Uncle Tom's Cabin part 2

WEBVTT
Kind: captions
Language: en

00:00:00.285 --> 00:00:02.108
- [Voiceover] So Becca and
I have been talking about

00:00:02.108 --> 00:00:06.144
Uncle Tom's Cabin which is
this book from the 1850s

00:00:06.144 --> 00:00:10.490
that Abraham Lincoln actually
said started the Civil War.

00:00:10.490 --> 00:00:13.406
So how did this book start a war?

00:00:13.406 --> 00:00:16.065
So in this video we'll tell
you more about the plot.

00:00:16.065 --> 00:00:18.413
But in the previous video
we kind of discussed

00:00:18.413 --> 00:00:20.467
what was going on in
the country at the time

00:00:20.467 --> 00:00:23.039
and Harriet Beecher
Stowe again was from this

00:00:23.039 --> 00:00:24.546
abolitionist family.

00:00:24.546 --> 00:00:27.358
She was really deeply effected
by the Compromise of 1850

00:00:27.358 --> 00:00:31.369
and the Fugitive Slave Act,
and also by slave auctions.

00:00:31.369 --> 00:00:33.215
So this video will get
a little bit more into

00:00:33.215 --> 00:00:35.425
the heart of the plot
of the novel which does

00:00:35.425 --> 00:00:38.616
have to do with the
family being torn apart.

00:00:38.616 --> 00:00:41.917
- [Voiceover] So what was Uncle
Tom's Cabin actually about?

00:00:41.917 --> 00:00:44.641
Uncle Tom's Cabin was set
on a plantation in Kentucky

00:00:44.641 --> 00:00:48.309
and it starts out with this
kind of group of slaves

00:00:48.309 --> 00:00:52.707
that are about to be sold
to other plantation owners.

00:00:52.707 --> 00:00:56.216
So Eliza and her son actually run away.

00:00:56.216 --> 00:00:59.606
They run up North, so Eliza goes up North

00:00:59.606 --> 00:01:02.773
and Uncle Toms is sold down the river.

00:01:04.535 --> 00:01:06.707
- [Voiceover] So, Eliza
is trying to make sure

00:01:06.707 --> 00:01:10.338
that she and her son are
not separated by being sold.

00:01:10.338 --> 00:01:12.033
So she decides that she is going to escape

00:01:12.033 --> 00:01:15.021
and take her son with her up to Canada.

00:01:15.021 --> 00:01:17.228
But Uncle Tom, he's not actually related,

00:01:17.228 --> 00:01:19.902
he is sold in the opposite direction.

00:01:19.902 --> 00:01:22.054
He's getting farther away from freedom

00:01:22.054 --> 00:01:24.836
by heading down the Mississippi.

00:01:24.836 --> 00:01:27.557
When you think about
the geography of slavery

00:01:27.557 --> 00:01:30.155
it's a much more urban
environment than some of

00:01:30.155 --> 00:01:32.395
the more coastal areas.

00:01:32.395 --> 00:01:34.793
So you might be in
Charleston, or you might even

00:01:34.793 --> 00:01:36.906
be in Baltimore as an enslaved person.

00:01:36.906 --> 00:01:39.144
You might have a pretty
high degree of freedom

00:01:39.144 --> 00:01:42.590
and also a possibility of escape either

00:01:42.590 --> 00:01:45.402
by crossing the border or by boat.

00:01:45.402 --> 00:01:49.569
When you're sold into the
deep South area you are deep

00:01:50.635 --> 00:01:52.993
in plantation country
and there might not be

00:01:52.993 --> 00:01:57.084
another soul that you could
rely on to help you escape

00:01:57.084 --> 00:01:58.667
for 100, 200 miles.

00:02:00.992 --> 00:02:02.316
- [Voiceover] And I think
this is really something

00:02:02.316 --> 00:02:05.253
that Harriet Beecher Stowe
wants to help point out

00:02:05.253 --> 00:02:08.046
in the book, that there
was this sense of doom

00:02:08.046 --> 00:02:09.323
for Uncle Tom.

00:02:09.323 --> 00:02:11.420
However, his Christian
faith was the only thing

00:02:11.420 --> 00:02:13.003
that really kept him going.

00:02:13.003 --> 00:02:16.138
He bonded with this young
white woman that he met, Eva,

00:02:16.138 --> 00:02:17.717
just about their Christian faith.

00:02:17.717 --> 00:02:19.377
And really, reading
his Bible was the thing

00:02:19.377 --> 00:02:21.128
that got him up in the morning.

00:02:21.128 --> 00:02:25.053
So where were those feelings
about religion coming from?

00:02:25.053 --> 00:02:26.481
- [Voiceover] You can definitely see

00:02:26.481 --> 00:02:28.926
that Harriet Beecher Stowe is influenced

00:02:28.926 --> 00:02:31.686
by her own family's
faith which is influenced

00:02:31.686 --> 00:02:34.281
by the Second Great Awakening.

00:02:34.281 --> 00:02:36.470
The Second Great Awakening
was this flowering

00:02:36.470 --> 00:02:40.053
of religious belief in
the 1830s and 1840s.

00:02:41.424 --> 00:02:44.939
It was kind of a reaction
against the Era of Enlightenment

00:02:44.939 --> 00:02:47.350
which was what had inspired the founders

00:02:47.350 --> 00:02:51.531
of the United States to think
of a more humanist world,

00:02:51.531 --> 00:02:54.064
a more rational, scientific world.

00:02:54.064 --> 00:02:55.733
People start going to camp meetings,

00:02:55.733 --> 00:02:58.129
they have religious revivals.

00:02:58.129 --> 00:03:00.734
They experience religious conversions.

00:03:00.734 --> 00:03:04.740
And in this time period
there's a shift in thinking

00:03:04.740 --> 00:03:06.991
about God in the United States.

00:03:06.991 --> 00:03:08.511
If you think back to
the puritans they have

00:03:08.511 --> 00:03:12.823
this incredibly punitive
sort of Old Testament

00:03:12.823 --> 00:03:15.341
destroyer God, right?

00:03:15.341 --> 00:03:19.327
One of the most famous early
sermons in the United States

00:03:19.327 --> 00:03:21.699
is sinners in the hands of an angry God,

00:03:21.699 --> 00:03:24.782
that at any moment God might
release you into the flames.

00:03:24.782 --> 00:03:28.699
Well there's a new emphasis
on Christ-like love

00:03:29.624 --> 00:03:31.791
in the early 1830s, 1840s.

00:03:33.096 --> 00:03:36.760
New interpretation of
God as being forgiving

00:03:36.760 --> 00:03:41.410
and gentle, family oriented,
it's very Victorian.

00:03:41.410 --> 00:03:44.261
Where God was seen as this punisher

00:03:44.261 --> 00:03:47.922
who condemned most people to hell,

00:03:47.922 --> 00:03:50.730
in the Second Great Awakening
there's a new emphasis

00:03:50.730 --> 00:03:54.230
on a forgiving, kind family-oriented Jesus

00:03:56.742 --> 00:03:58.551
who will save everyone.

00:03:58.551 --> 00:04:01.901
That's very incompatible
with the ideas of slavery.

00:04:01.901 --> 00:04:04.463
- [Voiceover] Exactly, and I
think that Uncle Tom's Cabin

00:04:04.463 --> 00:04:06.048
can really be considered a part

00:04:06.048 --> 00:04:07.327
of the Second Great Awakening

00:04:07.327 --> 00:04:09.642
because of the way that it
points out these fundamental

00:04:09.642 --> 00:04:14.097
inconsistencies and contradictions
between Christian faith

00:04:14.097 --> 00:04:15.580
and human bondage.

00:04:15.580 --> 00:04:18.190
How could a religion that
says treat thy neighbor

00:04:18.190 --> 00:04:21.679
as thyself actually sanction slavery?

00:04:21.679 --> 00:04:25.401
- [Voiceover] So Uncle Tom is
this martyr character, right?

00:04:25.401 --> 00:04:28.651
He is a devout believer in Christianity

00:04:29.550 --> 00:04:33.122
and the forgiveness of God right up until

00:04:33.122 --> 00:04:34.614
his very end.

00:04:34.614 --> 00:04:37.676
So how does Uncle Tom's
Cabin actually end?

00:04:37.676 --> 00:04:40.977
- [Voiceover] Uncle Tom's
Cabin ends with Uncle Tom

00:04:40.977 --> 00:04:42.677
is beaten by his overseers.

00:04:42.677 --> 00:04:46.161
He's sold through this chain
of different slave families

00:04:46.161 --> 00:04:50.189
in the deep South and he
ends up with just a terrible,

00:04:50.189 --> 00:04:54.213
terrible slave holder
who requests his death,

00:04:54.213 --> 00:04:56.051
actually partially because he was reading

00:04:56.051 --> 00:04:59.134
all of this religious text.

00:04:59.134 --> 00:05:01.676
[Voiceover] And this slave
owner was named Simon Legree.

00:05:01.676 --> 00:05:04.105
And this name, Simon Legree,
has actually stuck with us

00:05:04.105 --> 00:05:07.944
in popular culture to
mean a really evil, cruel,

00:05:07.944 --> 00:05:09.687
punitive master.

00:05:09.687 --> 00:05:12.286
- [Voiceover] The rest of the
family actually meets back up.

00:05:12.286 --> 00:05:14.867
Eliza is reunited with
a bunch of other people

00:05:14.867 --> 00:05:17.027
that were on the original
plantation and they really

00:05:17.027 --> 00:05:20.711
think about Uncle Tom as this
martyr, the hero's death.

00:05:20.711 --> 00:05:25.190
He's looked at as this sacrifice
for the cause of freedom.

00:05:25.190 --> 00:05:29.349
- [Voiceover] Right, also
Uncle Tom, he dies never having

00:05:29.349 --> 00:05:31.118
renounced his Christian faith.

00:05:31.118 --> 00:05:34.535
His example of martyrdom
actually leads everyone

00:05:34.535 --> 00:05:37.385
who witnesses his death,
including Simon Legree,

00:05:37.385 --> 00:05:40.968
to convert to Christianity
and to vow never

00:05:42.216 --> 00:05:44.610
to hold slaves again.

00:05:44.610 --> 00:05:46.380
- [Voiceover] But I think
the ending of the book

00:05:46.380 --> 00:05:48.567
really points out this main theme within

00:05:48.567 --> 00:05:51.512
a lot of Second Great
Awakening texts which was that

00:05:51.512 --> 00:05:55.356
if you just paid attention
to how you are falling away

00:05:55.356 --> 00:05:58.237
from your Christian
commitments then you could get

00:05:58.237 --> 00:06:01.151
back on track and maybe
bring people together

00:06:01.151 --> 00:06:04.992
by utilizing Christian
faith in a productive

00:06:04.992 --> 00:06:06.762
and public way.

00:06:06.762 --> 00:06:09.156
- [Voiceover] The book
is published in 1852,

00:06:09.156 --> 00:06:10.656
then what happens?

00:06:11.750 --> 00:06:13.937
How do people receive this book?

00:06:13.937 --> 00:06:16.541
- [Voiceover] We'll talk about
this Tom-mania that ensues

00:06:16.541 --> 00:06:18.041
in the next video.


Uncle Tom's Cabin part 1

WEBVTT
Kind: captions
Language: en

00:00:00.218 --> 00:00:01.262
- [Voiceover] Hey, Becca.

00:00:01.262 --> 00:00:02.293
- [Voiceover] Hi, Kim.

00:00:02.293 --> 00:00:03.745
- [Voiceover] Alright, so
we're here to talk about

00:00:03.745 --> 00:00:05.316
Uncle Tom's Cabin, and I think this is

00:00:05.316 --> 00:00:08.508
such an interesting book
because when Abraham Lincoln

00:00:08.508 --> 00:00:11.291
met Harriet Beecher Stowe, he said to her,

00:00:11.291 --> 00:00:14.735
"So you're the little lady
that started this great war."

00:00:14.735 --> 00:00:18.220
He said Uncle Tom's Cabin
actually started the Civil War.

00:00:18.220 --> 00:00:21.425
So how does a book start a war?

00:00:21.425 --> 00:00:22.731
- [Voiceover] I think that's
a really good question, Kim,

00:00:22.731 --> 00:00:24.472
and these next two
videos are gonna help us

00:00:24.472 --> 00:00:27.194
understand a little bit
more why Lincoln said that.

00:00:27.194 --> 00:00:29.477
How does a little book start a war?

00:00:29.477 --> 00:00:31.744
So this book was written
by Harriet Beecher Stowe,

00:00:31.744 --> 00:00:33.327
here she is, Stowe,

00:00:34.188 --> 00:00:35.657
and Harriet Beecher Stowe was born

00:00:35.657 --> 00:00:38.652
in Litchfield, Connecticut to this kind of

00:00:38.652 --> 00:00:40.552
great abolitionist family.

00:00:40.552 --> 00:00:42.379
So what's abolitionism, Kim?

00:00:42.379 --> 00:00:45.192
- [Voiceover] Well
abolitionism was the belief

00:00:45.192 --> 00:00:47.832
in mostly in the early 19th century

00:00:47.832 --> 00:00:51.187
that slavery should be ended immediately.

00:00:51.187 --> 00:00:53.914
So there were varieties of beliefs

00:00:53.914 --> 00:00:57.143
about the institution of
slavery in early America.

00:00:57.143 --> 00:00:59.365
Some people obviously
were very pro-slavery

00:00:59.365 --> 00:01:02.434
believe that it was a natural institution

00:01:02.434 --> 00:01:04.201
sanctioned by the Bible.

00:01:04.201 --> 00:01:06.472
Some people, like Abraham Lincoln,

00:01:06.472 --> 00:01:08.408
at least early in his political career,

00:01:08.408 --> 00:01:10.604
just wanted slavery to stay where it was,

00:01:10.604 --> 00:01:13.593
and those were what we
would call free-soilers,

00:01:13.593 --> 00:01:16.148
or anti-slavery advocates.

00:01:16.148 --> 00:01:17.826
They said, "Alright, we
can't get rid of slavery

00:01:17.826 --> 00:01:19.082
"in the South.

