Comparing mitosis and meiosis

WEBVTT
Kind: captions
Language: en

00:00:00.330 --> 00:00:02.675
- [Voiceover] Before we
go in-depth on meiosis,

00:00:02.675 --> 00:00:04.384
I want to do a very high level overview

00:00:04.440 --> 00:00:06.529
comparing mitosis to meiosis.

00:00:07.249 --> 00:00:10.238
So, in mitosis, this is all a review,

00:00:10.271 --> 00:00:11.953
if you've watched the mitosis video,

00:00:12.691 --> 00:00:14.566
in mitosis, we start with a cell,

00:00:16.884 --> 00:00:19.277
that has a diploid number of chromosomes.

00:00:19.277 --> 00:00:22.244
I'll just write 2n to show
it has a diploid number.

00:00:22.454 --> 00:00:24.753
For human beings, this
would be 46 chromosomes.

00:00:29.576 --> 00:00:33.425
46 for humans, you get 23
chromosomes from your mother,

00:00:33.659 --> 00:00:36.718
23 chromosomes from your
father or you can say

00:00:36.906 --> 00:00:41.906
you have 23 homologous pairs,
which leads to 46 chromosomes.

00:00:42.034 --> 00:00:44.588
Now after the process of mitosis happens

00:00:44.588 --> 00:00:47.072
and you have your
cytokinesis and all the rest,

00:00:47.072 --> 00:00:49.515
you end up with two cells that each

00:00:49.518 --> 00:00:52.380
have the same genetic
information as the original.

00:00:53.325 --> 00:00:55.995
So you now have two cells

00:00:56.605 --> 00:00:59.617
that each have the diploid
number of chromosomes.

00:00:59.617 --> 00:01:04.006
So, 2n and 2n.

00:01:04.236 --> 00:01:05.807
And now each of these cells

00:01:05.901 --> 00:01:07.554
are just like this cell was,

00:01:08.980 --> 00:01:10.452
it can go through interphase again.

00:01:10.452 --> 00:01:14.882
It grows and it can replicate its DNA

00:01:15.020 --> 00:01:16.846
and centrosomes and grow some more

00:01:16.846 --> 00:01:19.285
then each of these can
go through mitosis again.

00:01:19.285 --> 00:01:21.198
And this is actually how

00:01:21.198 --> 00:01:23.628
most of the cells in your body grow.

00:01:23.628 --> 00:01:27.783
This is how you turn from a
single cell organism into you,

00:01:27.783 --> 00:01:30.428
or for the most part, into you.

00:01:30.958 --> 00:01:32.046
So that is mitosis.

00:01:32.166 --> 00:01:33.239
It's a cycle.

00:01:33.239 --> 00:01:35.035
After each of these
things go through mitosis,

00:01:36.305 --> 00:01:37.459
they can then go through

00:01:37.459 --> 00:01:38.600
the entire cell cycle again.

00:01:38.600 --> 00:01:40.852
Let me write this a little bit neater.

00:01:40.852 --> 00:01:43.267
Mitosis, that s was a
little bit hard to read.

00:01:43.927 --> 00:01:45.795
Now what happens in meiosis?

00:01:45.975 --> 00:01:48.657
What happens in meiosis?

00:01:48.657 --> 00:01:49.681
I'll do that over here.

00:01:50.701 --> 00:01:53.691
In meiosis, something
slightly different happens

00:01:54.231 --> 00:01:55.757
and it happens in two phases.

00:01:58.183 --> 00:02:00.316
You will start with a cell

00:02:00.316 --> 00:02:02.426
that has a diploid number of chromosomes.

00:02:03.566 --> 00:02:05.755
So you will start with a cell

00:02:06.035 --> 00:02:09.188
that has a diploid number of chromosomes.

00:02:09.576 --> 00:02:12.081
And in it's interphase, it
also replicates its DNA.

00:02:12.587 --> 00:02:15.049
And then it goes through
something called Meiosis One.

00:02:15.589 --> 00:02:16.790
And in Meiosis One,

00:02:16.790 --> 00:02:18.846
what you end up with is two cells

00:02:18.887 --> 00:02:21.668
that now have haploid
number of chromosomes.

00:02:23.004 --> 00:02:25.657
So you end up with two cells,

00:02:26.124 --> 00:02:28.894
You now have two cells that each have a

00:02:28.894 --> 00:02:30.458
haploid number of chromosomes.

00:02:31.207 --> 00:02:34.135
So you have n and you have n.

00:02:34.345 --> 00:02:35.758
So if we're talking about human beings,

00:02:35.758 --> 00:02:37.244
you have 46 chromosomes here,

00:02:37.244 --> 00:02:40.935
and now you have 23
chromosomes in this nucleus.

00:02:40.935 --> 00:02:42.909
And now you have 23 in this nucleus.

00:02:42.909 --> 00:02:44.558
But you're still not done.

00:02:45.058 --> 00:02:47.371
Then each of these will
go through a phase,

00:02:47.371 --> 00:02:49.689
which I'll talk about
in a second, which is

00:02:49.813 --> 00:02:52.161
very similar to mitosis, which will

00:02:53.351 --> 00:02:55.791
duplicate this entire cell into two.

00:02:56.141 --> 00:02:57.466
So actually, let me do it like this.

00:02:57.869 --> 00:02:59.017
So now, this one,

00:03:01.234 --> 00:03:04.473
you're going to have four cells

00:03:04.572 --> 00:03:06.740
that each have the haploid number

00:03:06.740 --> 00:03:10.983
that each have the haploid
number of chromosomes.

00:03:11.554 --> 00:03:13.864
And they don't all
necessarily have the same

00:03:13.864 --> 00:03:15.023
genetic informatioin anymore.

00:03:15.023 --> 00:03:17.151
Because as we go through this first phase,

00:03:17.151 --> 00:03:19.767
right over here of meiosis,
and this first phase

00:03:19.767 --> 00:03:23.157
here you go from diploid to
haploid, right over here,

00:03:23.157 --> 00:03:24.515
this is called Meiosis One.

