Ozone layer and eukaryotes show up in the Proterozoic eon

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.650
We left off in the last
video in the Archean Eon.

00:00:03.650 --> 00:00:05.700
And "arche" comes
from ancient Greek.

00:00:05.700 --> 00:00:08.010
It means beginning or origin.

00:00:08.010 --> 00:00:11.530
And it's the eon in which either
life first started to exist

00:00:11.530 --> 00:00:13.870
or at least it first start
to somewhat flourish.

00:00:13.870 --> 00:00:16.170
It's possible that
maybe life first

00:00:16.170 --> 00:00:18.470
started to exist at the
end of the Hadeon Eon.

00:00:18.470 --> 00:00:20.920
And, of course, this
boundary is vague.

00:00:20.920 --> 00:00:23.210
And the Archean Eon
is also the first eon

00:00:23.210 --> 00:00:26.750
where we still have
rocks from that time.

00:00:26.750 --> 00:00:28.750
So we are able to find
rocks that we can date

00:00:28.750 --> 00:00:31.340
to be roughly 3.8 billion years.

00:00:31.340 --> 00:00:33.560
Now, the other really
interesting thing

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that happened in
the Archean Eon,

00:00:34.935 --> 00:00:37.870
and it really has
pretty profound effects

00:00:37.870 --> 00:00:40.640
once we get into
the Proterozoic Eon,

00:00:40.640 --> 00:00:45.795
is that you started to have
cyanobacteria produce oxygen.

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And we said in the last video
that they were producing

00:00:48.950 --> 00:00:50.960
oxygen, but most
of that oxygen was

00:00:50.960 --> 00:00:53.690
being absorbed by
iron in the oceans.

00:00:53.690 --> 00:00:57.230
But what happens is we enter
into the Proterozoic Eon.

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And Proterozoic Eon,
it's right over here.

00:00:59.817 --> 00:01:01.150
I don't know if you can see it .

00:01:01.150 --> 00:01:06.350
I'll rewrite it, Proterozoic.

00:01:06.350 --> 00:01:11.730
We're now in the
Proterozoic Eon that

00:01:11.730 --> 00:01:15.460
starts up about 2.5
billion years ago.

00:01:15.460 --> 00:01:18.940
And "proterozoic" comes from
the Greek for earlier life.

00:01:22.404 --> 00:01:23.570
And I'm not a Greek scholar.

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So any of you Greeks
out there, forgive me

00:01:26.190 --> 00:01:28.840
if I'm not getting the
translation exactly right.

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But what's really interesting
about the Proterozoic Eon

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is that that oxygen
that was being produced

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by the cyanobactera,
and at some point

00:01:38.300 --> 00:01:42.020
begins to saturate the iron
and any other molecule that

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could have absorbed it before.

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And once it saturates,
it starts to get

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released into the atmosphere.

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So the oxygen starts
to get released

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and accumulate in
the atmosphere.

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And we think this
started to happen

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about 2.4 billion years ago.

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So 2.4 billion years
ago, oxygen begins

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to accumulate in the atmosphere.

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And, of course,
these dates, they

00:02:05.700 --> 00:02:08.090
might be moved around a
few hundred million years

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as we get more and more data.

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But this is the
current understanding

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of when things happened.

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And maybe we'll look at
the geological record,

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or the fossil record, and
we'll move these things around

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in the future.

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I can only imagine
that 50 years from now,

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or 100 years from now,
if someone is still

00:02:21.530 --> 00:02:23.130
watching this
video, a lot of this

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might say, hey,
we found out later

00:02:24.870 --> 00:02:27.140
that oxygen started to
accumulate in the atmosphere

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earlier or later
or that eukaryotes

00:02:29.130 --> 00:02:30.270
occurred earlier or later.

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But this, as far
as I can tell, is

00:02:31.900 --> 00:02:33.690
our best current understanding.

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But the 2.4 billion
years ago, oxygen

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begins to accumulate
in the atmosphere.

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And what's interesting about
this is once it accumulates

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and once it gets kind of a
critical amount of oxygen

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in the atmosphere--
and I touched

00:02:54.010 --> 00:02:57.580
on this in the last video--
about 2.3 billion years ago,

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we have something called
the Great Oxygenation Event,

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sometimes called the
Oxygen Catastrophe.

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And this is right here.

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They marked it this right
here, 2.3 billion years ago

00:03:09.070 --> 00:03:13.780
or 2,300 million years ago,
atmosphere becomes oxygen rich.

00:03:13.780 --> 00:03:17.470
And it's not as oxygen rich
as our current atmosphere.

00:03:17.470 --> 00:03:20.220
But it becomes
oxygen rich enough

00:03:20.220 --> 00:03:22.640
that at least the
environment becomes

00:03:22.640 --> 00:03:27.009
suitable for eurkaryotic
organisms or eukaryotic cells.

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Now the other interesting
thing, and we might not

00:03:29.050 --> 00:03:31.626
care so much about it
because we needed the oxygen,

00:03:31.626 --> 00:03:33.250
is that we think that
this was actually

00:03:33.250 --> 00:03:37.070
the greatest extinction event
in the history of Earth.

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That's why it's called
the Oxygen Catastrophe.

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So this right over here,
2.3 billion years ago-- I

00:03:42.730 --> 00:03:45.670
shouldn't giggle about it--
this is a serious matter.

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This is the greatest extinction
event in Earth's history,

00:03:55.550 --> 00:03:56.860
in History.

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And I'll do history
with a capital H,

00:03:59.770 --> 00:04:01.700
in the history of the Earth.

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And that's because
the cyanobacteria

00:04:03.980 --> 00:04:05.190
is producing all this oxygen.

00:04:05.190 --> 00:04:07.170
It eventually
saturates the iron.

00:04:07.170 --> 00:04:08.789
It accumulates in the air.

00:04:08.789 --> 00:04:10.330
Once it gets to
enough concentration,

00:04:10.330 --> 00:04:11.663
it begins to actually suffocate.

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It's poisonous.

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It's poisonous to most
of the other organisms

00:04:16.339 --> 00:04:18.850
on the planet that
were anaerobic,

00:04:18.850 --> 00:04:22.320
that did not need
oxygen, that actually

00:04:22.320 --> 00:04:24.970
found oxygen poisonous.

00:04:24.970 --> 00:04:26.420
But since we have
oxygen, there's

00:04:26.420 --> 00:04:29.100
two interesting things that
happened once that oxygen

00:04:29.100 --> 00:04:32.120
accumulated, other than
causing this mass extinction

00:04:32.120 --> 00:04:33.990
event-- actually three
interesting things.

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Two of them are crucial
to us eventually

00:04:38.210 --> 00:04:40.140
showing up on this planet.

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The first is is that it became
suitable now for eukaryotes

00:04:44.370 --> 00:04:45.110
to exist.

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Eukaryotic organisms,
remember, these

00:04:46.970 --> 00:04:51.910
are organisms that have nuclear
membranes around their DNA.

00:04:51.910 --> 00:04:55.760
Most eukaryotes have other
organelles, like mitochondria.

00:04:55.760 --> 00:04:56.940
They need oxygen.

00:05:02.540 --> 00:05:04.010
You can go to the
biology playlist.

00:05:04.010 --> 00:05:05.840
We actually talk
about respiration

00:05:05.840 --> 00:05:07.760
that occurs in the mitochondria.

00:05:07.760 --> 00:05:11.090
And that's obviously a
process that needs oxygen.

00:05:11.090 --> 00:05:13.979
So one, we have, now that
oxygen is in the atmosphere,

00:05:13.979 --> 00:05:15.770
we're starting to have
an environment where

00:05:15.770 --> 00:05:17.850
eukaryotes could at least exist.

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And based on the fossil
records, and when

00:05:19.820 --> 00:05:22.300
we look at how the DNA
has changed over time--

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and we'll do multiple
videos of that-- we

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think that the first
eukaryotes showed up

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about 2.2 billion years ago.

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Although there's
some debate here.

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There's some evidence it might
have been a little earlier,

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some evidence it might
have a little later.

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I'm sure that number
will be refined.

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But give or take a
few hundred millions

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of years, one prokaryote got
engulfed by another prokaryote,

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and said, hey, we do pretty
well living together.

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I mean the current theory is
that mitochondria is actually

00:05:55.590 --> 00:05:58.730
descended from kind of an
ancient prokaryotic cell,

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an ancient bacteria.

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It actually has its own DNA.

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And actually, your
mitochondrial DNA

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is passed down from your mother,
and your mother's mother,

00:06:06.466 --> 00:06:08.380
and your mother's mother,
so on and so forth.

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So it's kind of like
another little animal

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living inside of a larger cell.

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And we are eukaryotes.

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We needed this to happen.

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The human body, we're not
just one eukaryotic cell.

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We're made up of trillions.

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The estimates are 50 to 100
trillion eukaryotic cells.

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So these are our
ancestors that had

00:06:27.850 --> 00:06:30.000
to come into being at that time.

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And once again, all of this is
happening inside of the oceans.

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Now, the other interesting
thing that happened--

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remember, we're being bombarded
with UV radiation from the Sun.

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So if you're on the land-- let
me draw the land and the ocean.

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So here is the ocean and
then here is the land.

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Here is the land, right over
there in yellow, constantly

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being bombarded
with UV radiation.

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And "UV" stands for ultraviolet.

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So I'm drawing it in purple.

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But it's even more
violet than purple.

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So it's constantly
being bombarded

00:07:05.060 --> 00:07:08.530
by with ultraviolet
radiation from the Sun,

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which is very inhospitable
to DNA and to life.

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And so the only
life at this point

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could occur in the ocean,
where it was protected

00:07:16.350 --> 00:07:20.070
to some degree from the
ultraviolet radiation.

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The land was just open to it.

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Anything on the land would
have just gotten irradiated.

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It's DNA would get mutated.

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It just would not
be able to live.

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So what happened, and what
I guess has to happen,

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and the reason why we
are able to live on land

00:07:33.890 --> 00:07:36.250
now is that we have
an ozone layer.

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We have an ozone layer up
in the upper atmosphere

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that helps absorb, that blocks
most of the UV radiation

00:07:46.260 --> 00:07:47.540
from the Sun.

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And now that oxygen
began to accumulate,

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we have the Oxygen Catastrophe.

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Oxygen accumulates
in the atmosphere.

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Some of that oxygen goes
into the upper atmosphere.

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So we're now in this time
period right over here.

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It goes into the
upper atmosphere.

00:08:01.215 --> 00:08:03.500
It actually really
reacts with the UV light

00:08:03.500 --> 00:08:07.390
to turn into ozone, which
then can help actually

00:08:07.390 --> 00:08:08.940
block the UV light.

00:08:08.940 --> 00:08:13.310
And I'll do another video maybe
on the ozone/oxygen cycle.

00:08:13.310 --> 00:08:16.060
So this oxygen
production, it's crucial,

00:08:16.060 --> 00:08:19.370
one, to having an ozone
layer so that eventually life

00:08:19.370 --> 00:08:21.030
can exist on the land.

00:08:21.030 --> 00:08:24.190
And it's also crucial
because eukaryotic organisms

00:08:24.190 --> 00:08:25.802
need that oxygen.

00:08:25.802 --> 00:08:27.260
Now, the third
thing that happened,

00:08:27.260 --> 00:08:30.950
and this is also pretty
significant event,

00:08:30.950 --> 00:08:33.200
we believe that that
oxygen that started

00:08:33.200 --> 00:08:36.240
to accumulate in the
atmosphere, reacted with methane

00:08:36.240 --> 00:08:37.710
in the atmosphere.

00:08:37.710 --> 00:08:39.700
So it reacted with methane.

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And methane is an
ozone-- not an ozone.

00:08:43.970 --> 00:08:45.590
It's a greenhouse gas.

00:08:45.590 --> 00:08:47.920
It helps retain heat
in the atmosphere.

00:08:47.920 --> 00:08:49.590
And once it reacts
with the oxygen

00:08:49.590 --> 00:08:52.310
and starts dropping out of
the atmosphere as methane,

00:08:52.310 --> 00:08:54.210
we believe the
Earth cooled down.

00:08:54.210 --> 00:08:56.510
And it entered its
first, and some people

00:08:56.510 --> 00:08:59.590
believe it's longest,
snowball period.

00:08:59.590 --> 00:09:01.490
So that's what they
talk about right here

00:09:01.490 --> 00:09:06.170
on this diagram, the
first snowball Earth.

00:09:06.170 --> 00:09:08.640
It's sometimes called
the Huronic glaciation.

00:09:08.640 --> 00:09:10.400
And that happened
because we weren't

00:09:10.400 --> 00:09:13.370
able to retain our heat,
if that theory is correct.

00:09:13.370 --> 00:09:16.910
And so the whole-- as the
theory goes-- the whole Earth

00:09:16.910 --> 00:09:20.010
essentially just iced over.

00:09:20.010 --> 00:09:24.210
So as we go through
the Proterozoic Eon,

00:09:24.210 --> 00:09:28.620
I guess the big markers of it is
it's the first time that we now

00:09:28.620 --> 00:09:30.760
have an oxygen-rich atmosphere.

