Simplifying quotient of powers (rational exponents)

WEBVTT
Kind: captions
Language: en

00:00:00.411 --> 00:00:02.141
- [Voiceover] So we have an
interesting equation here.

00:00:02.141 --> 00:00:03.940
Let's see if we can solve for k,

00:00:03.940 --> 00:00:07.819
and we're going to assume
that m is greater than zero.

00:00:07.819 --> 00:00:09.907
Like always, pause the video.

00:00:09.907 --> 00:00:13.903
Try it out on your own, and
then I will do it with you.

00:00:13.903 --> 00:00:16.559
All right, let's work
on this a little bit.

00:00:16.559 --> 00:00:18.486
So you can imagine that the key to this

00:00:18.486 --> 00:00:20.785
is to simplify it using our knowledge

00:00:20.785 --> 00:00:21.922
of exponent properties,

00:00:21.922 --> 00:00:23.884
and there's a couple of
ways to think about it.

00:00:23.884 --> 00:00:26.067
First, we can look at this
rational expression here,

00:00:26.067 --> 00:00:28.076
m to the 7/9 power divided

00:00:28.076 --> 00:00:30.033
by m to the 1/3 power.

00:00:30.033 --> 00:00:31.937
And the key realization here is that

00:00:31.937 --> 00:00:36.764
if I have x to the a

00:00:36.764 --> 00:00:40.247
over x to the b, that
this is going to be equal

00:00:40.247 --> 00:00:43.892
to x to the a minus b power.

00:00:43.892 --> 00:00:46.249
And actually comes
straight out of the notion

00:00:46.249 --> 00:00:49.291
that x to the a over x to the b,

00:00:49.291 --> 00:00:51.833
x to the a over x to the b,

00:00:51.833 --> 00:00:54.132
is the same thing as x to the a

00:00:54.132 --> 00:00:57.545
times one over x to the b,

00:00:57.545 --> 00:01:02.072
which is the same thing
as x to the a times...

00:01:02.072 --> 00:01:04.092
One over x to the b, that's the same thing

00:01:04.092 --> 00:01:06.449
as x to the negative b,

00:01:06.449 --> 00:01:08.109
which is going to be the same thing as...

00:01:08.109 --> 00:01:10.071
If I have a base to one exponent times

00:01:10.071 --> 00:01:11.954
the same base to another exponent,

00:01:11.954 --> 00:01:13.565
that's the same thing as that base

00:01:13.565 --> 00:01:16.804
to the sum of the
exponents, a plus negative b

00:01:16.804 --> 00:01:19.625
which is just gonna be a minus b.

00:01:19.625 --> 00:01:21.169
So, we got to the same place.

00:01:21.169 --> 00:01:24.605
So, we can re-write this as...

00:01:24.605 --> 00:01:28.285
So, we can re-write this
part as being equal to

00:01:28.285 --> 00:01:31.513
m to the 7/9 power

00:01:31.513 --> 00:01:35.416
minus 1/3 power is equal to,

00:01:35.936 --> 00:01:40.893
is equal to m to the k over nine.

00:01:40.893 --> 00:01:42.611
And I think you see where this is going.

00:01:42.611 --> 00:01:45.180
What is 7/9 minus 1/3?

00:01:45.180 --> 00:01:46.771
Well, 1/3 is the same thing,

00:01:46.771 --> 00:01:48.117
if we want to have a common denominator,

00:01:48.117 --> 00:01:50.998
1/3 is the same thing as 3/9.

00:01:50.998 --> 00:01:53.749
So, I can re-write this as 3/9.

00:01:53.749 --> 00:01:57.829
So 7/9 minus 3/9 is going to be 4/9.

00:01:57.834 --> 00:02:01.541
So, this is the same thing as m to the...

00:02:02.211 --> 00:02:05.242
M to the 4/9 power

00:02:05.682 --> 00:02:08.690
is going to be equal to

00:02:08.690 --> 00:02:11.900
m to the k-ninths power.

00:02:11.900 --> 00:02:15.248
So, 4/9 must be the
same thing as k-ninths.

00:02:15.248 --> 00:02:19.531
So, we can say 4/9 is equal to k-ninths.

00:02:19.531 --> 00:02:22.975
Four over nine is equal to k over nine,

00:02:22.975 --> 00:02:25.037
which tells us that k

00:02:25.037 --> 00:02:28.047
must be equal to four,

00:02:28.047 --> 00:02:30.015
and we're all done.


Evaluating quotient of fractional exponents

WEBVTT
Kind: captions
Language: en

00:00:00.331 --> 00:00:01.352
- [Voiceover] Let's see
if we can figure out

00:00:01.352 --> 00:00:06.352
what 256 to the four-sevenths power,

00:00:06.402 --> 00:00:11.229
divided by two to the
four-sevenths power is,

00:00:11.229 --> 00:00:13.398
and like always, pause the
video and see if you can

00:00:13.398 --> 00:00:14.809
figure this out.

00:00:15.290 --> 00:00:16.637
All right, let's work
through this together,

00:00:16.637 --> 00:00:18.854
and at first you might
find this kind of daunting.

00:00:18.854 --> 00:00:20.352
Especially when you see
something like two to the

00:00:20.352 --> 00:00:22.616
four-sevenths power or is that even,

00:00:22.616 --> 00:00:24.496
that's not going to be a
whole number, how do I,

00:00:24.496 --> 00:00:26.176
how do I do this, especially
without a calculator.

00:00:26.176 --> 00:00:28.513
And I should've said, do
this without a calculator.

00:00:28.513 --> 00:00:31.327
But then the key is to
see that we can use our

00:00:31.327 --> 00:00:33.602
exponent properties to
simplify this a little bit

00:00:33.602 --> 00:00:35.683
so that we can do this on paper.

00:00:35.683 --> 00:00:37.913
And the main property
that may jump out at you

00:00:37.913 --> 00:00:40.944
is if I have something, if I have,

00:00:40.944 --> 00:00:45.944
if I have x to the a power, over y

00:00:46.053 --> 00:00:50.678
to the a power, this is
the same thing as x over y,

00:00:51.248 --> 00:00:53.160
to the a power.

00:00:53.160 --> 00:00:54.634
And in our situation right over here,

00:00:54.634 --> 00:00:57.954
256 would be x, two would be y,

00:00:57.954 --> 00:01:01.158
and then a is four-sevenths,
so we can rewrite this,

00:01:01.158 --> 00:01:02.853
this is going to be equal to

00:01:03.383 --> 00:01:08.342
this is equal to 256, over two,

00:01:08.762 --> 00:01:12.869
to the four-sevenths
power, and so this is nice.

00:01:12.869 --> 00:01:14.471
We're already able to simplify this,

00:01:14.471 --> 00:01:19.137
because we know 256
divided by two, is 128.

00:01:19.137 --> 00:01:24.137
So this is 128 to the four-sevenths power.

00:01:25.326 --> 00:01:28.197
Now this might also seem
a little bit difficult,

00:01:28.197 --> 00:01:31.587
how do I raise 128 to a fractional power?

00:01:31.587 --> 00:01:32.702
But we just have to remind ourselves,

00:01:32.702 --> 00:01:36.463
this is the same thing, this
is the same thing as 128

00:01:36.463 --> 00:01:38.878
to the one-seventh power.

00:01:38.878 --> 00:01:41.420
Then raised to the fourth power.

00:01:41.420 --> 00:01:43.243
We could also view it the
other way around, we could say

00:01:43.243 --> 00:01:47.143
that this is also 128 to the fourth,

00:01:47.503 --> 00:01:50.614
to the fourth power, and
then raise that to the

00:01:50.614 --> 00:01:53.794
one-seventh, but
multiplying 128 four times,

00:01:53.794 --> 00:01:55.872
that's going to be very
computationally intensive,

00:01:55.872 --> 00:01:57.614
and then you have to find
the seventh root of that.

00:01:57.614 --> 00:01:59.251
That seems pretty difficult,
so we don't want to

00:01:59.251 --> 00:02:02.397
go in that way, but if we can
get the smaller number first,

00:02:02.397 --> 00:02:05.166
what is 128 to the one-seventh power?

00:02:05.166 --> 00:02:08.126
Then that might be easier to
raise to the fourth power.

00:02:08.126 --> 00:02:11.309
Now when you look at this,
and knowing that probably,

00:02:11.759 --> 00:02:13.605
the question writer in this
case, I'm the person who

00:02:13.605 --> 00:02:15.521
presented it with you is,
telling you that you're not

00:02:15.521 --> 00:02:18.283
going to use a calculator is,
it's a pretty good clue that,

00:02:18.283 --> 00:02:20.197
all right, this is probably going to be a,

00:02:20.197 --> 00:02:21.305
this is probably going to
be something that I can

00:02:21.305 --> 00:02:22.547
figure out on my own,

00:02:22.547 --> 00:02:26.680
and you might recognize
128 as a power of two,

00:02:26.680 --> 00:02:30.441
and maybe two to the seventh
is 128, and we can verify that.

00:02:30.441 --> 00:02:34.319
So let's see, two to the first is two.

00:02:34.319 --> 00:02:38.382
Four, eight, 16,

00:02:38.382 --> 00:02:42.594
32, 64, 128.

00:02:42.594 --> 00:02:44.963
Two times two is four, times two is eight,

00:02:44.963 --> 00:02:48.251
times two is 16, times
two is 32, times 2 is 64,

00:02:48.251 --> 00:02:49.853
times two is 128.

00:02:49.853 --> 00:02:52.732
So, two to the seventh power

00:02:52.732 --> 00:02:56.517
is equal to 128, or another way of saying

00:02:56.517 --> 00:02:59.373
this exact same thing is that 128,

00:02:59.373 --> 00:03:02.112
128 is equal to

00:03:02.112 --> 00:03:04.272
or 128 to the one-seventh power,

00:03:05.052 --> 00:03:07.859
is equal to two.

00:03:07.859 --> 00:03:10.030
Or you could even say
that the seventh root,

00:03:10.630 --> 00:03:14.314
the seventh root of 128, is equal,

00:03:14.314 --> 00:03:17.077
is equal to two.

00:03:17.077 --> 00:03:19.348
So, we can simplify this.

00:03:19.348 --> 00:03:21.334
This is two, so our whole
expression is now just

00:03:21.334 --> 00:03:23.110
two to the fourth power.

00:03:23.110 --> 00:03:25.008
Well, that's just two times two,

00:03:25.008 --> 00:03:26.430
times two, times two.

00:03:26.430 --> 00:03:30.250
So, that's two to the fourth power.

00:03:31.180 --> 00:03:33.036
Two to the fourth power, which is just

00:03:33.036 --> 00:03:35.263
going to be equal to 16.

00:03:35.263 --> 00:03:39.303
That's two, times two, times two, times

00:03:39.303 --> 00:03:40.511
two, right over there.

00:03:40.511 --> 00:03:41.472
And so we're done!

00:03:41.472 --> 00:03:43.858
This crazy,
complicated-looking expression,

00:03:43.858 --> 00:03:45.042
it is simplified

00:03:45.042 --> 00:03:47.370
to 16.


Rewriting square root of fraction

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:02.485
- [Voiceover] So we have
here the square root,

00:00:02.485 --> 00:00:04.917
the principal root, of one two-hundredth.

00:00:04.917 --> 00:00:06.984
And what I want to do is simplify this.

00:00:06.984 --> 00:00:09.485
When I say "simplify it" I really mean,

00:00:09.485 --> 00:00:10.986
if there's any perfect squares here that

00:00:10.986 --> 00:00:15.986
I can factor out to take it
out from under the radical.

00:00:16.018 --> 00:00:17.318
And so I encourage you to pause the video

00:00:17.318 --> 00:00:19.884
and see if you can do that.

00:00:19.884 --> 00:00:21.351
Alright so there's a couple of ways that

00:00:21.351 --> 00:00:22.351
you could approach this.

00:00:22.351 --> 00:00:24.924
One way is to say, Well this is going to

00:00:24.924 --> 00:00:26.551
be the same thing as the square root of

00:00:26.551 --> 00:00:30.284
one over the square root of 200.

00:00:30.284 --> 00:00:32.517
The square root of one is just one

00:00:32.517 --> 00:00:36.151
over the square root of 200.

00:00:36.151 --> 00:00:37.451
And there's a couple of ways to try to

00:00:37.451 --> 00:00:39.851
simplify the square root of 200.

00:00:39.851 --> 00:00:42.685
I'll do it a couple of ways here.

00:00:42.685 --> 00:00:44.617
Square root of 200.

00:00:44.617 --> 00:00:46.351
You could realize that, OK, look

00:00:46.351 --> 00:00:48.484
100 is a perfect square.

00:00:48.484 --> 00:00:50.051
And it goes into 200.

00:00:50.051 --> 00:00:53.953
So this is the same
thing as two times 100.