00:01:19.082 --> 00:01:22.401
"It's too entrenched
there as an institution,

00:01:22.401 --> 00:01:25.020
"but we can make sure
that it does not spread

00:01:25.020 --> 00:01:27.124
"to any of the Western territories

00:01:27.124 --> 00:01:29.411
"that we might settle in the future."

00:01:29.411 --> 00:01:34.272
But abolitionists were these
strongest opponents of slavery.

00:01:34.272 --> 00:01:38.590
They said that slavery
should be ended today

00:01:38.590 --> 00:01:41.876
everywhere in the United
States and the world,

00:01:41.876 --> 00:01:46.043
and that it is an immoral,
un-Christian institution.

00:01:47.193 --> 00:01:50.552
So these Western territories
were a really big part

00:01:50.552 --> 00:01:54.117
of the increasing tension
over the institution

00:01:54.117 --> 00:01:56.344
of slavery in the 1850s.

00:01:56.344 --> 00:01:59.272
So in 1848, the United States won

00:01:59.272 --> 00:02:02.603
the Mexican-American War
and they got a whole bunch

00:02:02.603 --> 00:02:05.861
of new territory that
had once been Mexico,

00:02:05.861 --> 00:02:09.151
and these will become the states of Texas,

00:02:09.151 --> 00:02:12.132
and Oklahoma, and many of the sort of

00:02:12.132 --> 00:02:14.743
Midwestern states we have today,

00:02:14.743 --> 00:02:18.029
but this now threatened
the balance of power

00:02:18.029 --> 00:02:21.706
between those slave-holding
states in US Congress

00:02:21.706 --> 00:02:24.036
and those that were free states,

00:02:24.036 --> 00:02:26.702
so now everyone is wondering is slavery

00:02:26.702 --> 00:02:28.436
going to spread to the West?

00:02:28.436 --> 00:02:30.525
Should slavery spread to the West?

00:02:30.525 --> 00:02:32.781
- [Voiceover] And this
kind of anxiety about

00:02:32.781 --> 00:02:36.256
the Western expansion of
slavery was more tense

00:02:36.256 --> 00:02:39.476
and became more sectionally divided

00:02:39.476 --> 00:02:41.508
after the Compromise of 1850.

00:02:41.508 --> 00:02:45.894
So the Compromise of 1850
happened right here in 1850,

00:02:45.894 --> 00:02:47.227
(laughter)

00:02:47.227 --> 00:02:50.324
and the Compromise of
1850, I like to think of it

00:02:50.324 --> 00:02:53.892
kind of like a band-aid
over this sectional tension,

00:02:53.892 --> 00:02:55.407
so I'll draw you guys a little band-aid.

00:02:55.407 --> 00:02:57.981
- [Voiceover] This is like
a gaping wound, right,

00:02:57.981 --> 00:03:00.382
and the Compromise of 1850 is just like

00:03:00.382 --> 00:03:02.135
this tiny, little band-aid that's kind of

00:03:02.135 --> 00:03:05.359
holding this dam together
to mix my metaphors.

00:03:05.359 --> 00:03:06.536
- [Voiceover] The Compromise of 1850

00:03:06.536 --> 00:03:09.588
actually admitted
California as a free state,

00:03:09.588 --> 00:03:12.349
which was a really big win
for the North, obviously.

00:03:12.349 --> 00:03:13.212
- [Voiceover] Right, lots of gold.

00:03:13.212 --> 00:03:15.428
- [Voiceover] But it
also had a really strong

00:03:15.428 --> 00:03:17.821
Fugitive Slave Act, so this was a really

00:03:17.821 --> 00:03:21.645
kind of critical part of
the Compromise of 1850,

00:03:21.645 --> 00:03:24.093
and this was a big win for the South.

00:03:24.093 --> 00:03:26.127
So why was it a big win?

00:03:26.127 --> 00:03:27.895
- [Voiceover] Well the Fugitive Slave Act

00:03:27.895 --> 00:03:31.139
said that if a marshal was in your town

00:03:31.139 --> 00:03:34.893
requesting your help in
rounding up an escaped slave,

00:03:34.893 --> 00:03:38.904
you had to help that marshal
or face charges yourself.

00:03:38.904 --> 00:03:41.684
So this meant that any time that someone

00:03:41.684 --> 00:03:45.525
who was enslaved in the South
made a run for the North,

00:03:45.525 --> 00:03:48.927
a run for Canada as many
of the enslaved people did,

00:03:48.927 --> 00:03:52.186
anyone in the North might be drafted

00:03:52.186 --> 00:03:54.883
to help return that person to the South.

00:03:54.883 --> 00:03:55.813
- [Voiceover] And if they didn't,

00:03:55.813 --> 00:03:58.332
they were oftentimes
fined, and this really made

00:03:58.332 --> 00:04:01.300
all Northerners participatory in slavery,

00:04:01.300 --> 00:04:03.786
even if they weren't
slaveholders themselves

00:04:03.786 --> 00:04:05.871
or living on a plantation in the South,

00:04:05.871 --> 00:04:08.466
Northerners were participating in the way

00:04:08.466 --> 00:04:12.134
that slavery was held
together by disallowing

00:04:12.134 --> 00:04:14.752
runaway slaves from continuing their lives

00:04:14.752 --> 00:04:16.443
in free territories.

00:04:16.443 --> 00:04:17.875
- [Voiceover] So you could imagine how

00:04:17.875 --> 00:04:21.068
this might really galvanize
a Northern audience

00:04:21.068 --> 00:04:23.673
into action about slavery because before,

00:04:23.673 --> 00:04:26.649
you might think, "Well,
I don't like slavery,

00:04:26.649 --> 00:04:29.418
"but what does it have
to do with me, right?

00:04:29.418 --> 00:04:32.755
"I'm just a grain miller
living in Pennsylvania.

00:04:32.755 --> 00:04:33.970
"None of my business.

00:04:33.970 --> 00:04:36.554
"I don't like it, but I
can't do anything about it,

00:04:36.554 --> 00:04:38.296
"and it's not my fault."

00:04:38.296 --> 00:04:41.431
Now all of a sudden, if an escaped slave

00:04:41.431 --> 00:04:46.117
comes past your house and a
marshal follows him or her,

00:04:46.117 --> 00:04:49.323
now you've got to be a person
to round that person up,

00:04:49.323 --> 00:04:51.871
and so that means you have to participate

00:04:51.871 --> 00:04:54.190
in slavery directly, and so you might find

00:04:54.190 --> 00:04:56.638
yourself thinking, "You know
what, I refuse to do that,

00:04:56.638 --> 00:04:59.337
"and that means that I
really do hate slavery."

00:04:59.337 --> 00:05:01.092
- [Voiceover] And this was
definitely the sentiment

00:05:01.092 --> 00:05:05.074
that Stowe and her family had
on the Underground Railroad.

00:05:05.074 --> 00:05:08.246
So Stowe lived on a stop in
the Underground Railroad,

00:05:08.246 --> 00:05:11.199
and that was this passageway
for Southern slaves

00:05:11.199 --> 00:05:14.259
to get to the North, and
Stowe and her husband

00:05:14.259 --> 00:05:16.187
actually helped a lot of runaway slaves.

00:05:16.187 --> 00:05:17.488
- [Voiceover] So the Underground Railroad

00:05:17.488 --> 00:05:19.419
wasn't like a literal railroad, right?

00:05:19.419 --> 00:05:20.896
I mean that would be pretty sweet

00:05:20.896 --> 00:05:24.298
if there were a railroad
that went under the ground

00:05:24.298 --> 00:05:26.107
all the way up to Canada,
but it was more like

00:05:26.107 --> 00:05:29.405
a sort of an informal network of people

00:05:29.405 --> 00:05:32.140
who might help escaped slaves, direct them

00:05:32.140 --> 00:05:35.378
to food and shelter, and
just kind of send them

00:05:35.378 --> 00:05:38.026
along to the next waypost on their trip,

00:05:38.026 --> 00:05:39.764
either to the North or to Canada.

00:05:39.764 --> 00:05:41.659
- [Voiceover] And so when
the Fugitive Slave Act

00:05:41.659 --> 00:05:44.012
was passed with the Compromise of 1850,

00:05:44.012 --> 00:05:47.356
the band-aid, this really
upset Harriet Beecher Stowe

00:05:47.356 --> 00:05:49.435
and really was one of the main catalysts

00:05:49.435 --> 00:05:51.389
for her writing this book.

00:05:51.389 --> 00:05:53.073
She also witnessed a slave auction,

00:05:53.073 --> 00:05:55.376
and this Harriet Beecher Stowe wrote about

00:05:55.376 --> 00:05:58.587
as just this terrible kind of scene

00:05:58.587 --> 00:06:00.774
of a family being just torn apart,

00:06:00.774 --> 00:06:03.486
and this was a really common
practice within slavery,

00:06:03.486 --> 00:06:06.181
that the unit of the
family was not respected

00:06:06.181 --> 00:06:09.327
as slaveholders wanted
to sell their slaves

00:06:09.327 --> 00:06:12.147
to different plantations
throughout the South,

00:06:12.147 --> 00:06:15.213
and the slave auction
really became the basis

00:06:15.213 --> 00:06:17.661
for the plot of Uncle Tom's Cabin.

00:06:17.661 --> 00:06:20.794
- [Voiceover] Slave auctions
were absolutely terrible.

00:06:20.794 --> 00:06:24.044
In fact, not long before the Civil War,

00:06:25.214 --> 00:06:29.016
the main slave auction
site in Washington, D.C.

00:06:29.016 --> 00:06:31.155
was just around the corner
from the White House,

00:06:31.155 --> 00:06:34.933
so imagine walking down the thoroughfare

00:06:34.933 --> 00:06:39.166
of this great democracy,
seeing the president's house,

00:06:39.166 --> 00:06:41.372
the seat of government,
and then turning a corner

00:06:41.372 --> 00:06:44.711
and seeing people being
sold off the block.

00:06:44.711 --> 00:06:46.971
You know Abraham Lincoln
saw a slave auction

00:06:46.971 --> 00:06:49.503
in New Orleans and he said
it was one of the things

00:06:49.503 --> 00:06:52.816
that most influenced him to hate slavery,

00:06:52.816 --> 00:06:55.579
just witnessing these
families being torn apart.

00:06:55.579 --> 00:06:59.928
And imagine either watching
a mother being sold

00:06:59.928 --> 00:07:03.885
away from her infant
children, or being that mother

00:07:03.885 --> 00:07:05.135
wondering what it would be like

00:07:05.135 --> 00:07:07.395
if you're ever going to see them again.

00:07:07.395 --> 00:07:08.970
- [Voiceover] I think that's
a really important point

00:07:08.970 --> 00:07:10.459
just to show that this was something

00:07:10.459 --> 00:07:12.517
that was happening all
around the United States

00:07:12.517 --> 00:07:15.432
and this was just abolitionist
fervor was bubbling up,

00:07:15.432 --> 00:07:19.059
and then in 1852, when
this book was published,

00:07:19.059 --> 00:07:21.965
it really set into motion this new wave

00:07:21.965 --> 00:07:25.061
of political rhetoric, and other novels,

00:07:25.061 --> 00:07:27.943
and just a lot of talk
about these fundamental

00:07:27.943 --> 00:07:32.592
contradictions between
Christianity and human bondage.

00:07:32.592 --> 00:07:34.178
- [Voiceover] And we'll get
to that in the next video.


Multiplying fractions by whole numbers word problem

WEBVTT
Kind: captions
Language: en

00:00:00.655 --> 00:00:03.634
- [Voiceover] Rishi spend
3/4 of an hour for two days

00:00:03.634 --> 00:00:05.553
working on his science project.

00:00:05.553 --> 00:00:08.353
Kyle spent 1/4 of an hour for six days

00:00:08.353 --> 00:00:10.510
working on his science project.

00:00:10.510 --> 00:00:14.385
Who spent more time on
his science project?

00:00:14.385 --> 00:00:17.434
So we wanna know who spent more time,

00:00:17.434 --> 00:00:19.178
and to do that we need to first know

00:00:19.178 --> 00:00:20.549
the amount of time Rishi and Kyle

00:00:20.549 --> 00:00:22.948
each spent on their science projects.

00:00:22.948 --> 00:00:27.746
So let's start with Rishi, and
see what we know about Rishi.

00:00:27.746 --> 00:00:31.306
Rishi spent 3/4 of an hour

00:00:31.306 --> 00:00:33.556
for two days, so two times,

00:00:34.941 --> 00:00:37.580
he spent 3/4 of an hour.

00:00:37.580 --> 00:00:39.091
Or another way we could say this,

00:00:39.091 --> 00:00:41.739
is he spent 3/4 of an hour once,

00:00:41.739 --> 00:00:46.213
plus on a second day he
spent another 3/4 of an hour.

00:00:46.213 --> 00:00:49.152
So 3/4 plus another 3/4

00:00:49.152 --> 00:00:53.319
will give us a total of
6/4, or six quarter hours,

00:00:54.772 --> 00:00:58.457
that Rishi spent on his science project.

00:00:58.457 --> 00:01:00.814
Now let's look at Kyle.

00:01:00.814 --> 00:01:04.923
Kyle spent a 1/4 of an
hour, so here's Kyle.

00:01:04.923 --> 00:01:07.185
Kyle spent 1/4 of an hour,

00:01:07.185 --> 00:01:08.913
but he did this for six days,

00:01:08.913 --> 00:01:13.173
so six times, he spent 1/4 of an hour.

00:01:13.173 --> 00:01:17.595
Or we could say that Kyle
spent 1/4 of an hour one day,

00:01:17.595 --> 00:01:20.602
plus 1/4 of an hour a second day,

00:01:20.602 --> 00:01:23.409
1/4 of an hour a third day,

00:01:23.409 --> 00:01:24.492
a fourth day,

00:01:25.436 --> 00:01:30.232
a fifth day, and then a sixth
day he spent 1/4 of an hour.

00:01:30.232 --> 00:01:32.095
So six times,

00:01:32.095 --> 00:01:35.468
one, two, three, four, five, six,

00:01:35.468 --> 00:01:36.793
he spent a quarter,

00:01:36.793 --> 00:01:39.437
or 1/4 of an hour working on his project.