00:09:30.760 --> 00:09:34.300
It's the first time
that eukaryotes can now

00:09:34.300 --> 00:09:36.400
come into existence
because they now

00:09:36.400 --> 00:09:39.920
have oxygen to, I guess
we could say, breathe.

00:09:39.920 --> 00:09:43.260
And the other big thing is now
this is where the ozone forms.

00:09:43.260 --> 00:09:46.840
So this kind of sets the
stage for in the next eon,

00:09:46.840 --> 00:09:52.690
for animals or living things, to
eventually get on to the land.

00:09:52.690 --> 00:09:55.690
And we'll talk about
that in the next video.


Beginnings of life

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Kind: captions
Language: en

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We finished off the last
video in the Hadean Eon.

00:00:05.170 --> 00:00:09.120
It was named for Hades or
the ancient Greek underworld.

00:00:09.120 --> 00:00:10.900
Hades is also the
name of the god that

00:00:10.900 --> 00:00:15.970
ran the Greek underworld,
Zeus's oldest brother.

00:00:15.970 --> 00:00:18.210
And it was an appropriate
name, although the idea of,

00:00:18.210 --> 00:00:19.960
the ancient Greek
notion of the underworld

00:00:19.960 --> 00:00:23.160
isn't exactly the more
modern notion of Hell.

00:00:23.160 --> 00:00:25.450
But it was a
hellish environment.

00:00:25.450 --> 00:00:27.090
You had all this
lava flowing around.

00:00:27.090 --> 00:00:30.840
You had things impacting
the Earth from space.

00:00:30.840 --> 00:00:32.570
And as far as we
can tell right now,

00:00:32.570 --> 00:00:36.110
it was completely
inhospitable to life.

00:00:36.110 --> 00:00:38.570
And to make matters worse,
even though the Earth started

00:00:38.570 --> 00:00:41.120
to cool down a little
bit, maybe the crust

00:00:41.120 --> 00:00:42.850
became a little bit more solid.

00:00:42.850 --> 00:00:45.510
Maybe the collisions started
to happen less and less,

00:00:45.510 --> 00:00:48.120
as we started to go a few
hundred million years fast

00:00:48.120 --> 00:00:48.740
forward.

00:00:48.740 --> 00:00:52.110
After the Theia rammed
into the early earth

00:00:52.110 --> 00:00:55.350
and formed the moon, there was
something called the Late Heavy

00:00:55.350 --> 00:00:56.180
Bombardment.

00:00:56.180 --> 00:00:59.880
And right now, the consensus
is that life, whatever

00:00:59.880 --> 00:01:01.470
we are descended
from, would have

00:01:01.470 --> 00:01:04.190
had to come about after
the Late Heavy Bombardment.

00:01:04.190 --> 00:01:06.847
Because this was a time where
so many things from outer space

00:01:06.847 --> 00:01:09.430
were hitting Earth, that it was
so violent, that it might have

00:01:09.430 --> 00:01:12.820
killed off any kind of
primitive, self-replicating

00:01:12.820 --> 00:01:16.550
organisms or molecules that
might have existed before it.

00:01:16.550 --> 00:01:19.500
And I won't go into the physics
of the Late Heavy Bombardment.

00:01:19.500 --> 00:01:23.040
But we believe that it happened,
because Uranus and Neptune--

00:01:23.040 --> 00:01:26.450
so if this is the sun right
here-- that is the sun.

00:01:26.450 --> 00:01:29.970
This is the asteroid belt.

00:01:29.970 --> 00:01:34.730
That's outside the orbits
of the inner, rocky planets.

00:01:34.730 --> 00:01:41.282
That Uranus and Neptune,
their orbits moved outward.

00:01:41.282 --> 00:01:42.990
And I'm not going to
go into the physics.

00:01:42.990 --> 00:01:47.270
But what that caused
is, gravitationally, it

00:01:47.270 --> 00:01:50.070
caused a lot of the asteroids
in the asteroid belt

00:01:50.070 --> 00:01:53.640
to move inward and start
impacting the inner planets.

00:01:53.640 --> 00:01:56.570
And of course, Earth was
one of the inner planets.

00:01:56.570 --> 00:01:59.645
And I should make the sun like
orange or something, not blue.

00:01:59.645 --> 00:02:01.270
I don't want you to
think that's Earth.

00:02:01.270 --> 00:02:02.430
And it also impacted the moon.

00:02:02.430 --> 00:02:03.846
And it's more
obvious on the moon,

00:02:03.846 --> 00:02:06.270
because the moon does
not have an atmosphere

00:02:06.270 --> 00:02:08.320
to kind of smooth
over the impact.

00:02:08.320 --> 00:02:11.070
So the consensus is that
only after the Late Heavy

00:02:11.070 --> 00:02:14.440
Bombardment was Earth
kind of ready for life.

00:02:14.440 --> 00:02:16.770
And we believe
that the first life

00:02:16.770 --> 00:02:22.400
formed 3.824 billion years ago.

00:02:22.400 --> 00:02:25.549
Remember, g for giga,
for billion years ago.

00:02:25.549 --> 00:02:27.340
And when we talk about
life at this period,

00:02:27.340 --> 00:02:31.450
we're not talking about
squirrels or panda bears.

00:02:31.450 --> 00:02:33.830
We're talking about
extremely simple life forms.

00:02:33.830 --> 00:02:35.470
We're talking about prokaryotes.

00:02:35.470 --> 00:02:38.700
And let me give you a little
primer on that right now,

00:02:38.700 --> 00:02:43.050
though we go into much more
detail in the biology playlist.

00:02:43.050 --> 00:02:45.210
We're talking about prokaryotes.

00:02:45.210 --> 00:02:47.443
And I'll compare
them to eukaryotes.

00:02:52.890 --> 00:02:56.850
Prokaryotes are,
for the most part,

00:02:56.850 --> 00:03:00.340
unicellular organisms
that have no nucleuses.

00:03:00.340 --> 00:03:03.259
They also don't have
any other membrane-bound

00:03:03.259 --> 00:03:05.675
what we call organelles or
these little parts of the cells

00:03:05.675 --> 00:03:08.860
that perform specific
functions, like mitochondria.

00:03:08.860 --> 00:03:11.340
So their DNA is just
kind of floating around.

00:03:11.340 --> 00:03:14.030
So let me draw this
character's DNA.

00:03:14.030 --> 00:03:16.880
So it's just floating
around, just like that.

00:03:16.880 --> 00:03:19.820
And prokaryote literally
means before kernel

00:03:19.820 --> 00:03:21.570
or before a nucleus.

00:03:21.570 --> 00:03:27.460
Eukaryotes do have a nucleus,
where all of their DNA is.

00:03:27.460 --> 00:03:30.690
So this is the nuclear membrane.

00:03:30.690 --> 00:03:35.480
And then all of its DNA is
floating inside of the nucleus.

00:03:35.480 --> 00:03:38.470
And then it also has other
membrane-bound organelles.

00:03:38.470 --> 00:03:41.054
Mitochondria is kind of
the most famous of them.

00:03:41.054 --> 00:03:42.720
So it also has things
like mitochondria.

00:03:42.720 --> 00:03:45.340
We'll learn more about
that in future videos.

00:03:45.340 --> 00:03:47.170
Mitochondria, we
believe, is essentially

00:03:47.170 --> 00:03:52.130
one prokaryote crawling inside
of another prokaryote and kind

00:03:52.130 --> 00:03:54.390
of starting to become
a symbiotic organism

00:03:54.390 --> 00:03:55.212
with each other.

00:03:55.212 --> 00:03:56.670
But I won't go into
that right now.

00:03:56.670 --> 00:03:58.461
But when we talk about
life at this period,

00:03:58.461 --> 00:04:00.010
we're talking about prokaryotes.

00:04:00.010 --> 00:04:02.860
And we still have
prokaryotes on the planet.

00:04:02.860 --> 00:04:06.600
Bacteria and archaea are
examples of prokaryotes.

00:04:06.600 --> 00:04:09.800
And just to give you a little
bit of a tidbit right here,

00:04:09.800 --> 00:04:12.160
this kind of shows our
current understanding

00:04:12.160 --> 00:04:14.990
of where we think things
branched off from.

00:04:14.990 --> 00:04:19.260
So at this point of the
tree is some common ancestor

00:04:19.260 --> 00:04:22.019
to prokaryotes and eukaryotes.

00:04:22.019 --> 00:04:23.550
So these are the
prokaryotes right

00:04:23.550 --> 00:04:26.510
over here, the bacteria
and the archaea.

00:04:26.510 --> 00:04:28.720
And here is the eukaryote.

00:04:28.720 --> 00:04:31.890
And this first living
thing, or this first set

00:04:31.890 --> 00:04:34.780
of living things, we
think might have just

00:04:34.780 --> 00:04:37.280
been some type of
self-replicating molecules.

00:04:37.280 --> 00:04:39.800
And slowly, some membrane
might have come around,

00:04:39.800 --> 00:04:41.900
and it became a little
bit more organized.

00:04:41.900 --> 00:04:45.670
DNA, RNA-- maybe RNA was that
original self-replicating

00:04:45.670 --> 00:04:48.310
molecule-- became
the method of kind

00:04:48.310 --> 00:04:51.159
of transmitting information
from one generation to the next.

00:04:51.159 --> 00:04:52.700
So it's really still
an open question

00:04:52.700 --> 00:04:56.100
of exactly what
that first life is

00:04:56.100 --> 00:04:58.180
or even how do you
define that first life.

00:04:58.180 --> 00:05:01.930
But based on studying the
genetic makeup of molecules

00:05:01.930 --> 00:05:05.830
of current organisms, this is
how we think the tree of life

00:05:05.830 --> 00:05:06.640
came about.

00:05:06.640 --> 00:05:08.890
So we have one common ancestor.

00:05:08.890 --> 00:05:10.130
Then they broke apart.

00:05:10.130 --> 00:05:11.900
And then the archaea
and eukaryotes

00:05:11.900 --> 00:05:15.215
have a common ancestor that's
different from the bacteria.

00:05:15.215 --> 00:05:17.090
And we'll talk more
about that in the future.

00:05:17.090 --> 00:05:19.173
And this right here, just
so you can visualize it,

00:05:19.173 --> 00:05:20.970
this is an example of bacteria.

00:05:20.970 --> 00:05:23.465
This is E. coli Or
Escherichia coli.

00:05:23.465 --> 00:05:24.840
It's just an
example of bacteria.

00:05:24.840 --> 00:05:26.506
It comes in a bunch
of shapes and forms.

00:05:26.506 --> 00:05:29.670
But it's a
prokaryotic life form.

00:05:29.670 --> 00:05:34.260
And the earliest life forms,
we also think were anaerobes.

00:05:34.260 --> 00:05:37.070
These are things
that did not-- one,

00:05:37.070 --> 00:05:38.400
that they did not need oxygen.

00:05:38.400 --> 00:05:41.920
And they, for the most part,
found oxygen poisonous.

00:05:41.920 --> 00:05:44.400
And the earliest life
forms also probably

00:05:44.400 --> 00:05:46.320
did not perform photosynthesis.

00:05:46.320 --> 00:05:49.480
They might have gotten their
energy from other sources,

00:05:49.480 --> 00:05:53.470
chemically, from this kind of
extremely volatile environment

00:05:53.470 --> 00:05:54.786
that they were in at that time.

00:05:54.786 --> 00:05:57.160
So if we fast forward a little
bit-- and this is actually

00:05:57.160 --> 00:05:58.750
a major event in the
history of Earth.

00:05:58.750 --> 00:06:01.230
And these are huge time
scales we're talking about.

00:06:01.230 --> 00:06:03.370
Remember, I'm kind of
just nonchalantly saying,

00:06:03.370 --> 00:06:05.880
oh, 4.6 billion years
ago to 3.8 billion.

00:06:05.880 --> 00:06:09.340
Oh, that's just
800 million years.

00:06:09.340 --> 00:06:11.850
Remember-- and I'll
talk about this.

00:06:11.850 --> 00:06:14.720
Grass has only existed
for 50 million years.

00:06:14.720 --> 00:06:16.770
This is 800 million years.

00:06:16.770 --> 00:06:21.770
Humans and chimpanzees only
diverged 5 million years ago.

00:06:21.770 --> 00:06:25.950
This is 800 million years
we're talking about,

00:06:25.950 --> 00:06:28.650
from ancient Greece
to now, we're

00:06:28.650 --> 00:06:33.390
only talking about 2,500 years.

00:06:33.390 --> 00:06:37.000
You multiply that
times 1,000, you

00:06:37.000 --> 00:06:39.840
get 2 and 1/2 a million years.

00:06:39.840 --> 00:06:42.890
And this is 800 million
years we're talking about.

00:06:42.890 --> 00:06:46.846
So these are extremely
huge periods of time.

00:06:46.846 --> 00:06:48.220
And that's why we
call them eons.

00:06:48.220 --> 00:06:51.510
Eons are 500 million
to a billion years.