00:00:53.953 --> 00:00:56.851
And so the square root of
200 is the square root of

00:00:56.851 --> 00:01:01.717
two times 100, which is the same thing as

00:01:01.717 --> 00:01:06.384
the square root of two times
the square root of 100.

00:01:06.384 --> 00:01:09.451
And we know that the
square root of 100 is 10.

00:01:09.451 --> 00:01:11.352
So it's the square root of two times 10

00:01:11.352 --> 00:01:14.585
or we could write this as
10 square roots of two.

00:01:14.585 --> 00:01:15.984
That's one way to approach it.

00:01:15.984 --> 00:01:18.018
But if it didn't jump out
at you immediately that

00:01:18.018 --> 00:01:21.784
you have this large perfect
square that is a factor of 200,

00:01:21.784 --> 00:01:24.084
you could just start with small numbers.

00:01:24.084 --> 00:01:26.460
You could say, alright, let me do this

00:01:26.460 --> 00:01:28.550
alternate method in a different color.

00:01:28.550 --> 00:01:30.485
You could say, ah it's the same color that

00:01:30.485 --> 00:01:31.885
I've been doing before.

00:01:31.885 --> 00:01:36.518
(laughing) You could say that
the square root of 200, say

00:01:36.518 --> 00:01:38.284
Well it's divisible by two.

00:01:38.284 --> 00:01:40.685
So it's two times 100.

00:01:40.685 --> 00:01:43.285
And if 100 didn't jump out
at you as a perfect square,

00:01:43.285 --> 00:01:46.784
you could say, Well that's
just going to be two times 50.

00:01:46.784 --> 00:01:48.251
Well I can still divide two into that.

00:01:48.251 --> 00:01:50.951
That's two times 25.

00:01:50.951 --> 00:01:52.884
Let's see, and 25, if
that doesn't jump out at

00:01:52.884 --> 00:01:56.388
you as a perfect square,
you could say that

00:01:56.388 --> 00:01:58.584
that's not divisible by
two, not divisible by three,

00:01:58.584 --> 00:02:01.051
four, but it is divisible by five.

00:02:01.051 --> 00:02:04.351
That is five times five.

00:02:04.351 --> 00:02:06.851
And to identify the perfect
squares you would say,

00:02:06.851 --> 00:02:09.151
Alright, are there any factors where

00:02:09.151 --> 00:02:10.918
I have at least two of them?

00:02:10.918 --> 00:02:13.651
Well I have two times two here.

00:02:13.651 --> 00:02:18.651
And I also have five times five here.

00:02:18.818 --> 00:02:21.118
So I can rewrite the square root of 200 as

00:02:21.118 --> 00:02:23.918
being equal to the square root of

00:02:23.918 --> 00:02:27.917
two times two.

00:02:27.917 --> 00:02:31.684
Let me just write it all out.

00:02:31.684 --> 00:02:33.451
Actually I think I'm
going to run out of space.

00:02:33.451 --> 00:02:36.284
So the square root, give
myself more space under

00:02:36.284 --> 00:02:39.784
the radical, square root of

00:02:39.784 --> 00:02:44.018
two times two times five times

00:02:44.018 --> 00:02:49.018
five times two.

00:02:50.847 --> 00:02:51.984
And I wrote it in this order so

00:02:51.984 --> 00:02:54.151
you can see the perfect squares here.

00:02:54.151 --> 00:02:56.351
Well this is going to
be the same thing as the

00:02:56.351 --> 00:02:58.418
square root of two times two.

00:02:58.418 --> 00:03:01.518
This second method is a
little bit more monotonous,

00:03:01.518 --> 00:03:05.434
but hopefully you see that it works,

00:03:05.434 --> 00:03:06.917
(laughing) I guess is one
way to think about it.

00:03:06.917 --> 00:03:09.384
And they really, they boil
down to the same method.

00:03:09.384 --> 00:03:11.051
We're still going to
get to the same answer.

00:03:11.051 --> 00:03:15.285
So square root of two times
two times the square root

00:03:15.285 --> 00:03:19.417
times the square root of five times five,

00:03:22.687 --> 00:03:27.687
times the square root of two.

00:03:27.835 --> 00:03:30.385
Well the square root of two
times two is just going to be,

00:03:30.385 --> 00:03:31.584
this is just two.

00:03:31.584 --> 00:03:33.051
Square root of five times five,

00:03:33.051 --> 00:03:35.784
well that's just going to be five.

00:03:35.784 --> 00:03:38.417
So you have two times five
times the square root of two,

00:03:38.417 --> 00:03:43.417
which is 10 times the square root of two.

00:03:44.155 --> 00:03:46.017
So this right over here,
square root of 200,

00:03:46.017 --> 00:03:48.485
we can rewrite as 10 square roots of two.

00:03:48.485 --> 00:03:52.151
So this is going to be equal to one

00:03:52.151 --> 00:03:55.417
over 10 square roots of two.

00:03:55.417 --> 00:03:58.384
Now some people don't like having a

00:03:58.384 --> 00:04:00.384
radical in the denominator
and if you wanted to

00:04:00.384 --> 00:04:02.251
get rid of that, you could multiply both

00:04:02.251 --> 00:04:05.451
the numerator and the denominator
by the square root of two.

00:04:05.451 --> 00:04:07.217
'Cause notice we're
just multiplying by one,

00:04:07.217 --> 00:04:08.264
we're expressing one as square root of

00:04:08.264 --> 00:04:09.684
two over square root of two,

00:04:09.684 --> 00:04:10.983
and then what that does is

00:04:10.983 --> 00:04:13.351
we rewrite this as the
square root of two over

00:04:13.351 --> 00:04:14.617
10 times the square root of two times

00:04:14.617 --> 00:04:15.551
the square root of two.

00:04:15.551 --> 00:04:17.351
Well the square root of two
times the square root of two

00:04:17.351 --> 00:04:18.985
is just going to be two.

00:04:18.985 --> 00:04:21.784
So it's going to be 10
times two which is 20.

00:04:21.784 --> 00:04:26.024
So it could also be written like that.

00:04:26.024 --> 00:04:29.451
So hopefully you found that helpful.

00:04:29.451 --> 00:04:31.085
In fact, even this one, you
could write if you want to

00:04:31.085 --> 00:04:32.452
visualize it slightly differently,

00:04:32.452 --> 00:04:34.218
you could view it as one twentieth

00:04:34.218 --> 00:04:36.018
times the square root of two.

00:04:36.018 --> 00:04:40.079
So these are all the same thing.


Solving quadratics using structure

WEBVTT
Kind: captions
Language: en

00:00:00.375 --> 00:00:02.233
- [Voiceover] So let's
try to find the solutions

00:00:02.233 --> 00:00:03.789
to this equation right over here.

00:00:03.789 --> 00:00:06.459
We have the quantity two
X minus three squared,

00:00:06.459 --> 00:00:08.990
and that is equal to four X minus six,

00:00:08.990 --> 00:00:10.383
and I encourage you to pause the video

00:00:10.383 --> 00:00:11.567
and give it a shot.

00:00:11.567 --> 00:00:12.821
And I'll give you a little bit of a hint,

00:00:12.821 --> 00:00:14.469
you could do this in the traditional way

00:00:14.469 --> 00:00:16.164
of expanding this out,

00:00:16.164 --> 00:00:20.019
and then turning it into kind
of a classic quadratic form,

00:00:20.019 --> 00:00:23.280
but there might be a faster
or a simpler way to do this

00:00:23.280 --> 00:00:25.927
if you really pay
attention to the structure

00:00:25.927 --> 00:00:28.982
of both sides of this equation.

00:00:30.218 --> 00:00:33.262
Well let's look at this, we
have two X minus three squared

00:00:33.262 --> 00:00:34.328
on the left-hand side,

00:00:34.328 --> 00:00:37.253
on the right-hand side
we have four X minus six.

00:00:37.253 --> 00:00:39.900
Well four X minus six,
that's just two times

00:00:39.900 --> 00:00:41.250
two X minus three,

00:00:41.250 --> 00:00:42.258
let me be clear there,

00:00:42.258 --> 00:00:43.313
so this is the same thing

00:00:43.313 --> 00:00:48.259
as two X minus three squared is equal to,

00:00:48.259 --> 00:00:50.395
four X minus six, if I factor out a two,

00:00:50.395 --> 00:00:54.714
that's two times two X minus three.

00:00:54.714 --> 00:00:55.921
And so this is really interesting,

00:00:55.921 --> 00:00:58.150
we have something squared is equal

00:00:58.150 --> 00:01:01.215
to two times that something.

00:01:01.215 --> 00:01:03.444
So if we can solve for the something,

00:01:03.444 --> 00:01:04.837
let me be very clear here,

00:01:04.837 --> 00:01:07.879
so the stuff in blue squared

00:01:07.879 --> 00:01:10.990
is equal to two times the stuff in blue.

00:01:10.990 --> 00:01:12.244
So if we can solve for what

00:01:12.244 --> 00:01:14.148
the stuff in blue could be equal to,

00:01:14.148 --> 00:01:15.611
then we could solve for X,

00:01:15.611 --> 00:01:17.074
and I'll show you that right now.

00:01:17.074 --> 00:01:19.953
So let's say, let's just
replace two X minus three,

00:01:19.953 --> 00:01:21.207
we'll do a little bit of a substitution,

00:01:21.207 --> 00:01:24.731
let's replace that with P.

00:01:24.731 --> 00:01:26.521
So let's say that P

00:01:28.260 --> 00:01:31.734
is equal to two X minus three.

00:01:31.734 --> 00:01:34.520
Well then this equation
simplifies quite nicely,

00:01:34.520 --> 00:01:38.746
the left-hand side becomes P squared,

00:01:38.746 --> 00:01:40.556
P squared is equal

00:01:40.556 --> 00:01:45.037
to two times P,

00:01:45.037 --> 00:01:48.753
'cause once again two X minus three is P,

00:01:48.753 --> 00:01:50.930
two times P.

00:01:50.930 --> 00:01:53.577
And now we just have to solve for P.

00:01:53.577 --> 00:01:56.224
And I'll switch to just one color now.

00:01:56.224 --> 00:01:57.780
So we can write this as,

00:01:57.780 --> 00:02:00.194
if we subtract two P from both sides,

00:02:00.194 --> 00:02:04.095
we can get P squared minus two P

00:02:04.095 --> 00:02:06.301
is equal to zero,

00:02:06.301 --> 00:02:07.996
and we can factor out a P,

00:02:07.996 --> 00:02:11.897
so we get P times P minus two

00:02:11.897 --> 00:02:13.731
is equal to zero.

00:02:13.731 --> 00:02:15.612
And we've seen this shown multiple times,

00:02:15.612 --> 00:02:18.096
if I have the product of two
things and they equal to zero,

00:02:18.096 --> 00:02:20.348
at least one of them
needs to be equal to zero,

00:02:20.348 --> 00:02:23.413
so either P is equal to zero,

00:02:23.413 --> 00:02:27.500
or P minus two is equal to zero.

00:02:27.500 --> 00:02:29.009
Well if P minus two is equal to zero,

00:02:29.009 --> 00:02:31.401
then that means P is equal to two.

00:02:31.401 --> 00:02:33.281
So either P equals zero,

00:02:33.281 --> 00:02:35.046
or P equals two.

00:02:35.046 --> 00:02:36.323
Well we're not quite done yet,

00:02:36.323 --> 00:02:37.763
because we wanted to solve for X,

00:02:37.763 --> 00:02:38.924
and not for P.

00:02:38.924 --> 00:02:41.013
But luckily we know that two X minus three

00:02:41.013 --> 00:02:42.499
is equal to P.

00:02:42.499 --> 00:02:44.249
So now we could say either

00:02:46.242 --> 00:02:50.709
two X minus three is going
to be equal to this P value,

00:02:50.709 --> 00:02:52.497
is going to be equal to zero,

00:02:52.497 --> 00:02:57.115
or two X minus three is going
to be equal to this P value,

00:02:57.115 --> 00:02:59.112
is going to be equal to two.

00:02:59.112 --> 00:03:01.155
And so this is pretty
straightforward to solve,

00:03:01.155 --> 00:03:02.455
add three to both sides,

00:03:02.455 --> 00:03:04.731
you get two X is equal to three.

00:03:04.731 --> 00:03:07.378
Divide both sides by two,

00:03:07.378 --> 00:03:10.954
and we get X is equal to 3/2,

00:03:10.954 --> 00:03:13.995
or over here if we add
three to both sides,

00:03:13.995 --> 00:03:16.874
we get two X is equal to five.

00:03:16.874 --> 00:03:18.964
Divide both sides by two,

00:03:18.964 --> 00:03:23.770
and you get X is equal to 5/2.

00:03:23.770 --> 00:03:26.673
So these are the possible solutions,

00:03:26.673 --> 00:03:28.205
and this is pretty neat.