00:03:25.385 --> 00:03:28.245
Meiosis One,

00:03:29.205 --> 00:03:32.551
you're essentially splitting
the homologous pairs

00:03:32.551 --> 00:03:34.618
and so this one might get some of the

00:03:36.738 --> 00:03:38.116
ones that you originally
got from your father,

00:03:38.116 --> 00:03:39.567
and some that you originally
got from your mother,

00:03:39.867 --> 00:03:41.587
some that you originally
got from your father,

00:03:41.587 --> 00:03:43.437
some that you originally
got from your mother,

00:03:43.437 --> 00:03:45.938
they split randomly,
but each homogolous pair

00:03:46.898 --> 00:03:48.665
gets split up.

00:03:48.992 --> 00:03:51.437
And then in this phase, Meiosis Two,

00:03:51.992 --> 00:03:54.587
so this phase right over
here is called Meiosis Two,

00:03:55.483 --> 00:03:56.586
it's very similar

00:03:56.936 --> 00:04:00.604
to mitosis, except your now
dealing with cells that start off

00:04:00.604 --> 00:04:02.485
with the haploid number.

00:04:02.485 --> 00:04:05.155
It's important to realize
that meiosis is not a cycle.

00:04:05.155 --> 00:04:08.513
These cells that you have
over here, these are gametes.

00:04:09.153 --> 00:04:10.263
This are sex cells.

00:04:10.845 --> 00:04:12.182
These are gametes.

00:04:12.675 --> 00:04:14.858
This can now be used in fertilization.

00:04:14.858 --> 00:04:18.387
If we're talking about, if
you're male, this is happening

00:04:18.387 --> 00:04:22.449
in your testes, and these
are going to be sperm cells

00:04:22.467 --> 00:04:26.317
If you are female, this is
happening in your ovaries

00:04:26.317 --> 00:04:28.146
and these are going to be egg cells.

00:04:28.450 --> 00:04:32.804
If you a tree, this could be
pollen or it could be an ovul.

00:04:34.115 --> 00:04:36.542
But these are used for fertilization.

00:04:37.197 --> 00:04:39.775
These will fuse together
in sexual reproduction

00:04:39.775 --> 00:04:44.775
to get to a fertilized
egg, which then can undergo

00:04:44.813 --> 00:04:48.180
mitosis to create an
entirely new organism.

00:04:48.180 --> 00:04:49.598
So not a cycle here,

00:04:49.662 --> 00:04:52.827
although these will find sex cells

00:04:52.827 --> 00:04:57.470
from another organism and fuse with them

00:04:57.488 --> 00:04:59.229
and those can turn into another organism.

00:04:59.229 --> 00:05:01.730
And I guess the whole
circle of life starts again.

00:05:01.730 --> 00:05:04.005
But it's not the case with
mitosis where this can keep

00:05:04.005 --> 00:05:05.445
going and going, going.

00:05:05.445 --> 00:05:10.132
This cell is just like this
cell, while these sex cells

00:05:10.132 --> 00:05:12.473
are differeent than this
one right over here.

00:05:12.473 --> 00:05:14.748
Now, where does this happen in the body?

00:05:14.748 --> 00:05:16.582
We've talked about this
in previous videos.

00:05:16.582 --> 00:05:19.949
These are your somatic
cells right over here.

00:05:19.949 --> 00:05:22.503
These are the ones that make
up the bulk of your body,

00:05:22.503 --> 00:05:23.734
somatic cells.

00:05:23.904 --> 00:05:25.452
And where is this happening?

00:05:25.496 --> 00:05:27.582
Well, this is happening in germ cells,

00:05:27.582 --> 00:05:29.764
As we mentioned, if you're
male it's in your tesis

00:05:29.764 --> 00:05:32.428
and if you're female it's in your ovaries.

00:05:32.767 --> 00:05:36.025
And germ cells actually
can undergo mitosis

00:05:36.075 --> 00:05:38.540
to produce other germ
cells that have a diploid

00:05:38.540 --> 00:05:42.046
number of chromosomes, or they can undergo

00:05:42.046 --> 00:05:46.179
meiosis in order to
produce sperm or egg cells

00:05:46.179 --> 00:05:48.338
in order to produce gametes.


Mitosis

WEBVTT
Kind: captions
Language: en

00:00:00.226 --> 00:00:03.069
- In the previous video,
we talked about interphase

00:00:03.069 --> 00:00:07.248
which is the bulk of a
cell's life cycle as it grows

00:00:07.248 --> 00:00:12.046
and its DNA replicates,
and it grows some more.

00:00:12.046 --> 00:00:14.759
And now, we're gonna talk
about the actual cell division.

00:00:14.759 --> 00:00:16.710
We're gonna talk about mitosis.

00:00:16.710 --> 00:00:20.053
And if you wanna be precise,
mitosis is the process

00:00:20.053 --> 00:00:24.279
by which this one nucleus
will turn into two nuclei

00:00:24.279 --> 00:00:27.959
that each have the original
genetic information.

00:00:27.959 --> 00:00:31.790
Now, as we exit mitosis,
we get into cytokineses

00:00:31.790 --> 00:00:36.231
which will then split each of the nuclei

00:00:36.231 --> 00:00:39.772
into a separate cell when
we split the cytoplasm

00:00:39.772 --> 00:00:40.608
right over here.

00:00:40.608 --> 00:00:44.280
We split or the cell as
it turns into to cells.

00:00:44.280 --> 00:00:47.552
Well, let's see how all of this happens.

00:00:48.515 --> 00:00:51.194
So the first phase, and I'll
leave the end of interphase

00:00:51.194 --> 00:00:52.158
right over here.

00:00:52.158 --> 00:00:56.198
We have this big cell, our
DNA has been replicated.

00:00:56.198 --> 00:00:59.240
We have two centrosomes right over here.

00:00:59.240 --> 00:01:04.240
The first phase of
mitosis involves the cell

00:01:04.394 --> 00:01:06.052
and I might draw a little
bit smaller just so I have

00:01:06.052 --> 00:01:07.712
enough space here.

