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.