00:03:28.205 --> 00:03:29.280
This one right over here,

00:03:29.280 --> 00:03:30.295
you could almost do this in your head,

00:03:30.295 --> 00:03:32.292
it was nice and simple.

00:03:32.292 --> 00:03:34.382
Well if you were to expand this out,

00:03:34.382 --> 00:03:35.612
and then subtract this,

00:03:35.612 --> 00:03:37.934
it would have been a much more complex set

00:03:37.934 --> 00:03:39.629
of operations that you would have done.

00:03:39.629 --> 00:03:41.672
You still would have hopefully
gotten to the right answer,

00:03:41.672 --> 00:03:42.926
but it would have just
taken a lot more steps.

00:03:42.926 --> 00:03:45.178
But here we could appreciate some patterns

00:03:45.178 --> 00:03:46.734
that we saw in our equations,

00:03:46.734 --> 00:03:49.079
and namely, we have
this thing being squared

00:03:49.079 --> 00:03:50.774
and then we have two
times that same thing,

00:03:50.774 --> 00:03:52.528
two times two X minus three.


Time dilation

WEBVTT
Kind: captions
Language: en

00:00:00.244 --> 00:00:01.544
- [Voiceover] So let's revisit a scenario

00:00:01.544 --> 00:00:03.100
that we have seen in several videos,

00:00:03.100 --> 00:00:04.285
especially the last video,

00:00:04.285 --> 00:00:07.186
where we tried to find this
neutral frame of reference.

00:00:07.186 --> 00:00:08.626
Let's say we're in spaceship A.

00:00:08.626 --> 00:00:10.902
We are in an inertial frame of reference.

00:00:10.902 --> 00:00:14.012
And let's say right at time equal zero

00:00:14.012 --> 00:00:15.289
in our frame of reference,

00:00:15.289 --> 00:00:18.100
spaceship B is exactly where we are,

00:00:18.100 --> 00:00:20.560
but it's traveling in
the positive x-direction

00:00:20.560 --> 00:00:22.766
at eight tenths the speed of light.

00:00:22.766 --> 00:00:24.643
We've already seen that we can overlay

00:00:24.643 --> 00:00:27.084
the spacetime diagrams for each of these

00:00:27.084 --> 00:00:28.501
frames of reference.

00:00:28.501 --> 00:00:32.501
If we do our axes,

00:00:32.501 --> 00:00:34.769
the x is one direction of spacetime

00:00:34.769 --> 00:00:37.161
that we associate with
the x-direction of space

00:00:37.161 --> 00:00:39.344
and then the ct, this vertical axis,

00:00:39.344 --> 00:00:41.132
this is another direction of spacetime

00:00:41.132 --> 00:00:44.985
that we perceive as the passage of time.

00:00:44.985 --> 00:00:48.655
Well, you can then overlay
B's frame of reference,

00:00:48.655 --> 00:00:53.655
and you kind of have these skewed axes.

00:00:53.670 --> 00:00:56.201
And you could look at a change in time.

00:00:56.201 --> 00:00:57.872
So for example, if you were to compare

00:00:57.872 --> 00:01:01.146
the event right when B passes us up to

00:01:01.146 --> 00:01:04.211
let's say some amount of time later,

00:01:04.211 --> 00:01:07.462
some amount of, so let's
say this right over there

00:01:07.462 --> 00:01:10.179
is the change in ct.

00:01:10.179 --> 00:01:11.385
In our frame of reference,

00:01:11.385 --> 00:01:14.637
these two events are happening
at the exact same place.

00:01:14.637 --> 00:01:16.912
They're just separated,
in our frame of reference,

00:01:16.912 --> 00:01:19.675
we would perceive them as
being separated by time.

00:01:19.675 --> 00:01:21.440
But if you want to say, well how much time

00:01:21.440 --> 00:01:24.807
seems to separate these events
in B's frame of reference?

00:01:24.807 --> 00:01:29.217
Well then, you would wanna go
parallel to the x prime axis.

00:01:29.217 --> 00:01:30.718
We're viewing B's frame of reference

00:01:30.718 --> 00:01:32.726
as a primed frame of reference

00:01:32.726 --> 00:01:35.928
and see where we intersect the ct prime,

00:01:35.928 --> 00:01:37.786
the ct prime axis.

00:01:37.786 --> 00:01:39.806
So this is our change in ct

00:01:39.806 --> 00:01:43.545
while this is our change in ct,

00:01:43.545 --> 00:01:46.842
this looks like our change in ct prime.

00:01:46.842 --> 00:01:48.792
And it looks longer
but we have to remember

00:01:48.792 --> 00:01:50.650
that we haven't scaled these things.

00:01:50.650 --> 00:01:52.390
And actually the scales change

00:01:52.390 --> 00:01:55.851
depending on the relative velocities.

00:01:55.851 --> 00:01:58.275
But we can actually verify algebraically

00:01:58.275 --> 00:02:00.642
that our change in ct
prime is going to be larger

00:02:00.642 --> 00:02:02.057
than our change in ct.

00:02:02.057 --> 00:02:03.922
And we just have to look
at Lorentz transformations

00:02:03.922 --> 00:02:05.410
to realize that.

00:02:05.410 --> 00:02:09.317
So our change in ct prime is going to be

00:02:09.317 --> 00:02:13.590
equal to the Lorentz factor
times our change in ct

00:02:13.590 --> 00:02:16.887
minus beta times our change in x.

00:02:16.887 --> 00:02:18.906
We've seen that multiple times before.

00:02:18.906 --> 00:02:20.463
Well our change in x,

00:02:20.463 --> 00:02:22.180
our change in x is zero.

00:02:22.180 --> 00:02:23.946
It looks stationary in
our frame of reference

00:02:23.946 --> 00:02:25.570
so that term is zero.

00:02:25.570 --> 00:02:29.100
So our change in ct prime
is going to be equal

00:02:29.100 --> 00:02:31.422
to the Lorentz factor times,

00:02:31.422 --> 00:02:33.210
I don't have to use this
parentheses anymore,

00:02:33.210 --> 00:02:37.063
times our change in ct.

00:02:37.063 --> 00:02:39.410
And our Lorentz factor is
going to be greater than one.

00:02:39.410 --> 00:02:41.196
I could actually calculate
that, let's do it.

00:02:41.196 --> 00:02:42.846
So the Lorentz factor,

00:02:42.846 --> 00:02:45.707
the Lorentz factor here.

00:02:45.707 --> 00:02:47.722
So gamma is going to be equal to one

00:02:47.722 --> 00:02:51.694
over the square root of one minus,

00:02:51.694 --> 00:02:56.694
well B's relative velocity so 0.8c

00:02:57.753 --> 00:03:00.841
over the speed of light squared.

00:03:00.841 --> 00:03:02.629
Well what is this going to simplify to?

00:03:02.629 --> 00:03:03.697
The c's cancel out.

00:03:03.697 --> 00:03:06.274
0.8 squared is 0.64.

00:03:06.274 --> 00:03:09.268
One minus that is 0.36.

00:03:09.268 --> 00:03:10.198
This is going to be,

00:03:10.198 --> 00:03:11.381
and then you take the square root of that,

00:03:11.381 --> 00:03:14.842
that's going to be one over 0.6

00:03:14.842 --> 00:03:16.839
which is equal to one over six tenths

00:03:16.839 --> 00:03:19.578
which is the same thing as 10 over six

00:03:19.578 --> 00:03:23.657
which is equal to the
same thing as five thirds

00:03:23.657 --> 00:03:26.767
which is equal to one and two thirds.

00:03:26.767 --> 00:03:28.625
So you can see our change in ct prime

00:03:28.625 --> 00:03:30.157
is going to be one and two thirds

00:03:30.157 --> 00:03:32.387
times the change in ct.

00:03:32.387 --> 00:03:33.362
Now you might wanna just say,

00:03:33.362 --> 00:03:35.684
well do these two look
like one and two thirds?

00:03:35.684 --> 00:03:37.263
And it might look a little bit like that

00:03:37.263 --> 00:03:38.702
the way I draw it, but you can't just

00:03:38.702 --> 00:03:40.514
look at it purely on,

00:03:40.514 --> 00:03:42.302
you can't just take a
ruler for this length

00:03:42.302 --> 00:03:43.370
and a ruler for this length

00:03:43.370 --> 00:03:44.924
because the scales are different

00:03:44.924 --> 00:03:46.689
and I haven't marked off the scales.

00:03:46.689 --> 00:03:48.709
Well this at least helps us visualize.

00:03:48.709 --> 00:03:52.263
But let's think about
the other way around.

00:03:52.263 --> 00:03:55.977
Let's imagine the change in ct prime

00:03:55.977 --> 00:03:58.590
between right where the spaceships pass by

00:03:58.590 --> 00:03:59.925
and a little bit later.

00:03:59.925 --> 00:04:01.318
Now, I'll do this in a different color

00:04:01.318 --> 00:04:04.615
'cause this is actually a
different event in spacetime

00:04:04.615 --> 00:04:07.657
than the one that we were
focusing on right now.

00:04:07.657 --> 00:04:10.188
And we're gonna be viewing it
from B's frame of reference

00:04:10.188 --> 00:04:13.369
so that's our change in ct prime.

00:04:13.369 --> 00:04:16.131
But what is going to be our change in ct?

00:04:16.131 --> 00:04:17.153
Well, to think about it,

00:04:17.153 --> 00:04:20.451
we could go parallel to the x-axis,

00:04:20.451 --> 00:04:23.701
the x-axis right over
there, so you'll parallel,

00:04:23.701 --> 00:04:27.904
parallel to the x-axis and
you get right over there

00:04:27.904 --> 00:04:30.226
and so our change in ct

00:04:30.226 --> 00:04:31.921
looks like it's more.

00:04:31.921 --> 00:04:34.057
And once again, we can
algebraically verify it

00:04:34.057 --> 00:04:35.797
by really doing the same thing.

00:04:35.797 --> 00:04:40.650
Our change in ct is going
to be equal to gamma

00:04:40.650 --> 00:04:43.692
times our change in ct prime

00:04:43.692 --> 00:04:48.127
minus beta times delta x prime.

00:04:48.127 --> 00:04:49.242
And if you were to actually,

00:04:49.242 --> 00:04:51.517
if we actually did have
a change in x prime here,

00:04:51.517 --> 00:04:53.815
the beta, the velocity now
has a different direction

00:04:53.815 --> 00:04:55.163
so it would all be the negative

00:04:55.163 --> 00:04:57.763
but our change in x prime
from B's point of view,

00:04:57.763 --> 00:05:00.318
these two events are
happening in the same place

00:05:00.318 --> 00:05:02.663
so our change in x prime is zero.

00:05:02.663 --> 00:05:07.663
So you have change in ct is
equal to gamma times change,

00:05:08.072 --> 00:05:09.628
once again I don't need my parentheses,

00:05:09.628 --> 00:05:12.600
times change in ct prime.

00:05:12.600 --> 00:05:14.458
And it's going to be the same gamma

00:05:14.458 --> 00:05:16.431
because remember, we're taking v over c

00:05:16.431 --> 00:05:18.080
and so whether it's
either v or negative v,

00:05:18.080 --> 00:05:20.170
when you square it, it gets the same value

00:05:20.170 --> 00:05:23.837
so once again, gamma is going
to be one and two thirds.

00:05:23.837 --> 00:05:26.670
So it's going to be one and two thirds.

00:05:26.670 --> 00:05:28.574
So it seems a little bit strange.

00:05:28.574 --> 00:05:30.177
You know, I have some passage of time

00:05:30.177 --> 00:05:31.895
in my frame of reference where it looks,

00:05:31.895 --> 00:05:33.381
you know, it's something
that looks stationary,

00:05:33.381 --> 00:05:34.542
two events that look
like they're happening

00:05:34.542 --> 00:05:37.026
in the same place but one after another.

00:05:37.026 --> 00:05:38.722
It looks like their change in time,

00:05:38.722 --> 00:05:41.461
it takes longer for those
two events to happen

00:05:41.461 --> 00:05:44.920
in the ct prime in the
moving frame of reference.

00:05:44.920 --> 00:05:47.574
But then, if we have some event

00:05:47.574 --> 00:05:49.704
that looks stationary in
B's frame of reference

00:05:49.704 --> 00:05:52.211
and they're separated
by a change in ct prime,

00:05:52.211 --> 00:05:55.625
it looks like the change in
ct between those two events

00:05:55.625 --> 00:05:57.134
is even larger.

00:05:57.134 --> 00:06:00.503
So it looks like this
somewhat bizarre phenomenon.