00:01:07.712 --> 00:01:09.256
So, involve.

00:01:09.256 --> 00:01:11.856
So, this is the cell right over here.

00:01:11.856 --> 00:01:13.981
So we're gonna go to this
phase right over here.

00:01:13.981 --> 00:01:16.813
And a few things start happening.

00:01:16.813 --> 00:01:20.993
One, the DNA, the chromosomes
go from being in their

00:01:20.993 --> 00:01:23.319
chromatin form where
they're all spread out

00:01:23.319 --> 00:01:25.722
to kind of a more condensed
form that you can actually see

00:01:25.722 --> 00:01:27.382
from a light microscope.

00:01:27.382 --> 00:01:30.505
So for example, that magenta
chromosome which is now made up

00:01:30.505 --> 00:01:33.384
of two sister chromatids
after replication,

00:01:33.384 --> 00:01:36.043
we talked about that
in the interphase video.

00:01:36.043 --> 00:01:39.280
It might look something like this in a ...

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It might look something like this

00:01:42.184 --> 00:01:45.183
if you were to look in a microscope.

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It's unlikely to be magenta but
it's gonna have kind of that

00:01:48.560 --> 00:01:50.802
classic chromosome shape
that you're used to seeing

00:01:50.802 --> 00:01:52.102
in textbooks.

00:01:52.102 --> 00:01:54.482
And it has the centromere that connects

00:01:54.482 --> 00:01:56.363
these two sister chromatids.

00:01:56.363 --> 00:01:59.764
Right now, both of these two
sister chromatids combined

00:01:59.764 --> 00:02:01.981
are considered to be one chromosome.

00:02:01.981 --> 00:02:04.489
Even though before replication
it was still considered,

00:02:04.489 --> 00:02:08.529
the magenta stuff was still
considered to be one chromosome.

00:02:08.529 --> 00:02:11.501
And we can draw the blue chromosome.

00:02:11.501 --> 00:02:13.045
Once again, it's now
in the condensed form.

00:02:13.045 --> 00:02:15.541
That's one sister
chromatid right over there.

00:02:15.541 --> 00:02:18.745
That's another sister chromatid.

00:02:18.745 --> 00:02:21.564
They are connected at the centromere.

00:02:23.752 --> 00:02:27.263
So they're condensing now
as we enter into mitosis.

00:02:27.263 --> 00:02:30.741
And the nuclear membrane

00:02:30.741 --> 00:02:34.514
starts to go away.

00:02:34.514 --> 00:02:37.822
So the nuclear membrane
is starting to go away.

00:02:37.822 --> 00:02:40.857
And these two centrosomes
are starting to migrate

00:02:40.857 --> 00:02:44.491
to opposite sides of the cell.

00:02:44.491 --> 00:02:47.230
So one of them is going
over here and one of them

00:02:47.230 --> 00:02:49.854
is maybe going to go over here.

00:02:49.854 --> 00:02:53.534
So they're migrating to
opposite sides of the cell.

00:02:53.534 --> 00:02:55.788
And it's pretty incredible.

00:02:55.788 --> 00:02:57.926
It's easy to say, "Oh, this
happens and then that happens."

00:02:57.926 --> 00:03:00.816
Remember, this cell doesn't have a brain.

00:03:00.816 --> 00:03:03.696
This is all happening through
chemical and thermodynamic

00:03:03.696 --> 00:03:07.445
reactions and the way, just
based on certain triggers

00:03:07.445 --> 00:03:09.255
of where the cell is and its life cycle.

00:03:09.255 --> 00:03:10.939
It's amazing that this is happening.

00:03:10.939 --> 00:03:12.436
It's amazing that the structures,

00:03:12.436 --> 00:03:14.431
and what sometimes we
consider to be a simple thing

00:03:14.431 --> 00:03:16.776
but it's actually
incredibly complex thing.

00:03:17.956 --> 00:03:20.678
It kind of knows what to do even though

00:03:20.678 --> 00:03:22.594
it has no intelligence here.

00:03:22.594 --> 00:03:23.801
And a lot of what I'm talking about,

00:03:23.801 --> 00:03:26.576
the general overview of the
process is well understood

00:03:26.576 --> 00:03:29.455
but scientists are still
understanding exactly

00:03:29.455 --> 00:03:32.438
how do the different things
happen when they should happen

00:03:32.438 --> 00:03:34.817
and by what mechanism and
what's actually happening.

00:03:34.817 --> 00:03:38.417
Sometimes, molecular or atomic basis.

00:03:38.417 --> 00:03:41.296
But anyway, this first phase of mitosis,

00:03:41.296 --> 00:03:43.560
the nuclear envelope, the nuclear membrane

00:03:43.560 --> 00:03:45.150
starts to disappear.

00:03:45.150 --> 00:03:48.354
The centrosomes migrate to
the opposite ends of the cell.

00:03:48.354 --> 00:03:51.953
And our DNA condenses into
kind of the condensed form

00:03:51.953 --> 00:03:53.752
of the chromosomes.

00:03:53.752 --> 00:03:55.690
And we call this prophase.

00:03:56.740 --> 00:03:58.769
Prophase.

00:03:58.769 --> 00:04:01.628
Prophase of mitosis.

00:04:01.628 --> 00:04:03.590
Now, in the next phase.

00:04:03.590 --> 00:04:07.734
In the next phase, let
me draw my cell again.

00:04:07.734 --> 00:04:09.498
Drawing that same green color.

00:04:10.571 --> 00:04:14.089
In the next phase,

00:04:14.089 --> 00:04:19.089
your nuclear membrane is now gone.

00:04:19.789 --> 00:04:22.579
And the chromosomes start lining up

00:04:22.579 --> 00:04:24.565
in the middle of the cells.

00:04:24.565 --> 00:04:27.790
So you have the blue one right over here,

00:04:27.790 --> 00:04:29.842
the blue one that's one sister chromatid.

00:04:29.842 --> 00:04:32.222
Here's another sister chromatid

00:04:32.222 --> 00:04:36.273
and they're connected at the centromere,

00:04:36.273 --> 00:04:39.257
not to be confused with a centrosome.