00:06:51.510 --> 00:06:55.760
Now, the dividing line between
the Hadean Eon and the Archean

00:06:55.760 --> 00:06:57.860
Eon-- and it's kind of
a fuzzy dividing line,

00:06:57.860 --> 00:07:01.500
but most people place it
about 3.8 billion years ago.

00:07:05.130 --> 00:07:08.380
It's kind of the earliest
rocks that we can observe.

00:07:08.380 --> 00:07:13.071
And so we have rocks from, that
are roughly 3.8 billion years

00:07:13.071 --> 00:07:13.570
ago.

00:07:13.570 --> 00:07:16.747
So we kind of put that as the
beginning of the Archean Eon.

00:07:16.747 --> 00:07:18.080
And so there's two things there.

00:07:18.080 --> 00:07:21.300
One, rocks have survived from
the beginning of the Archean

00:07:21.300 --> 00:07:21.960
Eon.

00:07:21.960 --> 00:07:24.190
And also, that's
roughly when we think

00:07:24.190 --> 00:07:26.790
that the first life existed.

00:07:26.790 --> 00:07:29.310
And so we're now
in the Archean Eon.

00:07:29.310 --> 00:07:30.760
And you might say,
oh, maybe Earth

00:07:30.760 --> 00:07:32.290
is a more pleasant place now.

00:07:32.290 --> 00:07:33.400
But it would not be.

00:07:33.400 --> 00:07:38.150
It still has no to little
oxygen in the environment.

00:07:38.150 --> 00:07:39.870
If you were to go to
Earth at that time,

00:07:39.870 --> 00:07:41.578
it might have looked
something like this.

00:07:41.578 --> 00:07:44.240
It would have a
been a reddish sky.

00:07:44.240 --> 00:07:47.251
You would have had nitrogen
and methane and carbon dioxide

00:07:47.251 --> 00:07:48.000
in the atmosphere.

00:07:48.000 --> 00:07:50.120
There would have been
nothing for you to breathe.

00:07:50.120 --> 00:07:52.882
There still would have been
a lot of volcanic activity.

00:07:52.882 --> 00:07:55.090
This right here, these are
pictures of stromatolites.

00:08:00.920 --> 00:08:04.104
And these are
formed from bacteria

00:08:04.104 --> 00:08:05.770
that are bringing in
sediment particles.

00:08:05.770 --> 00:08:08.270
And over time, these
things get built up.

00:08:08.270 --> 00:08:12.130
But the most significant
event in the Archean period,

00:08:12.130 --> 00:08:15.300
at least in my
humble opinion, was

00:08:15.300 --> 00:08:17.120
what we believe
started to happen

00:08:17.120 --> 00:08:20.760
about 3.5 billion years ago.

00:08:20.760 --> 00:08:24.690
And this is prokaryotes,
or especially bacteria,

00:08:24.690 --> 00:08:28.760
evolving to actually
utilize energy from the sun,

00:08:28.760 --> 00:08:31.030
to actually do photosynthesis.

00:08:31.030 --> 00:08:34.490
And the real fascinating
byproduct of that,

00:08:34.490 --> 00:08:36.940
other than the fact that they
can now use energy directly

00:08:36.940 --> 00:08:41.299
from the sun, is that it
started to produce oxygen,

00:08:41.299 --> 00:08:48.100
so starts to produce oxygen.

00:08:48.100 --> 00:08:50.270
And at first, this
oxygen, even though it

00:08:50.270 --> 00:08:52.960
was being produced
by the cyanobacteria,

00:08:52.960 --> 00:08:55.760
by this blue-green
bacteria, it really

00:08:55.760 --> 00:08:57.260
didn't accumulate
in the atmosphere.

00:08:57.260 --> 00:08:58.810
Because you had all
of this iron that

00:08:58.810 --> 00:09:00.460
was dissolved in the oceans.

00:09:00.460 --> 00:09:01.390
And let me be clear.

00:09:01.390 --> 00:09:04.120
All of the life that we're going
to be talking about for really

00:09:04.120 --> 00:09:07.480
the next several billion years,
it all occurred in the ocean.

00:09:07.480 --> 00:09:09.610
We had no ozone layer now.

00:09:09.610 --> 00:09:12.130
The land was being irradiated.

00:09:12.130 --> 00:09:15.810
The land was just a completely
inhospitable environment

00:09:15.810 --> 00:09:16.680
for life.

00:09:16.680 --> 00:09:18.770
So all of this was
occurring in the ocean.

00:09:18.770 --> 00:09:20.340
And so the first
oxygen that actually

00:09:20.340 --> 00:09:25.460
got produced, it actually,
instead of just being released

00:09:25.460 --> 00:09:27.360
into the atmosphere,
it ended up bonding

00:09:27.360 --> 00:09:30.172
with the iron that was dissolved
in the ocean at that time.

00:09:30.172 --> 00:09:31.630
So it actually
didn't have a chance

00:09:31.630 --> 00:09:33.280
to accumulate in the atmosphere.

00:09:33.280 --> 00:09:36.180
And when we fast forward
past the Archean period,

00:09:36.180 --> 00:09:39.190
we're going to see, that once
a lot of that iron was oxidized

00:09:39.190 --> 00:09:41.650
and the oxygen really
did start to get released

00:09:41.650 --> 00:09:44.784
in the atmosphere,
it actually had--

00:09:44.784 --> 00:09:46.450
it's funny to say--
a cataclysmic effect

00:09:46.450 --> 00:09:51.707
or a catastrophic effect
on the other anaerobic life

00:09:51.707 --> 00:09:52.790
on the planet at the time.

00:09:52.790 --> 00:09:53.670
And it's funny to
say that because it

00:09:53.670 --> 00:09:54.797
was a catastrophe for them.

00:09:54.797 --> 00:09:57.380
But it was kind of a necessary
thing that had to happen for us

00:09:57.380 --> 00:09:58.050
to happen.

00:09:58.050 --> 00:10:01.080
So for us, it was a blessing
that this cyanobacteria

00:10:01.080 --> 00:10:03.460
started to pump
out a lot of oxygen

00:10:03.460 --> 00:10:05.182
and eventually oxidized
all of the iron

00:10:05.182 --> 00:10:07.640
and eventually released a lot
of oxygen into the atmosphere

00:10:07.640 --> 00:10:10.220
and killed off all of
this anaerobic bacteria,

00:10:10.220 --> 00:10:14.840
so that eventually we
could-- us oxygen-breathing

00:10:14.840 --> 00:10:15.966
organisms could come about.

00:10:15.966 --> 00:10:17.756
But that's not going
to happen for a while.

00:10:17.756 --> 00:10:19.200
We still have a
few billion years

00:10:19.200 --> 00:10:23.020
before things start
flopping around on the land.

00:10:23.020 --> 00:10:25.560
Anyway, see you
in the next video.


Earth formation

WEBVTT
Kind: captions
Language: en

00:00:00.348 --> 00:00:03.808
What I'm going to attempt to do in the next two videos is really just give an overview

00:00:03.808 --> 00:00:08.333
of everything that's happened to Earth since it came into existence.

00:00:08.333 --> 00:00:13.468
We're going start really at the formation of Earth or the formation of our Solar system or the formation of

00:00:13.468 --> 00:00:17.000
the Sun, and our best sense of what actually happened is that

00:00:17.000 --> 00:00:20.733
there was a supernova in our vicinity of the galaxy,

00:00:20.733 --> 00:00:23.475
and this right here is a picture of a supernova remnant,

00:00:23.475 --> 00:00:25.960
actually, the remnant for Kepler's supernova.

00:00:25.960 --> 00:00:30.557
The supernova in this picture actually happened four hundred years ago in 1604,

00:00:30.557 --> 00:00:34.133
so right at the center a star essentially exploded

00:00:34.133 --> 00:00:37.867
and for a few weeks was the brightest object in the night sky,

00:00:37.867 --> 00:00:41.133
and it was observed by Kepler and other people in 1604,

00:00:41.133 --> 00:00:42.733
and this is what it looks like now.

00:00:42.733 --> 00:00:46.045
What we see is kinda the shockwave that's been traveling

00:00:46.045 --> 00:00:49.876
out for the past 400 years, so now it must be many

00:00:49.876 --> 00:00:52.547
light years across. It wasn't, obviously, matter wasn't

00:00:52.547 --> 00:00:55.133
traveling at the speed of light, but it must've been traveling

00:00:55.133 --> 00:01:00.209
pretty, pretty fast, at least relativistic speeds, a reasonable

00:01:00.209 --> 00:01:02.206
fraction of the speed of light.

00:01:02.206 --> 00:01:04.133
This has traveled a good bit out now, but what you can

00:01:04.133 --> 00:01:07.600
imagine is when you have the shockwave traveling out from

00:01:07.600 --> 00:01:12.667
a supernova, let's say you had a cloud of molecules,

00:01:12.667 --> 00:01:16.600
a cloud of gas, that before the shockwave came by just

00:01:16.600 --> 00:01:20.933
wasn't dense enough for gravity to take over,

00:01:20.933 --> 00:01:24.400
and for it to accrete, essentially, into a solar system.

00:01:24.400 --> 00:01:30.467
When the shockwave passes by it compresses all of this gas

00:01:30.467 --> 00:01:32.933
and all of this material and all of these molecules, so

00:01:32.933 --> 00:01:36.667
it now does have that critical density to form, to accrete

00:01:36.667 --> 00:01:39.533
into a star and a solar system.

00:01:39.533 --> 00:01:41.634
We think that's what's happened, and the reason why we

00:01:41.634 --> 00:01:46.467
feel pretty strongly that it must've been caused by a supernova

00:01:46.467 --> 00:01:49.714
is that the only way that the really heavy elements can form,

00:01:49.714 --> 00:01:52.600
or the only way we know that they can form is in kind of

00:01:52.600 --> 00:01:58.867
the heat of a supernova, and our uranium, the uranium that seems

00:01:58.867 --> 00:02:02.067
to be in our solar system on Earth, seems to have formed

00:02:02.067 --> 00:02:05.467
roughly at the time of the formation of Earth, at about

00:02:05.467 --> 00:02:07.547
four and a half billion years ago, and we'll talk

00:02:07.547 --> 00:02:10.400
in a little bit more depth in future videos on exactly how

00:02:10.400 --> 00:02:13.600
people figure that out, but since the uranium seems about

00:02:13.600 --> 00:02:16.626
the same age as our solar system, it must've been formed

00:02:16.626 --> 00:02:21.133
at around the same time, and it must've been formed by a supernova,

00:02:21.133 --> 00:02:23.801
and it must be coming from a supernova, so a supernova

00:02:23.801 --> 00:02:27.267
shockwave must've passed through our part of the universe,

00:02:27.267 --> 00:02:31.333
and that's a good reason for gas to get compressed and begin to accrete.

00:02:31.333 --> 00:02:34.600
So you fast-forward a few million years.

00:02:34.600 --> 00:02:36.867
That gas would've accreted into something like this.

00:02:36.867 --> 00:02:39.933
It would've reached the critical temperature, critical density

00:02:39.933 --> 00:02:43.867
and pressure at the center for ignition to occur, for fusion

00:02:43.867 --> 00:02:46.882
to start to happen, for hydrogen to start fusing into helium,

00:02:46.882 --> 00:02:52.267
and this right here is our early sun.

00:02:52.267 --> 00:02:56.400
Around the sun you have all of the gases and particles

00:02:56.400 --> 00:03:00.467
and molecules that had enough angular velocity to not fall into the sun,

00:03:00.467 --> 00:03:02.333
to go into orbit around the sun.

00:03:02.333 --> 00:03:04.533
They were actually supported by a little bit of pressure, too,

00:03:04.533 --> 00:03:07.667
because you can kinda view this as kind of a big cloud of gas,

00:03:07.667 --> 00:03:10.467
so they're always bumping into each other, but for the most part

00:03:10.467 --> 00:03:14.400
it was their angular velocity, and over the next tens of millions of years

00:03:14.400 --> 00:03:18.200
they'll slowly bump into each other and clump into each other.

00:03:18.200 --> 00:03:21.733
Even small particles have gravity, and they're gonna slowly

00:03:21.733 --> 00:03:26.533
become rocks and asteroids and, eventually, what we would call

00:03:26.533 --> 00:03:30.267
"planetesimals," which are, kinda view them as seeds

00:03:30.267 --> 00:03:35.203
of planets or early planets, and then those would have a reasonable amount

00:03:35.203 --> 00:03:37.836
of gravity and other things would be attracted to them

00:03:37.836 --> 00:03:39.736
and slowly clump up to them.

00:03:39.736 --> 00:03:42.772
This wasn't like a simple process, you know, you could imagine

00:03:42.772 --> 00:03:46.600
you might have one planetesimal form, and then there's another

00:03:46.600 --> 00:03:49.800
planetesimal formed, and instead of having a nice, gentle

00:03:49.800 --> 00:03:52.733
those two guys accreting into each other, they might have

00:03:52.733 --> 00:03:56.733
huge relative velocities and ram into each other, and then just,

00:03:56.733 --> 00:04:01.733
you know, shatter, so this wasn't just a nice, gentle process of constant accretion.