00:06:00.503 --> 00:06:02.382
And to help us reconcile these

00:06:02.382 --> 00:06:03.890
and to visualize a little bit better,

00:06:03.890 --> 00:06:07.746
actually even to be able
to put ct and ct prime

00:06:07.746 --> 00:06:10.020
on the same scale,

00:06:10.020 --> 00:06:12.575
we can look for this
neutral frame of reference

00:06:12.575 --> 00:06:15.361
which we did in actually
the last video I made.

00:06:15.361 --> 00:06:17.149
I don't know if it's the
last video you've seen.

00:06:17.149 --> 00:06:19.262
Or we could say look, if A and B

00:06:19.262 --> 00:06:22.256
are travelling with a
relative velocity of 0.8

00:06:22.256 --> 00:06:23.906
times of the speed of light
relative to each other,

00:06:23.906 --> 00:06:26.832
B is travelling 0.8c in
the positive x direction

00:06:26.832 --> 00:06:28.968
relative to a stationary A

00:06:28.968 --> 00:06:33.937
or A is travelling 0.8c in
the negative x direction

00:06:33.937 --> 00:06:35.608
relative to a stationary B.

00:06:35.608 --> 00:06:38.092
You can find a frame of reference

00:06:38.092 --> 00:06:40.693
where A and B were in
that frame of reference

00:06:40.693 --> 00:06:42.667
to a stationary observer
in that frame of reference,

00:06:42.667 --> 00:06:44.896
A and B are both travelling outwards

00:06:44.896 --> 00:06:46.614
at half the speed of light.

00:06:46.614 --> 00:06:47.797
And we figured that out where we

00:06:47.797 --> 00:06:50.560
did those videos on a
neutral frame of reference.

00:06:50.560 --> 00:06:51.721
And what's neat about that

00:06:51.721 --> 00:06:55.786
is if you make what looks like
a Minkowski spacetime diagram

00:06:55.786 --> 00:06:57.741
in that neutral frame of reference

00:06:57.741 --> 00:07:01.267
then the ct and ct prime, A
and B's frame of references,

00:07:01.267 --> 00:07:04.637
get equally skewed to the, you know,

00:07:04.637 --> 00:07:07.093
if you think about the time
axis to the left and the right.

00:07:07.093 --> 00:07:09.252
And since they're equally skewed,

00:07:09.252 --> 00:07:12.805
the time dilation relative
to this rest frame

00:07:12.805 --> 00:07:15.545
is the scaling is going to be the same.

00:07:15.545 --> 00:07:18.261
So we could put both of these on,

00:07:18.261 --> 00:07:20.374
both of these on the same scale.

00:07:20.374 --> 00:07:23.672
And you could see that this
neutral frame of reference,

00:07:23.672 --> 00:07:26.319
this ct prime prime frame of reference,

00:07:26.319 --> 00:07:27.503
I drew them over here,

00:07:27.503 --> 00:07:30.127
that if you look at either of
A or B's frame of reference,

00:07:30.127 --> 00:07:31.658
they're going to be in between

00:07:31.658 --> 00:07:33.121
A and B's frame of reference.

00:07:33.121 --> 00:07:34.422
But here it's the neutral one.

00:07:34.422 --> 00:07:36.244
It's the one where we're drawing the time

00:07:36.244 --> 00:07:38.392
or we're drawing the two
axes being perpendicular

00:07:38.392 --> 00:07:39.391
to each other.

00:07:39.391 --> 00:07:41.434
And now if you look at these two events,

00:07:41.434 --> 00:07:43.035
so if you look at this first event,

00:07:43.035 --> 00:07:45.103
where you have our delta ct,

00:07:45.103 --> 00:07:47.750
so you look at this first event

00:07:47.750 --> 00:07:50.722
where you have a delta in ct between

00:07:50.722 --> 00:07:53.137
right when the spaceships
pass and right over there.

00:07:53.137 --> 00:07:54.739
You see that if you were to look at that,

00:07:54.739 --> 00:07:57.710
if you were to look at
the ct prime coordinate

00:07:57.710 --> 00:08:01.750
for that event, when you go
parallel to the x prime axis,

00:08:01.750 --> 00:08:03.654
you get right over there.

00:08:03.654 --> 00:08:08.182
So this is, that is
your change in ct prime.

00:08:08.182 --> 00:08:11.223
And likewise, if you have that other event

00:08:11.223 --> 00:08:14.475
that I drew in that blue-green
color, right over here.

00:08:14.475 --> 00:08:17.191
And this is your change in ct prime.

00:08:17.191 --> 00:08:20.000
This is your change in ct prime,

00:08:20.000 --> 00:08:23.181
well then if you think of it
from A's frame of reference,

00:08:23.181 --> 00:08:25.736
well we just follow the
x-axis not the x prime.

00:08:25.736 --> 00:08:27.106
We go parallel to it.

00:08:27.106 --> 00:08:29.312
We end up right over there.

00:08:29.312 --> 00:08:31.493
Now what's really interesting about this,

00:08:31.493 --> 00:08:33.385
what's really interesting about this

00:08:33.385 --> 00:08:36.951
is from A's frame of reference,

00:08:36.951 --> 00:08:40.038
the yellow event happens
before the blue event.

00:08:40.038 --> 00:08:43.499
But in B's frame of
reference, the blue event

00:08:43.499 --> 00:08:45.728
happens before the yellow event.

00:08:45.728 --> 00:08:48.467
So really, really, really
fascinating things going on

00:08:48.467 --> 00:08:50.255
but what I really like about this diagram

00:08:50.255 --> 00:08:52.623
is that A and B's frame of reference

00:08:52.623 --> 00:08:53.993
are going to have the same scale

00:08:53.993 --> 00:08:56.902
since they're both equally
skewed to the left and the right

00:08:56.902 --> 00:08:58.382
if we're thinking about the time axis.

00:08:58.382 --> 00:09:01.492
And this type of diagram
is called a Loedel diagram.

00:09:01.492 --> 00:09:04.790
Loedel, Loedel diagram which is really

00:09:04.790 --> 00:09:07.461
a variation of a Minkowski diagram

00:09:07.461 --> 00:09:10.339
but it lets us really
appreciate the symmetry

00:09:10.339 --> 00:09:12.887
between these frames of reference.


Irregular plural nouns

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:01.636
- [Voiceover] Hello, grammarians.

00:00:01.636 --> 00:00:03.762
Welcome to Irregular Plurals, Part IV.

00:00:04.284 --> 00:00:07.883
The Mutant Plurals. (moans)

00:00:09.288 --> 00:00:13.328
Yes friends, these words
have mutant superpowers

00:00:13.328 --> 00:00:16.161
in that they can transform weirdly

00:00:16.161 --> 00:00:18.808
and obnoxiously, not
obeying any other rules

00:00:18.808 --> 00:00:20.248
of English pluralization.

00:00:20.393 --> 00:00:21.606
But here's the cool thing.

00:00:21.606 --> 00:00:24.520
There are only seven words
that behave this way.

00:00:24.836 --> 00:00:25.728
What way?

00:00:25.728 --> 00:00:27.574
I'll explain by writing all of them down.

00:00:27.574 --> 00:00:30.720
The words are foot,

00:00:30.720 --> 00:00:35.085
woman,

00:00:35.085 --> 00:00:37.651
man,

00:00:37.651 --> 00:00:40.368
tooth,

00:00:40.368 --> 00:00:43.131
goose,

00:00:43.131 --> 00:00:46.080
mouse,

00:00:46.080 --> 00:00:49.122
louse.

00:00:49.122 --> 00:00:51.003
Looks pretty straightforward, right?

00:00:51.328 --> 00:00:52.883
The thing about these words is that none

00:00:52.883 --> 00:00:55.403
of them take s as a plural.

00:00:56.169 --> 00:00:58.073
So the plural of foot is not foots,

00:00:58.073 --> 00:01:00.233
the plural of woman is not womans,

00:01:00.233 --> 00:01:02.206
the plural of tooth is not tooths.

00:01:03.716 --> 00:01:06.119
The reason these are called mutant plurals

00:01:06.119 --> 00:01:08.127
is because the vowel sound,

00:01:08.127 --> 00:01:10.240
the ooh, or the uh, or the ah, or the ooh,

00:01:10.240 --> 00:01:13.050
or the ooh, or the ou,
or the ou, turns into

00:01:13.050 --> 00:01:16.196
a different sound, turns
into a different vowel sound.

00:01:16.956 --> 00:01:20.329
So the plural of foot
is not foots, but feet.

00:01:21.328 --> 00:01:24.370
The plural of woman is
not womans, but women.

00:01:25.798 --> 00:01:29.246
The plural of man is not mans, but men.

00:01:29.878 --> 00:01:33.437
The plural of tooth is
not tooths, but teeth.

00:01:34.285 --> 00:01:37.210
The plural of goose is
not gooses, but geese.

00:01:38.597 --> 00:01:42.284
The plural of mouse is
not mouses, but mice.

00:01:43.445 --> 00:01:48.405
And the plural of louse
is not louses, but lice.

00:01:49.601 --> 00:01:51.354
You can see that mouse and louse actually

00:01:51.354 --> 00:01:53.553
change their end spelling as well

00:01:53.553 --> 00:01:56.128
from se to ce.

00:01:56.604 --> 00:01:58.195
Even though it's the same sound,

00:01:58.195 --> 00:02:01.689
louse, lice, for whatever reason,

00:02:01.689 --> 00:02:04.597
just from some quirk of our spelling history,

00:02:04.597 --> 00:02:06.484
not only do we change the vowels

00:02:06.484 --> 00:02:09.445
used here, we also change
the consonants you say.

00:02:10.478 --> 00:02:11.651
Why is this the case?

00:02:11.651 --> 00:02:12.846
I'm so glad you asked.

00:02:12.846 --> 00:02:14.611
I'm going to save that for another video.

00:02:15.250 --> 00:02:17.932
In the meantime, just these seven words.

00:02:18.210 --> 00:02:22.250
One, two, three, four, five,

00:02:22.250 --> 00:02:24.805
six, seven, are the only

00:02:24.805 --> 00:02:26.964
words in English that behave this way.

00:02:27.224 --> 00:02:28.404
So you're in luck.

00:02:28.404 --> 00:02:30.761
This is a handful of words to memorize.

00:02:31.167 --> 00:02:33.326
And provided you're not borrowing any one

00:02:33.326 --> 00:02:36.008
else' hats, ideally
you won't have to worry

00:02:36.008 --> 00:02:37.529
about lice very often.

00:02:37.999 --> 00:02:38.861
That's my hope.

00:02:38.861 --> 00:02:40.181
You can learn anything.

00:02:40.181 --> 00:02:40.912
David out.


BONUS VIDEO

WEBVTT
Kind: captions
Language: en

00:00:00.249 --> 00:00:01.096
- [Voiceover] Hello, grammarians!

00:00:01.096 --> 00:00:03.960
I wanted to talk to you
again about mutant plurals.

00:00:03.960 --> 00:00:06.056
So to review a mutant plural is,

00:00:06.056 --> 00:00:08.478
there are only seven of them in English,

00:00:08.478 --> 00:00:11.944
and they all change sound
when they pluralize.

00:00:11.944 --> 00:00:14.302
You don't add an -s, you don't add an -en,

00:00:14.302 --> 00:00:17.430
you don't change the ending,
you change the vowel,

00:00:17.430 --> 00:00:19.764
and there are only seven to go like this.

00:00:24.258 --> 00:00:26.492
There's man,

00:00:26.492 --> 00:00:28.855
woman,

00:00:28.855 --> 00:00:31.170
tooth,

00:00:31.170 --> 00:00:34.146
foot,

00:00:34.146 --> 00:00:36.858
mouse,

00:00:36.858 --> 00:00:40.779
louse,

00:00:40.779 --> 00:00:42.516
and goose.

00:00:44.736 --> 00:00:47.865
And these words become in the plural

00:00:47.865 --> 00:00:50.707
men,

00:00:50.707 --> 00:00:54.041
women,

00:00:54.041 --> 00:00:57.170
teeth,

00:00:57.170 --> 00:01:00.097
feet,

00:01:00.097 --> 00:01:03.515
mice,

00:01:03.515 --> 00:01:06.658
lice,

00:01:06.658 --> 00:01:08.499
and geese.

00:01:10.071 --> 00:01:14.022
Now the reason that we have
these seven weird mutant plurals

00:01:14.022 --> 00:01:17.131
in English is kind of
complicated, but I'm lucky enough

00:01:17.131 --> 00:01:19.726
to be able to work with
an actual linguist.

00:01:19.726 --> 00:01:20.492
Hello, Jake!

00:01:20.492 --> 00:01:21.755
- [Voiceover] Hey, grammarians!

00:01:21.755 --> 00:01:23.770
- [Voiceover] Jake, is it
true that you're a linguist?

00:01:23.770 --> 00:01:24.924
- [Voiceover] Yep, it's true.