00:04:39.257 --> 00:04:43.251
And then, here's the magenta, one of the

00:04:43.251 --> 00:04:45.932
magenta sister chromatids and another one.

00:04:47.005 --> 00:04:48.902
And once again, it's not
magenta in real life,

00:04:48.902 --> 00:04:52.106
I'm just making it a magenta
because it looks nice.

00:04:52.106 --> 00:04:54.486
That's the centromere right over there.

00:04:54.486 --> 00:04:58.526
Our centrosomes are at opposite ends

00:04:58.526 --> 00:05:00.685
of the cell at this point.

00:05:00.685 --> 00:05:02.078
At opposite ends of the cell.

00:05:02.078 --> 00:05:03.556
And you might have heard of the word,

00:05:03.556 --> 00:05:04.646
let me label this again.

00:05:04.646 --> 00:05:05.661
I labeled it in a previous video.

00:05:05.661 --> 00:05:08.018
That's centrosomes.

00:05:08.018 --> 00:05:13.018
Centrosomes where the two
sister chromatids are connected.

00:05:13.695 --> 00:05:16.129
That's a centromere.

00:05:16.129 --> 00:05:17.673
Centromere.

00:05:17.673 --> 00:05:20.715
Now, you might heard
of the word centrioles.

00:05:20.715 --> 00:05:22.607
Centrioles are actually, they help,

00:05:22.607 --> 00:05:25.451
they are exist inside the centrosomes.

00:05:25.451 --> 00:05:28.415
They're these two kind of
cylindrical looking structures.

00:05:28.415 --> 00:05:31.934
Each of the centrosomes
have two centrioles.

00:05:31.934 --> 00:05:33.280
But for the sake of
this video, you can say,

00:05:33.280 --> 00:05:36.403
"Well, the centrioles are
just part of the centrosomes."

00:05:36.403 --> 00:05:38.980
But I'm just listing you all the words

00:05:38.980 --> 00:05:41.232
that involves centri something.

00:05:41.232 --> 00:05:44.553
Centrioles right over there
and you have two of them

00:05:44.553 --> 00:05:46.062
per centrosome.

00:05:46.062 --> 00:05:48.256
So hopefully, that helps
clarify some confusion.

00:05:48.256 --> 00:05:49.695
But these things line up.

00:05:49.695 --> 00:05:51.576
And a lot of what you're about
to see, this orchestration,

00:05:51.576 --> 00:05:54.316
these things moving around
in the cell, things lining up

00:05:54.316 --> 00:05:57.276
and soon things pulling apart,
these are all coordinated

00:05:57.276 --> 00:06:00.479
with actually a fairly
sophisticated mechanism of kind of a

00:06:00.479 --> 00:06:04.833
scaffold of this kind of
ropes, these microtubules.

00:06:04.833 --> 00:06:07.434
And the centrosomes' role, until now

00:06:07.434 --> 00:06:08.618
I've just been kind of drawing.

00:06:08.618 --> 00:06:09.535
I mean, what do they do?

00:06:09.535 --> 00:06:12.821
Well, the centrosomes'
role is these microtubules

00:06:12.821 --> 00:06:17.138
extend from them to each
other and to the centromeres

00:06:18.618 --> 00:06:23.499
of these chromosomes
and to a large degree,

00:06:23.499 --> 00:06:25.437
they're not the only actors here.

00:06:25.437 --> 00:06:28.258
They help pull and push
things in the right way.

00:06:28.258 --> 00:06:32.077
So these help keep make sure
that the two centrosomes

00:06:32.077 --> 00:06:33.494
push away from each other.

00:06:33.494 --> 00:06:35.468
And then as we'll see in the next phase,

00:06:35.468 --> 00:06:37.754
that they're able to
pull one of each of the

00:06:37.754 --> 00:06:41.249
sister chromatids from
each of the pairs towards

00:06:42.469 --> 00:06:43.563
each of them.

00:06:43.563 --> 00:06:45.722
So this right over here where you see

00:06:45.722 --> 00:06:50.622
where the nuclear envelope
is now gone, the chromosomes

00:06:50.622 --> 00:06:53.861
have been lined up just like
this and your centrosomes

00:06:53.861 --> 00:06:57.281
are on opposite side of your cell.

00:06:57.281 --> 00:06:58.918
We call this the metaphase.

00:06:59.713 --> 00:07:04.713
We call this the metaphase of mitosis.

00:07:04.825 --> 00:07:07.786
And then you can imagine
what's going to happen next.

00:07:07.786 --> 00:07:09.666
What's going to happen next?

00:07:09.666 --> 00:07:11.991
What's going to happen next is ...

00:07:11.991 --> 00:07:14.584
Let me, I don't wanna draw it
too big 'cause I wanna be able

00:07:14.584 --> 00:07:16.465
to fit it all in one page.

00:07:16.465 --> 00:07:20.947
What's going to happen next is that

00:07:20.947 --> 00:07:23.849
those microtubules are
going to start pulling

00:07:23.849 --> 00:07:25.474
on each of the sister chromatids.

00:07:25.474 --> 00:07:26.891
Let me draw that.

00:07:26.891 --> 00:07:31.891
So you have this
centrosome right over here.

00:07:32.115 --> 00:07:35.990
You have all these
microtubules in your cell.

00:07:36.866 --> 00:07:38.561
You have the centrosome right over here,

00:07:38.561 --> 00:07:40.244
all those microtubules.

00:07:40.244 --> 00:07:43.019
And this one is going to be pulling,

00:07:43.019 --> 00:07:46.060
is going to be getting
one of the blue chromatids

00:07:46.060 --> 00:07:48.301
to pull towards it or to move towards it.

00:07:48.301 --> 00:07:49.578
So let me draw that.

00:07:49.578 --> 00:07:52.248
So this is one blue
chromatid right over here.

00:07:52.248 --> 00:07:54.746
aAd this one is going to
be pulling the other blue

00:07:54.746 --> 00:07:56.826
chromatid towards it.

00:07:58.004 --> 00:07:59.966
And same thing for the magenta.

