Video: Introduction to Complex Numbers

In this video, we will learn how to deal with imaginary numbers, knowing that complex numbers are made of a real part and an imaginary part.

13:15

Video Transcript

In this video, we’re going to look at the concept of imaginary and complex numbers. To begin with, we’ll simply define what we mean by an imaginary number and a complex number. And we’ll look at when and why we might need to use them. Then we’ll discover how to complete simple calculations and manipulate these types of numbers.

When we begin learning about numbers, we learn about the set of natural numbers. These are sometimes called the counting numbers. They’re the numbers that can be used for counting and ordering. For example, there are three apples in the bowl, or history is my second favourite subject, behind maths of course. We then extend this idea and we learn to do simple addition and subtraction.

Now at this stage, we might struggle to understand a sum such as three minus five without understanding the concept of negative numbers. We then come across the concept of sharing. And we’re forced to introduce a further set of numbers when we begin to look at division. An equation such as two 𝑥 equals one has no whole number or integer solutions. So we introduce the idea of fractions or decimals.

Our understanding of numbers now includes rational numbers, numbers that can be written in the form 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers. And then we discover there are even more numbers than the counting numbers, negative numbers, and rational numbers. We learn about radicals and 𝜋. These are irrational numbers, numbers that cannot be written as some integer over another. Combine all these numbers and we have the set of real numbers. Well, great, that’s all we need, right?

Well, no, not quite. During our exploration of the concept of numbers, we will have come across equations that have no solution or at least those that we presume to have no solution. Take 𝑥 squared plus one equals zero for example. We know that, for any real value of 𝑥, 𝑥 squared will always be greater than or equal to zero. So this means that 𝑥 squared plus one must always be greater than or equal to one.

So as far as we’re concerned, the equation 𝑥 squared plus one equals zero really doesn’t make a lot of sense, yet. And we might even consider the graph of the equation 𝑦 equals 𝑥 squared plus one. It’s a parabola which intersects the 𝑦-axis at one. We see that there simply can’t be any real solutions to the equation 𝑥 squared plus one equals zero. This graph doesn’t intersect the 𝑥-axis. In fact, we’ve already extended our understanding of numbers from the counting numbers all the way through to irrational numbers. So what’s to stop us extending it just a little bit more?

Let’s imagine the equation 𝑥 squared plus one equals zero does have a solution. We could solve it like any other equation. We could subtract one from both sides to get 𝑥 squared is equal to negative one. And here, this right here is where we extend our understanding of numbers.

We introduce a new number, 𝑖, such that 𝑖 squared is equal to negative one. And we can now see that 𝑖 must be a solution to the equation 𝑥 squared equals negative one. And in fact, if we square-root this, we see that 𝑖 is equal to the square root of negative one. We call this the imaginary number.

This term was originally used because, at the time, nobody believed that any real-world use will be found for this number. It was viewed as a pretend number, invented only for the purpose of solving certain equations. But if we think about the different sets of numbers, they’re all made up. So why not invent a new one? And this one really stuck.

So we define 𝑖 as being the solution to the equation 𝑥 squared equals negative one. And it’s often referred to as the square root of negative one. Now since 𝑖 is not a real number, it and any real multiples of 𝑖 — that’s numbers of the form 𝑏𝑖, where 𝑏 is a real number — are called purely imaginary numbers. And just like the set of all real numbers is denoted by ℝ, the set of all imaginary numbers is denoted by 𝕀. So that’s our first definition.

And here we introduce a second definition. This definition is for complex numbers. And those are the result of adding real and imaginary numbers. These are of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 themselves are real numbers. And a set of all complex numbers is denoted by this letter ℂ. And for a complex number 𝑧 is equal to 𝑎 plus 𝑏𝑖, we say that the real part of 𝑧 is 𝑎 and the imaginary part is 𝑏. And be careful. The imaginary part is 𝑏, not 𝑏𝑖.

Now that we have all the relevant definitions we need, let’s look at how to form and manipulate these types of numbers.

What is the value of five 𝑖 squared?

To answer this question, we’ll recall our rules for simplifying algebraic expressions. For example, to simplify an expression of the form 𝑎 multiplied by 𝑏 to the power of 𝑛, we’d work out 𝑎 to the power of 𝑛 and we multiply that by 𝑏 to the power of 𝑛. In this case, five 𝑖 squared is the same as five squared multiplied by 𝑖 squared. And of course, five squared is 25. And 𝑖 is defined as the solution to the equation 𝑥 squared is equal to negative one. 𝑖 squared is equal to negative one. So we can write five 𝑖 squared as 25 multiplied by negative one. And since a positive multiplied by a negative is a negative, we see that five 𝑖 squared is equal to negative 25.

Evaluate three 𝑖 multiplied by seven 𝑖.

To answer this question, we’ll recall the fact that multiplication is commutative. It can be performed in any order. We can rewrite this product as three multiplied by seven multiplied by 𝑖 multiplied by 𝑖. Three multiplied by seven is 21, and 𝑖 multiplied by 𝑖 is 𝑖 squared. We’re not quite finished though.