00:01:24.924 --> 00:01:26.488
- [Voiceover] All right,
so Jake, what is the deal

00:01:26.488 --> 00:01:27.828
with feet?

00:01:27.828 --> 00:01:29.515
Where do these mutant plurals come from?

00:01:29.515 --> 00:01:33.640
If we take the word foot and
we drag it through history,

00:01:33.640 --> 00:01:36.646
how do we get to the plural as feet?

00:01:36.646 --> 00:01:38.028
What's the deal with that?

00:01:38.028 --> 00:01:39.917
- [Voiceover] So if you look
at a lot of Germanic languages

00:01:39.917 --> 00:01:42.982
that are around today, you
find similar words to the word,

00:01:42.982 --> 00:01:44.347
the English word foot.

00:01:44.347 --> 00:01:46.237
In German we have the word Fuss,

00:01:46.237 --> 00:01:48.390
in Dutch we have the word voet.

00:01:48.390 --> 00:01:50.062
And when you have a lot
of different languages

00:01:50.062 --> 00:01:51.904
with slight variations of a word,

00:01:51.904 --> 00:01:54.809
it means there is some
old, old word out there

00:01:54.809 --> 00:01:58.812
that all these words are coming from.

00:01:58.812 --> 00:01:59.998
So we can pretty much be sure

00:01:59.998 --> 00:02:02.782
that there's some Proto-Germanic word

00:02:02.782 --> 00:02:05.074
that sounds something like foot.

00:02:05.074 --> 00:02:06.967
Now back in those days

00:02:06.967 --> 00:02:08.753
there was a different
way to form the plural,

00:02:08.753 --> 00:02:11.916
and that was to add an i
sound at the end of a word.

00:02:11.916 --> 00:02:16.225
So if the word was foot, then
the plural was maybe footi.

00:02:16.225 --> 00:02:18.252
That means many foots.

00:02:19.434 --> 00:02:21.982
Now there's a tendency in language

00:02:21.982 --> 00:02:23.497
that you have to understand here.

00:02:23.497 --> 00:02:25.793
It's called vowel harmony.

00:02:25.793 --> 00:02:30.055
Basically it means that
vowels within a given word,

00:02:30.055 --> 00:02:31.847
they like to sound like each other.

00:02:31.847 --> 00:02:34.795
So if you have two syllables
those syllables will start,

00:02:34.795 --> 00:02:39.453
the vowels in those syllables
will start to converge.

00:02:39.453 --> 00:02:42.040
And in Germanic languages especially,

00:02:42.040 --> 00:02:44.708
there's one typical kind of vowel harmony

00:02:44.708 --> 00:02:46.121
which is--
- [Voiceover] Okay.

00:02:46.121 --> 00:02:49.422
- [Voiceover] When you
have two vowels in a word,

00:02:49.422 --> 00:02:51.697
the first vowel will try to sound

00:02:51.697 --> 00:02:53.542
more like the second vowel,

00:02:53.542 --> 00:02:56.248
if that second vowel is the i sound,

00:02:56.248 --> 00:03:00.596
just like in the plural
formation of nouns.

00:03:00.596 --> 00:03:03.482
- [Voiceover] So you're
saying that the suffix -i

00:03:03.482 --> 00:03:07.754
at the end of this proposed word footi,

00:03:08.994 --> 00:03:11.922
the u sound tried to sound
more like the i sound.

00:03:11.922 --> 00:03:12.563
- [Voiceover] Right.

00:03:12.563 --> 00:03:13.904
- [Voiceover] This is just
a pattern that we find

00:03:13.904 --> 00:03:15.549
in Germanic languages.

00:03:15.549 --> 00:03:16.159
- [Voiceover] Exactly.

00:03:16.159 --> 00:03:17.953
It's very prevalent in Germanic languages,

00:03:17.953 --> 00:03:20.203
also exists in some Romance languages,

00:03:20.203 --> 00:03:21.604
some tiny Romance languages.

00:03:21.604 --> 00:03:24.233
- [Voiceover] So what happens
when you combine the u sound

00:03:24.233 --> 00:03:24.976
with the i sound?

00:03:24.976 --> 00:03:26.094
What sound do you get?

00:03:26.094 --> 00:03:29.810
- [Voiceover] Well, strangely
you get the ü sound.

00:03:29.810 --> 00:03:31.261
- [Voiceover] Are you, what is that?

00:03:31.261 --> 00:03:32.719
- [Voiceover] The ü sound.

00:03:32.719 --> 00:03:33.638
- [Voiceover] Are you okay?

00:03:33.638 --> 00:03:34.757
- [Voiceover] I think I am okay.

00:03:34.757 --> 00:03:36.070
You know, we linguists have to deal

00:03:36.070 --> 00:03:37.256
with this kind of thing all the time.

00:03:37.256 --> 00:03:37.923
- [Voiceover] (laughs) Sorry.

00:03:37.923 --> 00:03:39.015
- [Voiceover] We have
very strong stomachs.

00:03:39.015 --> 00:03:40.225
- [Voiceover] Uh-huh.

00:03:40.225 --> 00:03:41.952
- [Voiceover] You still
get this sound in languages

00:03:41.952 --> 00:03:44.032
like German and Dutch.

00:03:44.032 --> 00:03:47.643
- [Voiceover] So then that's
what this ü is, right?

00:03:47.643 --> 00:03:50.536
So this sound, or I guess,

00:03:50.536 --> 00:03:53.306
so the plural of German Fuss

00:03:53.306 --> 00:03:55.943
is what you said?

00:03:55.943 --> 00:03:58.233
Füsse?

00:03:58.233 --> 00:04:00.409
Okay, so bring this home for me.

00:04:00.409 --> 00:04:04.827
So at some point in the
development of English,

00:04:04.827 --> 00:04:07.678
or of all these Germanic languages,

00:04:07.678 --> 00:04:11.734
we had this word foot and
then it turned into footi,

00:04:11.734 --> 00:04:12.904
and then it turned into what?

00:04:12.904 --> 00:04:14.961
It turned into füti?

00:04:14.961 --> 00:04:16.370
- [Voiceover] Something like füti, right.

00:04:16.370 --> 00:04:17.146
- [Voiceover] Okay.

00:04:17.146 --> 00:04:18.984
- [Voiceover] Now eventually that ü sound

00:04:18.984 --> 00:04:19.966
dropped out of English,

00:04:19.966 --> 00:04:21.945
which is why it's so
hard for us to pronounce,

00:04:21.945 --> 00:04:24.036
and it was replaced in
pretty much all cases

00:04:24.036 --> 00:04:26.334
with the ii sound.

00:04:26.334 --> 00:04:29.970
So we get the word feeti
and then the i drops off,

00:04:29.970 --> 00:04:31.736
and we're left with foot as the singular

00:04:31.736 --> 00:04:33.161
and feet as the plural.

00:04:33.161 --> 00:04:33.789
- [Voiceover] Cool.

00:04:33.789 --> 00:04:38.330
So it goes from foot to feeti.

00:04:38.330 --> 00:04:40.086
- [Voiceover] Now the
same exact thing happened

00:04:40.086 --> 00:04:44.025
with the word mouse, which
probably used to be moos,

00:04:44.025 --> 00:04:47.252
because in Frisian, which is
the closest cousin to English,

00:04:47.252 --> 00:04:49.454
the word for mouse still is Muus.

00:04:49.454 --> 00:04:50.186
- [Voiceover] Moos?

00:04:50.186 --> 00:04:51.852
- [Voiceover] And you find similar things

00:04:51.852 --> 00:04:53.883
in a lot of Germanic languages

00:04:53.883 --> 00:04:57.954
like muis in Dutch and Maus in German.

00:04:59.064 --> 00:05:03.609
The plural became muusi,
and then eventually meesi,

00:05:03.609 --> 00:05:05.000
then you get mees.

00:05:05.000 --> 00:05:06.773
Then a few hundred year later,

00:05:06.773 --> 00:05:10.558
during the Great Vowel Shift
that mees becomes mice.

00:05:10.558 --> 00:05:14.005
A lot of ii vowels in
the Great Vowel Shift

00:05:14.005 --> 00:05:17.748
about 500 years ago became ai vowels.

00:05:17.748 --> 00:05:19.523
- [Voiceover] So this is
sort of a broad pattern

00:05:19.523 --> 00:05:21.837
these mutant words all take.

00:05:21.837 --> 00:05:23.877
It's this umlaut mutation, right?

00:05:23.877 --> 00:05:27.768
Because these little, these
double dots that go over a,

00:05:27.768 --> 00:05:29.483
oh it looks like a smiley face,

00:05:29.483 --> 00:05:32.685
that go over a vowel
change its color, right?

00:05:32.685 --> 00:05:33.644
They change its meaning.

00:05:33.644 --> 00:05:35.661
- [Voiceover] And that process is called

00:05:35.661 --> 00:05:38.497
either umlaut mutation or i-mutation,

00:05:38.497 --> 00:05:41.914
not eye like the sight organ,
i like the letter in English.

00:05:41.914 --> 00:05:42.705
- [Voiceover] Cool.

00:05:42.705 --> 00:05:45.676
Well, I hope that cleared
some things up for you.

00:05:45.676 --> 00:05:47.240
You can learn anything.

00:05:47.240 --> 00:05:48.365
David out.

00:05:48.365 --> 00:05:49.728
- [Voiceover] Jake out.

00:05:49.728 --> 00:05:50.488
(clap)

00:05:50.488 --> 00:05:52.567
- [Voiceover] That was
a high five, not a slap.


Common and proper nouns

WEBVTT
Kind: captions
Language: en

00:00:00.437 --> 00:00:01.644
- [Voiceover] Hello, grammarians.

00:00:01.644 --> 00:00:03.563
I'd like to bring up the
idea of the difference

00:00:03.563 --> 00:00:07.245
between a common and a proper noun.

00:00:07.245 --> 00:00:08.740
So the difference between a common

00:00:08.740 --> 00:00:09.878
and a proper noun is simply

00:00:09.878 --> 00:00:12.056
the difference between
something with a name

00:00:12.056 --> 00:00:15.111
and a more generic version of that thing.

00:00:15.111 --> 00:00:17.549
I'll give you a couple of
examples right off the bat.

00:00:17.549 --> 00:00:21.090
So speaking generally, I am from a city.

00:00:22.628 --> 00:00:26.468
The specific city that
I'm from is Chicago.

00:00:28.391 --> 00:00:31.953
I could talk about a frog generally,

00:00:31.953 --> 00:00:33.885
but if I were speaking of a specific frog,

00:00:33.885 --> 00:00:35.720
I would say Kermit.

00:00:37.160 --> 00:00:39.665
The difference between a
common and a proper noun

00:00:39.665 --> 00:00:43.770
is merely the difference
between a general thing,

00:00:43.770 --> 00:00:45.967
so this side is more general,

00:00:47.559 --> 00:00:49.031
and a specific thing.

00:00:52.220 --> 00:00:53.649
It's a continuum.

00:00:55.663 --> 00:01:00.637
So if you are speaking
of, let's see, a river,

00:01:01.927 --> 00:01:03.859
any old river, that's a common noun,

00:01:03.859 --> 00:01:05.639
but if you're talking
about a specific river,

00:01:05.639 --> 00:01:07.420
and it's a named river here,

00:01:07.420 --> 00:01:11.202
that would be the Nile, say.

00:01:12.129 --> 00:01:13.783
You could talk about a mountain,

00:01:15.767 --> 00:01:17.186
and that would be a common noun,

00:01:17.186 --> 00:01:18.755
because there are many mountains,

00:01:18.755 --> 00:01:20.717
but if you wanted to talk
about a specific mountain,

00:01:20.717 --> 00:01:22.695
say Mount Kilimanjaro,

00:01:25.614 --> 00:01:26.649
in Tanzania,

00:01:27.952 --> 00:01:28.995
that's a proper noun.

00:01:28.995 --> 00:01:31.410
So here are the properties
of proper nouns.

00:01:31.410 --> 00:01:35.463
Proper nouns are always capitalized.

00:01:38.926 --> 00:01:41.099
And that means that instead of using

00:01:41.099 --> 00:01:42.850
a little letter A like that,

00:01:42.850 --> 00:01:46.519
you would instead use a
big letter A like that.

00:01:46.519 --> 00:01:48.691
Common nouns are only capitalized

00:01:48.691 --> 00:01:51.039
if you find them at the
beginning of sentences.

00:01:57.550 --> 00:01:58.962
So you might say,

00:02:00.825 --> 00:02:04.818
"Mountains are my favorite."

00:02:08.764 --> 00:02:11.881
But you would also say, "Kilimanjaro

00:02:15.767 --> 00:02:18.945
"is my favorite

00:02:21.653 --> 00:02:22.929
"mountain."

00:02:26.573 --> 00:02:31.550
And that is a lowercase,
non-capitalized M,

00:02:31.550 --> 00:02:33.810
as opposed to this one,
which is uppercase.

