www.mathcentre.ac.uk/.../func-ext-2.mp4

Pythagoras theorem is one of the
most fundamental theorems in

mathematics. It says that if you
have a right angle triangle like

this. Then the square of the
length of the hypotenuse is

equal to the sum of the squares
of the lengths of these other

two sides. And that's something
that has important in

trigonometry as well. For
example, if I have an angle here

theater, and if I choose the
unit so that the length of the

hypotenuse is one.

Then This distance is just the
cosine of the angle Theta Cause

Theta. And this distance is the
sign of the angle theater.

And so Pythagoras Theorem says
that the square of cosine, Cos

squared theater, plus the square
of the sign.

Sine squared theater

Equals 1.

Now this has applications in
the real world. If we want to

measure positions and lengths
using systems of coordinates.

So let's suppose that I have
a pencil.

And I imagine that my pencil is

positioned. Just here.

So it's exactly on the
hypotenuse of a triangle.

And it supposed that
I've got some

coordinate axes. The
usual X axis there and

AY axis here.

So now with my pencil back here

this point. Is the

point? XY.

And so. Pythagoras theorem tells
me that X squared plus Y

squared must equal 1.

But of course, in the real world
we don't have a given set of

coordinate axes. I might choose
one set. You might choose a

different set. Let's suppose
that you chose a set UV with the

same origin, but pointing in

different directions. So if I
draw a new set of axes.

I'll have my U access
going this way.

And my

viaccess
Going this

way.

And then with respect to these

new axes. The pencil tip
will be at the point.

UV.

So if I now drop perpendiculars.

Then this distances you

along here. This

distance. His V
and this new angle.

Is 5.

But the length of the pencil
still the same, it's still

one that hasn't changed.

And so we must have
caused square 5 plus sign

squared Phi equals 1.

Be cause you squared plus B
squared equals 1.

So called squared Theta plus
sign squared theater equals 1

Costco at 5 plus sign. Squared 5
equals 1. It doesn't matter what

these angles, Theta or Phi are.
This equation must always hold,

so it's not just an equation,
it's actually what we call an

identity. Something that's
always true for any angle

theater or any angle fire.

Well, that's two dimensions, but
of course we live in a 3

dimensional world. So how would
pythagoras theorem work there?

Well our pencil.

Would now be sticking
up out of the paper.

And we'd have to have not
just an X coordinate and AY

coordinate, but also as Ed
coordinate going vertically

upwards.

And then it would no longer be
true that the X squared plus Y

squared value will be the square
of the length of the pencil.

It would be a bit shorter
because it would be the distance

to where the shadow of the
pencil falls on the paper.

Instead, we have to add the
square of the vertical distance

as well to get us to the tip of

the pencil. So in three
dimensions we would have not X

squared plus Y squared equals
the length of the pencil squared

plucked, but instead X squared
plus Y squared.

Plus, Z squared equals the
length of the pencil squared.

And if the pencil happened to be
in three dimensions, but lying

flat in the plane.

That would mean that zed
was zero, and so in our

formula, if said is zero,
we just get the X squared

plus Y squared back again.

Well, that's three dimensions,
and what mathematicians like to

do, of course, is to generalize.
We've had two dimensions, 3

dimensions. What about four

dimensions? Well, that might
seem a bit physically unreal,

but in fact in the middle of the
19th century physicists did

start to use 4 dimensions to
describe what was going on in

the world. They have XY and Z as
the three dimensions of space.

They also have an extra fourth

dimension. T for time and in
fact this was very important in

Einstein's theory of Relativity,
which he published in 19105.

So how does that work?

Well, if we look
at what a four

dimensional length might be.

Well, it's Square would be X
squared plus Y squared plus set

squared. And then you might
think it would be plus T

squared, but actually there's a

little twist. Because we don't
have plus T squared.

In fact we have minus T squared
because time is physically

different from the spatial

coordinates. And really, in
there there's a hidden factor

of C squared. Really, it's X
squared plus Y squared +6

squared minus C squared T
squared. To get the

dimensions right and see is
the speed of light.

But I suppose that we've chosen
our units so that the speed of

light is actually equal to 1,
and so X squared plus Y squared

plus Z squared minus T squared
is the formula for the square of

the four dimensional length.

And now, as in the three
dimensional case, we could ask

what happens if we. Just look at
length line one particular

plane. Well, let's take the XY
plane then. That would mean that

zed and T were both 0.

And so we would get for
our formula X squared plus

Y squared.

But what if we look instead not
at the XY plane, but at the XT

plane. And then.

Y and said will be 0 and
we would have.

X squared not plus T squared.

But minus T squared.

And now that gives us a problem.

Give us a problem if we try
to express the angle between

our 4 dimensional length and
a fixed direction.

Because if we say that the angle

was theater. Then we would
be looking at Cos squared

Theta minus.

Sine squared Theta.

And the problem is that this is
not a constant for different

values of Theta. If there were a
plus there, it would be constant

with a minus. It's not a
constant, so that doesn't work.

We can't use trigonometric
functions to talk about these

angles. Instead, we have to use
new kinds of functions called

hyperbolic functions. And these
are similar to the trigonometric

functions, calls them sign, but
we write them with the letter H

on the end. So this one.

Is caution and this one is a bit
harder to pronounce, so I'll

pronounce it as shine.

And for the angle I'll put you
rather than theater, because

this isn't a real angle. It's a
hyperbolic angle, and it works

in a different sort of way.

And now.

We can use these to obtain a
formula like this one that

didn't work for the
trigonometric functions, but one

which does work with the

hyperbolic functions. The reason

it works? Is to do with where
these functions come from. The

trick functions come from.

A circle. Where is the

hyperbolic functions? Come

from. A hyperbola.

And if you know about the
equations of these curves, the

equation of a circle is X
squared plus Y squared equals

the radius squared.

But for I hyperbola like this.

The equation is similar, but
without a plus sign. Instead it

has a minus sign.

Uncaught a minus sign
is just what we want.

And we find that rather than
having this formula, which

doesn't work for trig functions,
the corresponding one for

hyperbolic functions, cosh
squared U minus.

Shine squared you.

Does work and that this is

actually equal? To one, no
matter what the hyperbolic angle

you might be.

So the way that scientists think
about the four dimensions of

space time is very similar to
the way in which they think

about 3 dimensions of ordinary

space. But there is this little
twist that in the formula for

the square of the length,
there's a minus sign in front of

tea, and that means that when
they look at rotations involving

the time coordinate, they're not
ordinary rotations using

trigonometric functions, they
are hyperbolic rotations using

these new hyperbolic functions.