In this explainer, we will learn how to deal with imaginary numbers, knowing that complex numbers are made of a real part and an imaginary part.
When we start learning about numbers, we start from the idea of counting numbers: . With counting numbers, we can perform operations of addition and multiplication. However, when we introduce subtraction, there are operations which do not make sense if we are solely restricted to dealing with counting numbers. For example, the expression cannot be evaluated within the counting numbers. This is when we introduce the idea of negative numbers. With both positive and negative numbers (the integers), every subtraction operation makes sense and we can solve a larger set of equations involving addition and subtraction.
In a similar way, when we try to introduce division to integers, we come across a similar problem; we find equations that have no integer solutions. For example, has no integer solution. At this point, we extend our concept of numbers to include fractions. Combining integers and fractions, we form the rational numbers. Now, the operation of division (for nonzero divisors) makes sense and we can solve a larger set of equations involving multiplication and division.
However, we soon find that there are other equations which have no solutions even when we have all the rational numbers at our disposal. For example, we can solve the equation , but an analogous equation has no rational solution. Hence, we introduce the notion of an irrational number. Combining the rational and irrational numbers, we form the real numbers. At this point, we might think the real numbers are all the numbers we need; the operations of addition, subtraction, multiplication, and division all make sense and we can even take the square root of any positive number.
We still find there are equations which have no real-number solutions. For example, let us consider the equation . We know that squaring a number always gives a nonnegative result. Therefore, for any real number ,
Hence, the equation has no real solutions. Furthermore, when we look at the graph, we can see that nowhere does the curve intersect the -axis.
Consequently, we confirm that the equation has no real solution. However, maybe we can extend the definition of a number (as we did when we introduced negative numbers or irrational numbers) in such a way that it makes sense to talk about the solutions of equations such as . How might we do this? First, let us rearrange the equation, by subtracting one from both sides:
Now, let us introduce a “new number” which we will denote which is defined by the property that . This might seem like a bit of a crazy idea, but as we will see, this “new number” opens up a whole new concept of what numbers are. Furthermore, it turns out to be massively important in many areas of advanced mathematics and has applications in areas as wide as signal processing, electrical engineering, quantum mechanics, and fluid dynamics.
This “new number” we have introduced is what mathematicians refer to as the imaginary number . Similar to the introduction of negative numbers, wide-spread adoption of imaginary numbers took time. In fact, the very term “imaginary,” which was coined by René Descartes in 1637, was used in a somewhat derogatory manner in opposition to the “real” numbers.
Definition: Imaginary Numbers
An imaginary number is a number of the form , where is a real number and is defined by
As we have previously mentioned, is a solution to the equation . It is not the only solution, however; similar to having the two solutions 1 and , has the two solutions and . Thus, we can think of as being one of the two square roots of .
As stated in the definition, any real multiple of (i.e., for some real number ) is also an imaginary number. These imaginary numbers can be used to determine solutions to equations of the form , since we have
So, is a solution to the equation (as is ).
Imaginary numbers have many of the same properties that real numbers have. For instance, we can add, subtract, multiply, and divide them. In particular, imaginary numbers can be added to real numbers, and we refer to the result as a complex number. Let us define this concept concretely.
Definition: Complex Numbers
A complex number is a number of the form , where both and are real numbers. 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 .
Note that the value of is and not . Hence, always gives us a real number. Let us now look at some examples to help us gain more familiarity with the concept of complex numbers.
In the first example, let’s determine the value of an expression involving an imaginary number using the rules of algebra.
Example 1: Arithmetic with Imaginary Numbers
What is the value of ?
Using the rules of algebra (either the rules of indices or commutativity of multiplication), we can rewrite
Now, recall that is defined as the solution to the equation . Hence, we have . Substituting this back into the equation gives
Now, let’s consider an example where we determine the square root of a negative number in terms of the imaginary unit .
Example 2: Square Roots of Negative Numbers
Express in terms of .
We can rewrite as . Hence,
We can evaluate this by taking the square root of each part separately. By definition, the square root of negative one is , so we have
We can express 54 as the product of prime factors as . Hence,
Note that it is good practice to write imaginary radicals in the form or ; if we write it can easily be confused with .
Be careful: When taking square roots of the product of complex numbers, we need to be careful. For all positive real numbers and , we know that . However, this is not true for complex numbers in general. When evaluating the square root of a negative number, it is legitimate to write .
Now, let’s consider an example where we will form a complex number using arithmetic.
Example 3: Forming Complex Numbers
Add 4 to .
Recall that the definition of a complex number is a number in the form , where and are real numbers. So we can simply add 4 to to get . This is a perfectly acceptable form for a complex number where and . However, we prefer to write this more succinctly as .
In the next example, we will consider the relationship between real and complex numbers in general.
Example 4: The Relationship between Real and Complex Numbers
Is the following statement true or false: Any real number is also a complex number?
Recall that the definition of a complex number is a number of the form , where . Since 0 is a real number, all numbers of the form are complex numbers. However, can simply be expressed as the real number . Therefore, we conclude that the statement “any real number is also a complex number” is true.
Finally, let’s consider an example where we determine the imaginary part of a given complex number.
Example 5: The Imaginary Part of a Complex Number
What is the imaginary part of the complex number ?
Remember, for a complex number , the imaginary part . The number we are given is , so we have and (be careful not to miss the minus sign). Hence, the imaginary part of is . Be careful not to make the mistake of giving the answer of ; the imaginary part of a real number is always a real number!
Let us finish by summarizing the main things we have learned about complex numbers in this explainer.
- We can extend the real numbers by introducing the concept of the imaginary number defined as the solution to the equation .
- Adding together real and imaginary numbers forms the complex numbers .
- Each complex number is a number of the form , where .
- For a complex number , we define the real and imaginary parts as and .