00:02:34.888 --> 00:02:36.700
So that's the difference between common

00:02:36.700 --> 00:02:37.847
and proper nouns.

00:02:37.847 --> 00:02:39.205
If you're talking about something general,

00:02:39.205 --> 00:02:40.382
it's a common noun.

00:02:40.382 --> 00:02:42.012
If you're talking about
something specific,

00:02:42.012 --> 00:02:43.461
it's a proper noun,

00:02:43.461 --> 00:02:44.819
and the difference between them is

00:02:44.819 --> 00:02:47.137
that you capitalize a proper noun.

00:02:48.426 --> 00:02:49.845
You can learn anything.

00:02:49.845 --> 00:02:50.970
David out.


Irregular plural nouns

WEBVTT
Kind: captions
Language: en

00:00:00.551 --> 00:00:01.712
- [Voiceover] Hello grammarians.

00:00:01.712 --> 00:00:03.639
I wanted to talk today
about a different kind

00:00:03.639 --> 00:00:04.557
of irregular plural.

00:00:04.557 --> 00:00:06.635
So we've been talking
about regular plurals

00:00:06.635 --> 00:00:08.481
where you take a word and you add an S.

00:00:08.481 --> 00:00:12.103
So for example, the word dog becomes dogs.

00:00:12.103 --> 00:00:13.435
You add an S.

00:00:13.435 --> 00:00:16.050
And this is the regular plural here.

00:00:18.110 --> 00:00:20.560
But I've been talking
about the irregular plural,

00:00:20.560 --> 00:00:23.555
the plural, the multiple form of a verb

00:00:23.555 --> 00:00:26.156
that is not regular, irregular.

00:00:26.156 --> 00:00:28.594
But today I figured we'd
talk about something called

00:00:28.594 --> 00:00:30.245
the base plural.

00:00:34.558 --> 00:00:37.553
Which I will illustrate for
you using our friend the sheep.

00:00:37.553 --> 00:00:40.793
Now, sheep is a very strange word in that

00:00:40.793 --> 00:00:43.637
it doesn't matter whether not
there's more than one of them.

00:00:43.637 --> 00:00:46.110
The form of the word always looks the same

00:00:46.110 --> 00:00:48.954
whether it's one sheep or two sheep.

00:00:48.954 --> 00:00:51.392
It's an irregular plural
you don't add an S to.

00:00:51.392 --> 00:00:55.152
This is called a base plural
'cause the base sheep,

00:00:55.152 --> 00:00:59.273
the thing that you would
normally add this particle S to

00:00:59.273 --> 00:01:03.232
doesn't change whether it's
singular sheep or plural sheep.

00:01:03.232 --> 00:01:06.408
So that's you know.

00:01:13.878 --> 00:01:15.846
There was one sheep...

00:01:20.751 --> 00:01:22.569
on the hill.

00:01:26.312 --> 00:01:28.362
There's a sentence.

00:01:29.435 --> 00:01:32.477
What if we put another little
baby sheep on that hill?

00:01:32.477 --> 00:01:34.393
A little lamb.

00:01:34.393 --> 00:01:36.881
Well now the sentence looks like this.

00:01:40.833 --> 00:01:42.360
Two sheep...

00:01:45.152 --> 00:01:46.923
on the hill.

00:01:48.751 --> 00:01:50.597
Now the only difference
between these two sentences

00:01:50.597 --> 00:01:53.233
is that there's one sheep and two sheep

00:01:53.233 --> 00:01:55.868
and therefore that means that the verb

00:01:55.868 --> 00:01:58.318
changes to a plural conjugation.

00:01:58.318 --> 00:02:00.199
So there was one sheep.

00:02:00.199 --> 00:02:02.463
There were two sheep on the hill.

00:02:02.463 --> 00:02:04.622
But everything else
stays exactly the same.

00:02:04.622 --> 00:02:07.234
One sheep, two sheep.

00:02:07.234 --> 00:02:09.638
This is very strange, it's a base plural.

00:02:09.638 --> 00:02:13.568
So in standard English,
the form is two sheep...

00:02:16.313 --> 00:02:19.849
and not two sheeps.

00:02:23.152 --> 00:02:23.953
You see.

00:02:23.953 --> 00:02:26.472
Now, there are more words
that do behave this way.

00:02:26.472 --> 00:02:28.603
So let's go investigate.

00:02:30.919 --> 00:02:33.357
So there are a small number of words

00:02:33.357 --> 00:02:35.795
that also behave this
way, the way sheep does,

00:02:35.795 --> 00:02:39.034
these weird sheep plurals,
these base plurals.

00:02:39.034 --> 00:02:40.358
One of them is fish.

00:02:40.358 --> 00:02:42.958
So you could say the fish
are plentiful this season,

00:02:42.958 --> 00:02:45.844
but you could also say,
you know, the fish...

00:02:47.440 --> 00:02:50.128
is delicious.

00:02:53.918 --> 00:02:57.192
You could say the bison migrate west

00:02:57.192 --> 00:03:00.350
or you could also say
the bison migrates west

00:03:00.350 --> 00:03:03.410
indicating a single bison, you see.

00:03:03.914 --> 00:03:05.876
Bison can be singular or plural.

00:03:05.876 --> 00:03:09.041
Fish can be singular or plural.

00:03:09.917 --> 00:03:13.156
As is so frequently the case,
there is a special exception

00:03:13.156 --> 00:03:16.000
regarding the word fishes,

00:03:16.000 --> 00:03:17.161
which you may have heard before,

00:03:17.161 --> 00:03:18.880
and fishes is a word that we would use

00:03:18.880 --> 00:03:22.645
when we're talking
about individual species

00:03:23.916 --> 00:03:25.327
of fish.

00:03:27.481 --> 00:03:28.874
And fish is the word that we would use

00:03:28.874 --> 00:03:31.887
to refer to individual fish.

00:03:33.436 --> 00:03:36.186
So let's say your uncle Marty
is a prodigious fisherman

00:03:36.186 --> 00:03:38.359
and he catches,

00:03:38.359 --> 00:03:39.834
he goes fly fishing one weekend,

00:03:39.834 --> 00:03:41.714
he comes back, he has 30 fish.

00:03:41.714 --> 00:03:45.327
Marty caught...

00:03:48.514 --> 00:03:51.572
30 fish.

00:03:51.572 --> 00:03:53.825
But let's say on the other hand

00:03:53.825 --> 00:03:58.189
your aunt Marta is a
prodigious marine biologist

00:03:58.189 --> 00:04:01.948
and she studies 30
different types of fish.

00:04:01.948 --> 00:04:06.639
You would say Marta studies...

00:04:08.317 --> 00:04:10.204
30 fishes.

00:04:14.157 --> 00:04:16.676
And that doesn't mean that she
studies 30 individual fish.

00:04:16.676 --> 00:04:19.799
That means she studies 30 types of fish.

00:04:19.799 --> 00:04:20.554
That's the difference.

00:04:20.554 --> 00:04:23.839
Fishes is referring to species.

00:04:23.839 --> 00:04:26.916
Fish refers to individuals.

00:04:26.916 --> 00:04:28.829
That's how you'd use them in the plural.

00:04:28.829 --> 00:04:30.875
So to review,

00:04:30.875 --> 00:04:35.008
there's this entire class
of words called base plurals

00:04:36.198 --> 00:04:39.356
where the word itself, the
base, doesn't take an S

00:04:39.356 --> 00:04:40.714
for the plural, it's just the same.

00:04:40.714 --> 00:04:42.920
The singular is the same as the plural.

00:04:42.920 --> 00:04:45.474
So that gives us words like sheep,

00:04:45.474 --> 00:04:47.224
fish,

00:04:47.224 --> 00:04:48.995
and bison.

00:04:50.382 --> 00:04:52.680
There aren't a ton of English
words that behave this way

00:04:52.680 --> 00:04:54.793
where the plural is the
same as the singular.

00:04:54.793 --> 00:04:55.989
I just wanted to make you aware

00:04:55.989 --> 00:04:57.638
of some of the most common ones.

00:04:57.638 --> 00:05:00.517
There are also more
examples in the exercises.

00:05:00.517 --> 00:05:03.315
So I just wanted you to be aware of them.

00:05:03.315 --> 00:05:04.720
You can learn anything.

00:05:04.720 --> 00:05:06.085
David out.


Concrete and abstract nouns

WEBVTT
Kind: captions
Language: en

00:00:00.214 --> 00:00:01.457
- [Voiceover] Hello grammarians.

00:00:01.457 --> 00:00:02.814
So today I'd like to talk to you

00:00:02.814 --> 00:00:06.467
about the idea of concrete
and abstract nouns,

00:00:06.467 --> 00:00:07.408
and before we do that,

00:00:07.408 --> 00:00:11.390
I'd like to get into some
word origins or etymology.

00:00:11.390 --> 00:00:13.873
So let's take each of these words in turn,

00:00:13.873 --> 00:00:17.427
because I think by digging
into what these words mean,

00:00:17.427 --> 00:00:19.227
literally what they mean
and where they come from,

00:00:19.227 --> 00:00:21.873
we'll get a better
understanding of this concept.

00:00:21.873 --> 00:00:24.136
So both of these words
come to us from Latin.

00:00:24.136 --> 00:00:26.691
Concrete comes to us
from the Latin concretus,

00:00:26.691 --> 00:00:29.105
which means to grow together.

00:00:29.105 --> 00:00:33.302
So this part of it means grown.

00:00:33.302 --> 00:00:36.136
And this part means together.

00:00:36.136 --> 00:00:37.351
It refers to something that, you know,

00:00:37.351 --> 00:00:40.555
has grown together and become thick

00:00:40.555 --> 00:00:43.456
and kind of hard to get
through and physical.

00:00:43.456 --> 00:00:46.474
The connotation here is that
this is a physical thing.

00:00:46.474 --> 00:00:50.773
Something that is concrete is physical.

00:00:50.773 --> 00:00:55.773
Abstract, on the other hand,
means to draw something away.

00:01:00.121 --> 00:01:05.112
So something that is abstract
is drawn away from the real,

00:01:05.112 --> 00:01:07.098
from the concrete, from the physical.

00:01:07.098 --> 00:01:09.498
So this is not physical.

00:01:13.671 --> 00:01:15.773
And we make this distinction in English

00:01:15.773 --> 00:01:17.846
when we're talking about nouns.

00:01:17.846 --> 00:01:19.197
Is it something that is concrete,

00:01:19.197 --> 00:01:21.449
is it something you can look at or pick up

00:01:21.449 --> 00:01:24.793
or smell or sense or
something that is abstract,

00:01:24.793 --> 00:01:26.569
something that isn't physical,

00:01:26.569 --> 00:01:28.530
but can still be talked about.

00:01:28.530 --> 00:01:33.222
So for example, the word sadness...

00:01:34.727 --> 00:01:36.577
Is a noun, right?

00:01:36.577 --> 00:01:37.557
This is definitely a noun.

00:01:37.557 --> 00:01:40.333
It's got this noun-making ending,

00:01:40.333 --> 00:01:42.934
this noun-forming suffix, ness.

00:01:42.934 --> 00:01:45.858
You know, we take the adjective sad

00:01:45.858 --> 00:01:50.235
and we toss this ness part
onto it, we've got a noun.

00:01:50.235 --> 00:01:52.452
But can you see sadness?

00:01:52.452 --> 00:01:53.696
Is it something you can pick up?

00:01:53.696 --> 00:01:56.574
Sure, you can tell by being, you know

00:01:56.574 --> 00:01:59.581
observant and empathetic
that your friend is sad,

00:01:59.581 --> 00:02:02.554
but it's not something you can pick up.

00:02:02.554 --> 00:02:06.858
You can't be like a
measurable degree of sad.

00:02:06.858 --> 00:02:08.936
You couldn't take someone's sadness,

00:02:08.936 --> 00:02:10.434
put it under a microscope and say

00:02:10.434 --> 00:02:14.370
"Oh, Roberta, you are
32 degrees microsad."

00:02:14.370 --> 00:02:16.795
You know, it's not something physical.

00:02:16.795 --> 00:02:18.653
Concrete things, on the other hand,

00:02:18.653 --> 00:02:22.751
are things that we can
see or count or measure.

00:02:22.751 --> 00:02:24.528
Just parts of the physical world.

00:02:24.528 --> 00:02:28.509
So anything you look at,
like a dog is concrete,

00:02:28.509 --> 00:02:33.509
a ball is concrete, a cliff is concrete.

00:02:35.005 --> 00:02:36.398
Happiness...

00:02:38.087 --> 00:02:39.045
Is abstract.

00:02:39.045 --> 00:02:40.862
The idea of freedom...

00:02:43.189 --> 00:02:44.210
Is abstract.