00:04:01.733 --> 00:04:04.533
It would actually have been a very violent process,

00:04:04.533 --> 00:04:08.467
actually happened early in Earth's history, and we actually think

00:04:08.467 --> 00:04:10.733
this is why the Moon formed, so at some point

00:04:10.733 --> 00:04:15.067
you fast-forward a little bit from this, Earth would have formed,

00:04:15.067 --> 00:04:20.400
I should say, the mass that eventually becomes our modern Earth

00:04:20.400 --> 00:04:24.133
would have been forming. Let me draw it over here.

00:04:24.133 --> 00:04:28.800
So, let's say that that is our modern Earth, and what we think

00:04:28.800 --> 00:04:31.762
happened is that another proto-planet or another,

00:04:31.762 --> 00:04:34.677
it was actually a planet because it was roughly the size of Mars,

00:04:34.677 --> 00:04:39.800
ran into our, what it is eventually going to become our Earth.

00:04:39.800 --> 00:04:41.800
This is actually a picture of it.

00:04:41.800 --> 00:04:46.867
This is an artist's depiction of that collision, where this planet

00:04:46.867 --> 00:04:50.800
right here is the size of Mars, and it ran into what would eventually become Earth.

00:04:50.800 --> 00:04:54.600
This we call Theia. This is Theia,

00:04:54.600 --> 00:04:58.933
and what we believe happened, and if you look up,

00:04:58.933 --> 00:05:01.600
if you go onto the Internet, you'll see some simulations

00:05:01.600 --> 00:05:04.267
that talk about this, is that we think it was a glancing blow.

00:05:04.267 --> 00:05:06.933
It wasn't a direct hit that would've just kinda shattered

00:05:06.933 --> 00:05:10.600
each of them and turned into one big molten ball.

00:05:10.600 --> 00:05:13.467
We think it was a glancing blow, something like this.

00:05:13.467 --> 00:05:16.933
This was essentially Earth. Obviously, Earth got changed

00:05:16.933 --> 00:05:20.933
dramatically once Theia ran into it, but Theia is

00:05:20.933 --> 00:05:23.133
right over here, and we think it was a glancing blow.

00:05:23.133 --> 00:05:27.733
It came and it hit Earth at kind of an angle, and then it

00:05:27.733 --> 00:05:31.267
obviously the combined energies from that interaction

00:05:31.267 --> 00:05:35.067
would've made both of them molten, and frankly

00:05:35.067 --> 00:05:36.887
they probably already were molten because you had

00:05:36.887 --> 00:05:40.267
a bunch of smaller collisions and accretion events and

00:05:40.267 --> 00:05:43.200
little things hitting the surface, so probably both of them

00:05:43.200 --> 00:05:47.800
during this entire period, but this would've had a

00:05:47.800 --> 00:05:50.898
glancing blow on Earth and essentially splashed a bunch

00:05:50.898 --> 00:05:53.667
of molten material out into orbit.

00:05:53.667 --> 00:05:57.067
It would've just come in, had a glancing blow on Earth,

00:05:57.067 --> 00:05:59.733
and then splashed a bunch of molten material,

00:05:59.733 --> 00:06:02.800
some of it would've been captured by Earth, so this is

00:06:02.800 --> 00:06:08.133
the before and the after, you can imagine, Earth is

00:06:08.133 --> 00:06:12.800
kind of this molten, super hot ball, and some of it

00:06:12.800 --> 00:06:17.800
just gets splashed into orbit from the collision.

00:06:17.800 --> 00:06:19.400
Let me just see if I can draw Theia here, so

00:06:19.400 --> 00:06:26.400
Theia has collided, and it is also molten now because

00:06:26.400 --> 00:06:28.800
huge energies, and it splashes some of it into orbit.

00:06:28.800 --> 00:06:31.733
If we fast-forward a little bit, this stuff that got splashed

00:06:31.733 --> 00:06:37.800
into orbit, it's going in that direction, that becomes

00:06:37.800 --> 00:06:41.600
our Moon, and then the rest of this material eventually

00:06:41.600 --> 00:06:46.000
kind of condenses back into a spherical shape and is what

00:06:46.000 --> 00:06:50.933
we now call our Earth.

00:06:50.933 --> 00:06:53.133
So that's how we actually think right now that the Moon

00:06:53.133 --> 00:06:54.733
actually formed.

00:06:54.733 --> 00:06:58.000
Even after this happened, the Earth still had a lot more,

00:06:58.000 --> 00:07:00.467
I guess, violence to experience.

00:07:00.467 --> 00:07:04.533
Just to get a sense of where we are in the history of Earth,

00:07:04.533 --> 00:07:09.000
we're going to refer to this time clock a lot over the next few videos,

00:07:09.000 --> 00:07:12.867
this time clock starts right here at the formation of our solar system,

00:07:12.867 --> 00:07:16.600
4.6 billion years ago, probably coinciding with

00:07:16.600 --> 00:07:18.200
some type of supernova,

00:07:18.200 --> 00:07:21.800
and as we go clockwise on this diagram,

00:07:21.800 --> 00:07:24.067
we're moving forward in time, and we're gonna go

00:07:24.067 --> 00:07:26.800
all the way forward to the present period,

00:07:26.800 --> 00:07:29.267
and just so you understand some of the terminology,

00:07:29.267 --> 00:07:32.267
"Ga" means "billions of years ago"

00:07:32.267 --> 00:07:33.133
'G' for "Giga-"

00:07:33.133 --> 00:07:35.733
"Ma" means "millions of years ago"

00:07:35.733 --> 00:07:37.467
'M' for "Mega-"

00:07:37.467 --> 00:07:40.733
So where we are right now, the Moon has formed,

00:07:40.733 --> 00:07:42.533
and we're in what we call the Hadean period

00:07:42.533 --> 00:07:44.000
or actually I shouldn't say "period."

00:07:44.000 --> 00:07:45.733
It's the Hadean eon of Earth.

00:07:45.733 --> 00:07:48.200
"Period" is actually another time period,

00:07:48.200 --> 00:07:50.533
so let me make this very clear. It's the Hadean,

00:07:50.533 --> 00:07:57.133
we are in the Hadean eon, and an eon is kind of

00:07:57.133 --> 00:07:59.800
the largest period of time that we talk about, especially

00:07:59.800 --> 00:08:03.800
relative to Earth, and it's roughly 500 million to a billion years

00:08:03.800 --> 00:08:09.133
is an eon, and what makes the Hadean eon distinctive,

00:08:09.133 --> 00:08:12.200
well, from a geological point of view what makes it

00:08:12.200 --> 00:08:14.467
distinctive is really we don't have any rocks from the

00:08:14.467 --> 00:08:17.733
Hadean period. We don't have any kind of macroscopic-scale

00:08:17.733 --> 00:08:19.800
rocks from the Hadean period, and that's because

00:08:19.800 --> 00:08:24.000
at that time, we believe, the Earth was just this molten

00:08:24.000 --> 00:08:29.267
ball of kind of magma and lava, and it was molten

00:08:29.267 --> 00:08:31.733
because it was a product of all of these accretion events

00:08:31.733 --> 00:08:35.600
and all of these collisions and all this kinetic energy turning into heat.

00:08:35.600 --> 00:08:38.133
If you were to look at the surface of the Earth,

00:08:38.133 --> 00:08:40.133
if you were to be on the surface of the Earth during

00:08:40.133 --> 00:08:43.800
the Hadean eon, which you probably wouldn't want to be

00:08:43.800 --> 00:08:45.333
because you might get hit by a falling meteorite

00:08:45.333 --> 00:08:48.133
or probably burned by some magma, whatever,

00:08:48.133 --> 00:08:50.333
it would look like this, and you wouldn't be able to breathe anyway;

00:08:50.333 --> 00:08:53.067
this is what the surface of the Earth would look like.

00:08:53.067 --> 00:08:55.933
It would look like a big magma pool, and that's why we

00:08:55.933 --> 00:08:57.800
don't have any rocks from there because the rocks were

00:08:57.800 --> 00:09:03.733
just constantly being recycled, being dissolved and churned

00:09:03.733 --> 00:09:07.467
inside of this giant molten ball, and frankly

00:09:07.467 --> 00:09:10.533
the Earth still is a giant molten ball, it's just

00:09:10.533 --> 00:09:14.600
we live on the super-thin, cooled crust of that molten ball.

00:09:14.600 --> 00:09:17.202
If you go right below that crust, and we'll talk a little bit more

00:09:17.202 --> 00:09:18.025
about that in future videos,

00:09:18.025 --> 00:09:21.533
you will get magma, and if you go dig deeper,

00:09:21.533 --> 00:09:23.067
you'll have liquid iron.

00:09:23.067 --> 00:09:26.133
I mean, it still is a molten ball.

00:09:26.133 --> 00:09:27.800
And this whole period is just a violent,

00:09:27.800 --> 00:09:31.400
not only was Earth itself a volcanic, molten ball,

00:09:31.400 --> 00:09:34.933
it began to harden as you get into the late Hadean eon,

00:09:34.933 --> 00:09:39.959
but we also had stuff falling from the sky and constantly

00:09:39.959 --> 00:09:42.723
colliding with Earth, and really just continuing to add

00:09:42.723 --> 00:09:45.800
to the heat of this molten ball.

00:09:45.800 --> 00:09:47.600
Anyway, I'll leave you there,

00:09:47.600 --> 00:09:48.733
and, as you can imagine,

00:09:48.733 --> 00:09:50.200
at this point there was no, as far as we can tell,

00:09:50.200 --> 00:09:51.733
there was no life on Earth.

00:09:51.733 --> 00:09:54.533
Some people believe that maybe some life could've formed

00:09:54.533 --> 00:09:57.667
in the late Hadean eon, but for the most part

00:09:57.667 --> 00:10:01.267
this was just completely inhospitable for any life forming.

00:10:01.267 --> 00:10:04.000
I'll leave you there, and where we take up the next video,

00:10:04.000 --> 00:10:07.369
we'll talk a little bit about the Archean eon.


Adding numbers with different signs (example)

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:05.310
Find the sum 15 plus
negative 46 plus 29.

00:00:05.310 --> 00:00:07.630
Let's just think about
this first part over here

00:00:07.630 --> 00:00:13.060
where we have 15
plus negative 46.

00:00:13.060 --> 00:00:16.420
And we'll worry about
the plus 29 later on.

00:00:16.420 --> 00:00:23.680
So let's just do
15-- let me do that

00:00:23.680 --> 00:00:26.677
in a different color--
plus negative 46.

00:00:26.677 --> 00:00:27.760
I'll do it in that orange.

00:00:30.959 --> 00:00:33.250
Let me draw a number line
here, just so we can properly

00:00:33.250 --> 00:00:36.630
visualize what is going on.

00:00:36.630 --> 00:00:39.620
That's my number line.

00:00:39.620 --> 00:00:41.120
And we're starting at 15.

00:00:41.120 --> 00:00:44.790
So let's draw a 0 over here.

00:00:44.790 --> 00:00:47.270
And so we're starting at 15.

00:00:47.270 --> 00:00:49.080
15 could be right over here.

00:00:49.080 --> 00:00:51.930
So this is 15.

00:00:51.930 --> 00:00:56.356
Let me draw a big fat arrow
to signify this is 15.

00:00:56.356 --> 00:01:00.020
15 has an absolute value of
15, so the length of this arrow

00:01:00.020 --> 00:01:01.850
would be 15.

00:01:01.850 --> 00:01:07.310
Now we're adding
negative 46 to that 15.

00:01:07.310 --> 00:01:12.580
This is equivalent
to 15 minus 46,

00:01:12.580 --> 00:01:17.030
which means we are going to
move 46 spots to the left of 15.

00:01:17.030 --> 00:01:19.130
A negative sign or
a minus means we're

00:01:19.130 --> 00:01:21.350
moving to the left
on the number line.

00:01:21.350 --> 00:01:23.910
So we're going to
move 46 to the left.

00:01:23.910 --> 00:01:31.020
We're starting at 15, and we are
going to move 46 to the left.

00:01:31.020 --> 00:01:38.152
So the length of this arrow
right here is going to be 46,

00:01:38.152 --> 00:01:39.360
and we're moving to the left.

00:01:39.360 --> 00:01:43.370
This is the negative 46
that we're adding to the 15.

00:01:43.370 --> 00:01:47.220
We're going to end up
at some point over here.

00:01:47.220 --> 00:01:48.890
That point is clearly 0.

00:01:48.890 --> 00:01:51.466
Because we were 15 to the right.

00:01:51.466 --> 00:01:53.090
Now we're going to
move 46 to the left.

00:01:53.090 --> 00:01:55.360
So we're definitely going
to be to the left of 0.

00:01:55.360 --> 00:01:57.370
It's definitely going
to be a negative number.

00:01:57.370 --> 00:02:00.510
And we can even think
about the absolute value

00:02:00.510 --> 00:02:01.570
of that negative number.

