Hi, I'm Toby, and welcome back to the joy of mathematics. It's been a minute since I
filmed one of these, but I'm so glad that you're joining me again today for a relaxing journey
through mathematics. I'm going to run some things across the bottom of the screen that we will come
to understand in this video. But mostly, we will be learning how to escape from reality.
Some people do this through drawing, painting, reading or playing video games,
we are going to get a private tour of an imaginary world with the help of my imaginary friends.
Before I introduce you, let me tell you a story. When I was six years old,
I remember a kid in my class at school, saying that there were secret numbers that only he knew
about. He said that the teacher had told him about them after class, that they were less than zero,
and called negative numbers. I was doubtful. I hadn't ever heard of such things.
I thought that if it was true, then the teacher surely would have told all of us.
Maybe I felt a little hurt that she had picked a favourite student,
or that I was missing out on something. Perhaps that's why I still remember all these years later.
Of course, I eventually found out that negative numbers were real. But I had another similar
experience several years later, in high school math class, we had a relief teacher one day,
I think it was the careers counsellor. He spent the start of the hour following our
teachers lesson plan. But towards the end veered far enough off course, that he said, Have you
guys heard of imaginary numbers? We hadn't. With animated strokes, he drew a large axes,
and labelled one real and the other imaginary. After that class, I felt pretty Mind blown.
Maybe I was yet to really understand them. But that was the first I heard
of these mysterious new numbers. I had again experienced my view of numbers being expanded.
A long time later, I would realise that negative numbers and imaginary numbers
are in fact, similar in the way that they extend mathematics. They are both somewhat intangible
ideas that prove to be useful. Where the negatives come from asking what one minus two might equal
the imaginary unit i comes from asking what the square root of minus one might be if it existed.
Now it's in the best interests of you and I to acknowledge that we live in the real world. And
our actions have real consequences. So we should try to be nice to each other. I will introduce
you to my imaginary friends now. And whilst this isn't a popularity contest, we will want to keep
track of how many imaginary friends we have. At the moment we haven't met any. so we can write i
to the zero. And it's one of those quirky little mathematical facts that anything to
the power of zero is equal to one. This solution is our passport. And since one is a real number,
we're not going anywhere. And we have to stay out here in the real world. But meet my first
imaginary friends, Ignatio, with him, we have i to the one which is equal to i.
So without first friend, we can now travel from the real world to the imaginary world.
It's wonderful in here, but now I'm trapped. How can I go back to reality?
My second imaginary friend, Iguana can help. With two imaginary friends
we have negative one. I'm back in the real world. But this is the negative real world
and it's so sad out here. I could get help from a real friend who is negative such as minus one.
But there's not really anyone out here. I could instead get help from two more imaginary friends.
With three imaginary friends in total, we are taken back to the imaginary world.
But we end up being underground. And a fourth imaginary friend in total,
will bring us back out to the positive real world. Right where we started.
This is all you need to know, to travel between our world and the imaginary world,
we can show our story on a map, where we're jumping between the points one, i, minus one
and minus i. And that is because multiplying by i is a rotation of 90 degrees. If the friends
we met had been a gang of negative numbers, then we would have toggled between one and minus one.
This flipping shows that multiplying by minus one is a rotation of 180 degrees.
An imaginary friend takes us to a pit stop in between to have a timeout from reality. Because
multiplying by i is a rotation of 90 degrees, then if we do it twice, we get to negative one.
And if we do it four times, we get back where we started, we could of course go the other way
as well, we would see a clockwise rotation of 90 degrees if we multiplied by minus i.
This ability to multiply and move around in a circle is what makes imaginary numbers so good at
modelling things that are circular in nature, or that rotate. And despite the name, they are not
imaginary at all. Some people just have trouble seeing them. And whether you want to believe
in imaginary numbers or not. They are right there. A second dimension to every number.
It says mind boggling as decimals or a rationals would have been
to early mathematicians. Some never got to know there were numbers in between one and two.
Larger friends such as two i, three i, and four i, allow us to go anywhere on these axes.
But what about the rest of the plane? Well, yes, we can travel anywhere we like. It's just a little
bit more complicated. This is the complex plane, after all, in the rest of this region, we need
two parts, a real part and an imaginary part. To explain where we are. With one foot in each door
here at one plus i, we are halfway between worlds, a rotation of 45 degrees.
Let's see a use for multiplying complex numbers.
Say you go for a walk in the forest, but you end up getting a little lost.
At first you walk along the path laid out in the direction of eight plus two i.
But you realise this isn't right, and you want to change your track by 45 degrees. How would
you work this out? Well, normally, you could do some trigonometry, some sine and cosine. Maybe you
need to carry a protractor with you to work out the angles. Or you can multiply eight plus two i
by one plus i, which we already know is a rotation of 45 degrees. Eight plus two i times one plus i
is equal to eight plus eight i plus two i plus two i squared. Since i squared is equal to minus one,
that term becomes equal to minus two, and we end up with six plus 10 i. So walking along the path
laid out in the direction of six plus 10 i, will be an offset of 45 degrees from the original path.
Introducing imaginary friends has given us an extra dimension in which to travel and a useful
tool for performing rotations. These friends might become a dime a dozen as you go on and maths,
but each one is still special. I'm proud of the friend who appears in the wave function of
the electromagnetic wave. The electric field and magnetic fields are perpendicular to each other,
and their magnitude and time variation are correlated. by assigning the magnetic wave
to the imaginary part and the electric wave to the real part, we can explain these two
waves in one equation and capture the way that the sum of the two individual waves rotates.
To meet some more imaginary friends to head over to Brilliant, who have kindly sponsored this
video. Brilliant is a place to learn in a fun and challenging way. And while there you can meet my
pals Julia and Mandelbrot in a course on complex numbers, they will guide you through the rest of
the topic with visits to complex multiplication, polar coordinates and the most beautiful Euler's
formula. It's great that you're on YouTube now sparking your imagination by watching math videos.
But to solidify your knowledge as real, the best thing you can do is to try problems for yourself.
Brilliant's website and app are built off the principle that you learn best while doing. They
have something for everyone. So if you'd like to join me and a community of 8 million learners
and educators today, click the link in the description below or visit brilliant.org/tibees
Safe travels to your imaginary world. I hope you have an absolutely mathematical day.
Thank you to my Patreon supporters, and a special mention to today's Patreon cat of the day Mina.