00:07:59.966 --> 00:08:03.285
And same thing for the magenta.

00:08:03.285 --> 00:08:05.724
So that one's being pulled to that way.

00:08:05.724 --> 00:08:09.543
And this one is being pulled that way.

00:08:09.543 --> 00:08:12.746
And just in case you're
concerned about the,

00:08:12.746 --> 00:08:15.448
some of more of the words
of vocabulary involved,

00:08:15.448 --> 00:08:18.548
the point it was this
microtubules connect to

00:08:18.548 --> 00:08:19.952
what used to be sister chromatids,

00:08:19.952 --> 00:08:23.114
but now that they're apart,
we now call them chromosomes.

00:08:23.114 --> 00:08:25.585
When they emerge, this is
one chromosome and they have

00:08:25.585 --> 00:08:26.618
two sister chromatids.

00:08:26.618 --> 00:08:28.882
But now they're apart, we would
actually now consider this

00:08:28.882 --> 00:08:30.588
each an independent chromosome.

00:08:30.588 --> 00:08:33.781
So now, you actually have
four chromosomes over here.

00:08:33.781 --> 00:08:37.318
This point right over here,
we call it kinetochore.

00:08:37.318 --> 00:08:39.524
And even exactly how that interphase works

00:08:39.524 --> 00:08:42.682
and exactly how things move
is not fully understood.

00:08:42.682 --> 00:08:44.539
Some of this stuff is
understood but some of this

00:08:44.539 --> 00:08:46.699
is still an area of research.

00:08:46.699 --> 00:08:49.415
So something even as
basic as cell division,

00:08:49.415 --> 00:08:51.316
not so basic after all.

00:08:51.316 --> 00:08:53.939
So this right over here where
you can start to see the DNA

00:08:53.939 --> 00:08:56.598
kind of migrating to their
respective sides of the cell.

00:08:56.598 --> 00:08:58.919
We call this the anaphase.

00:08:58.919 --> 00:09:01.472
We call this the anaphase.

00:09:02.503 --> 00:09:06.584
And that takes us to the
last formal phase of mitosis

00:09:06.584 --> 00:09:08.522
and that is called telophase.

00:09:09.677 --> 00:09:12.173
Telophase.

00:09:12.173 --> 00:09:16.735
And in telophase, I'm gonna
do my best shot to draw it.

00:09:16.735 --> 00:09:19.580
And you could already see
I have started to draw that

00:09:19.580 --> 00:09:23.155
the cellular membrane is
starting to pinch in kind of in

00:09:23.155 --> 00:09:26.345
preparation for cytokineses,
in preparation for the cell

00:09:26.345 --> 00:09:27.645
splitting into two cells.

00:09:27.645 --> 00:09:29.838
So I'll do it a little bit more.

00:09:29.838 --> 00:09:33.078
Cytokineses is usually described
as kinda being a separate

00:09:33.078 --> 00:09:36.758
process in mitosis although
it's obviously they are,

00:09:36.758 --> 00:09:39.347
they kind of together
help the cell fully turn

00:09:39.347 --> 00:09:40.926
into two cells.

00:09:40.926 --> 00:09:44.316
So now in telophase, so you have this,

00:09:44.316 --> 00:09:45.840
what used to be a sister chromatid,

00:09:45.840 --> 00:09:47.991
now we can call it a
chromosome on its own.

00:09:47.991 --> 00:09:50.582
And we have this chromosome.

00:09:51.586 --> 00:09:55.628
And we have this chromosome

00:09:57.177 --> 00:09:58.060
right over there.

00:09:58.060 --> 00:10:00.296
We have to do it on both sides.

00:10:00.296 --> 00:10:04.133
So you're there and you have

00:10:05.446 --> 00:10:07.292
this right over here.

00:10:07.292 --> 00:10:10.125
And actually, let me draw
it a little bit different.

00:10:10.125 --> 00:10:12.783
Because at this phase, you're
really starting to unwind

00:10:12.783 --> 00:10:14.884
what happened at prophase.

00:10:14.884 --> 00:10:17.113
So prophase, you have the disappearance

00:10:17.113 --> 00:10:19.331
of the nuclear membrane and
you have the condensation

00:10:19.331 --> 00:10:21.103
of the chromosomes into this form here.

00:10:21.103 --> 00:10:24.144
Telophase, that's unwinding a little bit.

00:10:24.144 --> 00:10:26.683
So actually, let me draw
the two centrosomes.

00:10:26.683 --> 00:10:28.738
So you have one centrosome
with that same color that

00:10:28.738 --> 00:10:29.991
I had before.

00:10:29.991 --> 00:10:31.964
So you have one centrosome right here.

00:10:31.964 --> 00:10:34.449
You have another centrosome
right over there.

00:10:34.449 --> 00:10:37.734
And now, the DNA, this blue DNA,

00:10:37.734 --> 00:10:39.320
this chromosome is now here
but it's starting to get

00:10:39.320 --> 00:10:40.977
a little under wind a little bit,

00:10:40.977 --> 00:10:43.693
same thing on this side right over here.

00:10:43.693 --> 00:10:45.818
Magenta chromosome is
on here but once again,

00:10:45.818 --> 00:10:47.676
it's starting to get a little bit unwound.

00:10:47.676 --> 00:10:49.198
Same thing over here.

00:10:49.198 --> 00:10:54.198
And you start to have
nuclear membranes forming

00:10:54.645 --> 00:10:56.154
around the DNA.

00:10:56.154 --> 00:10:58.767
So once again, it's kind of redoing what

00:10:58.767 --> 00:11:01.553
was undone in prophase.

00:11:01.553 --> 00:11:04.153
Undone in prophase.

00:11:04.153 --> 00:11:05.897
And so, when you're done, essentially,

00:11:05.897 --> 00:11:07.816
you're gonna have these nuclear membranes.

00:11:07.816 --> 00:11:11.752
The DNA is gonna go back
to its chromatin form.

00:11:11.752 --> 00:11:14.028
And then you have cytokineses.

