To begin this video, I'd like to share a quotation I heard recently, from Marcus Du Sautoy, on
his TV Show, The Code. In it, he says that: "Calculating with imaginary numbers is the
mathematical equivalent of believing in fairies"
Personally I think that's a lovely metaphor. But in all honesty, it makes out imaginary
numbers to be something completely removed from reality. For me, the more I learn about
imaginary numbers, the more I realise that actually, they're not very imaginary at all,
and so this video is my attempt to explain the basic concepts behind imaginary numbers
in an easy way, and make them no less mysterious than, say, negative numbers. I'm not going
to prove anything here, just demonstrate how to learn about them in a hopefully un-confusing
way.
In this video, I'll be introducing the concept that rotating can be considered as a multiplication
by a number. This might well seem a bit weird, but if you accept it, it will soon hopefully
make a lot of sense.
To start with, let's imagine you're walking forwards with a speed of 1 unit.
Now let's say you want to turn around 180 degrees, and keep your same speed. What number
would you need to multiply your velocity by? Well I'd hope this answer is fairly obviously
-1. But now we need to distinguish between speed and velocity: speed is how fast you
are going, but velocity also indicates your direction - so whilst your speed is still
1, your velocity is now -1.
If you want to turn back around, you would again multiply by -1 to turn through 180 degrees.
Overall, we have multiplied by -1 twice, and done two 180 degree turns, so in total we
have multiplied by -1 squared, which equals 1, which naturally takes us back to the start.
If you were to multiply your velocity by -½ then you would not only turn around, but your
speed would also halve. The size (or in maths lingo, the magnitude) of the number you multiply
by will affect your speed - if the magnitude is bigger than 1, you will go faster, if it's
less than one you will go slower, and of course if it's 1, you will keep the same speed.
Okay, so that's all pretty boring and obvious so far: multiplying our velocity by any positive
or negative number will still give us a velocity going forwards or backwards, along our initial
1 dimensional line. Now how about we introduce a 2 dimensional world: what if we wanted to
turn 90 degrees?
Well let us imagine that there is a number that we can multiply by to rotate 90 degrees
- what do we know about it? Well we know that if we rotate 90 degrees twice we will have
a turn of 180 degrees - in other words, if we multiply by our number twice it is equivalent
to multiplying by -1. Mathematicians call this number "i" (or occasionally "j"). It
has the property that i² = -1. Now you may be thinking: in a 2D world, I can rotate 90
degrees in two different directions! So which is i? Well by convention, i is a turn of 90
degrees anti-clockwise. So what is a turn of 90 degrees clockwise called? Well, if you
think about it, a turn of 90 degrees clockwise is equivalent to a turn of 90 degrees anticlockwise
followed by a turn of 180 degrees. Or multiplying by i and then -1. So a rotation of 90 degrees
clockwise is just -i. Now some more maths lingo: A number which lies on the normal number
line (ie a negative, zero or positive number) is known as a real number. A number which
is just some multiple of i is an imaginary number. And a number which is the sum of both
a real number and an imaginary number is known as a complex number. For example, 2 - ½ i
would be a complex number. It has a real part of 2, and an imaginary part of -½.
So now comes an interesting question: what number do we multiply by to turn through 45
degrees? Well just like before, two turns of 45 degrees equal a turn of 90 degrees.
So the number we need to multiply by is actually the square root of i. The natural thought
now might well be: "Oh no! We need to introduce yet more numbers!" well actually, it's a bit
easier than that - the square root of i is actually a complex number, in other words
it's a number which is the sum of a real part, and an imaginary part.
So let's look at the diagram we have so far. We can draw this diagram, called an Argand
diagram, with 1, i, -1 and -i at 0, 90, 180 and 270 degrees anticlockwise from the start
position, just like the rotations they represent. But one of the amazing things about this is
that we can actually now find numbers representing different rotations by using this diagram.
The first thing to remember is that if we want to keep the same speed, the number we
multiply by must have a magnitude of 1. So we can draw a circle with radius 1, which
passes through all the numbers with a size, or magnitude, of 1.
So if we want to find a number that represents a rotation of 45 degrees, it will lie on the
circle, and be at 45 degrees from the start position. I'll mark it on the diagram with
a cross. Now let's zoom in a bit. We need to find the real and imaginary parts of our
number- in other words, we need to find out how far our number is horizontally from zero
(which I have marked in blue and called a), and how far it is vertically from zero (which
I have marked in red and called b). These measurements will give us the real part and
the imaginary part of our number respectively. Notice that the triangle here is right-angled,
so we can apply Pythagoras to it, giving us the equation a² + b² = 1, because the radius
has length 1. In this case, we have a 45 degree angle, which means that our triangle is also
isosceles, and so a = b. This gives us the equation that 2a² = 1, so a² = ½ and a is √½.
b is of course also √½. And so, the number that we need to multiply by to rotate 45 degrees is √½ + √½ i.
Now we can quickly verify that this number is actually the root of i, by multiplying
itself, using the standard rules for expanding out brackets. So doing this, this gives us
four separate terms from the expansion. Now of course, each of these terms contains √½
times √½, which of course equals ½. Also, we can notice that i² is of course -1, so
the ½ at the start cancels with the -½ at the end, leaving two terms of ½ i. These
add together to of course give us i.
This is a demonstration that using the diagram in this case gives us the right answer. In
actual fact the diagram always works, and this is one of the amazing properties of complex
numbers. I'll just point out here that you can find numbers representing other rotations
just as easily by using a bit of simple trigonometry!
So anyway, I hope you've enjoyed this quick and slightly quirky introduction to complex
numbers - and I hope that this video has shown you that actually imaginary and complex numbers do
make sense in the physical world - or at least that don't make much less sense than negative
numbers do!
I hope you've enjoyed this video, and please recommend it to anyone you think might be
interested. Also, please leave me any feedback - positive or negative - below. Thanks for
listening.