00:01:39.437 --> 00:01:42.209
So if we have six 1/4s,

00:01:42.209 --> 00:01:45.459
then we have a total of 6/4.

00:01:47.648 --> 00:01:52.254
So Kyle spent 6/4, or six quarter hours.

00:01:52.254 --> 00:01:53.866
And now back to our question,

00:01:53.866 --> 00:01:56.054
now that we know how
much each of them spent,

00:01:56.054 --> 00:01:59.943
who spent more time working
on his science project?

00:01:59.943 --> 00:02:04.059
Rishi with 6/4 hours,
or Kyle with 6/4 hours?

00:02:04.059 --> 00:02:05.291
And the answer here, of course,

00:02:05.291 --> 00:02:09.198
is that these are equal,
these are the same.

00:02:09.198 --> 00:02:13.002
Rishi and Kyle spent
the same amount of time,

00:02:13.002 --> 00:02:15.002
so we could say the same

00:02:16.274 --> 00:02:20.175
amount of time working on each
other their science projects.


Common fractions (halves)

WEBVTT
Kind: captions
Language: en

00:00:00.924 --> 00:00:03.447
- [Voiceover] Let's try to show that 0.5

00:00:03.447 --> 00:00:05.323
is equal to one half.

00:00:05.323 --> 00:00:08.468
We often hear people use
these interchangeably.

00:00:08.468 --> 00:00:09.858
They use them back and forth.

00:00:09.858 --> 00:00:11.533
Maybe someone would say,

00:00:11.533 --> 00:00:13.607
I have a 2.5 pound rat

00:00:13.607 --> 00:00:14.881
and then someone would say back to them,

00:00:14.881 --> 00:00:18.146
wow, a two and a half
pound rat, that's enormous.

00:00:18.146 --> 00:00:21.046
If we get past thinking about
how enormous that rat is,

00:00:21.046 --> 00:00:24.635
what we can notice is they first said 2.5

00:00:24.635 --> 00:00:26.956
and then they changed
it to two and a half.

00:00:26.956 --> 00:00:29.384
They use them interchangeably
because they're equal

00:00:29.384 --> 00:00:30.477
and that's fine.

00:00:30.477 --> 00:00:31.724
You can use these back and forth

00:00:31.724 --> 00:00:34.054
because they are equal to each other.

00:00:34.054 --> 00:00:37.304
.5 and one half are equivalent or equal

00:00:38.845 --> 00:00:40.995
but let's not just accept
that that's true, let's see,

00:00:40.995 --> 00:00:42.057
let's see if we can prove it.

00:00:42.057 --> 00:00:43.581
Let's try to show that.

00:00:43.581 --> 00:00:44.848
So the first way we can show it

00:00:44.848 --> 00:00:47.298
is down here with a number line.

00:00:47.298 --> 00:00:51.415
We have a number line
that goes from zero to one

00:00:51.415 --> 00:00:55.034
and it's divided into
one, two, three, four,

00:00:55.034 --> 00:00:58.867
five, six, seven, eight,
nine, 10 equal pieces

00:00:59.761 --> 00:01:02.049
so that means each of these pieces

00:01:02.049 --> 00:01:05.632
is one of the 10 equal
pieces or one tenth.

00:01:07.598 --> 00:01:10.556
So looking back at this decimal up here,

00:01:10.556 --> 00:01:13.556
this five is in the tenths place

00:01:13.556 --> 00:01:16.639
so we can also say 0.5 is five tenths

00:01:18.214 --> 00:01:20.201
is another way to read that decimal

00:01:20.201 --> 00:01:22.534
or we could say five tenths.

00:01:24.181 --> 00:01:27.120
So if this distance here is one tenth,

00:01:27.120 --> 00:01:30.037
if we travel another one,
we've gone two of the tenths,

00:01:30.037 --> 00:01:32.312
two of the 10 equal spaces.

00:01:32.312 --> 00:01:33.731
So to get to five tenths,

00:01:33.731 --> 00:01:37.713
we're going to travel
one, two, three, four,

00:01:37.713 --> 00:01:39.875
five of the tenths

00:01:39.875 --> 00:01:42.875
and right here, we have five tenths.

00:01:44.405 --> 00:01:46.033
We could write is as a fraction

00:01:46.033 --> 00:01:49.410
or we could write is as a decimal.

00:01:49.410 --> 00:01:51.539
So we found five tenths
on this number line

00:01:51.539 --> 00:01:54.303
but let's see if that really
is the same as one half.

00:01:54.303 --> 00:01:57.303
This distance here from zero to one,

00:01:58.354 --> 00:02:01.207
if we were to block that off,

00:02:01.207 --> 00:02:03.122
going to mark it off,

00:02:03.122 --> 00:02:06.671
this point right here
where we said five tenths,

00:02:06.671 --> 00:02:10.226
this distance, this whole
distance that we traveled

00:02:10.226 --> 00:02:11.316
is this much

00:02:11.316 --> 00:02:14.387
and we can see here that this is in fact

00:02:14.387 --> 00:02:17.595
half of the distance between zero and one.

00:02:17.595 --> 00:02:19.412
This is halfway.

00:02:19.412 --> 00:02:22.829
So 0.5 and one half are at the same point

00:02:23.962 --> 00:02:25.356
on a number line

00:02:25.356 --> 00:02:28.870
so that's one one to
show they're equivalent.

00:02:28.870 --> 00:02:30.137
Another way we can show

00:02:30.137 --> 00:02:31.977
that five tenths and one half are equal

00:02:31.977 --> 00:02:33.977
is with a fraction model.

00:02:33.977 --> 00:02:35.850
So here we have a rectangle

00:02:35.850 --> 00:02:37.487
and let's try to show one half.

00:02:37.487 --> 00:02:39.673
Let's start with one half this time.

00:02:39.673 --> 00:02:42.959
So if we split this rectangle in half,

00:02:42.959 --> 00:02:44.106
here we're trying to show half.

00:02:44.106 --> 00:02:47.207
One out of two equal pieces.

00:02:47.207 --> 00:02:49.498
So we'll shade one half,

00:02:49.498 --> 00:02:51.380
one of the two pieces.

00:02:51.380 --> 00:02:53.375
So this is one half and now,

00:02:53.375 --> 00:02:57.425
let's try to show that this
is equal to five tenths.

00:02:57.425 --> 00:03:01.199
Remember, this five is in tenths place.

00:03:01.199 --> 00:03:03.036
So now, instead of halves,

00:03:03.036 --> 00:03:05.957
we're going to divide our
rectangle into tenths,

00:03:05.957 --> 00:03:06.957
five tenths,

00:03:08.865 --> 00:03:10.282
so let's do that.

00:03:11.504 --> 00:03:12.574
Here we go.

00:03:12.574 --> 00:03:15.311
So we'll make 10 equal pieces for tenths

00:03:15.311 --> 00:03:18.244
and we will shade five of them.

00:03:18.244 --> 00:03:21.222
We want to see five of the tenths

00:03:21.222 --> 00:03:22.811
so let's do that.

00:03:22.811 --> 00:03:24.144
One, two, three,

00:03:27.349 --> 00:03:28.516
four and five.

00:03:30.126 --> 00:03:31.516
Here, we've shown five tenths

00:03:31.516 --> 00:03:35.355
and we can see that it
is equal to one half.

00:03:35.355 --> 00:03:39.377
One half and five tenths
cover the same amount

00:03:39.377 --> 00:03:40.866
so they are equal.

00:03:40.866 --> 00:03:42.093
So we've shown two ways now.

00:03:42.093 --> 00:03:44.300
One half and five tenths
were at the same spot

00:03:44.300 --> 00:03:47.140
on a number line which means they're equal

00:03:47.140 --> 00:03:49.496
and they cover the same amount of area

00:03:49.496 --> 00:03:51.838
which is another way
to show they're equal.

00:03:51.838 --> 00:03:54.421
So one half is definitely equal

00:03:55.876 --> 00:03:57.709
to 0.5 or five tenths.


Rewriting fractions as decimals

WEBVTT
Kind: captions
Language: en

00:00:00.734 --> 00:00:03.888
- [Voiceover] Let's write
29/100 as a decimal.

00:00:03.888 --> 00:00:06.555
So we'll start with 29 over 100,

00:00:07.958 --> 00:00:09.624
and let's start breaking that down

00:00:09.624 --> 00:00:11.316
till we can get to place values

00:00:11.316 --> 00:00:13.483
'cause place values
will help us to convert

00:00:13.483 --> 00:00:15.711
to write this as a decimal.

00:00:15.711 --> 00:00:18.128
29/100 we can break that down

00:00:18.128 --> 00:00:19.961
into 20/100

00:00:21.362 --> 00:00:23.685
plus 9/100.

00:00:23.685 --> 00:00:25.847
20/100 plus nine more hundredths

00:00:25.847 --> 00:00:28.401
is equal to 29/100.

00:00:28.401 --> 00:00:30.355
So we've just broken it down some here.

00:00:30.355 --> 00:00:32.571
It's still equivalent to 29/100.

00:00:32.571 --> 00:00:36.558
We're just taking it apart
trying to get to place values.

00:00:36.558 --> 00:00:39.873
Going from there, we
can simplify even more.

00:00:39.873 --> 00:00:42.035
20/100 can be simplified.

00:00:42.035 --> 00:00:45.229
We could divide 20 and 100 by 10.

00:00:45.229 --> 00:00:49.396
So we'll say 2/10 plus 9/100.

00:00:52.227 --> 00:00:56.214
And now, if you can hear,
the tenths, hundredths,

00:00:56.214 --> 00:00:59.559
these are place values we
should recognize from decimals.

00:00:59.559 --> 00:01:03.963
So what we've shown here is that 29/100

00:01:03.963 --> 00:01:05.796
is equal to two-tenths

00:01:10.228 --> 00:01:11.978
plus nine-hundredths.

00:01:16.560 --> 00:01:19.773
Here we have two-tenths,
which matches 2/10,

00:01:19.773 --> 00:01:24.438
and nine-hundreths, like
our 9/100 right there.

00:01:24.438 --> 00:01:29.015
And so we can take those
and think about place value.

00:01:29.015 --> 00:01:33.374
If we have our decimal,
let's first put our decimal.

00:01:33.374 --> 00:01:35.230
Over here to the left of the decimal's

00:01:35.230 --> 00:01:37.182
our ones place value.

00:01:37.182 --> 00:01:40.357
We don't have any ones
here so we'll put a zero,

00:01:40.357 --> 00:01:43.602
zero ones, but then next is tenths,

00:01:43.602 --> 00:01:47.769
then we have our tenths and we have 2/10.

00:01:49.002 --> 00:01:51.231
And after tenths, the
place value after tenths

00:01:51.231 --> 00:01:55.296
is hundredths, and here, we
see, we have nine hundredths.

00:01:55.296 --> 00:01:58.740
So we'll put a nine in
the hundredths place.

00:01:58.740 --> 00:02:00.157
And so down here,

00:02:01.474 --> 00:02:03.307
0.29 or 29 hundredths,

00:02:05.767 --> 00:02:09.410
is equivalent to the fraction, 29/100.

00:02:09.410 --> 00:02:11.123
And just listen to how those are said.

00:02:11.123 --> 00:02:13.727
It's kind of a clue that we did well.

00:02:13.727 --> 00:02:17.810
29 hundredths is equivalent to 29/100.

00:02:19.138 --> 00:02:22.148
Both the fraction and the
decimal are read the same way.

00:02:22.148 --> 00:02:23.392
So that's a pretty good clue

00:02:23.392 --> 00:02:25.915
that we probably rewrote these correctly.

00:02:25.915 --> 00:02:29.082
So 0.29 is equal to 29/100.

00:02:32.588 --> 00:02:34.334
Let's try one more, let's try something

00:02:34.334 --> 00:02:37.084
let's say maybe 53/10.

00:02:41.729 --> 00:02:44.570
So again we're gonna break this down.

00:02:44.570 --> 00:02:48.785
We'll have 50/10 plus how
many more tenths are left?

00:02:48.785 --> 00:02:52.245
There's still three more tenths.

00:02:52.245 --> 00:02:56.742
And again, we can simplify,
we can simplify this.

00:02:56.742 --> 00:02:59.909
50/10 can be divided by 10.

00:03:00.916 --> 00:03:04.812
So instead of 50/10, we'll say 5/1,

00:03:04.812 --> 00:03:08.312
plus, we still have those 3/10.

00:03:10.489 --> 00:03:12.762
And now we've got this to place value.

00:03:12.762 --> 00:03:15.179
Five ones, is 5/1,

00:03:17.130 --> 00:03:19.213
plus your 3/10

00:03:22.094 --> 00:03:23.427
is three-tenths.

00:03:25.967 --> 00:03:28.878
So thinking, let's put our decimal,

00:03:28.878 --> 00:03:32.030
over here to the left of
the decimal's our ones,

00:03:32.030 --> 00:03:34.272
and we have five ones.

00:03:34.272 --> 00:03:36.688
And to the right of the decimal is tenths,

00:03:36.688 --> 00:03:38.874
and we have three tenths.

00:03:38.874 --> 00:03:41.457
So 5.3 or five and three tenths

00:03:43.177 --> 00:03:46.365
is equal to 53/10.

00:03:46.365 --> 00:03:48.701
To write 53/10 as a decimal,

00:03:48.701 --> 00:03:52.868
we will get 5.3 because it was
five ones and three-tenths,

00:03:54.168 --> 00:03:57.304
which we showed right here,
five and three tenths.


Identifying hundredths on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.831 --> 00:00:03.399
- [Voiceover] Where is the
point on the number line?

00:00:03.399 --> 00:00:06.196
Here we have a number
line that starts at 1.5

00:00:06.196 --> 00:00:10.316
or 1 5/10 and goes to 1 7/10.

00:00:10.316 --> 00:00:13.783
The distance between these
larger blue tick marks

00:00:13.783 --> 00:00:17.550
is 1/10 because we go
from 1 5/10 to 1 6/10,

00:00:17.550 --> 00:00:22.110
so that went up at 1/10
and then up to 1 7/10.