00:11:14.028 --> 00:11:16.535
And cytokineses is the process
by which this gets fully

00:11:16.535 --> 00:11:20.064
pinched together and you
have two separate cells.

00:11:20.064 --> 00:11:23.535
And some folks will say, "Oh,
it kind of begins in anaphase

00:11:23.535 --> 00:11:25.984
"and finishes after telophase."

00:11:25.984 --> 00:11:28.574
But it's kind of happening
near the end of mitosis

00:11:28.574 --> 00:11:29.653
in parallel with it.

00:11:29.653 --> 00:11:31.752
So this is cytokineses.

00:11:32.721 --> 00:11:34.267
Cytokineses.

00:11:35.124 --> 00:11:37.424
Cytokineses right over here.

00:11:37.424 --> 00:11:41.045
This essentially is how this larger cell

00:11:41.045 --> 00:11:45.086
that had two nuclei, how
this divides, fully divides

00:11:45.086 --> 00:11:47.685
into two cells.

00:11:47.685 --> 00:11:52.166
And at that point,
you're back to this phase

00:11:52.166 --> 00:11:53.386
of the cell cycle.

00:11:53.386 --> 00:11:56.021
Now, each of these two cells

00:11:56.021 --> 00:11:58.883
are going to go through interphase, G1,

00:11:58.883 --> 00:12:02.582
S-phase replicate their DNA,
and G2 phase goes some more

00:12:02.582 --> 00:12:05.055
and then it go through mitosis again.

00:12:05.055 --> 00:12:07.597
And then these two will
turn into four cells.

00:12:07.597 --> 00:12:09.786
Anyway, hopefully, you enjoyed that.


Interphase

WEBVTT
Kind: captions
Language: en

00:00:00.278 --> 00:00:01.045
- [Voiceover] Let's talk a little bit

00:00:01.045 --> 00:00:02.391
about the life cycle of a cell.

00:00:02.391 --> 00:00:05.271
In particular, we're gonna
talk about interphase.

00:00:05.271 --> 00:00:08.568
The interphase part of
the life cycle of a cell.

00:00:08.568 --> 00:00:10.773
And as we'll see,
interphase is where a cell

00:00:10.773 --> 00:00:12.817
spends most of its life.

00:00:12.817 --> 00:00:15.626
Let's draw a timeline for a cell.

00:00:15.626 --> 00:00:18.993
So let's say this is a new cell

00:00:18.993 --> 00:00:22.963
and it will go through interphase.

00:00:23.453 --> 00:00:24.604
So I'm gonna make it like a cycle

00:00:24.604 --> 00:00:26.531
so it's gonna go back on itself.

00:00:26.531 --> 00:00:28.853
At some point, so all
of that is interphase.

00:00:28.853 --> 00:00:30.989
At some point it will be ready to divide

00:00:30.989 --> 00:00:33.404
and it will undergo mitosis.

00:00:33.404 --> 00:00:36.492
Mitosis is, more formally,
it's the process by which

00:00:36.492 --> 00:00:39.533
the nucleus turns into two nuclei,

00:00:39.533 --> 00:00:41.623
but then that's obviously
needed for cell division.

00:00:41.623 --> 00:00:43.968
So this is mitosis right here in green.

00:00:43.968 --> 00:00:46.035
So as you see, and this isn't precise,

00:00:46.035 --> 00:00:48.658
a cell spends most of
its life in interphase

00:00:48.658 --> 00:00:50.934
and that's where it's just
kind of living as a cell.

00:00:50.934 --> 00:00:53.116
It's living, growing, producing proteins,

00:00:53.116 --> 00:00:55.740
whatever other functions it has

00:00:55.740 --> 00:00:58.201
and mitosis, it's a
shorter part of its life,

00:00:58.201 --> 00:00:59.920
a small fraction, a very interesting part.

00:00:59.920 --> 00:01:03.774
It's necessary in order
for the cell to replicate,

00:01:03.774 --> 00:01:06.026
but you see it's a much smaller fraction.

00:01:06.026 --> 00:01:09.439
But what I wanna focus on
in this video is interphase.

00:01:09.439 --> 00:01:12.685
To do that, let's draw ourselves a cell.

00:01:13.224 --> 00:01:14.873
So let me copy and paste.

00:01:15.483 --> 00:01:17.389
So this right over here,

00:01:17.697 --> 00:01:20.381
actually let me, I did
that just to save time.

00:01:20.381 --> 00:01:23.191
So let's say this is a cell, so green.

00:01:23.191 --> 00:01:26.325
I have it's nuclear membrane,
or not nuclear membrane,

00:01:26.325 --> 00:01:28.067
I have its cell membrane.

00:01:28.067 --> 00:01:30.923
Inside of that, of course,
you have all of the,

00:01:30.923 --> 00:01:35.535
all of the cytosol, and then
this, in this orangeish color,

00:01:35.535 --> 00:01:38.368
I have the nuclear membrane
that defines the nucleus.

00:01:38.368 --> 00:01:41.154
And then inside of that I have the DNA.

00:01:41.154 --> 00:01:45.310
And you might be used to
seeing DNA all tightly bound,

00:01:45.310 --> 00:01:48.398
or chromosomes all tightly bound like that

00:01:48.798 --> 00:01:53.798
and like that or like this,
this would be another chromosome

00:01:53.878 --> 00:01:55.946
right over here in magenta.

00:01:55.951 --> 00:01:58.482
But during interphase, the chromosomes

00:01:58.482 --> 00:02:01.245
aren't tightly bound like that
so that they're easy to see

00:02:01.245 --> 00:02:05.541
from a traditional or a
simple light microscope.

00:02:05.541 --> 00:02:07.422
For most of a cell's life, the chromosomes

00:02:07.422 --> 00:02:09.256
are completely unwound.

00:02:09.256 --> 00:02:11.346
They are in their chromatin form.

00:02:11.346 --> 00:02:13.575
So they are in their chromatin form.

00:02:13.575 --> 00:02:15.014
It's actually hard to see if you have

00:02:15.014 --> 00:02:19.704
just a simple microphone
(laughing) a simple microscope.