00:02:44.210 --> 00:02:47.134
Though the presence of
freedom in your life

00:02:47.134 --> 00:02:49.888
may manifest in physical objects, like

00:02:49.888 --> 00:02:52.848
"Oh, my parents let me have
the freedom to eat ice cream."

00:02:52.848 --> 00:02:57.525
Ice cream is, you know, a concrete noun.

00:02:57.525 --> 00:02:59.894
But freedom, the thing
that allows you, you know,

00:02:59.894 --> 00:03:03.004
the permission that you get from

00:03:03.004 --> 00:03:07.400
your parents to have ice cream.

00:03:07.400 --> 00:03:10.178
That's not a physical object.

00:03:10.178 --> 00:03:11.699
So that's basically the difference.

00:03:11.699 --> 00:03:14.753
So a concrete noun is a physical object

00:03:14.753 --> 00:03:16.900
and an abstract noun is not.

00:03:16.900 --> 00:03:18.088
This is why I really wanted to hit

00:03:18.088 --> 00:03:23.088
the idea that a noun can be a
person, place, thing or idea,

00:03:23.713 --> 00:03:25.973
because nouns can be ideas,

00:03:25.973 --> 00:03:28.316
and those ideas tend to be abstract.

00:03:28.316 --> 00:03:32.576
Sadness, happiness, freedom,
permission, liberty, injustice.

00:03:32.576 --> 00:03:37.510
All of these are abstract ideas.

00:03:37.510 --> 00:03:38.995
That's the difference.

00:03:38.995 --> 00:03:40.516
You can learn anything.

00:03:40.516 --> 00:03:41.777
David out.


Finding an in-between frame of reference

WEBVTT
Kind: captions
Language: en

00:00:00.413 --> 00:00:02.818
- [Voiceover] Let's say I'm
person A here in my ship

00:00:02.818 --> 00:00:05.882
traveling through the universe
at a constant velocity,

00:00:05.882 --> 00:00:08.091
so that is person A right over there.

00:00:08.091 --> 00:00:10.333
Let me write it a little bit bigger.

00:00:12.004 --> 00:00:13.406
Person A.

00:00:13.406 --> 00:00:17.478
And let's say that I
have a friend, Person B,

00:00:17.478 --> 00:00:19.183
and they are in another ship.

00:00:19.183 --> 00:00:21.878
And in my frame of reference,
so this is Person B,

00:00:21.878 --> 00:00:25.438
in my frame of reference,
they are traveling,

00:00:25.438 --> 00:00:28.094
so their velocity vector looks like this,

00:00:28.094 --> 00:00:31.854
they are traveling at 8/10
of the speed of light,

00:00:31.854 --> 00:00:35.247
0.8 c.

00:00:35.247 --> 00:00:36.456
And so once again this is all given

00:00:36.456 --> 00:00:39.929
in A's frame of reference
and my frame of reference.

00:00:39.929 --> 00:00:42.013
Now the question I have for you is

00:00:42.013 --> 00:00:45.606
surely there must be some third party,

00:00:45.606 --> 00:00:50.575
let's call them C for convenience.

00:00:50.575 --> 00:00:52.569
Surely there must be some third party

00:00:52.569 --> 00:00:54.165
or some third party frame of reference

00:00:54.165 --> 00:00:56.696
that we could imagine it
might be someone in some ship,

00:00:56.696 --> 00:01:00.214
where their velocity in
my frame of reference

00:01:00.214 --> 00:01:03.696
is in between being
stationary and traveling

00:01:03.696 --> 00:01:06.472
0.8 times the velocity of light.

00:01:06.472 --> 00:01:09.613
And even more, from
their frame of reference,

00:01:09.613 --> 00:01:13.289
A and B should be leaving
or going outward from them

00:01:13.289 --> 00:01:15.097
at the same velocity.

00:01:15.097 --> 00:01:16.615
So what am I talking about?

00:01:16.615 --> 00:01:19.203
So, this is...

00:01:20.261 --> 00:01:21.486
we're gonna call,

00:01:21.486 --> 00:01:22.876
I'll just call this
the frame of reference.

00:01:22.876 --> 00:01:25.979
So this first row is
A's frame of reference

00:01:25.979 --> 00:01:27.758
that we just described.

00:01:27.758 --> 00:01:31.212
And C is going to be going
with some velocity, V,

00:01:31.212 --> 00:01:34.761
away from A.

00:01:34.761 --> 00:01:37.739
So some velocity, V, we
haven't figured it out yet.

00:01:37.739 --> 00:01:40.111
Now let's think about
C's frame of reference.

00:01:40.111 --> 00:01:44.065
So frame of reference C, so
C and C's frame of reference

00:01:44.065 --> 00:01:46.231
is just going to be stationary,

00:01:46.231 --> 00:01:50.039
and we want to figure
out a V so that A and B

00:01:50.039 --> 00:01:54.263
are moving away from C
at the same velocity.

00:01:54.263 --> 00:01:59.114
So in this frame of
reference, A will look like,

00:01:59.114 --> 00:02:02.778
A will look like it's moving to the left

00:02:02.778 --> 00:02:05.151
with a velocity of negative,

00:02:05.151 --> 00:02:06.702
here it's magnitude is the same,

00:02:06.702 --> 00:02:08.171
but just in the other direction.

00:02:08.171 --> 00:02:12.723
And B will also be moving
away with velocity V.

00:02:12.723 --> 00:02:17.187
So B, right over here, is
going to be moving away

00:02:17.187 --> 00:02:19.275
with velocity V.

00:02:19.275 --> 00:02:20.952
So this is a really interesting question.

00:02:20.952 --> 00:02:23.684
Can we figure out what V is going to be?

00:02:23.684 --> 00:02:25.736
If we were dealing with
the Galilean world,

00:02:25.736 --> 00:02:28.134
you might say well V is
just going to be half way

00:02:28.134 --> 00:02:29.652
in between these two things.

00:02:29.652 --> 00:02:31.383
If we were just on the
highway in the Galilean

00:02:31.383 --> 00:02:35.423
or Newtonian world, and B
is going 80 miles per hour,

00:02:35.423 --> 00:02:38.604
and A is stationary, well
then if C goes half way,

00:02:38.604 --> 00:02:41.802
if C is going 40 miles per hour,
then from C's point of view

00:02:41.802 --> 00:02:44.585
it looks like A is going
backwards at 40 miles per hour,

00:02:44.585 --> 00:02:47.569
and it'll look like B is going
forward at 40 miles per hour.

00:02:47.569 --> 00:02:49.981
But we know by now that we don't deal,

00:02:49.981 --> 00:02:53.257
we aren't living in a
Newtonian or Galilean universe,

00:02:53.257 --> 00:02:57.351
we're living in one defined
by special relativity.

00:02:57.351 --> 00:02:59.568
So I encourage you to pause the video

00:02:59.568 --> 00:03:02.883
and figure out what this in
between frame of reference,

00:03:02.883 --> 00:03:06.278
what it's velocity needs
to be relative to A.

00:03:06.278 --> 00:03:09.016
And I'll give you a hint,
it's going to involve

00:03:09.016 --> 00:03:12.338
the Einstein Velocity Addition Formula.

00:03:12.338 --> 00:03:13.769
So let's work through this together.

00:03:13.769 --> 00:03:15.082
So I'm just gonna write down

00:03:15.082 --> 00:03:17.657
the Einstein Velocity Addition Formula.

00:03:17.657 --> 00:03:20.863
So it tells us that the change

00:03:20.863 --> 00:03:24.465
in x prime

00:03:24.465 --> 00:03:27.730
with respect to t prime is equal to

00:03:27.730 --> 00:03:29.969
u minus v

00:03:29.969 --> 00:03:33.238
over one minus uv

00:03:33.238 --> 00:03:35.422
over c squared.

00:03:35.422 --> 00:03:37.482
Now let's think about
how we might apply it.

00:03:37.482 --> 00:03:39.606
And the trick here is
to really think about it

00:03:39.606 --> 00:03:43.180
from C's frame of reference,

00:03:43.180 --> 00:03:46.461
it's to think about it from
C's frame of reference.

00:03:46.461 --> 00:03:49.521
So if you think about it
from C's frame of reference,

00:03:49.521 --> 00:03:54.062
you could say, you could
say that v right over there,

00:03:54.062 --> 00:03:56.005
let me put this in a different color,

00:03:56.005 --> 00:03:59.967
you could say that v is the velocity

00:03:59.967 --> 00:04:03.022
that A is moving away from C at,

00:04:03.022 --> 00:04:07.989
so velocity A

00:04:07.989 --> 00:04:09.463
moving...

00:04:10.425 --> 00:04:13.324
from C.

00:04:13.324 --> 00:04:17.293
And then you could say that u

00:04:17.293 --> 00:04:19.863
is the velocity that B is moving from C.

00:04:19.863 --> 00:04:23.343
Remember, we're dealing
in C's frame of reference.

00:04:23.343 --> 00:04:25.108
So velocity...

00:04:26.947 --> 00:04:30.011
velocity that B moving...

00:04:31.360 --> 00:04:32.594
from...

00:04:34.024 --> 00:04:36.772
from C, actually let me
make everything upper case.

00:04:36.772 --> 00:04:38.831
This should be upper case A,

00:04:38.831 --> 00:04:40.873
and this should be upper case C.

00:04:40.873 --> 00:04:45.536
And in that case, what is delta
x prime over delta t prime?

00:04:45.536 --> 00:04:49.572
Well, that would be the
velocity that B is moving

00:04:49.572 --> 00:04:52.026
away from A in A's frame of reference.

00:04:52.026 --> 00:04:53.300
I know that this can get
a little bit confusing,

00:04:53.300 --> 00:04:54.565
but I really want you to pause it,

00:04:54.565 --> 00:04:57.854
watch it in slow mo, really
think about what we're doing.

00:04:57.854 --> 00:05:01.509
I'm kind of starting now,
I know I started this video

00:05:01.509 --> 00:05:03.108
in A's frame of reference,
and this is really

00:05:03.108 --> 00:05:05.764
the trick of the problem
is I'm now shifting

00:05:05.764 --> 00:05:07.477
over to C's frame of reference.

00:05:07.477 --> 00:05:11.238
I'm like, okay, V is the
velocity A is moving away from C,

00:05:11.238 --> 00:05:15.408
B is the velocity, or u is the
velocity B is moving from C,

00:05:15.408 --> 00:05:17.797
and in that case we can view delta x prime

00:05:17.797 --> 00:05:21.520
over delta t prime as the
velocity that B is moving

00:05:21.520 --> 00:05:24.956
away from A in A's frame of reference.

00:05:24.956 --> 00:05:27.422
So B moving...

00:05:28.685 --> 00:05:31.325
moving from A

00:05:31.325 --> 00:05:32.878
in A's frame of reference.

00:05:32.878 --> 00:05:37.046
So in A's reference.

00:05:37.046 --> 00:05:40.049
And both of these are in C's reference.

00:05:40.049 --> 00:05:41.601
So let me write that down.

00:05:41.601 --> 00:05:45.776
In C's reference,

00:05:45.776 --> 00:05:48.489
in C's

00:05:48.489 --> 00:05:52.438
in C's reference.

00:05:52.438 --> 00:05:53.596
I really want you to think about this,

00:05:53.596 --> 00:05:56.145
this is a little confusing,
but hopefully this helps you

00:05:56.145 --> 00:05:59.090
appreciate how this Einstein
Addition, Velocity Addition

00:05:59.090 --> 00:06:00.599
can be valuable.

00:06:00.599 --> 00:06:04.394
Well now, we can substitute what we know.

00:06:04.394 --> 00:06:07.875
We know the velocity B
is moving away from A,

00:06:07.875 --> 00:06:10.740
it is 0.8 c.

00:06:10.740 --> 00:06:14.091
So we can write, we can write

00:06:14.091 --> 00:06:17.174
0.8 c

00:06:17.174 --> 00:06:19.626
is going to be equal to,

00:06:19.626 --> 00:06:23.635
is going to be equal to u,
the velocity B is moving

00:06:23.635 --> 00:06:26.184
from C in C's frame of reference.

00:06:26.184 --> 00:06:28.110
Well we say that this
is just going to be B,

00:06:28.110 --> 00:06:29.867
that's going to be a positive v.

00:06:29.867 --> 00:06:33.267
So that right over there is
going to be a positive v.

00:06:34.357 --> 00:06:36.488
Let me do that in,

00:06:36.488 --> 00:06:38.001
let me do that in the same color.

00:06:38.001 --> 00:06:39.598
It's going to be a positive v.

00:06:39.598 --> 00:06:43.749
And then from that, we are
going to subtract the velocity A

00:06:43.749 --> 00:06:47.230
is moving from C in
C's frame of reference.

00:06:47.230 --> 00:06:50.906
So A is moving from C with
the velocity negative v.