00:00:22.110 --> 00:00:26.277
This distance here is 1/10
or we could write that as

00:00:30.135 --> 00:00:33.218
0.1 or as a fraction 1/10.

00:00:34.353 --> 00:00:36.139
That distance between each of these

00:00:36.139 --> 00:00:39.300
blue large tick marks is 1/10,

00:00:39.300 --> 00:00:43.868
but we want to know what is
this green point right here.

00:00:43.868 --> 00:00:46.342
To figure that out we also
need to figure out what do

00:00:46.342 --> 00:00:50.509
these black tick marks represent,
these smaller distances.

00:00:51.674 --> 00:00:55.219
From here to here is 1/10,
and within that 1/10th

00:00:55.219 --> 00:00:58.899
there are one, two,
three, four, five, six,

00:00:58.899 --> 00:01:01.899
seven, eight, nine, 10 equal spaces.

00:01:03.023 --> 00:01:07.190
Each of these little spaces
is 1/10 of this distance

00:01:10.252 --> 00:01:14.278
because it's one of the 10 equal spaces.

00:01:14.278 --> 00:01:18.445
This, right here, is 1/10th of this 1/10th

00:01:21.446 --> 00:01:23.772
because the large distance is 1/10th

00:01:23.772 --> 00:01:26.947
and this is 1/10 of that 1/10th.

00:01:26.947 --> 00:01:30.826
Another way we could say that is 1/100.

00:01:30.826 --> 00:01:34.369
1/10th of 1/10th, if you
take 1/10 and divide it

00:01:34.369 --> 00:01:38.585
into 10 pieces, now you have 1/100.

00:01:38.585 --> 00:01:42.129
This distance is 1/100 which
means that each of these

00:01:42.129 --> 00:01:45.863
distances, this is another
1/100 and another 1/100,

00:01:45.863 --> 00:01:49.610
so how many 1/100 till
we get to our point?

00:01:49.610 --> 00:01:50.443
Let's see.

00:01:50.443 --> 00:01:55.284
1/100, two, three, four, five, six, seven,

00:01:55.284 --> 00:01:57.710
7/100 is what it took us to get there,

00:01:57.710 --> 00:02:01.877
so we could say 7/100, the
fraction or the decimal, 0.07

00:02:03.643 --> 00:02:06.560
with a seven in the hundreds place.

00:02:07.647 --> 00:02:10.128
Looking at the whole number
line now, putting the whole

00:02:10.128 --> 00:02:14.295
thing together, we
started at 1.5 or 1 5/10,

00:02:17.241 --> 00:02:21.408
and we went another 7/100, so
we can write that as +7/100.

00:02:26.679 --> 00:02:29.688
We have 1 5/10 plus 7/100

00:02:29.688 --> 00:02:33.688
which is a total of 1 57/100 or 1.57.

00:02:36.703 --> 00:02:39.536
Our point, right here, is at 1.57.

00:02:43.066 --> 00:02:47.233
1 5/10 plus 7/100 gets us to 1 57/100.


Graphing hundredths from 0 to 0.1

WEBVTT
Kind: captions
Language: en

00:00:00.427 --> 00:00:03.623
- [Voiceover] Graph
0.04 on the number line.

00:00:03.623 --> 00:00:05.369
So here we have this number line

00:00:05.369 --> 00:00:09.917
that goes from zero to 0.1, or 1/10,

00:00:09.917 --> 00:00:11.884
and between zero and 1/10,

00:00:11.884 --> 00:00:14.999
we have one, two, three, four, five, six,

00:00:14.999 --> 00:00:18.703
seven, eight, nine, ten equal spaces.

00:00:18.703 --> 00:00:23.566
So each of these spaces
represents 1/10th of the distance,

00:00:23.566 --> 00:00:26.149
is one out of ten equal spaces.

00:00:26.986 --> 00:00:31.956
Or we could say, it is
1/10th of this tenth,

00:00:31.956 --> 00:00:35.298
it's a tenth of the way to 1/10th.

00:00:35.298 --> 00:00:36.456
That's a little bit tricky,

00:00:36.456 --> 00:00:38.138
maybe let's pause and
look at that visually

00:00:38.138 --> 00:00:42.034
and think about what does a
tenth of a tenth really mean.

00:00:42.034 --> 00:00:44.425
So here I have already drawn

00:00:44.425 --> 00:00:47.733
a picture that's divided
into 10 equal pieces.

00:00:47.733 --> 00:00:51.900
And if we shade one of them,
then we're shading 1/10th.

00:00:53.034 --> 00:00:55.204
This right here is 1/10th.

00:00:55.204 --> 00:00:58.830
And this amount right here is
what the number line showed,

00:00:58.830 --> 00:01:01.835
the whole number line showed 1/10th.

00:01:01.835 --> 00:01:05.698
But then we had divided that
tenth into 10 equal pieces,

00:01:05.698 --> 00:01:09.576
so if we split that 10
into 10 equal pieces,

00:01:09.576 --> 00:01:12.478
we're gonna have something like this.

00:01:12.478 --> 00:01:17.058
So here again this top row would
represent the entire tenth,

00:01:17.058 --> 00:01:19.128
from the whole number line,

00:01:19.128 --> 00:01:23.295
but each little piece would
be 1/10th of that whole tenth.

00:01:26.938 --> 00:01:29.342
Or, so this is what each little piece

00:01:29.342 --> 00:01:31.673
on the number line represents,

00:01:31.673 --> 00:01:35.653
or we could say that's 1/100th,

00:01:35.653 --> 00:01:38.057
a tenth of a tenth is a 1/100th,

00:01:38.057 --> 00:01:40.930
'cause when you divide
10 into 10 equal pieces,

00:01:40.930 --> 00:01:43.574
you're gonna end up with 100 pieces.

00:01:43.574 --> 00:01:47.407
So a tenth of a tenth is a 1/100th.

00:01:48.414 --> 00:01:50.177
So looking back at the number line,

00:01:50.177 --> 00:01:53.463
now we know that this distance,

00:01:53.463 --> 00:01:57.662
a tenth of our tenth, is 1/100th.

00:01:57.662 --> 00:01:59.829
And we want to graph 0.04.

00:02:02.357 --> 00:02:06.550
Well this four right here,
let's think about place value.

00:02:06.550 --> 00:02:10.063
The zero's in the ones,
the next zero in the tenth,

00:02:10.063 --> 00:02:12.396
and this four is in the hundreds,

00:02:12.396 --> 00:02:16.366
so we could call this four hundredths,

00:02:16.366 --> 00:02:19.616
or even the fraction, 4/100.

00:02:21.000 --> 00:02:24.401
So if one of these lengths is 1/100

00:02:24.401 --> 00:02:26.464
and we wanna go 4/100,

00:02:26.464 --> 00:02:28.990
then we're gonna need to
go four of these lengths.

00:02:28.990 --> 00:02:32.018
One, two, three, four of these lengths

00:02:32.018 --> 00:02:35.435
would be four of the hundredths, or 0.04.


Identifying tenths on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.825 --> 00:00:03.064
- [Voiceover] Where is the
point on the number line?

00:00:03.064 --> 00:00:05.985
Well, here it is, here is the point.

00:00:05.985 --> 00:00:08.167
But I'm guessing that
they're asking not literally

00:00:08.167 --> 00:00:09.904
just to find it and look at it

00:00:09.904 --> 00:00:13.307
but what number is this point graphed at.

00:00:13.307 --> 00:00:15.898
Where is this on the number line?

00:00:15.898 --> 00:00:17.645
So, one thing we know pretty quickly

00:00:17.645 --> 00:00:20.119
is the number is between three and four.

00:00:20.119 --> 00:00:23.505
It's greater than three
but it's not quite four.

00:00:23.505 --> 00:00:25.642
But to figure out how
much greater than three

00:00:25.642 --> 00:00:29.776
we need to know what these
black tick marks represent.

00:00:29.776 --> 00:00:31.902
So, between three and four there is

00:00:31.902 --> 00:00:35.207
one, two, three, four, five, six,

00:00:35.207 --> 00:00:38.290
seven, eight, nine, ten equal spaces.

00:00:39.295 --> 00:00:41.514
So, each of these distances,

00:00:41.514 --> 00:00:44.172
each of these equal spaces,

00:00:44.172 --> 00:00:45.172
is one tenth

00:00:47.513 --> 00:00:50.863
or one tenth of the distance
between three and four.

00:00:50.863 --> 00:00:53.613
It's one out of ten equal spaces.

00:00:55.355 --> 00:00:57.314
So, if that's one tenth

00:00:57.314 --> 00:01:00.501
and this next space is another one tenth.

00:01:00.501 --> 00:01:03.096
And then we have to travel one more tenth

00:01:03.096 --> 00:01:05.284
to get to our point.

00:01:05.284 --> 00:01:08.273
So, we went three, we know it's three.

00:01:08.273 --> 00:01:09.651
Plus,

00:01:09.651 --> 00:01:11.568
one, two, three tenths.

00:01:13.878 --> 00:01:16.462
Three and three tenths.

00:01:16.462 --> 00:01:19.519
Or, let's write this as a decimal,

00:01:19.519 --> 00:01:21.316
let's look at it as a decimal.

00:01:21.316 --> 00:01:25.980
If we wanted, we could
have our ones place value

00:01:25.980 --> 00:01:29.703
and then after the ones,
the decimal and the tenths.

00:01:29.703 --> 00:01:32.983
So, for the ones, there's three ones.

00:01:32.983 --> 00:01:35.558
And how many tenths did we see here?

00:01:35.558 --> 00:01:37.748
There were three tenths.

00:01:37.748 --> 00:01:41.221
So, either way we can say
three and three tenths

00:01:41.221 --> 00:01:43.896
or three and three tenths.

00:01:43.896 --> 00:01:46.626
Our decimal, our point

00:01:46.626 --> 00:01:49.793
is 3.3 on the number line.


Graphing tenths from 0 to 1

WEBVTT
Kind: captions
Language: en

00:00:00.804 --> 00:00:03.372
- [Voiceover] Graph
0.6 on the number line.

00:00:03.372 --> 00:00:06.600
Down here we have a number
line that goes from zero to one

00:00:06.600 --> 00:00:10.820
and it's split into one, two, three, four,

00:00:10.820 --> 00:00:14.737
five, six, seven, eight,
nine, 10 equal spaces.

00:00:15.857 --> 00:00:18.849
These tick marks, there's
10 of these spaces

00:00:18.849 --> 00:00:21.178
all the way from zero to one.

00:00:21.178 --> 00:00:24.659
So between any two of these spaces,

00:00:24.659 --> 00:00:27.851
for example, between these
first two tick marks,

00:00:27.851 --> 00:00:31.184
could be represented by 1/10.

00:00:32.224 --> 00:00:35.307
It's one, one of the 10 equal spaces.

00:00:37.256 --> 00:00:40.330
It's 1/10 of the distance.

00:00:40.330 --> 00:00:44.267
Now the number we're
supposed to graph is 0.6.

00:00:44.267 --> 00:00:47.850
Well this six is in
the tenths place value,

00:00:51.240 --> 00:00:53.318
so another way we could read this decimal

00:00:53.318 --> 00:00:57.401
is 6 tenths, or as the fraction, 6/10.

00:01:01.241 --> 00:01:05.408
So if one of these spaces
represents one out of 10

00:01:06.496 --> 00:01:10.330
equal spaces, and we want 6/10,

00:01:10.330 --> 00:01:13.877
then we need to travel six
of the 10 equal spaces,

00:01:13.877 --> 00:01:16.072
6/10 of the way.

00:01:16.072 --> 00:01:18.084
Six of the 10 spaces.

00:01:18.084 --> 00:01:21.476
So we can start at the
beginning and go one, two,

00:01:21.476 --> 00:01:23.393
three, four, five, six.

00:01:24.365 --> 00:01:28.532
And graph right here with
a point and this is 6/10.


Labeling voltages

WEBVTT
Kind: captions
Language: en

00:00:00.659 --> 00:00:01.492
- [Voiceover] In this video,

00:00:01.492 --> 00:00:03.316
I want to do a
demonstration of the process

00:00:03.316 --> 00:00:05.190
of labeling voltages on a circuit

00:00:05.190 --> 00:00:07.719
that we're about to analyze.

00:00:07.719 --> 00:00:09.957
This is something that
sometimes causes stress,

00:00:09.957 --> 00:00:11.982
or confusion, and I want to just basically

00:00:11.982 --> 00:00:15.299
try to get out of that
stressful situation,

00:00:15.299 --> 00:00:16.798
so the first thing I want to do,

00:00:16.798 --> 00:00:19.548
is remind ourselves of the convention,

00:00:19.548 --> 00:00:22.809
the sign convention
for passive components,

00:00:22.809 --> 00:00:25.957
so we said, If we have a
resistor that we draw this way,

00:00:25.957 --> 00:00:29.935
and we label the voltage
plus or minus V on it,

00:00:29.935 --> 00:00:32.743
then when we label the current arrow,

00:00:32.743 --> 00:00:35.883
we want to label the current arrow

00:00:35.883 --> 00:00:40.098
so it goes into the positive
terminal of the component,

00:00:40.098 --> 00:00:42.456
in this case a resistor,

00:00:42.456 --> 00:00:44.372
so just another quick example,

00:00:44.372 --> 00:00:47.254
if I draw the resistor sideways like this,

00:00:47.254 --> 00:00:50.461
if I label the voltages,
the minus one on this side,

00:00:50.461 --> 00:00:53.083
and the plus one on this side,

00:00:53.083 --> 00:00:56.583
then when I go to apply the current arrow,

00:01:01.210 --> 00:01:04.252
I put the current arrow into
the positive voltage sign,

00:01:04.252 --> 00:01:07.212
so this here is the sign convention

00:01:07.212 --> 00:01:09.158
for an individual component.

00:01:09.158 --> 00:01:10.942
How do we label the voltage,

00:01:10.942 --> 00:01:14.725
and the current together to be consistent?

