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- [Voiceover] So we've
already been able to explore

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a lot with our little thought experiment

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where I am floating in space.

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I'm at the center of
my frame of reference,

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and right at time equals zero

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in my frame of reference,

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a friend comes in a
spaceship passing me by

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with a velocity v,

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I'll say the magnitude is v,

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and it's going in the
positive x direction.

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We're just going to focus,

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and we have been focusing,

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just on the x dimension for simplicity.

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And we've thought about reconciling space

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and time in my frame of reference

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relative to her frame of reference.

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The conundrum that we've faced

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in previous videos is how do we reconcile

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that the speed of light is always

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going to be the same in
every frame of reference?

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And to reconcile them, we
had to essentially come

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up with the idea of spacetime.

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I should say it even faster, spacetime.

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Spacetime, let me write it out, spacetime.

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And the first time I
heard about spacetime,

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I assumed that people were just talking

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about space and time as independent things

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and just plotting your
point in space and time.

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But when people talk about spacetime,

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they're really talking about
this continuum of one thing,

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and we're just talking
about different directions

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in spacetime.

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They could have called
this something else.

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They could have called this spime,

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or tace, or stace,

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or a lot of different things.

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But this is spacetime,

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and it's this idea that
it's this continuum.

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And when we started to
make spacetime diagrams,

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we realized in order
for the speed of light

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to be absolute, that time and space

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weren't as independent of each other

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as we thought, and they
weren't as absolute

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as we thought.

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And we constructed these Minkowsky

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spacetime diagrams for each

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of our frames of reference.

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So my frame of reference,

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the spacetime diagram,

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is here in white,

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and for my friend's frame of reference,

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her spacetime diagram is here

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in this blue color.

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The angle formed between these axes,

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between the x and the x prime axis

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and the ct and the ct prime axis,

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this angle alpha here,

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this is going to be dependent on her

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relative velocity in
my frame of reference.

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So if her velocity is v,

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or the magnitude of her velocity is v

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in my frame of reference,

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this angle we've already seen

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is going to be the inverse tangent

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or the arctangent of the ratio

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between her relative velocity

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and the speed of light.

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So this is going to be
equal to the inverse tangent

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of v over c.

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So the faster she goes,

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these two things are going
to start squeezing together,

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and if somehow she were
to approach the speed

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of light, they would both
approach a 45 degree angle

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and actually start to coincide

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if they were actually able to approach

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the speed of light.

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And that's all interesting already,

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this idea that space and time are not

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as independent, that it's all a continuum

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called spacetime, but some
of you have probably said,

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well, I want to deal with some more

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tangible numbers here.

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For example, if this event right over here

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in spacetime, we can think about it

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from my frame of reference,

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and we can think about it

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from her frame of reference.

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In my frame of reference,

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I would view the coordinates here.

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This coordinate would be x,

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and this coordinate right over here

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would be ct.

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We had a whole video on why we think

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of time in terms of ct.

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The units here literally would be meters.

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We could think of it as
light-meters if we like.

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So that would be the coordinates

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in my frame of reference.

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Well, what would be the coordinates

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in her frame of reference?

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Well, we've already
thought about how to read

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these Minkowski spacetime diagrams.

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To find her x prime coordinate,

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we would just go parallel
to the ct prime axis.

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So that would be the x prime coordinate

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in her frame of reference.

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And to figure out the ct prime coordinate,

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we would just go parallel
to the x prime axis.

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So this would be the ct prime coordinate.

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Now how do you actually go in between,

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transform, from x to x prime

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and from ct to ct prime?

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And to do that, we're going to introduce

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in this video the Lorentz transformations.

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And what they do is they allow us to do

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exactly what we just needed to do.

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They'll allow us to go from x, ct

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to x prime

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and ct prime.

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And to help us think about it,

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I'm going to introduce some variables,

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and hopefully it will show the symmetry

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of the Lorentz transformations.

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You might see them written in other ways

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in other sources, and we'll reconcile all

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of those in the future.

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But the Lorentz transformations,

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we'll start with what we
call the Lorentz factor

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because this shows up a lot

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in the transformation.

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So I'll just define this ahead of time.

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So the Lorentz factor,
denoted by the Greek letter

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gamma, lowercase gamma,

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it is equal to one over the square root

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of one minus v squared over c squared.

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Now sometimes you might
even see it written like,

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well, I'll write it another way.

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Sometimes you might see
it written as gamma,

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let me do it in that same color,

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same reddish color,

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gamma is equal to one over the square root

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of one minus beta squared.

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You might say, well, what is beta?

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Well, beta is another
variable that shows up

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a lot when we're thinking
about special relativity.

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And beta is just the ratio

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between the relative velocity,

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her relative velocity in
my frame of reference,

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and the speed of light.

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It shows up a lot,

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even this angle alpha here,

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we could have said this
is the inverse tangent

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of beta.

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This becomes a little bit
simpler when you write

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the Lorentz factor.

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And when we look at the
actual transformation

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between the coordinates,
we'll see that beta

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becomes useful again, at least the way

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I like to write it.

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So if we want to think about what x prime

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is going to be, so we can write

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x prime is going to be equal

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to the Lorentz factor,

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let me do it in that red color,

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is going to be equal to the Lorentz factor

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times x

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minus, and now we're going to say

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beta times ct,

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and ct prime

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is going to be equal to
the Lorentz factor gamma,

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let me do that same color again,

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switching colors is sometimes difficult,

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gamma times, and we'll see it's just

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the other way around.

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It is going to be c,

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let me do it in that green color,

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ct

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minus, you might even guess what I'm

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about to write based on the symmetry

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that we see here,

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ct minus beta times x.

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And I really want you to appreciate this

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because it really shows
that space and time

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are really just different directions

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in the spacetime and
there's a nice symmetry

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to them right over here.

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Notice we have an x and we have an x.

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We have a ct and we have a ct.

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So when we're thinking about x prime,

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it's x minus beta times ct,

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and we're scaling it
by the Lorentz factor,

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and then when we're thinking about time,

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well we do it the other way around.

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We're still scaled by the Lorentz factor,

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but now it is ct minus beta times x.

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Now this might all seem like Greek to you,

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and we actually are using Greek letters,

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but in the next video,

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I'll actually use some
sample numbers here,

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and you'll see that evaluating these

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is just a little bit of
straightforward algebra.