00:02:01.570 --> 00:02:03.220
We can just visualize it.

00:02:03.220 --> 00:02:05.790
This yellow arrow
has a length of 15.

00:02:05.790 --> 00:02:09.000
This orange arrow
has a length of 46.

00:02:09.000 --> 00:02:11.100
The blue arrow that
I'm about to draw,

00:02:11.100 --> 00:02:12.890
which is the sum
of these two, is

00:02:12.890 --> 00:02:18.310
going to have this
length right over here.

00:02:18.310 --> 00:02:21.340
And just visually,
how can we figure out

00:02:21.340 --> 00:02:23.410
the length of this
blue part if we

00:02:23.410 --> 00:02:25.020
know the length of
this orange part

00:02:25.020 --> 00:02:27.320
and we know the length
of this yellow part?

00:02:27.320 --> 00:02:28.600
Well, it's just going
to be the difference.

00:02:28.600 --> 00:02:30.683
It's just going to be the
difference of these two.

00:02:30.683 --> 00:02:32.980
So the absolute
value of the sum is

00:02:32.980 --> 00:02:36.140
going to be the difference
between this length, 46,

00:02:36.140 --> 00:02:38.850
and 15.

00:02:38.850 --> 00:02:42.300
Let me just figure
that out, 46 minus 15.

00:02:42.300 --> 00:02:43.740
6 minus 5 is 1.

00:02:43.740 --> 00:02:45.790
4 minus 1 is 3.

00:02:45.790 --> 00:02:48.140
So the length of this
is going to be 31,

00:02:48.140 --> 00:02:52.100
and it's going to be
31 to the left of 0.

00:02:52.100 --> 00:02:56.720
So this is going to be
negative 31 right over here.

00:02:56.720 --> 00:03:00.910
We know that this first part
over here is negative 31,

00:03:00.910 --> 00:03:05.080
and then to that we
are going to add 29.

00:03:05.080 --> 00:03:06.657
So we're going to add 29.

00:03:06.657 --> 00:03:07.990
Let me do that in another color.

00:03:07.990 --> 00:03:09.200
What does that mean?

00:03:09.200 --> 00:03:11.840
That means that we're going
to start at negative 31,

00:03:11.840 --> 00:03:13.940
and we're going to
move 29 to the right.

00:03:13.940 --> 00:03:19.070
We're adding 29, so we're
going to move 29 to the right.

00:03:19.070 --> 00:03:20.810
So maybe that gets
us right about there.

00:03:20.810 --> 00:03:24.470
I'm trying to draw an
arrow of length 29.

00:03:24.470 --> 00:03:27.730
I can draw a cleaner
looking arrow than that.

00:03:27.730 --> 00:03:29.910
I'll do it right
over here, actually.

00:03:29.910 --> 00:03:35.400
So then we're going to
move 29 over to the right.

00:03:35.400 --> 00:03:38.350
That's the 29 part.

00:03:38.350 --> 00:03:41.600
Now this is a positive
29, and so how

00:03:41.600 --> 00:03:42.990
do we figure out what this is?

00:03:46.320 --> 00:03:48.570
This is 29 right here
that we're adding.

00:03:48.570 --> 00:03:51.990
This is going to land us right
over here on the number line.

00:03:51.990 --> 00:03:54.160
So how do we figure out
what number that is?

00:03:54.160 --> 00:03:56.000
Well, once again, we
can just visualize it.

00:03:56.000 --> 00:03:58.541
And eventually you won't have
to draw number lines and stuff,

00:03:58.541 --> 00:04:01.290
but I think it'll
be useful here.

00:04:01.290 --> 00:04:02.540
We're starting at negative 31.

00:04:02.540 --> 00:04:05.720
We're adding 29 to it, so it's
going to make it less negative.

00:04:05.720 --> 00:04:08.690
But we're still
adding less than 31,

00:04:08.690 --> 00:04:10.750
so we're not going to get
all the way back to 0.

00:04:10.750 --> 00:04:12.583
We're still going to
have a negative number.

00:04:15.950 --> 00:04:18.000
But how could we figure
out the absolute value

00:04:18.000 --> 00:04:19.870
of that negative
number, its distance?

00:04:19.870 --> 00:04:22.410
Well, once again, that little
white part right there,

00:04:22.410 --> 00:04:24.720
that white part plus
this 29, is going

00:04:24.720 --> 00:04:28.334
to equal 31, if you just
think of absolute value,

00:04:28.334 --> 00:04:29.750
if we don't think
about the signs,

00:04:29.750 --> 00:04:31.166
if we just think
about the length.

00:04:31.166 --> 00:04:34.100
Or another way to think
about it, 31 minus 29

00:04:34.100 --> 00:04:36.640
will give us the length
of that white part.

00:04:36.640 --> 00:04:38.600
And of course, it's
going to be negative,

00:04:38.600 --> 00:04:40.330
because the negative
number here is

00:04:40.330 --> 00:04:41.740
larger than the positive number.

00:04:41.740 --> 00:04:43.480
And we're adding the 2.

00:04:43.480 --> 00:04:49.642
If we do 31 minus 29, you
could borrow and all of that,

00:04:49.642 --> 00:04:51.600
but that's clearly just
going to be equal to 2.

00:04:51.600 --> 00:04:53.320
You could say that's 11.

00:04:53.320 --> 00:04:54.580
This is a 2.

00:04:54.580 --> 00:04:56.600
You subtract, this
is equal to 2.

00:04:56.600 --> 00:05:00.010
But since it's
negative 31 plus 29,

00:05:00.010 --> 00:05:01.350
it's going to be a negative 2.

00:05:01.350 --> 00:05:02.808
We're still going
to stay negative.

00:05:02.808 --> 00:05:05.630
We haven't moved far to
the right enough to pass 0.

00:05:05.630 --> 00:05:08.265
So this right here is
going to be negative 2.

00:05:08.265 --> 00:05:09.640
Or another way to
think about it,

00:05:09.640 --> 00:05:12.170
the length of this white
bar, the absolute value,

00:05:12.170 --> 00:05:13.466
is going to be 2.

00:05:13.466 --> 00:05:15.710
2 plus 29 is 31.

00:05:15.710 --> 00:05:18.940
But we're operating to the
left of 0, so it's negative 31.

00:05:18.940 --> 00:05:20.250
This is negative 2.

00:05:20.250 --> 00:05:22.590
Anyway, hopefully you
found that useful.

00:05:22.590 --> 00:05:24.300
And let me make it clear.

00:05:24.300 --> 00:05:31.545
Our final answer is negative 2.


Decimals and fractions on a number line

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:04.204
- [Voiceover] Plot the following
numbers on the number line.

00:00:04.204 --> 00:00:07.705
The first number we have here is five,

00:00:07.705 --> 00:00:10.070
and so five is five to the right of zero,

00:00:10.070 --> 00:00:14.671
five is right over there. That's our five.

00:00:14.671 --> 00:00:19.172
Then we get 1/3. 1/3.

00:00:19.172 --> 00:00:22.289
So 1/3 is between zero and one.

00:00:22.289 --> 00:00:24.256
We can actually split this into thirds.

00:00:24.256 --> 00:00:28.089
So that would be 1/3, 2/3,
and then 3/3, which is one,

00:00:28.089 --> 00:00:31.553
so 1/3 is going to sit right over there.

00:00:31.553 --> 00:00:35.454
It's 1/3 of the way from
zero to one, that's 1/3.

00:00:35.454 --> 00:00:38.053
Let me write that. That's
1/3 right over there.

00:00:38.053 --> 00:00:42.938
Then we have negative 1.2. I'll
do that in this blue color.

00:00:42.938 --> 00:00:47.303
Negative 1.2. So, negative
one is right over here.

00:00:47.303 --> 00:00:49.802
This is more negative than negative one.

00:00:49.802 --> 00:00:54.635
It's negative 1.2. It's negative
one, and then another .2,

00:00:54.635 --> 00:00:57.319
so it's going to be right over here.

00:00:57.319 --> 00:01:02.319
This is negative 1.2.

00:01:02.520 --> 00:01:06.868
Zero is pretty straight forward.
Zero is right over there.

00:01:06.868 --> 00:01:09.501
It's even labeled for us at zero.

00:01:09.501 --> 00:01:11.553
Five was labeled for us too at five.

00:01:11.553 --> 00:01:14.635
Then we have negative two and 1/4.

00:01:14.635 --> 00:01:17.301
So let's go to negative two.

00:01:17.301 --> 00:01:19.936
Negative two is here, and it's going to be

00:01:19.936 --> 00:01:22.037
more negative than negative two.

00:01:22.037 --> 00:01:25.762
It's negative two and then
another another negative 1/4.

00:01:25.822 --> 00:01:27.824
So it's negative two, and then we go

00:01:27.824 --> 00:01:30.396
1/4 of the way to negative three.

00:01:30.396 --> 00:01:34.885
So negative 2 and 1/4 is
going to be right over here.

00:01:34.885 --> 00:01:38.567
So negative two and 1/4.

00:01:38.567 --> 00:01:41.984
And then finally we have 4.1. 4.1.

00:01:41.984 --> 00:01:43.667
So four is right over here.

00:01:43.667 --> 00:01:46.969
.1 is another tenth greater than four,

00:01:46.969 --> 00:01:49.070
another tenth on the way to five.

00:01:49.070 --> 00:01:53.247
So four and 1/10 is going
to be right over here. 4.1.

00:01:53.801 --> 00:01:56.833
4.1. And we are done.


Absolute value of integers

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.900
Find the absolute value
of x when x is equal to 5,

00:00:03.900 --> 00:00:09.240
x is equal to negative 10,
and x is equal to negative 12.

00:00:09.240 --> 00:00:13.530
So the absolute value,
the way of writing it

00:00:13.530 --> 00:00:15.810
is almost more complicated
than what it really is.

00:00:15.810 --> 00:00:19.535
The absolute value is really
just the distance of x from 0.

00:00:25.150 --> 00:00:28.460
So let me just draw a fast
number line over here.

00:00:28.460 --> 00:00:31.510
So let's just put
0 right over here,

00:00:31.510 --> 00:00:33.880
since we're thinking
about the distance from 0.

00:00:33.880 --> 00:00:36.150
So let's just think
about the absolute value.

00:00:36.150 --> 00:00:37.650
Let's think about
the absolute value

00:00:37.650 --> 00:00:39.640
of x when x is equal to 5.

00:00:39.640 --> 00:00:42.390
So that's equivalent to
the absolute value of 5.

00:00:42.390 --> 00:00:44.530
We just substituted 5 for x.

00:00:44.530 --> 00:00:48.590
The absolute value of 5 is
the distance of 5 from the 0.

00:00:48.590 --> 00:00:51.570
So you go 1, 2, 3, 4, 5.

00:00:51.570 --> 00:00:55.350
5 is exactly 5 to
the right of 0.

00:00:55.350 --> 00:00:59.300
So the absolute
value of 5 is just 5.

00:00:59.300 --> 00:01:00.760
Now I think you
already get to see

00:01:00.760 --> 00:01:02.670
this is a pretty
straightforward concept.

00:01:02.670 --> 00:01:07.960
Now let's do something a
little more interesting,

00:01:07.960 --> 00:01:10.480
the absolute value of x when
x is equal to negative 10.

00:01:10.480 --> 00:01:13.120
So let's just put
negative 10 in for x.

00:01:13.120 --> 00:01:16.000
This is the distance that
negative 10 is from 0.

00:01:16.000 --> 00:01:19.110
And so let's just go negative
1, negative 2, negative 3,

00:01:19.110 --> 00:01:22.620
negative 4, negative 5, negative
6, negative 7, negative 8,

00:01:22.620 --> 00:01:23.920
negative 9, negative 10.

00:01:23.920 --> 00:01:26.020
I should extend the
number line more.

00:01:26.020 --> 00:01:28.470
So this right here
is negative 10.

00:01:28.470 --> 00:01:31.770
So how far is it away from 0?

00:01:31.770 --> 00:01:34.590
Well, it's 10 to the left of 0.

00:01:34.590 --> 00:01:36.780
So you put a 10 here.

00:01:36.780 --> 00:01:38.670
And so in general,
absolute value

00:01:38.670 --> 00:01:40.460
will always be a
positive quantity.

00:01:40.460 --> 00:01:43.160
And when we're rethinking
about just absolute values

00:01:43.160 --> 00:01:45.830
of just numbers, it's
just going to be,

00:01:45.830 --> 00:01:48.310
really, the positive
version of that number.

00:01:48.310 --> 00:01:49.590
Let's do one more.

00:01:49.590 --> 00:01:53.020
Well, they tell us to do one
more, the absolute value of x

00:01:53.020 --> 00:01:54.920
when x is equal to negative 12.

00:01:54.920 --> 00:01:57.912
So we have the absolute
value of negative 12.

00:01:57.912 --> 00:01:59.620
We don't even to look
at the number line.

00:01:59.620 --> 00:02:01.545
It's just going to be
the positive version

00:02:01.545 --> 00:02:02.740
of negative 12.