00:01:14.725 --> 00:01:16.776
Now we're going to to over
and analyze a circuit.

00:01:16.776 --> 00:01:18.460
I've drawn a circuit here.

00:01:18.460 --> 00:01:20.136
It's two identical circuits,

00:01:20.136 --> 00:01:21.871
and we're going to do
it two different ways,

00:01:21.871 --> 00:01:24.370
with two different voltage labels,

00:01:24.370 --> 00:01:25.835
so the first thing we do, of course,

00:01:25.835 --> 00:01:26.875
when we analyze a circuit,

00:01:26.875 --> 00:01:30.578
is we set up the variables
that we want to talk about.

00:01:30.578 --> 00:01:31.800
We'll do this side first.

00:01:31.800 --> 00:01:33.550
I'll label this as i,

00:01:35.383 --> 00:01:37.966
and that's a choice I can make,

00:01:39.385 --> 00:01:41.804
and then I'm going to
label the voltages too,

00:01:41.804 --> 00:01:45.101
and I'm going to choose to
label the voltages like this,

00:01:45.101 --> 00:01:48.925
plus and minus V, and we'll call this V1,

00:01:48.925 --> 00:01:51.758
and this will be plus or minus V2,

00:01:53.766 --> 00:01:54.972
so first, let's carry through

00:01:54.972 --> 00:01:56.885
and do the analysis of this circuit,

00:01:56.885 --> 00:01:59.463
and I'll give some
component values to this.

00:01:59.463 --> 00:02:01.463
We'll call this 10 ohms,

00:02:02.330 --> 00:02:04.247
and this one's 20 ohms.

00:02:07.738 --> 00:02:09.020
Now I'm going to do an application

00:02:09.020 --> 00:02:11.270
of Kirchhoff's voltage law,

00:02:12.420 --> 00:02:13.712
around this loop,

00:02:13.712 --> 00:02:15.455
and we'll see how it turns out.

00:02:15.455 --> 00:02:17.555
All right, so Kirchhoff's voltage law

00:02:17.555 --> 00:02:20.122
will start at this node here,

00:02:20.122 --> 00:02:22.481
and will go around the circuit this way,

00:02:22.481 --> 00:02:23.539
and we'll do some equations.

00:02:23.539 --> 00:02:25.872
We'll say, plus three volts,

00:02:27.540 --> 00:02:28.616
when we go through this device,

00:02:28.616 --> 00:02:30.823
we get a voltage rise of three,

00:02:30.823 --> 00:02:32.460
then we get a voltage drop,

00:02:32.460 --> 00:02:33.891
because we go from plus to minus,

00:02:33.891 --> 00:02:36.724
we get a voltage drop of minus V1,

00:02:37.776 --> 00:02:40.693
and then we get another drop of V2,

00:02:41.667 --> 00:02:43.500
minus V2, equals zero.

00:02:46.274 --> 00:02:50.383
That's our KVL equation
for this circuit over here,

00:02:50.383 --> 00:02:53.106
so let's keep going with this analysis.

00:02:53.106 --> 00:02:55.556
Three volts, minus V1

00:02:55.556 --> 00:02:56.723
is i times R1,

00:02:59.043 --> 00:02:59.960
i times 10,

00:03:02.318 --> 00:03:04.485
and V2 is i times 20 ohms,

00:03:10.435 --> 00:03:13.935
and that equals zero, so let's keep going.

00:03:14.834 --> 00:03:16.751
Three minus i times 10,

00:03:22.025 --> 00:03:23.775
plus 20, equals zero,

00:03:26.203 --> 00:03:29.536
and that means that i equals minus three

00:03:31.186 --> 00:03:34.152
over, minus three goes to this side,

00:03:34.152 --> 00:03:38.329
10 plus 20 is 30, and the
minus sign goes with the 30,

00:03:38.329 --> 00:03:39.162
minus 30,

00:03:41.487 --> 00:03:43.237
so i equals plus amp,

00:03:48.914 --> 00:03:52.108
so we solve for i, and
let's just pick out V1.

00:03:52.108 --> 00:03:53.533
Let's solve for V1,

00:03:53.533 --> 00:03:57.116
and we said earlier
that V1 was i times R1,

00:03:58.454 --> 00:04:01.204
so V1 equals i, which is .1 amps,

00:04:04.337 --> 00:04:05.504
times 10 ohms,

00:04:08.069 --> 00:04:09.402
equals one volt.

00:04:13.881 --> 00:04:15.464
If we do it for V2,

00:04:17.656 --> 00:04:19.585
that equals .1 amps,

00:04:19.585 --> 00:04:23.002
times 20 ohms, and that equals two volts,

00:04:24.211 --> 00:04:25.748
and I can do one last little check,

00:04:25.748 --> 00:04:28.057
I can go back, and I can see KVL,

00:04:28.057 --> 00:04:31.019
I could do a check to see if
this equation came out right,

00:04:31.019 --> 00:04:34.102
so three volts, minus V1 is one volt,

00:04:37.717 --> 00:04:39.384
and V2 is two volts,

00:04:42.142 --> 00:04:43.892
and that equals zero,

00:04:44.895 --> 00:04:46.355
and I get to put a check mark here,

00:04:46.355 --> 00:04:48.635
because yes, it's equal to zero.

00:04:48.635 --> 00:04:50.116
That was a real quick analysis

00:04:50.116 --> 00:04:53.152
of a simple two resistor circuit,

00:04:53.152 --> 00:04:55.296
and we got all the voltages and currents.

00:04:55.296 --> 00:04:56.998
Now I'm going to go do
the same thing again,

00:04:56.998 --> 00:04:59.095
but this time I'm going
to do the voltage labels

00:04:59.095 --> 00:05:00.884
a little bit odd.

00:05:00.884 --> 00:05:01.757
What I'm going to do here,

00:05:01.757 --> 00:05:04.174
is I'm going to say two plus.

00:05:05.635 --> 00:05:08.192
I'm going to define V1
to be in that direction,

00:05:08.192 --> 00:05:11.275
and we'll keep V2 like it was before,

00:05:13.865 --> 00:05:16.179
and now I need a current variable,

00:05:16.179 --> 00:05:19.236
and I'm going to call my
current variable i here,

00:05:19.236 --> 00:05:21.457
just like we did before.

00:05:21.457 --> 00:05:22.922
Now at this point, you might say,

00:05:22.922 --> 00:05:24.163
Willie, you did it wrong.

00:05:24.163 --> 00:05:24.996
You did it wrong.

00:05:24.996 --> 00:05:26.764
That's not the right voltage label,

00:05:26.764 --> 00:05:27.762
but I want to show you

00:05:27.762 --> 00:05:29.214
that I'm going to get the same answer,

00:05:29.214 --> 00:05:33.164
even though I did it this unusual way,

00:05:33.164 --> 00:05:36.141
so let's do the same KVL
analysis on this circuit,

00:05:36.141 --> 00:05:37.511
and what I want to show you,

00:05:37.511 --> 00:05:39.713
is that the arithmetic
that we're about to do

00:05:39.713 --> 00:05:42.355
takes care of the science just fine.

00:05:42.355 --> 00:05:43.772
All right, so KVL

00:05:46.329 --> 00:05:48.383
on this circuit says that

00:05:48.383 --> 00:05:50.628
we'll start at the same place,

00:05:50.628 --> 00:05:53.378
and go around the same direction,

00:05:54.394 --> 00:05:57.639
so this says they're three volts,

00:05:57.639 --> 00:05:59.196
a voltage rise of three volts,

00:05:59.196 --> 00:06:01.458
we go in the minus sign,
and out the positive sign,

00:06:01.458 --> 00:06:03.920
so that's a voltage rise.

00:06:03.920 --> 00:06:07.245
Now we get over to R1, and we
go in the minus sign of R1,

00:06:07.245 --> 00:06:08.810
and out the plus sign,

00:06:08.810 --> 00:06:10.310
so that's plus V1,

00:06:13.301 --> 00:06:16.610
that's different than
we had last time, right?

00:06:16.610 --> 00:06:19.232
Last time we had minus V1 here, see,

00:06:19.232 --> 00:06:21.567
and now we have plus V1.

00:06:21.567 --> 00:06:24.261
This is going to work out okay though.

00:06:24.261 --> 00:06:27.483
Now we go in the plus sign of V2,

00:06:27.483 --> 00:06:29.941
and we come out the minus sign,

00:06:29.941 --> 00:06:34.236
so that's a voltage drop,
so we do a subtraction,

00:06:34.236 --> 00:06:35.986
and that equals zero.

00:06:38.190 --> 00:06:39.850
All right, we've got different equations,

00:06:39.850 --> 00:06:42.602
but we've got different definitions of V1,

00:06:42.602 --> 00:06:44.462
so now I want to write these V terms

00:06:44.462 --> 00:06:46.296
in terms of the resistance value,

00:06:46.296 --> 00:06:47.741
and the current value,

00:06:47.741 --> 00:06:51.908
and this is where we use the
sign convention carefully,

00:06:53.237 --> 00:06:56.613
so now we need to include
a term to represent R1,

00:06:56.613 --> 00:06:58.306
to use Ohm's law here.

00:06:58.306 --> 00:06:59.426
Now we have to be careful,

00:06:59.426 --> 00:07:02.194
this is one point we have to be careful,

00:07:02.194 --> 00:07:07.085
the current is going in the
negative terminal of R1,

00:07:07.085 --> 00:07:09.594
so we're going to say

00:07:09.594 --> 00:07:11.427
VR1 equals negative i,

00:07:17.054 --> 00:07:17.887
times R1.

00:07:20.638 --> 00:07:21.991
Does that make sense?

00:07:21.991 --> 00:07:23.604
If we define our current variable

00:07:23.604 --> 00:07:25.116
to be going in the negative sign,

00:07:25.116 --> 00:07:27.947
then Ohm's law picks
up this negative sign,

00:07:27.947 --> 00:07:30.439
to make it come out right,

00:07:30.439 --> 00:07:33.356
so down here, we plug in minus iR1,

00:07:36.307 --> 00:07:38.474
which is minus i times 10,

00:07:39.551 --> 00:07:41.327
that's a difference,

00:07:41.327 --> 00:07:42.631
and then we go through V2,

00:07:42.631 --> 00:07:45.606
and V2 is the same as it
was in the other equation,

00:07:45.606 --> 00:07:46.523
minus V2 is

00:07:48.464 --> 00:07:50.464
i times R2, which is 20,

00:07:52.978 --> 00:07:55.061
and that all equals zero.

00:07:57.160 --> 00:07:58.988
Even right now, if you
look at this equation,

00:07:58.988 --> 00:08:01.380
you see this minus sign,
that snuck in here,

00:08:01.380 --> 00:08:04.375
because of our good use
of the sign convention

00:08:04.375 --> 00:08:06.073
for passive componets.

00:08:06.073 --> 00:08:08.415
That makes this equation

00:08:08.415 --> 00:08:10.832
look just like this one here,

00:08:14.098 --> 00:08:17.098
so let's continue with the analysis.

00:08:19.865 --> 00:08:22.615
Just need a little bit more room,

00:08:23.639 --> 00:08:25.222
three volts minus i

00:08:30.806 --> 00:08:32.223
times 10 plus 20,

00:08:37.605 --> 00:08:38.605
equals zero.

00:08:39.488 --> 00:08:41.211
Now we have the same equation as before,

00:08:41.211 --> 00:08:44.491
so we're going to get the same i,

00:08:44.491 --> 00:08:47.950
i equals, three goes to the other side,

00:08:47.950 --> 00:08:50.450
and becomes minus three,

00:08:50.450 --> 00:08:51.995
10 plus 20 is 30,

00:08:51.995 --> 00:08:53.429
with the minus sign.

00:08:53.429 --> 00:08:54.512
It goes over,

00:09:01.387 --> 00:09:02.637
same as before,

00:09:05.517 --> 00:09:07.790
so those came out the same,

00:09:07.790 --> 00:09:09.434
so now let's go check the voltages,

00:09:09.434 --> 00:09:11.173
see if we can compute the same voltages,

00:09:11.173 --> 00:09:12.484
and what he have to notice here,

00:09:12.484 --> 00:09:14.260
is our reference direction,

00:09:14.260 --> 00:09:16.559
the original reference
direction we have for V

00:09:16.559 --> 00:09:19.392
generated a minus sign
when we used Ohm's law,

00:09:19.392 --> 00:09:22.651
so we keep doing that, it's okay,

00:09:22.651 --> 00:09:24.651
so V1, equals minus iR1,

00:09:29.074 --> 00:09:31.574
equals minus .1 amps times 10,

00:09:34.578 --> 00:09:37.161
and that equals minus one volt,

00:09:38.060 --> 00:09:40.143
and V2 equals i times R2,

00:09:43.981 --> 00:09:46.898
which is equal to .1 times 20 ohms,

00:09:49.309 --> 00:09:51.892
and that equals plus two volts,

00:09:53.097 --> 00:09:54.243
and now the difference,

00:09:54.243 --> 00:09:55.076
we see the difference here.

00:09:55.076 --> 00:09:58.243
Here is the difference that showed up.

00:10:01.097 --> 00:10:05.734
V1 in this circuit has the
positive voltage at the top,

00:10:05.734 --> 00:10:09.797
it came out with a value of plus one volt,

00:10:09.797 --> 00:10:12.766
and when we flipped over V1,

00:10:12.766 --> 00:10:14.493
negative one volt here,

00:10:14.493 --> 00:10:16.455
means that this terminal of the resistor,

00:10:16.455 --> 00:10:21.237
is one volt below this
terminal of the resistor,

00:10:21.237 --> 00:10:24.118
and that means exactly the
same thing in this case,

00:10:24.118 --> 00:10:25.442
as it does in this case,

00:10:25.442 --> 00:10:29.257
so these two things mean the same thing,

00:10:29.257 --> 00:10:33.424
and of course, the voltage on
number two came out the same,

00:10:36.801 --> 00:10:38.409
so the purpose of the demonstration

00:10:38.409 --> 00:10:39.634
was just to show you,

00:10:39.634 --> 00:10:42.403
that no matter which way
you name the voltages,

00:10:42.403 --> 00:10:44.788
somewhere in here, like right here,

00:10:44.788 --> 00:10:46.369
Kirchhoff's voltage law will take care

00:10:46.369 --> 00:10:47.780
of keeping the sign right,

00:10:47.780 --> 00:10:50.987
and you end up with the
same answer at the end,

00:10:50.987 --> 00:10:52.306
so when you're faced with the problem

00:10:52.306 --> 00:10:54.146
of labeling a circuit like this,

00:10:54.146 --> 00:10:57.357
don't stress out about
trying to guess ahead,

00:10:57.357 --> 00:11:00.002
what the sign of the voltage
is going to turn out.

