WEBVTT

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Pythagoras theorem is one of the
most fundamental theorems in

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

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

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

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

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

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

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hypotenuse is one.

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

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

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

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squared theater, plus the square
of the sign.

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Sine squared theater

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Equals 1.

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

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measure positions and lengths
using systems of coordinates.

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So let's suppose that I have
a pencil.

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And I imagine that my pencil is

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positioned. Just here.

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

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And it supposed that
I've got some

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coordinate axes. The
usual X axis there and

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AY axis here.

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So now with my pencil back here

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this point. Is the

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point? XY.

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

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squared must equal 1.

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

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

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

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same origin, but pointing in

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

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I'll have my U access
going this way.

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And my

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viaccess
Going this

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way.

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And then with respect to these

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

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UV.

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So if I now drop perpendiculars.

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Then this distances you

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along here. This

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distance. His V
and this new angle.

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Is 5.

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

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one that hasn't changed.

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And so we must have
caused square 5 plus sign

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squared Phi equals 1.

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Be cause you squared plus B
squared equals 1.

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So called squared Theta plus
sign squared theater equals 1

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

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

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

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identity. Something that's
always true for any angle

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theater or any angle fire.

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

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dimensional world. So how would
pythagoras theorem work there?

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Well our pencil.

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Would now be sticking
up out of the paper.

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

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coordinate, but also as Ed
coordinate going vertically

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upwards.

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

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

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

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

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

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as well to get us to the tip of

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

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squared plus Y squared equals
the length of the pencil squared

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plucked, but instead X squared
plus Y squared.

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

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

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flat in the plane.

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

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

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plus Y squared back again.

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

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

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dimensions. What about four

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

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

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

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

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They also have an extra fourth

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

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

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So how does that work?

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Well, if we look
at what a four

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dimensional length might be.

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

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

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squared, but actually there's a

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

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In fact we have minus T squared
because time is physically

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different from the spatial

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

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

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squared minus C squared T
squared. To get the

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dimensions right and see is
the speed of light.

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

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

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

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the four dimensional length.

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

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

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

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zed and T were both 0.

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And so we would get for
our formula X squared plus

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Y squared.

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

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plane. And then.

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Y and said will be 0 and
we would have.

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X squared not plus T squared.

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But minus T squared.

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And now that gives us a problem.

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

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our 4 dimensional length and
a fixed direction.

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Because if we say that the angle

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was theater. Then we would
be looking at Cos squared

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Theta minus.

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Sine squared Theta.

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

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

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

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We can't use trigonometric
functions to talk about these

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

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hyperbolic functions. And these
are similar to the trigonometric

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

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on the end. So this one.

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

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pronounce it as shine.

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

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

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in a different sort of way.

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And now.

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

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didn't work for the
trigonometric functions, but one

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which does work with the

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hyperbolic functions. The reason

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

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trick functions come from.

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A circle. Where is the

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hyperbolic functions? Come

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from. A hyperbola.

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

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equation of a circle is X
squared plus Y squared equals

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the radius squared.

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But for I hyperbola like this.

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

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has a minus sign.

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Uncaught a minus sign
is just what we want.

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And we find that rather than
having this formula, which

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

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hyperbolic functions, cosh
squared U minus.

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Shine squared you.

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Does work and that this is

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

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you might be.

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So the way that scientists think
about the four dimensions of

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

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about 3 dimensions of ordinary

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

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

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

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the time coordinate, they're not
ordinary rotations using

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trigonometric functions, they
are hyperbolic rotations using

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these new hyperbolic functions.