00:02:02.740 --> 00:02:04.920
It's just going
to be equal to 12.

00:02:04.920 --> 00:02:08.779
And this is just saying that
negative 12 is 12 away from 0.

00:02:08.779 --> 00:02:10.070
And we could draw it over here.

00:02:10.070 --> 00:02:10.979
This is negative 11.

00:02:10.979 --> 00:02:13.080
Negative 12 is right over here.

00:02:13.080 --> 00:02:21.396
It is 1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12 away from 0.


How to evaluate an expression with variables

WEBVTT
Kind: captions
Language: en

00:00:00.620 --> 00:00:03.820
A local hospital is holding
a raffle as a fundraiser.

00:00:03.820 --> 00:00:06.390
The individual cost of
participating in the raffle

00:00:06.390 --> 00:00:10.810
is given by the following
expression-- 5t plus 3,

00:00:10.810 --> 00:00:14.610
or 5 times t plus 3,
where t represents

00:00:14.610 --> 00:00:17.320
the number of tickets
someone purchases.

00:00:17.320 --> 00:00:22.330
Evaluate the expression when t
is equal to 1, t is equal to 8,

00:00:22.330 --> 00:00:25.790
and t is equal to 10.

00:00:25.790 --> 00:00:27.620
So let's first
take the situation

00:00:27.620 --> 00:00:31.530
where t is equal to 1.

00:00:31.530 --> 00:00:33.360
Then this expression
right over here

00:00:33.360 --> 00:00:39.750
becomes-- and I'll use that
same color-- becomes 5 times 1.

00:00:39.750 --> 00:00:43.680
5 times 1 plus 3.

00:00:43.680 --> 00:00:47.850
5 times 1 plus 3, and we know
from order of operations,

00:00:47.850 --> 00:00:50.640
you do the multiplication
before you do the addition.

00:00:50.640 --> 00:00:54.870
So this will be 5
times 1 is 5 plus 3,

00:00:54.870 --> 00:00:59.300
and then this is
clearly equal to 8.

00:00:59.300 --> 00:01:02.360
Now let's do it when
t is equal to 8.

00:01:02.360 --> 00:01:07.300
So when t is equal to 8,
this expression becomes--

00:01:07.300 --> 00:01:16.990
and I'll do the same colors
again-- 5 times 8 plus 3.

00:01:16.990 --> 00:01:19.690
Same color of green.

00:01:19.690 --> 00:01:25.650
And once again, 5 times 8 is
40, and then we have the plus 3,

00:01:25.650 --> 00:01:28.730
there so this is equal to 43.

00:01:28.730 --> 00:01:30.270
And so we have the
last situation,

00:01:30.270 --> 00:01:31.900
with t is equal to 10.

00:01:31.900 --> 00:01:34.440
I'll do that in blue.

00:01:34.440 --> 00:01:36.660
So we have 5 times 10.

00:01:36.660 --> 00:01:41.670
So 5t is 5 times 10.

00:01:41.670 --> 00:01:43.330
Instead of a t, put a 10 there.

00:01:43.330 --> 00:01:47.249
5 times 10 plus 3.

00:01:47.249 --> 00:01:49.040
That's a slightly
different shade of green,

00:01:49.040 --> 00:01:50.248
but I think you get the idea.

00:01:50.248 --> 00:01:52.970
5 times 10 is 50.

00:01:52.970 --> 00:01:55.310
We do 50, and then we're going
to have to add 3 to that,

00:01:55.310 --> 00:01:58.400
and that is equal to 53.

00:01:58.400 --> 00:02:00.190
And we're done.


Opposite of a given number

WEBVTT
Kind: captions
Language: en

00:00:01.834 --> 00:00:04.992
Identify each number in the left column

00:00:04.992 --> 00:00:09.172
with its opposite in the right column

00:00:09.172 --> 00:00:11.355
They are taking opposites, these are integers

00:00:11.355 --> 00:00:14.373
so they are taking about the negative

00:00:14.373 --> 00:00:17.995
of that number.

00:00:17.995 --> 00:00:27.191
So if we are looking at 3 its opposite in this context

00:00:27.191 --> 00:00:32.206
we have 0 here, 3 is here to the right

00:00:32.206 --> 00:00:35.457
so we go to the left of 0 for opposite

00:00:35.457 --> 00:00:41.448
and we move the same distance 3 , so we have -3

00:00:41.448 --> 00:00:44.188
So now 3 and -3 are opposites

00:00:44.188 --> 00:00:56.959
No lets do 100, and -100 is the opposite

00:00:56.959 --> 00:01:04.203
We have 12 , and -12 is the opposite

00:01:04.203 --> 00:01:17.021
then we have -55, the opposite is positive +55 or 55

00:01:17.021 --> 00:01:33.182
opposite of -3 is the positive 3 , almost done

00:01:33.182 --> 00:01:42.934
opposite of -5 is the positive 5

00:01:42.934 --> 00:01:53.104
opposite of 21 is -21

00:01:53.104 --> 99:59:59.999
and then opposite of -12 is positive 12


Adding fractions with different signs

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.362
Find the sum 3 and 1/8 plus
3/4 plus negative 2 and 1/6.

00:00:05.362 --> 00:00:06.820
Let's just do the
first part first.

00:00:06.820 --> 00:00:08.240
It's pretty straightforward.

00:00:08.240 --> 00:00:10.620
We have two positive numbers.

00:00:10.620 --> 00:00:13.190
Let me draw a number line.

00:00:13.190 --> 00:00:16.219
So let me draw a number line.

00:00:16.219 --> 00:00:17.270
And I'll try to focus in.

00:00:17.270 --> 00:00:19.300
So we're going to
start at 3 and 1/8.

00:00:19.300 --> 00:00:20.950
So let's make this 0.

00:00:20.950 --> 00:00:28.020
So you have 1, 2, 3,
and then you have 4.

00:00:28.020 --> 00:00:30.440
3 and 1/8 is going to
be right about there.

00:00:30.440 --> 00:00:32.299
So let me just draw
its absolute value.

00:00:32.299 --> 00:00:40.710
So this 3 and 1/8 is going to
be 3 and 1/8 to the right of 0.

00:00:40.710 --> 00:00:45.430
So it's going to be exactly that
distance from 0 to the right.

00:00:45.430 --> 00:00:48.020
So this right here, the
length of this arrow,

00:00:48.020 --> 00:00:50.194
you could view it as 3 and 1/8.

00:00:50.194 --> 00:00:52.860
Now whenever I like to deal with
fractions, especially when they

00:00:52.860 --> 00:00:55.040
have different denominators
and all of that,

00:00:55.040 --> 00:00:57.330
I like to deal with them
as improper fractions.

00:00:57.330 --> 00:00:59.080
It makes the addition
and the subtraction,

00:00:59.080 --> 00:01:01.163
and, actually, the
multiplication and the division

00:01:01.163 --> 00:01:01.770
a lot easier.

00:01:01.770 --> 00:01:06.580
So 3 and 1/8 is the same
thing as 8 times 3 is 24,

00:01:06.580 --> 00:01:09.220
plus 1 is 25 over 8.

00:01:09.220 --> 00:01:14.010
So this is 25 over 8, which is
the same thing as 3 and 1/8.

00:01:14.010 --> 00:01:16.770
Another way to think
about it, 3 is 24 over 8.

00:01:16.770 --> 00:01:20.030
And then you add 1/8 to
that, so you get 25 over 8.

00:01:20.030 --> 00:01:22.000
So this is our starting point.

00:01:22.000 --> 00:01:25.000
Now to that, we are
going to add 3/4.

00:01:25.000 --> 00:01:27.000
We are going to add 3/4.

00:01:27.000 --> 00:01:29.610
So we're going to
move another 3/4.

00:01:29.610 --> 00:01:33.080
We are going to
move another 3/4.

00:01:33.080 --> 00:01:34.480
It's hard drawing these arrows.

00:01:34.480 --> 00:01:38.190
We're going to move
another 3/4 to the right.

00:01:38.190 --> 00:01:40.060
So this right here,
the length of this

00:01:40.060 --> 00:01:42.120
that we're moving
to the right is 3/4.

00:01:42.120 --> 00:01:43.886
So plus 3/4.

00:01:43.886 --> 00:01:45.010
Now where does this put us?

00:01:45.010 --> 00:01:47.440
Well, both of these
are positive integers.

00:01:47.440 --> 00:01:48.440
So we can just add them.

00:01:48.440 --> 00:01:50.106
We just have to find
a like denominator.

00:01:50.106 --> 00:01:52.910
So we have 25 over 8.

00:01:52.910 --> 00:01:57.880
We have 25 over 8 plus 3/4.

00:01:57.880 --> 00:02:01.150
That's the same thing as we need
to find a common denominator

00:02:01.150 --> 00:02:01.650
here.

00:02:01.650 --> 00:02:04.020
The common denominator, or
the least common multiple

00:02:04.020 --> 00:02:05.250
of 4 and 8 is 8.

00:02:05.250 --> 00:02:07.350
So it's going to be
something over 8.

00:02:07.350 --> 00:02:09.560
To get from 4 to 8,
we multiply by 2.

00:02:09.560 --> 00:02:11.680
So we have to multiply
3 by 2 as well.

00:02:11.680 --> 00:02:12.820
So you get 6.

00:02:12.820 --> 00:02:16.450
So 3/4 is the same thing as 6/8.

00:02:16.450 --> 00:02:18.550
If we have 25/8 and
we're adding 6/8

00:02:18.550 --> 00:02:23.990
to that, that gives
us 25 plus 6 is 31/8.

00:02:23.990 --> 00:02:28.320
So this number right over here,
this number right over here,

00:02:28.320 --> 00:02:29.205
is 31/8.

00:02:32.950 --> 00:02:35.670
And it makes sense
because 32/8 would be 4.

00:02:35.670 --> 00:02:38.050
So it should be a little
bit less than four.

00:02:38.050 --> 00:02:43.400
So this number right
over here is 31/8.

00:02:43.400 --> 00:02:46.250
Or the length of this arrow, the
absolute value of that number,

00:02:46.250 --> 00:02:51.020
is 31/8, a little
bit less than 4.

00:02:51.020 --> 00:02:56.830
If you wanted to write that as a
mixed number, it would be what?

00:02:56.830 --> 00:02:59.439
It would be 3 and 7/8.

00:02:59.439 --> 00:03:00.730
So that's that right over here.

00:03:00.730 --> 00:03:03.250
This is 31/8.

00:03:03.250 --> 00:03:05.010
That's that part
right over there.

00:03:05.010 --> 00:03:10.610
Now to that, we want to
add a negative 2 and 1/6.

00:03:10.610 --> 00:03:12.700
So we're going to add
a negative number.

00:03:12.700 --> 00:03:16.210
So think about what negative
2 and 1/6 is going to be like.

00:03:16.210 --> 00:03:19.420
So let me do this in a
new color, do it in pink.

00:03:19.420 --> 00:03:21.470
Negative 2 and 1/6.

00:03:21.470 --> 00:03:23.054
So we're going to
subtract, or I guess

00:03:23.054 --> 00:03:25.178
we're going to say we're
going to add a negative 1.

00:03:25.178 --> 00:03:27.670
We're going to add a negative
2 and then a negative 1/6.

00:03:27.670 --> 00:03:29.100
So let me draw.

00:03:29.100 --> 00:03:31.785
So negative 2 and 1/6, we
could literally draw like this.

00:03:35.840 --> 00:03:39.180
Negative 2 and 1/6 we
could draw with an arrow

00:03:39.180 --> 00:03:41.180
that looks something like that.

00:03:41.180 --> 00:03:43.655
So this is negative 2 and 1/6.

00:03:43.655 --> 00:03:45.530
Now, there's a couple
ways to think about it.

00:03:45.530 --> 00:03:47.170
You could just say,
hey, look, when

00:03:47.170 --> 00:03:49.090
you add this arrow,
this thing that's

00:03:49.090 --> 00:03:51.089
moving to the left-- we
could put it over here,

00:03:51.089 --> 00:03:53.130
and you would get straight
to negative 2 and 1/6.

00:03:53.130 --> 00:03:55.160
But we're adding this
negative 2 and 1/6.

00:03:55.160 --> 00:03:57.760
It's the same thing
as subtracting

00:03:57.760 --> 00:03:58.920
a positive 2 and 1/6.

00:03:58.920 --> 00:04:01.030
We're moving 2 and
1/6 to the left.

00:04:01.030 --> 00:04:06.520
And we're going to end up with
a number whose absolute value is

00:04:06.520 --> 00:04:08.130
going to look
something like that.

00:04:08.130 --> 00:04:09.921
And it's actually going
to be to the right.

00:04:09.921 --> 00:04:12.095
So it's not going to only
be its absolute value.

00:04:12.095 --> 00:04:13.470
Well, its absolute
value is going

00:04:13.470 --> 00:04:16.119
to be the number since it's
going to be a positive number.

00:04:16.119 --> 00:04:17.660
So let's just think
about what it is.