00:02:19.704 --> 00:02:21.933
It's all unwound, you
just have the proteins

00:02:21.933 --> 00:02:24.743
and the DNA, it's all tangled together.

00:02:24.743 --> 00:02:26.693
Now there's one other
thing that I drew here.

00:02:28.753 --> 00:02:29.889
You might say, why am I drawing it

00:02:29.889 --> 00:02:32.536
when I haven't drawn most
of the other organelles?

00:02:32.536 --> 00:02:35.415
But I'm drawing this thing,
which is called a centrosome,

00:02:35.415 --> 00:02:39.548
'cause it's going to be important for,

00:02:39.548 --> 00:02:43.750
it's going to be important
for when we go into mitosis.

00:02:43.750 --> 00:02:46.170
Now, this drawing as
well, you might say, wait,

00:02:46.170 --> 00:02:48.933
doesn't a cell, at least a human cell

00:02:50.993 --> 00:02:52.826
that has a diploid number of chromosomes,

00:02:52.826 --> 00:02:54.335
and once again, if we're
not talking about sex cells,

00:02:54.335 --> 00:02:56.262
we're talking about
just our somatic cells,

00:02:56.262 --> 00:02:58.212
doesn't it have to have 46 chromosomes?

00:02:58.212 --> 00:03:00.000
It looks like you only drew two.

00:03:00.000 --> 00:03:01.811
And it is true, I only
drew two chromosomes

00:03:01.811 --> 00:03:04.180
for the sake of simplicity,
we're just going to assume

00:03:04.180 --> 00:03:06.618
that this is the cell of some
organism that's much simpler,

00:03:06.618 --> 00:03:09.538
that it only has two chromosomes.

00:03:09.891 --> 00:03:12.562
So anyway, this is the
new cell right over here.

00:03:12.562 --> 00:03:14.832
It is going to grow.

00:03:15.023 --> 00:03:18.088
So it is going to grow, it's
going to take in nutrients

00:03:18.088 --> 00:03:20.285
from its environment,
and it's going to grow

00:03:20.285 --> 00:03:22.276
as we would expect it to.

00:03:22.328 --> 00:03:24.186
So that's that right over there.

00:03:24.186 --> 00:03:27.297
And then let me give it its
nucleus and its centrosome

00:03:27.297 --> 00:03:28.917
just like that.

00:03:29.038 --> 00:03:33.705
And this phase, this phase,
where it is just growing

00:03:33.705 --> 00:03:37.281
from this new cell, this is,

00:03:37.281 --> 00:03:39.371
this phase right over here,

00:03:39.371 --> 00:03:42.993
is the G1 phase, the G1,
actually I'm gonna do that

00:03:42.993 --> 00:03:44.363
in a different color
since I'm already using

00:03:44.363 --> 00:03:45.983
that green so much.

00:03:46.197 --> 00:03:49.610
This is the G1 phase and so
that might look something

00:03:50.610 --> 00:03:53.200
like this, different
cells are going to do this

00:03:53.200 --> 00:03:56.219
for different periods
of time, the G1 phase.

00:03:56.219 --> 00:03:58.262
But then you can imagine,
well look, it's going to need

00:03:58.262 --> 00:04:00.537
to replicate some of the,
or, it's gonna replicate

00:04:00.537 --> 00:04:05.088
the information inside of,
or that's coded by the DNA

00:04:05.088 --> 00:04:09.286
at some point, and actually,
this happens before mitosis.

00:04:09.314 --> 00:04:11.213
So let's depict that.

00:04:11.590 --> 00:04:13.488
So let me draw,

00:04:14.864 --> 00:04:18.086
let me draw the nucleus
and the centrosome again.

00:04:19.461 --> 00:04:20.344
Let me draw that again.

00:04:20.344 --> 00:04:24.054
Let me draw the cellular membrane.

00:04:24.245 --> 00:04:27.769
This nice healthy growing cell.

00:04:28.262 --> 00:04:31.187
And now, its DNA is
actually going to replicate.

00:04:31.187 --> 00:04:35.622
So instead of having one copy of its DNA,

00:04:35.622 --> 00:04:38.176
it's essentially going
to go to two copies.

00:04:38.176 --> 00:04:39.825
But I wanna be very very careful now.

00:04:39.825 --> 00:04:42.588
So if I draw that magenta
chromosome up here,

00:04:42.588 --> 00:04:45.206
so once again it's all unwound like that.

00:04:45.281 --> 00:04:48.044
When it replicates, it's
going to create a copy

00:04:48.044 --> 00:04:50.830
of its DNA, and once again,
I'm not doing justice

00:04:50.830 --> 00:04:55.097
for how much DNA, how much
DNA there actually is.

00:04:55.196 --> 00:04:57.750
But it was one chromosome
before, it was one chromosome

00:04:57.750 --> 00:05:00.652
when it was just like this,
and it's still one chromosome,

00:05:00.652 --> 00:05:03.113
even though it's copied
its genetic material.

00:05:03.113 --> 00:05:04.707
Let me draw this a little bit neater.

00:05:04.707 --> 00:05:07.238
So this is one chromosome right over here.

00:05:07.238 --> 00:05:09.653
And that one chromosome, after it's copied

00:05:09.653 --> 00:05:13.896
all of its genetic material,
it is still one chromosome.

00:05:14.087 --> 00:05:16.804
Now how do we, but there's
two copies over here,

00:05:16.804 --> 00:05:18.963
what do we call these two copies?

00:05:18.963 --> 00:05:22.307
Well, each of these two
copies are called a chromatid

00:05:22.307 --> 00:05:25.808
and these two right over here,
these are sister chromatids.

00:05:26.766 --> 00:05:31.766
Sister, sister chromatids.

00:05:32.087 --> 00:05:35.476
But either way, this is one
chromosome right over here.

00:05:35.476 --> 00:05:37.916
Chromosome, chromosome.

00:05:38.665 --> 00:05:40.592
And this is also, so
this is one chromosome

00:05:40.592 --> 00:05:44.214
right over there, and that
is also one chromosome.