00:11:00.002 --> 00:11:01.813
You just need to pick an orientation,

00:11:01.813 --> 00:11:02.692
and go with it,

00:11:02.692 --> 00:11:04.532
and the arithmetic will take care of

00:11:04.532 --> 00:11:06.571
the positive signs,
and the negative signs.


Multiplying monomials

WEBVTT
Kind: captions
Language: en

00:00:00.258 --> 00:00:02.649
- [Voiceover] Alright, in this video,

00:00:02.649 --> 00:00:06.482
we're gonna be multiplying
monomials together.

00:00:08.194 --> 00:00:12.084
Let me give you an example of a monomial.

00:00:12.084 --> 00:00:14.584
4x squared, that's a monomial.

00:00:15.700 --> 00:00:17.359
Now, why?

00:00:17.359 --> 00:00:21.526
Well, mono means one, which
refers to the number of terms.

00:00:24.892 --> 00:00:28.157
So this 4x squared, this is all one term.

00:00:28.157 --> 00:00:30.687
So we're gonna be working
with things like that.

00:00:30.687 --> 00:00:33.291
What won't we be working with?

00:00:33.291 --> 00:00:36.208
Well what about 4x squared plus 5x.

00:00:37.170 --> 00:00:38.975
How many terms are there?

00:00:38.975 --> 00:00:41.491
4x squared's the first
term, 5x is the second term,

00:00:41.491 --> 00:00:43.658
so this is not a monomial,

00:00:44.538 --> 00:00:48.705
this is actually called a
binomial, because bi means two.

00:00:50.503 --> 00:00:54.180
Like your bicycle's got
two wheels, for example.

00:00:54.180 --> 00:00:56.081
So not yet, go on to the future videos

00:00:56.081 --> 00:00:57.383
if you're ready for binomials.

00:00:57.383 --> 00:00:58.550
But we're just gonna be working

00:00:58.550 --> 00:01:00.803
with multiplying monomials together.

00:01:00.803 --> 00:01:03.886
So can we grab an example to look at.

00:01:06.465 --> 00:01:10.033
By the end of this video,
it should be very easy

00:01:10.033 --> 00:01:13.866
for you to multiply this
monomial, 5x squared,

00:01:15.926 --> 00:01:18.135
by this monomial.

00:01:18.135 --> 00:01:21.334
And I'm actually just gonna
give you the answer right here.

00:01:21.334 --> 00:01:22.972
And then I'm gonna slowly
walk you through some other

00:01:22.972 --> 00:01:25.269
questions that will lead us to why.

00:01:25.269 --> 00:01:28.936
But the answer to this
is 20x to the eighth.

00:01:31.977 --> 00:01:33.376
20x to the eighth.

00:01:33.376 --> 00:01:35.392
Take a look at that, see if
you can notice a pattern.

00:01:35.392 --> 00:01:37.785
What did we do with the five
and the four to get the 20?

00:01:37.785 --> 00:01:40.780
What did we do with the two
and the six to get the eight?

00:01:40.780 --> 00:01:43.467
That's getting a little
ahead of ourselves though.

00:01:43.467 --> 00:01:44.897
Before we can dive in there,

00:01:44.897 --> 00:01:48.086
let's remember some of
the exponent properties.

00:01:48.086 --> 00:01:49.393
A very specific exponent property

00:01:49.393 --> 00:01:51.723
that you should've seen before.

00:01:51.723 --> 00:01:55.280
If we look at five squared
times five to the fourth power,

00:01:55.280 --> 00:01:57.236
what's that going to equal?

00:01:57.236 --> 00:01:59.814
Well, if you remember
your exponent property,

00:01:59.814 --> 00:02:03.795
we'll do a quick reminder
here, I always add my exponent.

00:02:03.795 --> 00:02:06.072
So five squared times
five to the fourth power

00:02:06.072 --> 00:02:08.734
is equal to five to the sixth power.

00:02:08.734 --> 00:02:10.553
What about three to the fourth power

00:02:10.553 --> 00:02:13.639
times three to the fifth power?

00:02:13.639 --> 00:02:16.747
Well, again, I always add my exponents.

00:02:16.747 --> 00:02:20.049
Four plus five is three
to the ninth power,

00:02:20.049 --> 00:02:23.596
and my base of three stays the same.

00:02:23.596 --> 00:02:25.558
Great, so if you remember that,

00:02:25.558 --> 00:02:27.321
now we're ready to really start

00:02:27.321 --> 00:02:29.646
multiplying monomials that are new to you.

00:02:29.646 --> 00:02:31.402
And the new thing there
is that we are going

00:02:31.402 --> 00:02:33.802
to have variables involved.

00:02:33.802 --> 00:02:37.969
So let's start, let's take a
look at two monomials here.

00:02:39.462 --> 00:02:44.215
The first monomial is 4x,
and the second one is just x.

00:02:44.215 --> 00:02:47.819
And the four, I don't have
another number to multiply by,

00:02:47.819 --> 00:02:48.840
just got the four.

00:02:48.840 --> 00:02:51.335
And can I simplify x times x?

00:02:51.335 --> 00:02:54.252
Well, that's equal to x squared.

00:02:55.388 --> 00:02:56.697
Remember if I just have a variable,

00:02:56.697 --> 00:02:59.740
and there's no exponent there,

00:02:59.740 --> 00:03:01.562
it's equivalent to having a one,

00:03:01.562 --> 00:03:04.610
so x to the first power
times x to the first power,

00:03:04.610 --> 00:03:07.247
I add my exponents like
we just talked about,

00:03:07.247 --> 00:03:09.630
and one plus one is equal to two.

00:03:09.630 --> 00:03:13.297
Great, so let's move
on to another one here.

00:03:15.235 --> 00:03:17.068
If I have 4t times 3t.

00:03:19.058 --> 00:03:22.990
Well, four times three
is gonna be equal to 12,

00:03:22.990 --> 00:03:26.486
so I've combined my coefficients.

00:03:26.486 --> 00:03:30.688
And then t times t, again,
think of a one being there,

00:03:30.688 --> 00:03:32.771
is going to be t squared.

00:03:35.588 --> 00:03:38.027
So the answer here is 12t squared.

00:03:38.027 --> 00:03:40.234
So let's keep going,

00:03:40.234 --> 00:03:42.145
and once you get into the rhythm of these,

00:03:42.145 --> 00:03:44.202
they become pretty alright.

00:03:44.202 --> 00:03:48.199
So what if I had 4p to
the fifth power times,

00:03:48.199 --> 00:03:50.866
let's say 5p to the third power.

00:03:52.459 --> 00:03:54.274
What would that equal?

00:03:54.274 --> 00:03:56.304
Well you're gonna notice a pattern here

00:03:56.304 --> 00:03:57.565
that we've been pickin' up on,

00:03:57.565 --> 00:04:00.641
which is that I'm always gonna
multiply my coefficients,

00:04:00.641 --> 00:04:04.058
so four times five, is going to equal 20.

00:04:05.741 --> 00:04:09.162
And I'm always going to add my exponents.

00:04:09.162 --> 00:04:12.162
So p to the fifth and p to the third

00:04:14.649 --> 00:04:16.732
is p to the eighth power.

00:04:18.972 --> 00:04:21.764
so I multiply four and five til we get 20,

00:04:21.764 --> 00:04:25.264
I add five and three to get eight.

00:04:25.264 --> 00:04:27.033
And if you really wanted
to see why that is,

00:04:27.033 --> 00:04:28.867
let's really dive in
here and let's break down

00:04:28.867 --> 00:04:31.819
this first term, let's
break down 4p to the fifth.

00:04:31.819 --> 00:04:33.889
I can write that out as four times p,

00:04:33.889 --> 00:04:38.056
times p, times p, times p,
times p, that's five of 'em.

00:04:39.460 --> 00:04:40.737
That's four and five p's.

00:04:40.737 --> 00:04:42.683
And then that second term I can write as

00:04:42.683 --> 00:04:45.766
times five times p, times p, times p.

00:04:47.146 --> 00:04:50.695
What I'm gonna do is I'm
gonna group my numbers,

00:04:50.695 --> 00:04:52.145
cause I can work with numbers together,

00:04:52.145 --> 00:04:55.480
so let's put four times
five at the very front,

00:04:55.480 --> 00:04:58.961
and then it just becomes a
matter of how many p's do I have?

00:04:58.961 --> 00:05:01.387
We'll put all of those together as well.

00:05:01.387 --> 00:05:05.137
So I had five p's, so
there's the first five,

00:05:06.113 --> 00:05:08.163
and then I had three more.

00:05:08.163 --> 00:05:11.532
And we can simplify this
crazy looking expression

00:05:11.532 --> 00:05:14.849
by just multiplying my four
and my five to be my 20,

00:05:14.849 --> 00:05:18.438
and then writing this with an exponent,

00:05:18.438 --> 00:05:20.507
that's the beauty of exponents,
that's why we have 'em,

00:05:20.507 --> 00:05:23.468
is we can write a crazy
expression like that

00:05:23.468 --> 00:05:24.677
as p to the eighth,

00:05:24.677 --> 00:05:26.911
and you'll notice that this is, of course,

00:05:26.911 --> 00:05:29.593
what we got the first time.

00:05:29.593 --> 00:05:30.426
So great.

00:05:34.707 --> 00:05:36.979
What about 5y to the sixth times

00:05:36.979 --> 00:05:39.646
negative 3y to the eighth power?

00:05:45.410 --> 00:05:49.577
Again, multiply the
coefficients, add the exponents,

00:05:52.663 --> 00:05:55.746
and I've got a simplified expression.

00:05:56.655 --> 00:05:59.227
Let's get really crazy here,
let's have a little fun.

00:05:59.227 --> 00:06:01.301
So we've noticed the pattern,
let's have a little fun.

00:06:01.301 --> 00:06:04.128
Just saying, I can, I can do more.

00:06:04.128 --> 00:06:07.128
Negative 9x to the fifth power times

00:06:13.369 --> 00:06:16.536
negative three, use parentheses there,

00:06:17.662 --> 00:06:19.153
when you have a negative in front,

00:06:19.153 --> 00:06:20.417
you always wanna use parentheses.

00:06:20.417 --> 00:06:23.167
Let's do x to the 107th power.

00:06:25.178 --> 00:06:27.255
If I would have showed you
this before this video,

00:06:27.255 --> 00:06:28.637
you would have said oh my goodness,

00:06:28.637 --> 00:06:30.740
there's nothing I can do, I'm boxed,

00:06:30.740 --> 00:06:32.470
there's no way out.

00:06:32.470 --> 00:06:36.083
But now you know that it's as
simple as follow the rules.

00:06:36.083 --> 00:06:38.787
We're going to multiply the coefficients,

00:06:38.787 --> 00:06:42.166
negative nine times negative three is 27.

00:06:42.166 --> 00:06:45.229
Two negatives is a positive
and nine times three is 27.

00:06:45.229 --> 00:06:47.771
I'm gonna add my powers.

00:06:47.771 --> 00:06:51.188
Five plus 107 is a hundred, ooh, not two,

00:06:54.435 --> 00:06:58.109
that was almost a mistake I made there.

00:06:58.109 --> 00:07:00.667
Let's get rid of that, give
me a second chance here.

00:07:00.667 --> 00:07:02.660
Life's all about second chances,

00:07:02.660 --> 00:07:04.410
five plus 107 is 112.

00:07:06.695 --> 00:07:09.424
And so, this crazy expression,

00:07:09.424 --> 00:07:12.651
which is two monomials, here's the first,

00:07:12.651 --> 00:07:16.865
here's the second, when
we multiply and simplify

00:07:16.865 --> 00:07:20.511
we get another monomial,
which is 27x to the 112th.

00:07:20.511 --> 00:07:24.251
I'm gonna leave you on a cliffhanger here.

00:07:24.251 --> 00:07:27.591
Which, I'm gonna show you a problem.

00:07:27.591 --> 00:07:30.296
What variable should we use?

00:07:30.296 --> 00:07:31.785
You notice I've been trying
to vary the variables up

00:07:31.785 --> 00:07:34.705
to show you that it just doesn't matter.

00:07:34.705 --> 00:07:35.959
That's an ugly five,
let's get rid of that.

00:07:35.959 --> 00:07:38.547
Give me a second chance with that one too.

00:07:38.547 --> 00:07:41.797
So let's look at 5x to the third power,

00:07:45.942 --> 00:07:48.275
times 4x to the sixth power.

00:07:51.211 --> 00:07:53.393
And I'm gonna show you a wrong answer.

00:07:53.393 --> 00:07:55.736
I had a student that asked to do this,

00:07:55.736 --> 00:07:58.448
and here's the wrong
answer that they gave me.

00:07:58.448 --> 00:08:01.281
They told me 9x to the 18th power.

00:08:03.425 --> 00:08:04.424
That's terribly wrong.

00:08:04.424 --> 00:08:05.422
What did they do wrong?

00:08:05.422 --> 00:08:07.161
What did they do wrong?

00:08:07.161 --> 00:08:08.719
I want you to think to yourself,

00:08:08.719 --> 00:08:10.048
what have we been talking about?

00:08:10.048 --> 00:08:11.724
What did they do with the five
and the four to get the nine?

00:08:11.724 --> 00:08:13.336
What should they have done?

00:08:13.336 --> 00:08:15.934
What did they do with the three
and the six to get the 18,

00:08:15.934 --> 00:08:18.243
and what should they have done?