00:04:17.660 --> 00:04:18.959
This value right
here, which is going

00:04:18.959 --> 00:04:20.625
to be the answer to
our problem, is just

00:04:20.625 --> 00:04:25.140
going to be the difference
of 31/8 and 2 and 1/6.

00:04:25.140 --> 00:04:27.230
And it's the positive
difference because we're

00:04:27.230 --> 00:04:28.560
dealing with a positive number.

00:04:28.560 --> 00:04:30.100
So we just take 31/8.

00:04:30.100 --> 00:04:33.790
And from that, we will
subtract 2 and 1/6.

00:04:33.790 --> 00:04:34.730
So let's do this.

00:04:34.730 --> 00:04:42.780
So this orange value is going
to be 31/8 minus 2 and 1/6.

00:04:42.780 --> 00:04:46.600
So 2 and 1/6 is the
same thing as 6 times 2

00:04:46.600 --> 00:04:51.500
is 12 plus 1 is 13.

00:04:51.500 --> 00:04:53.651
Minus 13/6.

00:04:53.651 --> 00:04:55.150
And this is equal
to, once again, we

00:04:55.150 --> 00:04:57.740
need to get a common
denominator over here.

00:04:57.740 --> 00:05:03.600
And it looks like 24 will be
the common denominator, 24.

00:05:03.600 --> 00:05:06.040
And let me make it very clear.

00:05:06.040 --> 00:05:07.920
This is the 31/8.

00:05:07.920 --> 00:05:10.220
And this is the 2 and 1/6.

00:05:10.220 --> 00:05:12.580
This right here
is the 2 and 1/6.

00:05:12.580 --> 00:05:15.100
So 31/8 over 24.

00:05:15.100 --> 00:05:17.930
You have to multiply by 3
to get to the 24 over here.

00:05:17.930 --> 00:05:20.510
So we multiply by 3 on the 31.

00:05:20.510 --> 00:05:22.650
That gives us 93.

00:05:22.650 --> 00:05:27.210
And then to go from 6 to 24,
you have to multiply by 4.

00:05:27.210 --> 00:05:29.050
We do that in another color.

00:05:29.050 --> 00:05:32.880
You have to multiply
it by 4, so we

00:05:32.880 --> 00:05:35.740
have to multiply by
4 up here as well.

00:05:35.740 --> 00:05:37.870
So 4 times 13, let's see.

00:05:37.870 --> 00:05:40.286
4 times 10 is 40.

00:05:40.286 --> 00:05:42.180
4 times 3 is 12.

00:05:42.180 --> 00:05:44.180
So that's 52.

00:05:44.180 --> 00:05:49.970
So this is going to be equal
to 93 minus 52 over 24.

00:05:49.970 --> 00:05:55.220
And that is-- so 93 minus 52.

00:05:55.220 --> 00:05:56.620
3 minus 2 is 1.

00:05:56.620 --> 00:05:59.010
9 minus 5 is 4.

00:05:59.010 --> 00:06:04.729
So it is 41/24 and positive.

00:06:04.729 --> 00:06:07.270
And you can see that here just
by looking at the number line.

00:06:07.270 --> 00:06:11.450
This right here is 41 over 24.

00:06:11.450 --> 00:06:13.500
And it should be a
little bit less than 2

00:06:13.500 --> 00:06:16.940
because 2 would be 48 over 24.

00:06:16.940 --> 00:06:18.340
So this would be 48 over 24.

00:06:18.340 --> 00:06:21.750
And it makes sense that we're
a little bit less than that.


Identity property of 0

WEBVTT
Kind: captions
Language: en

00:00:01.277 --> 00:00:11.382
Evaluate 0 plus y plus -7 when y is equal to -3 and y is equal to 0 and y is equal to 7.

00:00:11.382 --> 00:00:17.647
So let's take the first situation where y is equal to -3...

00:00:17.647 --> 00:00:31.805
Then this expression right here would be... 0 plus -3 (because that's our "y" now) plus -7.

00:00:31.805 --> 00:00:36.647
Now 0 plus or minus anything won't change its value

00:00:36.647 --> 00:00:39.375
So you can really just ignore the 0 here.

00:00:39.375 --> 00:00:50.994
This is going to be the same thing, this is going to be the exact same thing, as -3 plus -7.

00:00:53.102 --> 00:00:55.631
-3 plus -7

00:00:55.631 --> 00:00:58.827
Now we could draw a number line here, just to help us visualize it

00:00:58.827 --> 00:01:02.084
But even if we didn't have the number line, we could say "Look, we're already 3 below zero...

00:01:02.084 --> 00:01:06.873
We're going to go another 7 to the left... another 7 more negative."

00:01:06.873 --> 00:01:11.007
So we're 3 away from zero, we're going to go 7 more away from zero

00:01:11.007 --> 00:01:13.749
So we're going to be 10 to the left of zero, or -10.

00:01:13.749 --> 00:01:15.389
Or another way to think of it...

00:01:15.389 --> 00:01:18.726
Lemme draw the number line there (always better to have the visual).

00:01:18.726 --> 00:01:21.270
So we're starting (this is zero)

00:01:21.270 --> 00:01:26.239
We're starting at -3, and to that we're adding -7

00:01:26.239 --> 00:01:29.675
So we're starting at -3, the absolute value is -3[sic]

00:01:29.675 --> 00:01:33.460
is our starting point. To that we're adding another negative 7

00:01:33.460 --> 00:01:35.226
We're going to move another 7 to the left.

00:01:35.226 --> 00:01:40.447
So we're adding (let me draw this)... we're adding a negative 7 right over here

00:01:40.447 --> 00:01:43.081
We're adding a negative 7 right over here

00:01:43.081 --> 00:01:45.672
So what's the length of this over here?

00:01:45.672 --> 00:01:50.397
The absolute value of -7 is equal to 7... that's the length of this arrow.

00:01:50.397 --> 00:01:53.461
But we're moving it to the left. The abolsute value of this

00:01:53.461 --> 00:01:59.699
arrow right here (the absolute value of negative 3) is 3.

00:01:59.699 --> 00:02:03.809
So we're already 3 to the left, now we're moving 7 more to left

00:02:03.809 --> 00:02:06.804
So now we're going to be 10 to the left.

00:02:06.804 --> 00:02:09.730
We are going to be 10 to left.

00:02:09.730 --> 00:02:12.470
10 to the left.

00:02:12.470 --> 00:02:18.625
This is literally, this is equal to the absolute value of -7...

00:02:18.625 --> 00:02:20.759
Let me do it in the order that we wrote it in the problem...

00:02:20.759 --> 00:02:26.402
So this is equal to the absolute value of -3 (you'd want to use the same colors)

00:02:26.402 --> 00:02:37.547
The absolute value of -3 plus the absolute value of -7.

00:02:37.547 --> 00:02:44.328
But we're to the left of zero (we've been moving to the left) so it's the <i>negative</i> of that

00:02:44.328 --> 00:02:46.951
which is -10

00:02:46.951 --> 00:02:50.620
So if your signs are the same, you can just take the absolute value of them

00:02:50.620 --> 00:02:56.490
say "ok, that's how far we're going to move, total. and we're that far from zero to the left"

00:02:56.490 --> 00:02:58.956
So the answer here is -10

00:02:58.956 --> 00:03:01.348
That was when y is -3.

00:03:01.348 --> 00:03:04.250
Let's think about when y is equal to 0.

00:03:04.250 --> 00:03:14.003
When y is equal to 0, this expression up here becomes 0 plus 0 (our y here is going to be zero)...

00:03:14.003 --> 00:03:16.278
(We'll do it in that same blue color.)

00:03:16.278 --> 00:03:22.524
.... plus 0 plus -7.

00:03:22.524 --> 00:03:26.031
Now, if you add 0 to anything it's not going to change the value.

00:03:26.031 --> 00:03:30.690
So this thing over here, this is pretty straight-forward, these don't matter

00:03:30.690 --> 00:03:33.649
It's just going to be equal to -7

00:03:33.649 --> 00:03:37.044
Now let's do the last one (when y is equal to 7)

00:03:37.044 --> 00:03:49.645
It's going to be 0 plus (our y is equal to 7) so 0 plus 7 and then we have plus -7

00:03:49.645 --> 00:03:51.642
Now there's a couple of ways to do this...

00:03:51.642 --> 00:03:57.215
You could literally just say "now adding a negative number is equivalent to subtracting the number".

00:03:57.215 --> 00:04:01.905
You could say this is equivalent to 0 plus 7 minus 7

00:04:01.905 --> 00:04:05.180
Plus -7 is the same thing as subtracting a 7

00:04:05.180 --> 00:04:10.288
The zero doesn't matter, so this is equal to 7 minus 7... which is equal to zero!

00:04:10.288 --> 00:04:12.169
Another way to think about it...

00:04:12.169 --> 00:04:16.325
Let's draw a number line

00:04:16.325 --> 00:04:18.368
So let's say that this is zero

00:04:18.368 --> 00:04:27.378
We're starting off at 7. We're 7 to the right of zero.

00:04:27.378 --> 00:04:32.564
And to that we're adding a -7.

00:04:32.564 --> 00:04:36.550
So we're going to move 7 to the left of where we were.

00:04:36.550 --> 00:04:39.619
7 to the left of 7 which was 7 to the right of 0!

00:04:39.619 --> 00:04:43.365
This gets us back to 0.

00:04:43.400 --> 99:59:59.999
So the answer is... zero.


Multiplying numbers with different signs

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:06.820
Find the product 4 times
negative 5 times negative 7.5.

00:00:06.820 --> 00:00:09.600
So we're just multiplying
three numbers right here,

00:00:09.600 --> 00:00:11.470
some are positive,
some are negative.

00:00:11.470 --> 00:00:13.200
We could actually do
this in any order,

00:00:13.200 --> 00:00:14.700
but let's just go left to right.

00:00:14.700 --> 00:00:17.565
So let's start with the
4 times the negative 5.

00:00:17.565 --> 00:00:19.440
And then we'll multiply
whatever we get there

00:00:19.440 --> 00:00:21.950
times negative 7.5.

00:00:21.950 --> 00:00:23.460
So let's just
rewrite the problems.

00:00:23.460 --> 00:00:29.367
We have 4 times negative
5, times negative 7.5.

00:00:29.367 --> 00:00:31.200
I really don't have to
write that dot there.

00:00:31.200 --> 00:00:33.900
If you just have 4 and then
in parentheses a negative 5,

00:00:33.900 --> 00:00:35.964
that means 4 times negative 5.

00:00:35.964 --> 00:00:37.630
Now, the first thing
you need to realize

00:00:37.630 --> 00:00:40.590
is that a positive
times a positive

00:00:40.590 --> 00:00:42.220
is obviously a positive.

00:00:42.220 --> 00:00:44.940
A positive times a negative,
or a negative times a positive,

00:00:44.940 --> 00:00:46.481
when you have
different signs, you're

00:00:46.481 --> 00:00:47.815
going to end up with a negative.

00:00:47.815 --> 00:00:50.189
If you have same signs, even
if you have a negative times

00:00:50.189 --> 00:00:52.570
a negative, you're going
to have a positive as well.

00:00:52.570 --> 00:00:54.550
So let's just work that
out in this problem.

00:00:54.550 --> 00:00:56.870
So you have 4 times negative 5.

00:00:56.870 --> 00:00:58.970
4 times 5 would be 20.

00:00:58.970 --> 00:01:00.550
Let me do this in
a different color.

00:01:00.550 --> 00:01:03.090
So 4 times 5 would be 20.

00:01:03.090 --> 00:01:06.320
But we have a positive
times a negative,

00:01:06.320 --> 00:01:07.650
so it's going to be a negative.

00:01:07.650 --> 00:01:09.970
Different signs mean we're
going to have a negative.

00:01:09.970 --> 00:01:16.120
So it becomes negative
20 times negative 7.5.

00:01:16.120 --> 00:01:17.650
So let's figure
out what that is.

00:01:17.650 --> 00:01:19.700
We have to multiply
20 times 7.5.

00:01:19.700 --> 00:01:21.800
We could do that out here.

00:01:21.800 --> 00:01:25.460
So 7.5 times 20.

00:01:25.460 --> 00:01:27.460
0 times 5 is 0.

00:01:27.460 --> 00:01:29.690
0 times 7 is 0.

00:01:29.690 --> 00:01:31.376
Put a 0 here, because
we're obviously

00:01:31.376 --> 00:01:32.500
not going to multiply by 2.

00:01:32.500 --> 00:01:33.874
We're actually
multiplying by 20.

00:01:33.874 --> 00:01:36.720
So 2 times 5 is 10.

00:01:36.720 --> 00:01:41.340
2 times 7 is 14, plus 1 is 15.

00:01:41.340 --> 00:01:44.290
And we get a 0, we
get a 0, we get a 5.

00:01:44.290 --> 00:01:45.300
I'm just adding here.

00:01:45.300 --> 00:01:46.354
You get a 1.

00:01:46.354 --> 00:01:47.770
And then you count
the numbers you

00:01:47.770 --> 00:01:49.430
have behind the decimal point.