00:06:50.906 --> 00:06:53.302
I know it's kind of
confusing to replace a v

00:06:53.302 --> 00:06:55.922
with a negative v, but this
is a generalized formula,

00:06:55.922 --> 00:06:57.275
while this is the actually value

00:06:57.275 --> 00:06:59.037
that we're using in this case.

00:06:59.037 --> 00:07:02.871
So minus negative V all of that over

00:07:02.871 --> 00:07:06.791
all of that over one minus,

00:07:06.791 --> 00:07:11.736
all of that over one
minus the velocity of u

00:07:11.736 --> 00:07:14.018
times the velocity of v, well,

00:07:14.018 --> 00:07:17.039
or that's just going to
be, that's going to be v

00:07:17.039 --> 00:07:20.677
times negative v or negative v squared.

00:07:20.677 --> 00:07:24.007
So I'll just write that
as negative v squared

00:07:24.007 --> 00:07:26.048
over c squared.

00:07:26.048 --> 00:07:27.436
And just like that we have set it up

00:07:27.436 --> 00:07:28.913
so we can solve for v.

00:07:28.913 --> 00:07:31.662
And the key realization
is is that we said,

00:07:31.662 --> 00:07:34.285
"Okay, there must be some spaceship C

00:07:34.285 --> 00:07:36.169
"that defines a frame of reference

00:07:36.169 --> 00:07:39.432
"where A and B are moving away from it

00:07:39.432 --> 00:07:41.846
"with velocities of equal magnitude."

00:07:41.846 --> 00:07:43.359
And so we use that information

00:07:43.359 --> 00:07:45.406
to go into C's frame of reference

00:07:45.406 --> 00:07:48.279
and use the Einstein
Velocity Addition Formula.

00:07:48.279 --> 00:07:51.385
But instead of knowing what,

00:07:51.385 --> 00:07:53.678
instead of knowing what these are,

00:07:53.678 --> 00:07:55.300
and then solving for this,

00:07:55.300 --> 00:07:57.066
we know what this is, and we're assuming

00:07:57.066 --> 00:07:58.908
that these two have the same magnitude,

00:07:58.908 --> 00:08:01.524
and we're able to solve for v.

00:08:01.524 --> 00:08:03.530
And so let's do that right now.

00:08:03.530 --> 00:08:05.166
In fact, I will do that in the next video,

00:08:05.166 --> 00:08:06.270
so that we will have enough time.

00:08:06.270 --> 00:08:07.664
And I encourage you to solve for v

00:08:07.664 --> 00:08:10.586
on your own before you
watch the next video.


Calculating neutral velocity

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.862
- [Voiceover] Alright, we can now do

00:00:01.862 --> 00:00:03.626
the math to solve for v.

00:00:03.626 --> 00:00:06.320
So let me just simplify
the right hand side

00:00:06.320 --> 00:00:08.636
of this equation, v minus negative v.

00:00:08.687 --> 00:00:11.074
Well, that's just going to be two v.

00:00:11.230 --> 00:00:14.446
One minus negative of v
squared over c squared.

00:00:14.446 --> 00:00:18.539
Well that's just one plus
positive v squared over c squared.

00:00:18.939 --> 00:00:20.559
And let's see, what can we do next?

00:00:20.947 --> 00:00:24.227
Well we can multiply,
we want to solve for v.

00:00:24.510 --> 00:00:26.642
We can multiply both
sides of this equation by

00:00:26.740 --> 00:00:30.182
one plus v squared over c squared.

00:00:30.710 --> 00:00:31.819
So let's do that.

00:00:31.860 --> 00:00:34.931
One plus v squared over c squared.

00:00:35.539 --> 00:00:37.508
On the right hand side, those cancel.

00:00:37.793 --> 00:00:40.781
And on the left hand side,
we can distribute the 0.8

00:00:41.215 --> 00:00:43.904
and we will get 0.8,

00:00:44.119 --> 00:00:46.616
or we can distribute
the 0.8 c, I should say,

00:00:46.616 --> 00:00:51.616
0.8 c plus 0.8 c v squared over c squared.

00:00:53.034 --> 00:00:54.996
Well, the c in the
numerator is going to cancel

00:00:54.996 --> 00:00:56.384
with one of these c's in the denominator.

00:00:56.471 --> 00:01:01.471
So it's going to be plus
0.8 v squared over c.

00:01:04.203 --> 00:01:08.214
And then, it is going
to be equal to two v.

00:01:08.255 --> 00:01:10.885
And remember, we want to solve for v.

00:01:11.018 --> 00:01:13.404
So we're setting up
essentially a quadratic in v.

00:01:15.347 --> 00:01:19.349
So let's now find
ourselves some real estate,

00:01:19.349 --> 00:01:21.397
so let's go right over there.

00:01:21.809 --> 00:01:23.777
Let's subtract two v from both sides.

00:01:23.828 --> 00:01:26.296
And actually, I'm going to
write it in order of degree.

00:01:26.348 --> 00:01:31.348
So on the left hand side, I
have 0.8 over c v squared,

00:01:34.324 --> 00:01:37.615
and then minus two v, I
subtracted two v from both sides,

00:01:37.784 --> 00:01:41.515
Plus 0.8 c,

00:01:41.585 --> 00:01:46.585
plus 0.8 c is equal to zero.

00:01:48.058 --> 00:01:49.399
I guess if we want to
simplify a little bit,

00:01:49.405 --> 00:01:52.104
we can multiply both sides by c.

00:01:54.077 --> 00:01:55.833
If we multiply both sides
by c, this will become

00:01:55.898 --> 00:01:59.847
0.8 v squared

00:02:00.178 --> 00:02:05.178
minus two c v plus 0.8c
squared equals zero.

00:02:08.223 --> 00:02:10.801
And we could keep trying to
algebraically manipulate this,

00:02:10.801 --> 00:02:13.233
but we could just go straight
to the quadratic formula here

00:02:13.238 --> 00:02:14.313
to solve for v.

00:02:14.957 --> 00:02:16.576
I'll do that in a different
color just for kicks.

00:02:16.982 --> 00:02:19.956
v is going to be equal to negative b.

00:02:20.286 --> 00:02:23.113
So this right over here is b.

00:02:25.799 --> 00:02:27.235
This is our a.

00:02:27.983 --> 00:02:30.136
And this is our c.

00:02:30.258 --> 00:02:32.550
We're talking about
the different variables

00:02:32.555 --> 00:02:34.455
using the quadratic formula.

00:02:35.098 --> 00:02:36.753
So v is going to be negative b.

00:02:36.817 --> 00:02:38.448
So the negative of negative two c.

00:02:38.534 --> 00:02:41.884
So it's going to be two c,
plus or minus the square root

00:02:42.074 --> 00:02:46.551
of b squared, so negative
two c squared is positive

00:02:46.684 --> 00:02:51.486
four c squared, minus four, times a,

00:02:51.583 --> 00:02:56.583
so minus four times 0.8 times
c, so times 0.8 c squared.

00:03:03.951 --> 00:03:08.951
All of that over two a,
so 0.8 times two is 1.6.

00:03:10.796 --> 00:03:13.496
Now let's see this is
going to be equal to two c

00:03:13.756 --> 00:03:16.084
plus or minus the square root --

00:03:16.322 --> 00:03:18.905
now I can factor out a four c squared.

00:03:19.154 --> 00:03:24.154
Four c squared times one
minus, 0.8 times 0.8 is 0.64.

00:03:29.080 --> 00:03:32.129
All I did was factor out a
four c squared from both terms.

00:03:32.400 --> 00:03:35.786
Of course, all of that over 1.6

00:03:36.162 --> 00:03:37.642
Scroll down a little bit.

00:03:37.962 --> 00:03:39.383
And this is going to be equal to --

00:03:39.400 --> 00:03:40.730
this is all algebra at this point --

00:03:40.782 --> 00:03:42.844
two c plus or minus --

00:03:42.952 --> 00:03:45.885
if I take the four c
squared out of the radical,

00:03:45.993 --> 00:03:48.646
its going to be two c.

00:03:48.872 --> 00:03:52.083
So two c plus or minus two
c times the square root

00:03:52.112 --> 00:03:53.883
of one minus 0.64.

00:03:54.040 --> 00:03:56.450
Well, that's 0.36.

00:03:56.953 --> 00:03:58.278
Things are getting nicely simple now.

00:03:58.278 --> 00:04:00.686
All over 1.6.

00:04:01.133 --> 00:04:05.005
The square root of 0.36, the
principle root of that 0.6.

00:04:05.312 --> 00:04:07.890
So that's 0.6.

00:04:08.904 --> 00:04:10.001
And move down a little bit.

00:04:10.333 --> 00:04:12.741
So this is going to be equal to two c --

00:04:13.049 --> 00:04:14.517
and I can see the end --

00:04:14.604 --> 00:04:19.604
plus or minus 1.2 c, all of that over 1.6.

00:04:23.335 --> 00:04:25.581
So we got two possible values for v,

00:04:25.667 --> 00:04:29.730
but we can tell if we add this up here,

00:04:29.730 --> 00:04:34.014
you're going to end up
with 3.2 c divided by 1.6,

00:04:34.014 --> 00:04:36.528
which is going to be a speed, a velocity,

00:04:36.534 --> 00:04:38.407
greater than the speed of light.

00:04:38.716 --> 00:04:41.149
So we can rule out the
positive version of it.

00:04:41.293 --> 00:04:42.599
So we know the answer's going

00:04:42.627 --> 00:04:44.084
have to be negative version of it.

00:04:44.114 --> 00:04:47.998
So two c minus 1.2 c over 1.6,

00:04:48.177 --> 00:04:53.177
which is equal to 0.8 c over 1.6.

00:04:54.754 --> 00:04:59.754
Well 0.8 is half of 1.6, so this is 0.5 c.

00:05:00.626 --> 00:05:02.803
That was really, really, really cool.

00:05:02.947 --> 00:05:04.254
Because what do we know now?

00:05:04.468 --> 00:05:09.096
We know now that A in
A's frame of reference

00:05:09.275 --> 00:05:11.301
feels like it is stationary.

00:05:11.493 --> 00:05:14.121
We have A's friend moving
with a relative velocity

00:05:14.150 --> 00:05:15.724
of eight-tenths of the speed of light.

00:05:15.973 --> 00:05:17.802
There could be a third party, C,

00:05:17.808 --> 00:05:20.403
that defines a frame of
reference moving away from A,

00:05:20.408 --> 00:05:23.038
with the velocity of
half the speed of light.

00:05:24.054 --> 00:05:26.393
Half the speed of light.

00:05:26.630 --> 00:05:30.397
And from A's frame of reference,
it looks like C's velocity

00:05:30.426 --> 00:05:32.522
is closer to B's than it is to A's.

00:05:32.608 --> 00:05:34.849
But we're dealing in this relativistic,

00:05:34.849 --> 00:05:39.469
this Einstein-ian world
because if we then go into

00:05:39.469 --> 00:05:42.321
C's frame of reference, it
actually looks like A and B

00:05:42.347 --> 00:05:46.731
are both leaving C with a speed of 0.5 c.

00:05:49.692 --> 00:05:52.351
So here you can view the
velocity as positive 0.5 c,

00:05:52.576 --> 00:05:57.576
and here you could say the
velocity is negative 0.5 c.

00:06:01.097 --> 00:06:01.951
This was really cool.

00:06:01.979 --> 00:06:03.523
We were able to find a frame of reference

00:06:03.523 --> 00:06:05.085
that you could think of
it as right in between.

00:06:05.440 --> 00:06:07.146
Now what's going to be
really cool about this,

00:06:07.146 --> 00:06:08.684
is that we can actually --

00:06:08.794 --> 00:06:11.162
you know what some people
don't like about Minkowski

00:06:11.162 --> 00:06:14.350
space-time diagrams is
that it looks asymmetric.

00:06:14.541 --> 00:06:17.455
Even though B is moving
in A's frame of reference

00:06:17.455 --> 00:06:19.777
with a velocity of positive eight-tenths

00:06:19.777 --> 00:06:20.798
of the speed of light,

00:06:20.798 --> 00:06:22.237
if you were in B's frame of reference,

00:06:22.237 --> 00:06:25.994
A is moving to the left
with a speed of 0.8 times c.

00:06:26.532 --> 00:06:29.568
But now we can define a
neutral frame of reference,

00:06:29.620 --> 00:06:31.594
where they're both moving
in different directions

00:06:31.594 --> 00:06:33.109
with the same speed.

00:06:33.219 --> 00:06:36.435
Which actually makes
interpreting the space-time

00:06:36.435 --> 00:06:37.764
diagram a little bit easier.

00:06:37.781 --> 00:06:39.337
We will do that in future videos.

00:06:39.337 --> 00:06:42.350
But this was a fun
problem in it of itself.