00:08:18.243 --> 00:08:21.743
That's multiplying monomials by monomials.


Relation of null space to linear independence of columns

WEBVTT
Kind: captions
Language: en

00:00:00.360 --> 00:00:02.824
- [Voiceover] So I have
the matrix A over here,

00:00:02.824 --> 00:00:05.407
and A has m rows and n columns,

00:00:07.736 --> 00:00:10.986
so we could call this an m by n matrix.

00:00:12.897 --> 00:00:15.256
And what I want to do in this video,

00:00:15.256 --> 00:00:19.423
is relate the linear independence,
or linear dependence,

00:00:20.497 --> 00:00:24.664
of the column vectors of
A, to the null space of A.

00:00:25.813 --> 00:00:26.646
So what, first of all

00:00:26.646 --> 00:00:28.850
what I am talking about as column vectors?

00:00:28.850 --> 00:00:31.148
Well as you can see
there's n columns here,

00:00:31.148 --> 00:00:35.315
and we could view each of those
as an m-dimensional vector.

00:00:36.746 --> 00:00:38.600
And so, let me do it this way,

00:00:38.600 --> 00:00:41.706
so you can view this one right over here,

00:00:41.706 --> 00:00:44.123
we could write that as V one,

00:00:45.058 --> 00:00:45.891
V one,

00:00:46.840 --> 00:00:50.623
this next one over here,
this would be V two,

00:00:50.623 --> 00:00:51.949
V two,

00:00:51.949 --> 00:00:54.378
and you would have n of these,
because we have n columns,

00:00:54.378 --> 00:00:58.128
and so this one right
over here would be V n,

00:01:00.381 --> 00:01:01.214
V sub n.

00:01:03.148 --> 00:01:07.714
And so we could rewrite A, we
could rewrite the matrix A,

00:01:07.714 --> 00:01:09.381
the m by n matrix A,

00:01:10.339 --> 00:01:12.401
I'm bolding it to show
that that's a matrix,

00:01:12.401 --> 00:01:14.318
we could rewrite it as,

00:01:15.595 --> 00:01:19.067
so let me do it the same way, so,

00:01:19.067 --> 00:01:21.945
draw my little brackets there,

00:01:21.945 --> 00:01:23.947
we can write it, just express it,

00:01:23.947 --> 00:01:26.323
in terms of its column vectors,

00:01:26.323 --> 00:01:28.155
cos we could just say
well this is going to be

00:01:28.155 --> 00:01:29.988
V one for that column,

00:01:31.075 --> 00:01:32.908
V one for that column,

00:01:34.457 --> 00:01:37.374
V two for this column, all the way,

00:01:38.685 --> 00:01:39.980
we're gonna have n columns,

00:01:39.980 --> 00:01:43.569
so you're gonna have V
n, for the nth column.

00:01:43.569 --> 00:01:47.330
And remember, each of these,
are going to have m terms,

00:01:47.330 --> 00:01:50.084
or I should say, m components in them.

00:01:50.084 --> 00:01:53.134
These are m-dimensional column vectors.

00:01:53.134 --> 00:01:54.476
Now what I want to do, I said I want to

00:01:54.476 --> 00:01:58.578
relate the linear
independence of these vectors,

00:01:58.578 --> 00:02:00.607
to the null space of A.

00:02:00.607 --> 00:02:04.530
So let's remind ourselves what
the null space of A even is.

00:02:04.530 --> 00:02:06.447
So the null space of A,

00:02:08.827 --> 00:02:10.494
the null space of A,

00:02:12.446 --> 00:02:17.027
is equal to, or I could
say it's equal to the set,

00:02:17.027 --> 00:02:19.527
it's the set of all vectors x,

00:02:21.230 --> 00:02:23.230
that are members of R n,

00:02:24.943 --> 00:02:27.831
and I'm gonna double down
on why I'm saying R n,

00:02:27.831 --> 00:02:30.664
in a second, such that, such that,

00:02:32.291 --> 00:02:34.124
if I take my matrix A,

00:02:35.148 --> 00:02:37.356
if I take my matrix A,

00:02:37.356 --> 00:02:41.021
and multiply it by one of those x's,

00:02:41.021 --> 00:02:43.008
by one of those x's,

00:02:43.008 --> 00:02:47.175
I'm going to get, I'm going
to get the zero vector.

00:02:48.666 --> 00:02:52.930
So, why do the, why does x
have to be a member of R n?

00:02:52.930 --> 00:02:56.045
Well just for the matrix
multiplication to work,

00:02:56.045 --> 00:03:00.748
for this to be, if this is m
by n, let me write this down,

00:03:00.748 --> 00:03:02.248
if this is m by n,

00:03:04.491 --> 00:03:06.801
well in order just to make
the matrix multiplication work

00:03:06.801 --> 00:03:09.509
or you could say the matrix
vector multiplication,

00:03:09.509 --> 00:03:11.759
this has to be an n by one,

00:03:13.423 --> 00:03:15.090
an n by one, vector,

00:03:16.986 --> 00:03:18.695
and so it's gonna have n components,

00:03:18.695 --> 00:03:21.324
so it's gonna be a member of R n.

00:03:21.324 --> 00:03:24.560
If this was m by A, well, or,
let me use a different letter,

00:03:24.560 --> 00:03:27.519
if this was m by, I don't know, seven,

00:03:27.519 --> 00:03:31.743
then this would be R seven,
that we would be dealing with.

00:03:31.743 --> 00:03:34.360
So that is the null space.

00:03:34.360 --> 00:03:36.327
So, another way of thinking about it is,

00:03:36.327 --> 00:03:37.617
well if I take my matrix A,

00:03:37.617 --> 00:03:40.144
and I multiply it by sum vector x,

00:03:40.144 --> 00:03:41.590
that's a member of this null space,

00:03:41.590 --> 00:03:43.375
I'm going to get the zero vector.

00:03:43.375 --> 00:03:44.859
So if I take my matrix A,

00:03:44.859 --> 00:03:47.428
which I've expressed here in
terms of its column vectors,

00:03:47.428 --> 00:03:49.761
multiply it by sum vector x,

00:03:51.116 --> 00:03:52.449
so sum vector x,

00:03:53.531 --> 00:03:57.389
and, actually let me make it clear that,

00:03:57.389 --> 00:03:59.916
it doesn't have to have the same,

00:03:59.916 --> 00:04:02.916
so, sum vector x right over here,

00:04:04.114 --> 00:04:06.281
we draw the other bracket,

00:04:07.939 --> 00:04:09.987
so this is the vector x, and so,

00:04:09.987 --> 00:04:12.954
it's going to have, it's a member of R n,

00:04:12.954 --> 00:04:14.255
so it's going to have n components,

00:04:14.255 --> 00:04:17.071
you're gonna have x one,
as the first component,

00:04:17.071 --> 00:04:20.154
x two, and go all the way, to x n.

00:04:22.424 --> 00:04:25.339
If you multiply, so if we say that this x

00:04:25.339 --> 00:04:26.506
is a member of

00:04:28.657 --> 00:04:30.824
the null space of A, then,

00:04:32.266 --> 00:04:36.274
this whole thing is going to
be equal to the zero vector,

00:04:36.274 --> 00:04:38.269
is going to be equal to the zero vector,

00:04:38.269 --> 00:04:39.827
and once again the zero vector,

00:04:39.827 --> 00:04:42.858
this is gonna be an m by one vector,

00:04:42.858 --> 00:04:47.488
so it's gonna look, actually
let me write it like this,

00:04:47.488 --> 00:04:51.599
it's gonna have the same
number of rows as A,

00:04:51.599 --> 00:04:54.019
so, I'll try to make it,

00:04:54.019 --> 00:04:57.436
the brackets roughly the same length, so,

00:04:58.409 --> 00:05:02.992
and there we go, try and
draw my brackets neatly,

00:05:02.992 --> 00:05:05.993
so you're gonna have m of these, one, two,

00:05:05.993 --> 00:05:09.326
and then go all the way to the mth zero.

00:05:10.170 --> 00:05:12.821
So, let's actually just multiply this out,

00:05:12.821 --> 00:05:16.042
using what we know of
matrix multiplication.

00:05:16.042 --> 00:05:18.029
And by the definition of
matrix multiplication,

00:05:18.029 --> 00:05:18.924
one way to view this,

00:05:18.924 --> 00:05:21.415
if you were to multiply our matrix A

00:05:21.415 --> 00:05:23.525
times our vector x here,

00:05:23.525 --> 00:05:28.088
you're going to get the
first column vector, V one,

00:05:28.088 --> 00:05:28.921
V one,

00:05:29.758 --> 00:05:33.175
times the first component here, x one,

00:05:34.830 --> 00:05:36.172
x one,

00:05:36.172 --> 00:05:39.674
plus, the second component
times the second column vector,

00:05:39.674 --> 00:05:41.174
x two times V two,

00:05:43.071 --> 00:05:43.904
V two,

00:05:45.309 --> 00:05:47.577
and we're gonna do that n times,

00:05:47.577 --> 00:05:49.160
so plus dot dot dot

00:05:51.097 --> 00:05:52.930
x sub n times V sub n,

00:05:55.118 --> 00:05:55.951
V sub n,

00:05:58.631 --> 00:06:01.491
and these all when you add them together

00:06:01.491 --> 00:06:04.908
are going to be equal to the zero vector.

00:06:07.453 --> 00:06:09.504
Now this should be, this,

00:06:09.504 --> 00:06:11.771
so it's gonna be equal to the zero vector,

00:06:11.771 --> 00:06:14.057
and now this should start
ringing a bell to you,

00:06:14.057 --> 00:06:17.006
when we looked at, when we
looked at linear independence

00:06:17.006 --> 00:06:18.420
we saw something like this,

00:06:18.420 --> 00:06:23.184
in fact we saw that these
vectors V, V sub one, V sub two,

00:06:23.184 --> 00:06:27.351
these n vectors, are linearly
independent if and only if,

00:06:29.413 --> 00:06:32.601
any linear, if and only
if the solution to this,

00:06:32.601 --> 00:06:35.576
or I guess you could say the
weights on these vectors,

00:06:35.576 --> 00:06:37.399
the only way to get this to be true

00:06:37.399 --> 00:06:42.139
is if x one, x two, x
n are all equal zero.

00:06:42.139 --> 00:06:43.792
So let me write this down.

00:06:43.792 --> 00:06:44.875
So V sub one,

00:06:47.119 --> 00:06:47.952
V sub two,

00:06:49.366 --> 00:06:51.283
all the way to V sub n,

00:06:52.796 --> 00:06:57.247
are linearly independent,
linearly independent,

00:06:57.247 --> 00:06:59.830
if and only if, if and only if,

00:07:01.215 --> 00:07:04.548
only solution, so let me, only solution,

00:07:07.775 --> 00:07:11.358
or you could say weights on
these vectors, to this equation,

00:07:11.358 --> 00:07:12.691
only solution is

00:07:14.959 --> 00:07:16.292
x one, x two,

00:07:18.105 --> 00:07:21.438
all the way to x n are equal to zero.

00:07:22.318 --> 00:07:23.707
So if the only solution here,

00:07:23.707 --> 00:07:26.333
if the only way to get this sum

00:07:26.333 --> 00:07:27.881
to be equal to the zero vector,

00:07:27.881 --> 00:07:30.733
is if x one, x two and x,
all the way through x n,

00:07:30.733 --> 00:07:31.814
are equal to zero,

00:07:31.814 --> 00:07:34.144
well that means that our
vectors V one, V two,

00:07:34.144 --> 00:07:37.226
all the way to V n, are
linearly independent,

00:07:37.226 --> 00:07:39.813
or vice versa, if they're
linearly independent,

00:07:39.813 --> 00:07:41.448
then the only solution to this,

00:07:41.448 --> 00:07:44.436
if we're solving for the
weights on those vectors,

00:07:44.436 --> 00:07:48.500
is if for x one, x two and
x n to be equal to zero.

00:07:48.500 --> 00:07:51.417
Remember, linear independence,
if you want to say,

00:07:51.417 --> 00:07:52.315
it's still mathematical,

00:07:52.315 --> 00:07:54.680
but a little bit more, common language is,

00:07:54.680 --> 00:07:56.821
if these vectors are linearly independent,

00:07:56.821 --> 00:08:01.061
that means that none of these
vectors can be constructed

00:08:01.061 --> 00:08:05.504
by linear combinations
of the other vectors,

00:08:05.504 --> 00:08:07.620
or, looking at it this way,

00:08:07.620 --> 00:08:09.532
this right over here is
a, you could view this

00:08:09.532 --> 00:08:12.768
as a linear combination
of all of the vectors,

00:08:12.768 --> 00:08:14.459
that the only way to get
this linear combination

00:08:14.459 --> 00:08:16.861
of all the vectors to be equal to zero,

00:08:16.861 --> 00:08:20.444
is if x one, x two,
all the way through x n

00:08:20.444 --> 00:08:21.531
are equal to zero,

00:08:21.531 --> 00:08:25.305
and we proved that in other
videos on linear independence.

00:08:25.305 --> 00:08:28.621
Well, if the only solution to this

00:08:28.621 --> 00:08:32.693
is all of the x one's through
x n's are equal to zero,

00:08:32.693 --> 00:08:34.682
that means that the null space,

00:08:34.682 --> 00:08:36.717
this is only going to
be true, you could say,

00:08:36.717 --> 00:08:39.717
if and only if, the null space of A,

00:08:41.450 --> 00:08:43.478
the null space of A,

00:08:43.478 --> 00:08:46.396
let me make sure it looks like
a matrix, I'm gonna bold it,

00:08:46.396 --> 00:08:50.471
the null space of A only
contains one vector,

00:08:50.471 --> 00:08:53.270
it only contains the zero vector.

00:08:53.270 --> 00:08:56.025
Remember, this is, if all
of these are gonna be zero,

00:08:56.025 --> 00:08:57.500
well then the only solution here

00:08:57.500 --> 00:08:59.644
is gonna be the zero vector,

00:08:59.644 --> 00:09:03.561
is going to be, is going
to be the zero vector.