𝑖 is not a variable like 𝑥 or 𝑦. We know that 𝑖 is the solution to the equation 𝑥 squared is equal to negative one. We can say that 𝑖 squared is equal to negative one or 𝑖 is equal to the square root of negative one. We’ll replace 𝑖 squared in our problem with negative one. And we can see that three 𝑖 multiplied by seven 𝑖 becomes 21 multiplied by negative one. 21 multiplied by negative one is negative 21. And we’ve evaluated three 𝑖 multiplied by seven 𝑖. It’s negative 21.

Express the square root of negative four in terms of 𝑖.

To answer this question, we’re going to need to rewrite negative four slightly. We write it as four multiplied by negative one. And why have we done this? Well, it means that we can rewrite the square root of negative four as the square root of four multiplied by negative one.

Now the laws of radicals tell us that, for positive real numbers 𝑎 and 𝑏, the square root of 𝑎𝑏 is the same as the square root of 𝑎 multiplied by the square root of 𝑏. Now whereas this isn’t true for all complex numbers in general, we can say that the square root of negative 𝑎 can be written as the square root of 𝑎 multiplied by negative one. And in turn, that can be written as the square root of 𝑎 multiplied by the square root of negative one.

This means we can write the square root of negative four as the square root of four multiplied by the square root of negative one. We know that the square root of four is two, and we also know that the square root of negative one is 𝑖. So we can say that the square root of negative four is two multiplied by 𝑖. And in fact, we’ll simplify this. And we see that the square root of negative four is two 𝑖.

Express the square root of negative 54 in terms of 𝑖.

To answer this question, we’re going to rewrite negative 54 slightly. We rewrite it as 54 multiplied by negative one. And this means we can say that the square root of negative 54 is the same as the square root of 54 multiplied by negative one. And let’s see why we do this in a more general form.

Now we can say that the square root of negative 𝑎 can be written as the square root of 𝑎 multiplied by negative one, which in turn can be written as the square root of 𝑎 multiplied by the square root of negative one. And if we recall that we say that 𝑖 is equal to the square root of negative one, the square root of negative 𝑎 must be equal to the square root of 𝑎 multiplied by 𝑖.

For our number, we can write it as the square root of 54 multiplied by the square root of negative one, which is the square root of 54 multiplied by 𝑖. We’re not quite finished though. We need to simplify root 54 as far as possible. There are a number of ways we can do this. We could consider 54 as a product of its prime factors. Or alternatively, we could find the largest factor of 54, which is also a square number.

In this case, the factor that we’re interested in is nine. So we say that the square root of 54 is equal to the square root of nine multiplied by six, or the square root of nine multiplied by the square root of six. But we, of course, know that the square root of nine is three. So we can say that the square root of 54 is equal to three root six. And we can see that the square root of negative 54 is three root six 𝑖.

Now it’s important to be careful here. Try to include the brackets as shown. If we were to write three root six 𝑖 without brackets, it could easily be confused for three multiplied by the square root of six 𝑖, which is a different solution altogether.

Add four to negative 𝑖.

We’ve been given a real and an imaginary number. And we’re looking to find their sum. What this question is really asking us to do is to form a complex number. Remember, a complex number, 𝑧, is of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are both real numbers and 𝑎 is the real component of 𝑧 and 𝑏 is the imaginary component.

Now if we add four to negative 𝑖, we get four plus negative 𝑖. But we can actually write this as simply four minus 𝑖. And we can now see that we have a complex number with a real component of four and an imaginary component of negative one.

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

We know that a complex number 𝑧 is of the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. Now in fact, we could say that a real number 𝑎 is of the form 𝑎 plus zero 𝑖. Zero is indeed a real number. And by its very definition, 𝑎 plus zero 𝑖 is a complex number.

So we have shown that any real number is also a complex number. We can say that the statement “Any real number is also a complex number” is true. It’s important to note though that the opposite statement is not true. We cannot say any complex number is also a real number, since a complex number has a real and an imaginary part. The only way that a complex number can also be a real number is if the imaginary part is zero.

What is the imaginary part of the complex number two minus two 𝑖?

A complex number is the result of adding a real and an imaginary number. And a complex number 𝑧 is in the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers. We say that the real part of this general complex number is 𝑎. And the imaginary part is essentially the coefficient of 𝑖. It’s 𝑏. Remember, it’s not 𝑏𝑖, just 𝑏 itself.

Let’s compare this general form to our complex number two minus two 𝑖. We can see that 𝑎 is equal to two and 𝑏 is equal to negative two. And this means that the real part of this complex number is two and the imaginary part is negative two.

So let’s recap what we’ve learnt today. We’ve extended our understanding of the set of all numbers to now include imaginary numbers. We have a new number 𝑖, which is defined as the solution to the equation 𝑥 squared is equal to negative one. And of course, we often say that 𝑖 is equal to the square root of negative one.

We’ve learnt that a number in the form 𝑎 plus 𝑏𝑖, where 𝑎 and 𝑏 are real numbers, is called a complex number. They’re found by adding a real and an imaginary number. And finally, we’ve seen that the real part of our complex number is 𝑎 and the imaginary part is 𝑏, not 𝑏𝑖.

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