Explainer: Introduction to Complex Numbers

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: 1,2,3,…. 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 3βˆ’5 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, 2π‘₯=1 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 π‘₯=4, but an analogous equation π‘₯=2 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 π‘₯+1=0. We know that squaring a number always gives a nonnegative result. Therefore, for any real number π‘₯, π‘₯+1β‰₯1.

Hence, the equation has no real solutions. Furthermore, when we look at the graph, we can see that nowhere does the curve intersect the π‘₯-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 π‘₯+1=0. How might we do this? First, let us rearrange the equation, by subtracting one from both sides: π‘₯=βˆ’1.

Now, let us introduce a β€œnew number” which we will denote 𝑖 which is defined by the property that 𝑖=βˆ’1. This might seem like a bit of a crazy idea, but as we will see, this β€œnew 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 β€œnew number” we have introduced is what mathematicians refer to as the imaginary number 𝑖. Similar to the introduction of negative numbers, wide-spread adoption of imaginary numbers took time. In fact, the very term β€œimaginary,” which was coined by RenΓ© Descartes in 1637, was used in a somewhat derogatory manner in opposition to the β€œreal” numbers.

Definition: Imaginary Numbers

The number 𝑖 is defined as the solution to the equation π‘₯=βˆ’1. Since, 𝑖 is not a real number, it is referred to as an imaginary number and all real multiples of 𝑖 (numbers of the form 𝑏𝑖, where 𝑏 is real) are called (purely) imaginary numbers. Often 𝑖 is referred to as the square root of negative one.

If we add together real and imaginary numbers, we refer to the result as a complex number.

Definition: Complex Numbers

A complex number is a number of the form π‘Ž+𝑏𝑖, where both π‘Ž and 𝑏 are real numbers. The set of all complex numbers is denoted by β„‚.

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

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

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

Note that the value of Im(𝑧) is 𝑏 and not 𝑏𝑖. Hence, Im(𝑧) 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.

Example 1: Arithmetic with Imaginary Numbers

What is the value of (5𝑖)?

Answer

Using the rules of algebra (either the rules of indices or commutativity of multiplication), we can rewrite (5𝑖)=5𝑖=25𝑖.

Now, recall that 𝑖 is defined as the solution to the equation π‘₯=βˆ’1. Hence, we have 𝑖=βˆ’1. Substituting this back into the equation gives 25𝑖=25(βˆ’1)=βˆ’25.

Hence, (5𝑖)=βˆ’25.

Example 2: Square Roots of Negative Numbers

Express βˆšβˆ’54 in terms of 𝑖.

Answer

We can rewrite βˆ’54 as 54Γ—(βˆ’1). Hence, βˆšβˆ’54=√54Γ—(βˆ’1).

We can evaluate this by taking the square root of each part separately. By definition, the square root of negative one is 𝑖, so we have βˆšβˆ’54=√54×𝑖.

We can express 54 as the product of prime factors as 54=2Γ—3. Hence, βˆšβˆ’54=√2Γ—3Γ—3×𝑖=ο€»3√6𝑖.

Note that it is good practice to write imaginary radicals in the form ο€»3√6𝑖 or 3π‘–βˆš6; if we write 3√6𝑖 it can easily be confused with 3√6𝑖.

Be careful: When taking square roots of the product of complex numbers, we need to be careful. For all positive real numbers π‘Ž and 𝑏, we know that βˆšπ‘Žπ‘=βˆšπ‘Žβˆšπ‘. However, this is not true for complex numbers in general. When evaluating the square root of a negative number, it is legitimate to write βˆšβˆ’π‘Ž=√(βˆ’1)Γ—π‘Ž=βˆšβˆ’1Γ—βˆšπ‘Ž=π‘–βˆšπ‘Ž.

Example 3: Forming Complex Numbers

Add 4 to βˆ’π‘–.

Answer

Recall that the definition of a complex number is a number in the form π‘Ž+𝑏𝑖, where π‘Ž and 𝑏 are real numbers. So we can simply add 4 to βˆ’π‘– to get 4+(βˆ’π‘–). This is a perfectly acceptable form for a complex number where π‘Ž=4 and 𝑏=βˆ’1. However, we prefer to write this more succinctly as 4βˆ’π‘–.

Example 4: The Relationship Between Real and Complex Numbers

Is the following statement true or false: Any real number is also a complex number?

Answer

Recall that the definition of a complex number is a number of the form π‘Ž+𝑏𝑖, where π‘Ž,π‘βˆˆβ„. Since 0 is a real number, all numbers of the form π‘Ž+0𝑖 are complex numbers. However, π‘Ž+0𝑖 can simply be expressed as the real number π‘Ž. Therefore, we conclude that the statement β€œany real number is also a complex number” is true.

Example 5: The Imaginary Part of a Complex Number

What is the imaginary part of the complex number 2βˆ’2𝑖?

Answer

Remember, for a complex number 𝑧=π‘Ž+𝑏𝑖, the imaginary part Im(𝑧)=𝑏. The number we are given is 2βˆ’2𝑖, so we have π‘Ž=2 and 𝑏=βˆ’2 (be careful not to miss the minus sign). Hence, the imaginary part of 2βˆ’2𝑖 is βˆ’2. Be careful not to make the mistake of giving the answer of βˆ’2𝑖; the imaginary part of a real number is always a real number!

Key Points

  1. We can extend the real numbers by introducing the concept of the imaginary number 𝑖 defined as the solution to the equation π‘₯=βˆ’1.
  2. Adding together real and imaginary numbers forms the complex numbers β„‚.
  3. Each complex number 𝑧 is a number of the form π‘Ž+𝑏𝑖, where π‘Ž,π‘βˆˆβ„.
  4. For a complex number 𝑧=π‘Ž+𝑏𝑖, we define the real and imaginary parts as Re(𝑧)=π‘Ž and Im(𝑧)=𝑏.

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