00:01:49.430 --> 00:01:52.420
We only have one digit
behind the decimal point

00:01:52.420 --> 00:01:53.470
in both numbers.

00:01:53.470 --> 00:01:56.300
So you only go one digit
behind the decimal point

00:01:56.300 --> 00:01:57.460
in our product.

00:01:57.460 --> 00:02:00.670
So the answer here is 150.

00:02:00.670 --> 00:02:04.990
So this is equal to 150.

00:02:04.990 --> 00:02:06.610
And now we just have
to think about is

00:02:06.610 --> 00:02:08.110
it going to be
positive or negative.

00:02:08.110 --> 00:02:10.400
We have a negative
times a negative.

00:02:10.400 --> 00:02:12.600
They are the same sign.

00:02:12.600 --> 00:02:15.850
So it's going to result
in a positive number.

00:02:15.850 --> 00:02:18.920
So our final answer is 150.

00:02:18.920 --> 00:02:21.350
And if we actually wanted
to keep the precision here,

00:02:21.350 --> 00:02:23.320
we could write 150.0.


Identity property of 1 (second example)

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.950
Evaluate 1x when x is equal to
negative 1, x is equal to 0,

00:00:04.950 --> 00:00:06.950
and x is equal to 1.

00:00:06.950 --> 00:00:10.590
So let's do the scenario
x is equal to negative 1.

00:00:10.590 --> 00:00:15.030
Then 1x becomes 1 times x,
which is now negative 1.

00:00:15.030 --> 00:00:19.759
So it's 1 times negative 1.

00:00:19.759 --> 00:00:21.800
Well, there's a couple of
ways to think about it.

00:00:21.800 --> 00:00:25.270
1 times anything is going
to be that anything.

00:00:25.270 --> 00:00:31.509
So 1 times negative 1 is going
to be equal to negative 1.

00:00:31.509 --> 00:00:33.300
Another way you could
have thought about it

00:00:33.300 --> 00:00:36.274
is you could've just looked
at the absolute value

00:00:36.274 --> 00:00:36.940
of both numbers.

00:00:36.940 --> 00:00:38.800
You'd say 1 times 1 is 1.

00:00:38.800 --> 00:00:40.010
They have different signs.

00:00:40.010 --> 00:00:41.218
So it's going to be negative.

00:00:41.218 --> 00:00:43.890
It would have gotten you
the same answer either way.

00:00:43.890 --> 00:00:46.350
Now we have the scenario
where x is equal to 0.

00:00:46.350 --> 00:00:51.610
So 1 times x is going to be
the same thing as 1 times 0.

00:00:51.610 --> 00:00:54.020
We've substituted a 0 for x.

00:00:54.020 --> 00:00:58.520
Once again, 1 times anything is
going to become that anything.

00:00:58.520 --> 00:01:01.270
So this is going to become 0.

00:01:01.270 --> 00:01:04.140
Another way to think about
it is 0 times anything

00:01:04.140 --> 00:01:06.010
is going to become 0.

00:01:06.010 --> 00:01:08.134
So either one of those
ways of thinking about it

00:01:08.134 --> 00:01:09.550
would have gotten
you this answer.

00:01:09.550 --> 00:01:11.220
1 times-- actually,
we could even

00:01:11.220 --> 00:01:15.752
write 1 times x is equal to x.

00:01:15.752 --> 00:01:17.460
So really, when you
evaluate any of these

00:01:17.460 --> 00:01:21.450
at x is equal to negative 1, or
x is equal to 0, or x equals 1,

00:01:21.450 --> 00:01:23.450
it's really just going
to equal that number

00:01:23.450 --> 00:01:26.730
because 1 times anything
is equal to that anything

00:01:26.730 --> 00:01:28.560
right over there.

00:01:28.560 --> 00:01:32.610
Just to make it clear, 0
times anything is equal to--

00:01:32.610 --> 00:01:35.770
and this x here is a variable
not a multiplication sign-- 0

00:01:35.770 --> 00:01:38.720
times anything is equal to 0.

00:01:38.720 --> 00:01:40.790
Just for good measure--
we kind of already

00:01:40.790 --> 00:01:43.530
went over the answer-- but
let's try x is equal to 1.

00:01:43.530 --> 00:01:48.330
So this will give us 1 times
1, which you've probably

00:01:48.330 --> 00:01:49.770
learned a while ago.

00:01:49.770 --> 00:01:51.830
But 1 times anything is
equal to that number.

00:01:51.830 --> 00:01:54.390
So it's equal to 1.


Order of operations example: putting it all together

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.420
Simplify negative 1
times this expression

00:00:03.420 --> 00:00:09.290
in brackets, negative 7 plus
2 times 3 plus 2 minus 5,

00:00:09.290 --> 00:00:11.140
in parentheses, squared.

00:00:11.140 --> 00:00:13.920
So this is an order
of operations problem.

00:00:13.920 --> 00:00:15.750
And remember, order
of operations,

00:00:15.750 --> 00:00:18.280
you always want to
do parentheses first.

00:00:18.280 --> 00:00:22.230
Parentheses first.

00:00:22.230 --> 00:00:24.870
Then you do exponents.

00:00:24.870 --> 00:00:25.540
Exponents.

00:00:25.540 --> 00:00:28.940
And there is an exponent in
this problem right over here.

00:00:28.940 --> 00:00:31.290
Then you want to
do multiplication.

00:00:31.290 --> 00:00:33.560
Multiplication and division.

00:00:33.560 --> 00:00:37.790
And then finally, you do
addition and subtraction.

00:00:37.790 --> 00:00:40.380
So let's just try to
tackle this as best we can.

00:00:40.380 --> 00:00:44.900
So first, let's do
the parentheses.

00:00:44.900 --> 00:00:47.780
We have a 3 plus 2
here in parentheses,

00:00:47.780 --> 00:00:52.156
so we can evaluate
that to be equal to 5.

00:00:52.156 --> 00:00:53.780
And let's see, we
could do other things

00:00:53.780 --> 00:00:55.363
in other parts of
this expression that

00:00:55.363 --> 00:00:58.190
won't affect what's going on
right here in the parentheses.

00:00:58.190 --> 00:01:01.280
We have this negative 5 squared.

00:01:01.280 --> 00:01:04.310
Or I should just, we're
subtracting a 5 squared.

00:01:04.310 --> 00:01:06.050
We want to do the
exponent before we

00:01:06.050 --> 00:01:08.390
worry about it being subtracted.

00:01:08.390 --> 00:01:13.380
So this 5 squared over
here we can rewrite as 25.

00:01:13.380 --> 00:01:15.600
And so let's not do
too many steps at once.

00:01:15.600 --> 00:01:18.970
So this whole thing will
simplify to negative 1.

00:01:18.970 --> 00:01:24.070
And then in brackets, we have
negative 7 plus 2 times 5.

00:01:27.000 --> 00:01:29.530
And then, 2 times 5.

00:01:29.530 --> 00:01:31.280
And then close brackets.

00:01:31.280 --> 00:01:32.285
Minus 25.

00:01:35.490 --> 00:01:38.869
Now, this thing-- we want
to do multiplication.

00:01:38.869 --> 00:01:39.910
You could say, hey, wait.

00:01:39.910 --> 00:01:40.940
I still have a parentheses here.

00:01:40.940 --> 00:01:41.990
Why don't I do that first?

00:01:41.990 --> 00:01:43.220
But when you just
evaluate what's

00:01:43.220 --> 00:01:45.360
inside of this parentheses,
you just get a negative 7.

00:01:45.360 --> 00:01:46.776
It doesn't really
change anything.

00:01:46.776 --> 00:01:49.847
So we can just leave this
here as a negative 7.

00:01:49.847 --> 00:01:50.680
And this expression.

00:01:50.680 --> 00:01:53.220
We do want to evaluate this
whole expression before we

00:01:53.220 --> 00:01:53.930
do anything else.

00:01:53.930 --> 00:01:56.429
I mean, we could distribute
this negative 1 and all of that,

00:01:56.429 --> 00:01:59.480
but let's just do straight
up order of operations here.

00:01:59.480 --> 00:02:01.230
So let's evaluate
this expression.

00:02:01.230 --> 00:02:03.950
We want to do multiplication
before we add anything.

00:02:03.950 --> 00:02:06.520
So we get 2 times
5 right over there.

00:02:06.520 --> 00:02:09.239
2 times 5 is 10.

00:02:09.239 --> 00:02:10.090
That is 10.

00:02:10.090 --> 00:02:11.465
So our whole
expression becomes--

00:02:11.465 --> 00:02:13.840
and normally, you wouldn't
have to rewrite the expression

00:02:13.840 --> 00:02:14.590
this many times.

00:02:14.590 --> 00:02:16.006
But we're going
to do it this time

00:02:16.006 --> 00:02:17.750
just to make sure no
one gets confused.

00:02:17.750 --> 00:02:24.670
So it becomes negative 1
times negative 7 plus 10.

00:02:24.670 --> 00:02:26.650
Plus 10.

00:02:26.650 --> 00:02:28.660
And we close our brackets.

00:02:28.660 --> 00:02:32.020
Minus 25.

00:02:32.020 --> 00:02:35.110
Now, we can evaluate
this pretty easily.

00:02:35.110 --> 00:02:37.980
Negative 7 plus 10.

00:02:37.980 --> 00:02:40.190
We're starting at negative 7.

00:02:40.190 --> 00:02:42.430
So I was going to draw
a number line there.

00:02:42.430 --> 00:02:45.710
So we're starting-- let
me draw a number line.

00:02:45.710 --> 00:02:48.640
So we're starting at negative 7.

00:02:48.640 --> 00:02:51.610
So the length of this
line is negative 7.

00:02:51.610 --> 00:02:53.650
And then, we're adding 10 to it.

00:02:53.650 --> 00:02:55.136
We're adding 10 to it.

00:02:55.136 --> 00:02:56.760
So we're going to
move 10 to the right.

00:02:56.760 --> 00:02:59.180
If we move 7 to the
right, we get back to 0.

00:02:59.180 --> 00:03:02.800
And then we're going to
go another 3 after that.

00:03:02.800 --> 00:03:05.820
So we're going to
go 7, 8, 9, 10.

00:03:05.820 --> 00:03:08.590
So that gets us to positive 3.

00:03:08.590 --> 00:03:11.250
Another way to
think about it is we

00:03:11.250 --> 00:03:14.590
are adding integers
of different signs.

00:03:14.590 --> 00:03:18.500
We can view the
sum as going to be

00:03:18.500 --> 00:03:20.490
the difference of the integers.

00:03:20.490 --> 00:03:23.710
And since the larger
integer is positive,

00:03:23.710 --> 00:03:25.470
our answer will be positive.

00:03:25.470 --> 00:03:27.940
So you could literally just
view this as 10 minus 7.

00:03:27.940 --> 00:03:31.100
10 minus 7 is 3.

00:03:31.100 --> 00:03:32.590
So this becomes a 3.

00:03:32.590 --> 00:03:35.990
And so our entire expression
becomes negative 1.

00:03:35.990 --> 00:03:37.960
Negative 1 times.

00:03:37.960 --> 00:03:39.880
And just to be clear,
brackets and parentheses

00:03:39.880 --> 00:03:41.130
are really the same thing.

00:03:41.130 --> 00:03:42.804
Sometimes people
will write brackets

00:03:42.804 --> 00:03:44.470
around a lot of
parentheses just to make

00:03:44.470 --> 00:03:45.900
it a little bit easier to read.

00:03:45.900 --> 00:03:48.150
But they're really just the
same thing as parentheses.

00:03:48.150 --> 00:03:50.620
So these brackets out here,
I could just literally

00:03:50.620 --> 00:03:52.230
write them like that.

00:03:52.230 --> 00:03:56.244
And then I have a
minus 25 out over here.

00:03:56.244 --> 00:03:58.660
Now, once again, you want to
do multiplication or division

00:03:58.660 --> 00:04:00.350
before we do addition
and subtraction.

00:04:00.350 --> 00:04:05.260
So let's multiply the negative
1 times 3 is negative 3.

00:04:05.260 --> 00:04:08.100
And now we need to
subtract our 25.

00:04:08.100 --> 00:04:10.880
So negative 3 minus 25.

00:04:10.880 --> 00:04:13.079
We are adding two
integers of the same sign.

00:04:13.079 --> 00:04:14.620
We're already at
negative 3 and we're

00:04:14.620 --> 00:04:16.980
going to become 25 more
negative than that.

00:04:16.980 --> 00:04:20.390
So you can view
this as we're moving

00:04:20.390 --> 00:04:22.270
25 more in the
negative direction.

00:04:22.270 --> 00:04:24.515
Or you could view it
as 3 plus 25 is 28.

00:04:24.515 --> 00:04:26.390
But we're doing it in
the negative direction,

00:04:26.390 --> 00:04:28.480
so it's negative 28.

00:04:28.480 --> 00:04:33.460
So this is equal to negative 28.

00:04:33.460 --> 00:04:35.650
And we are done.