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.