑦=54.

Adding two to both sides and simplifying gives 𝑦=134. Substituting this back into the equation for 𝑥 gives 𝑥=21342.=92.

Having found 𝑥 and 𝑦, we might be tempted to stop. However, the question actually asked us for 𝑧1 and 𝑧2. Hence, we still need to substitute these values back into the equations for 𝑧1 and 𝑧2 to finish. Starting with 𝑧1, we have 𝑧1=492+2134𝑖=18+132𝑖.

Similarly, 𝑧2=4134+292𝑖=13+92𝑖.

It is always a good practice to check your answer. Therefore, we subtract 𝑧1 from 𝑧2 and get 5+2𝑖 as expected.

Key Points

  1. Addition, subtraction, and equality of complex numbers are defined in an analogous way to addition, subtraction, and equality of polynomial expressions.
  2. By applying familiar rules of algebra, we can begin to work effectively with complex numbers.
  3. For complex numbers 𝑧1=𝑎+𝑏𝑖 and 𝑧2=𝑐+𝑑𝑖,
    1. 𝑧1 = 𝑧2 is equivalent to saying both their real and imaginary parts are equal or 𝑎=𝑐 and 𝑏=𝑑;
    2. 𝑧1±𝑧2=(𝑎±𝑐)+(𝑏±𝑑)𝑖;
    3. Re(𝑧1±𝑧2)=Re(𝑧1)±Re(𝑧2);
    4. Im(𝑧1±𝑧2)=Im(𝑧1)±Im(𝑧2).

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