00:09:04.554 --> 00:09:06.873
So the result that we're showing here is,

00:09:06.873 --> 00:09:11.040
if the column vectors of a
matrix are linearly independent,

00:09:13.560 --> 00:09:15.806
then the null space of that matrix

00:09:15.806 --> 00:09:18.485
is only going to consist
of the zero vector.

00:09:18.485 --> 00:09:20.162
Or you could go the other way.

00:09:20.162 --> 00:09:24.631
If the null space of a matrix
only contains the zero vector,

00:09:24.631 --> 00:09:27.903
well that means that the
columns of that matrix

00:09:27.903 --> 00:09:29.986
are linearly independent.


RC natural response derivation (2 of 3)

WEBVTT
Kind: captions
Language: en

00:00:00.497 --> 00:00:02.579
- [Voiceover] Now what I want
to to do for the RC circuit

00:00:02.579 --> 00:00:05.793
is a formal derivation of exactly what

00:00:05.793 --> 00:00:08.040
these two curves look like.

00:00:08.040 --> 00:00:11.847
And then we'll have a
precise definition of

00:00:11.847 --> 00:00:13.597
the natural response.

00:00:16.403 --> 00:00:20.403
What I want to do now is
draw our circuit again.

00:00:24.256 --> 00:00:26.006
There's R, there's C.

00:00:29.273 --> 00:00:31.537
And we have some initial voltage, we said,

00:00:31.537 --> 00:00:34.537
on the capacitor, which is V naught.

00:00:35.418 --> 00:00:38.335
And what we want to find is V of T.

00:00:43.640 --> 00:00:46.057
And there's R, and there's C.

00:00:47.944 --> 00:00:52.502
So let me also label the
currents in these guys.

00:00:52.502 --> 00:00:55.730
This is the current in the resistor,

00:00:55.730 --> 00:00:59.147
and this is the current in the capacitor.

00:01:01.041 --> 00:01:03.730
So now I'm gonna write
the two voltage current

00:01:03.730 --> 00:01:06.119
relationships for these two components.

00:01:06.119 --> 00:01:09.869
And Ohm's Law tells me
that equals one over R

00:01:12.852 --> 00:01:13.685
times V.

00:01:15.604 --> 00:01:19.187
And the current in a
capacitor, this is iR,

00:01:22.511 --> 00:01:26.261
current in a capacitor
equals C times dv, dt.

00:01:31.560 --> 00:01:33.687
That's the two currents in our devices.

00:01:33.687 --> 00:01:37.122
Now, gonna use Kirchhoff's
Current Law on this node

00:01:37.122 --> 00:01:39.744
right here, and that says
that all the currents

00:01:39.744 --> 00:01:43.673
flying out of that node
have to add up to zero.

00:01:43.673 --> 00:01:44.827
So let's do that.

00:01:44.827 --> 00:01:47.577
Let's say iC plus iR equals zero.

00:01:52.477 --> 00:01:55.560
This is KCL, Kirchhoff's Current Law.

00:01:58.748 --> 00:02:02.748
C dv, dt, plus one over
R, times V, equals zero.

00:02:09.903 --> 00:02:12.016
Now I'm just gonna divide through by C

00:02:12.016 --> 00:02:13.349
just to be tidy.

00:02:14.292 --> 00:02:18.125
Dv, dt plus one over RC,
times V, equals zero.

00:02:28.086 --> 00:02:30.871
So what we have here,
this expression here is

00:02:30.871 --> 00:02:34.954
referred to as an ordinary
differential equation.

00:02:36.251 --> 00:02:38.249
It has a derivative in it,
and that's what makes it

00:02:38.249 --> 00:02:41.371
a differential equation,
and you can go look up

00:02:41.371 --> 00:02:45.980
ODE and search for this in
Khan Academy and elsewhere

00:02:45.980 --> 00:02:48.870
and in the mathematical
videos, you'll find out

00:02:48.870 --> 00:02:52.199
different ways to solve this equation.

00:02:52.199 --> 00:02:54.164
Now we're gonna do the same thing here,

00:02:54.164 --> 00:02:58.370
and we're gonna try to find
an expression for V of t.

00:02:58.370 --> 00:03:02.787
We're gonna define V of
t in a way that makes

00:03:02.787 --> 00:03:04.457
this equation true.

00:03:04.457 --> 00:03:07.929
So we plug the function
in here, whatever it is,

00:03:07.929 --> 00:03:10.890
and then we take its derivative
and plug that in here

00:03:10.890 --> 00:03:14.593
and that has to make this
whole expression true.

00:03:14.593 --> 00:03:16.545
There's a couple of different
ways to do this, and the one

00:03:16.545 --> 00:03:18.730
that we're gonna use
here is we're gonna guess

00:03:18.730 --> 00:03:20.242
at an answer.

00:03:20.242 --> 00:03:23.348
So what we need is, we need
a function whose derivative

00:03:23.348 --> 00:03:26.825
looks kind of like
itself, because this thing

00:03:26.825 --> 00:03:28.658
has to add up to zero.

00:03:29.611 --> 00:03:33.778
So that means that the function
V and the function dv, dt

00:03:34.621 --> 00:03:37.032
they have to be of the
same form so that they have

00:03:37.032 --> 00:03:39.532
a chance of adding up to zero.

00:03:40.737 --> 00:03:42.977
The function that I know
whose derivative looks

00:03:42.977 --> 00:03:45.815
like itself is the exponential.

00:03:45.815 --> 00:03:49.201
So we're gonna propose that
there's gonna be an exponential

00:03:49.201 --> 00:03:52.402
and the way we do it is say,
we'll put in some constant here

00:03:52.402 --> 00:03:54.397
that we can adjust,
and then we'll put some

00:03:54.397 --> 00:03:55.897
constant out here.

00:03:59.211 --> 00:04:00.165
So this is our guess.

00:04:00.165 --> 00:04:03.904
Our guess is gonna be that
the solution for V of t

00:04:03.904 --> 00:04:06.802
is gonna be something of
the form, some constant

00:04:06.802 --> 00:04:10.135
times e to another constant, s, times t.

00:04:13.215 --> 00:04:15.951
The way we test our guess
is to put it back into

00:04:15.951 --> 00:04:18.350
the equation and see if it works.

00:04:18.350 --> 00:04:20.156
And if it does, then we made a good guess.

00:04:20.156 --> 00:04:21.578
If it doesn't work,
then we have to go back

00:04:21.578 --> 00:04:23.890
and try something else.

00:04:23.890 --> 00:04:25.226
So one of the things we
need, we're gonna plug

00:04:25.226 --> 00:04:27.142
v of t in right here.

00:04:27.142 --> 00:04:31.065
We're gonna need derivative
of v with respect to time.

00:04:31.065 --> 00:04:33.732
So let's take a derivative here.

00:04:37.006 --> 00:04:39.506
Equals d dt of k, e to the st.

00:04:45.009 --> 00:04:48.342
And that equals K, and the S comes down.

00:04:52.218 --> 00:04:54.218
And e to the st remains.

00:04:57.366 --> 00:05:00.432
So we just took the derivative
of our proposed solution

00:05:00.432 --> 00:05:01.984
and now we're gonna plug
it into the equation

00:05:01.984 --> 00:05:03.734
and see what happens.

00:05:09.691 --> 00:05:12.762
So now I'm gonna plug in
the derivative that we had,

00:05:12.762 --> 00:05:15.178
plus the proposed solution,
and see if our equation

00:05:15.178 --> 00:05:16.428
comes out true.

00:05:18.884 --> 00:05:21.717
So dv, dt is sk, e to the st, plus

00:05:28.201 --> 00:05:30.118
one over RC stays here.

00:05:32.141 --> 00:05:35.558
And we plug in V, and V was ke to the st.

00:05:38.785 --> 00:05:41.035
And that'll all equal zero.

00:05:43.778 --> 00:05:47.741
Now I'm gonna factor out,
here's the common term

00:05:47.741 --> 00:05:51.908
for each one of these, so
we'll just factor that out.

00:05:52.795 --> 00:05:56.962
So now we get ke to the st,
times S plus one over RC

00:06:06.335 --> 00:06:07.335
equals zero.

00:06:09.439 --> 00:06:12.258
Okay, now let's take
a little peek at this.

00:06:12.258 --> 00:06:14.783
We have to make this
zero by the way we pick

00:06:14.783 --> 00:06:15.616
K and S.

00:06:18.047 --> 00:06:20.199
How can we make this zero?

00:06:20.199 --> 00:06:22.204
There's three terms in
here, there's three terms

00:06:22.204 --> 00:06:23.168
in the product.

00:06:23.168 --> 00:06:25.585
There's this, this, and that.

00:06:26.788 --> 00:06:29.614
Any three of these terms could be zero.

00:06:29.614 --> 00:06:32.633
The first one, K is
zero, that's a boring one

00:06:32.633 --> 00:06:35.567
because it says if we have
a circuit with no energy

00:06:35.567 --> 00:06:37.505
it sits there with no energy.

00:06:37.505 --> 00:06:40.488
So that's not so interesting.

00:06:40.488 --> 00:06:44.830
Now, e to the st, we
can make that go to zero

00:06:44.830 --> 00:06:47.973
but that takes a long time
because we have to let

00:06:47.973 --> 00:06:49.390
t go to infinity.

00:06:50.571 --> 00:06:53.290
S would have to be a negative
number, and t would have to

00:06:53.290 --> 00:06:55.534
go to infinity, which
is a long time to wait

00:06:55.534 --> 00:06:57.839
for something to happen,
so that's not a very

00:06:57.839 --> 00:06:59.606
interesting solution either.

00:06:59.606 --> 00:07:00.927
So here's the interesting one.

00:07:00.927 --> 00:07:04.513
The interesting one is
when S plus RC equals zero

00:07:04.513 --> 00:07:08.596
so the interesting solution
is S plus one over RC

00:07:11.056 --> 00:07:15.223
equals zero, and that says that
S equals minus one over RC.

00:07:19.464 --> 00:07:22.466
That makes this solution true.

00:07:22.466 --> 00:07:26.631
And if that's the case,
now we can say v of t,

00:07:26.631 --> 00:07:28.412
we're making good progress here,

00:07:28.412 --> 00:07:31.912
k, some k, times e to the minus t over RC.

00:07:38.962 --> 00:07:41.068
Now something really
interesting just happened here.

00:07:41.068 --> 00:07:44.496
Look up here where we
have this RC term here.

00:07:44.496 --> 00:07:46.163
See this right here?

00:07:49.191 --> 00:07:52.859
This whole exponent has to
have no units by the time

00:07:52.859 --> 00:07:54.281
we evaluate it.

00:07:54.281 --> 00:07:58.031
That means that RC has
to have units of time.

00:08:02.214 --> 00:08:04.964
RC is a measure, it's in seconds.

00:08:09.558 --> 00:08:14.005
RC is the units of R times
C, ohms times farads,

00:08:14.005 --> 00:08:15.627
is actually seconds.

00:08:15.627 --> 00:08:18.764
This whole exponent has to
have no units by the time

00:08:18.764 --> 00:08:20.211
we evaluate it.

00:08:20.211 --> 00:08:23.996
And that means that RC
has to have units of time.

00:08:23.996 --> 00:08:26.921
And it's actually ohms times farads,

00:08:26.921 --> 00:08:28.921
comes out to be seconds.

00:08:31.127 --> 00:08:32.917
Now we solve for S, and that means we only

00:08:32.917 --> 00:08:34.167
have to find K.

00:08:35.277 --> 00:08:36.944
Let's work on K now.

00:08:42.523 --> 00:08:45.968
K is a variable here, time is a variable,

00:08:45.968 --> 00:08:47.541
and voltage is a variable.

00:08:47.541 --> 00:08:51.345
If we knew V and T at
the same time, we would

00:08:51.345 --> 00:08:53.627
be able to figure out K.

00:08:53.627 --> 00:08:55.829
And there's a time when we do know that.

00:08:55.829 --> 00:08:57.293
And that is t equals zero.

00:08:57.293 --> 00:09:01.675
At t equals zero we know that
the battery was hooked up

00:09:01.675 --> 00:09:04.242
and there's a bunch of
charge on the capacitor

00:09:04.242 --> 00:09:07.294
so we know that the
voltage on the capacitor

00:09:07.294 --> 00:09:10.211
at time zero, is equal to V naught.

00:09:13.275 --> 00:09:15.452
So now we have a pair of variables here.

00:09:15.452 --> 00:09:17.441
We know v and t at the same time, so let's

00:09:17.441 --> 00:09:20.015
plug that into our
proposed solution and see

00:09:20.015 --> 00:09:21.182
how k pops up.

00:09:25.363 --> 00:09:28.200
So we're gonna put these two
into our proposed solution

00:09:28.200 --> 00:09:30.891
and see what happens to k.

00:09:30.891 --> 00:09:35.383
V, at time equals zero,
is V naught, and that's

00:09:35.383 --> 00:09:39.133
equal to k times e to
the minus t, minus zero

00:09:43.179 --> 00:09:45.012
at time zero, over RC.

00:09:47.159 --> 00:09:50.742
Now this whole expression
is e to the zero,

00:09:52.455 --> 00:09:53.911
or one.

00:09:53.911 --> 00:09:57.049
So it drops out of the expression

00:09:57.049 --> 00:10:00.049
and all we get is k equals V naught.

00:10:01.222 --> 00:10:04.805
K is the starting
voltage on the capacitor.

00:10:06.424 --> 00:10:09.896
And that gives us to our
final answer, which is

00:10:09.896 --> 00:10:13.313
v of t equals V naught, starting voltage,

00:10:16.669 --> 00:10:19.002
e to the minus time over RC.

00:10:28.434 --> 00:10:30.371
We can go one small step
further and find out

00:10:30.371 --> 00:10:31.621
what is i of t.

00:10:34.051 --> 00:10:37.301
Well, i of t is equal to v of t over R,

00:10:40.067 --> 00:10:43.266
that's just Ohm's Law, and so i of t

00:10:43.266 --> 00:10:47.349
equals V naught over R,
e to the minus t over RC.

00:10:54.363 --> 00:10:58.073
That is the natural
response of an RC circuit.