00:05:44.214 --> 00:05:45.259
This whole thing right over here

00:05:45.259 --> 00:05:46.931
is also one chromosome.

00:05:46.931 --> 00:05:48.997
Later on, when we go
through mitosis, we'll see

00:05:48.997 --> 00:05:51.783
that these two sister
chromatids get split apart,

00:05:51.783 --> 00:05:53.269
they're no longer connected.

00:05:53.269 --> 00:05:55.615
At that point, we refer to each of them

00:05:55.615 --> 00:05:58.308
as an individual chromosome.

00:05:58.308 --> 00:06:00.189
Now you might be
wondering, is there a word

00:06:00.189 --> 00:06:01.085
for this place where

00:06:01.085 --> 00:06:03.570
these two sister chromatids are connected?

00:06:03.570 --> 00:06:05.845
And the answer is, yes, there is a word,

00:06:05.845 --> 00:06:07.958
and that word is centromere,

00:06:07.958 --> 00:06:10.791
not to be confused with centrosome.

00:06:10.791 --> 00:06:12.671
Let me give myself some space here.

00:06:12.671 --> 00:06:15.731
So that right over
here, that is a centro-,

00:06:16.549 --> 00:06:19.753
centromere, right over that.

00:06:19.753 --> 00:06:24.258
So we had one one magenta, or
we had this magenta chromosome

00:06:24.258 --> 00:06:27.503
right over here, and now it replicates.

00:06:27.555 --> 00:06:29.018
It's still one chromosome,

00:06:29.018 --> 00:06:31.223
although it has twice the
genetic material right now.

00:06:31.223 --> 00:06:34.056
You have these two sister
chromatids connected

00:06:34.056 --> 00:06:36.633
at the, say the centromere's
right over there.

00:06:36.633 --> 00:06:39.879
And that's also going to
happen for the blue chromosome.

00:06:40.000 --> 00:06:42.662
All this genetic material
is going to replicate,

00:06:42.662 --> 00:06:44.264
you're gonna have two copies of it.

00:06:44.264 --> 00:06:47.097
And so now it's gonna be made
up of two sister chromatids

00:06:47.097 --> 00:06:51.857
that are maybe connected
once again at a centromere.

00:06:51.857 --> 00:06:54.016
And also while all of this
replication is happening

00:06:54.016 --> 00:06:58.144
inside the nucleus, the
centrosome also duplicates.

00:06:58.405 --> 00:07:02.625
The centrosome also duplicates.

00:07:02.677 --> 00:07:04.534
And this process, the
part of the life cycle

00:07:04.534 --> 00:07:07.635
where all of this genetic
information is duplicating,

00:07:07.635 --> 00:07:12.018
we call that the S-phase,
S-phase for synthesis.

00:07:12.140 --> 00:07:14.955
So this is the synthesis phase.

00:07:14.955 --> 00:07:17.555
So that is the S-phase.

00:07:17.555 --> 00:07:19.204
And then before going into mitosis,

00:07:19.204 --> 00:07:21.242
there is one more growth phase.

00:07:21.433 --> 00:07:23.688
There is one more growth phase,

00:07:23.688 --> 00:07:26.005
and we call that G2.

00:07:26.312 --> 00:07:28.216
So we have one more growth phase,

00:07:28.216 --> 00:07:31.206
which we call G2.

00:07:31.397 --> 00:07:35.014
And then we are ready, so let
me just copy and paste this.

00:07:35.065 --> 00:07:38.729
Let me just do this, so
that's what we had before.

00:07:39.175 --> 00:07:42.583
Now we need to remember
that our DNA has replicated.

00:07:43.099 --> 00:07:47.296
Our DNA has replicated,
so let me draw that.

00:07:47.781 --> 00:07:50.608
Let me draw the two centromeres,

00:07:51.078 --> 00:07:53.354
one for each of the chromosomes.

00:07:53.354 --> 00:07:56.120
Let me draw the replicated,

00:07:56.120 --> 00:07:59.649
the duplicated centrosome,

00:07:59.649 --> 00:08:01.530
not to be confused with centromere.

00:08:01.530 --> 00:08:04.125
Now the cell has grown even more.

00:08:04.246 --> 00:08:08.333
The cell has grown, the
cell has grown even more.

00:08:08.333 --> 00:08:11.258
And once again, going from this to this,

00:08:11.258 --> 00:08:13.505
we call that the G2 phase.

00:08:13.627 --> 00:08:16.251
At this point, at the end of the G2 phase,

00:08:16.251 --> 00:08:19.496
this is now when we are ready,

00:08:19.664 --> 00:08:22.334
this is now, what if we do
this in a different color?

00:08:22.334 --> 00:08:26.183
This is now when we are ready for mitosis.


Comparing irrational numbers involving radicals

WEBVTT
Kind: captions
Language: en

00:00:00.392 --> 00:00:02.017
- I have six numbers here

00:00:02.017 --> 00:00:04.478
and you see that five
of them are irrational.

00:00:04.478 --> 00:00:09.239
They involve the square root
of a non-perfect square.

00:00:09.239 --> 00:00:13.696
Our goal in this video
is, without a calculator,

00:00:13.696 --> 00:00:17.645
see if we can sort these
numbers from least to greatest.

00:00:17.645 --> 00:00:18.503
And like always,

00:00:18.503 --> 00:00:21.615
pause this video and
see if you can do that.

00:00:21.615 --> 00:00:23.542
So I'll give you a hint.

00:00:23.542 --> 00:00:24.656
The hint is--

00:00:24.656 --> 00:00:26.211
it's very hard without a calculator.

00:00:26.211 --> 00:00:27.257
Square root of two

00:00:27.257 --> 00:00:28.743
is gonna be one point
something, something.

00:00:28.743 --> 00:00:29.417
Square root of three

00:00:29.417 --> 00:00:30.971
is gonna be one point
something, something.

00:00:30.971 --> 00:00:32.295
How do we do this?

00:00:32.295 --> 00:00:34.804
We just have to realize
that, if I have some number,

00:00:34.804 --> 00:00:38.797
let's say I have some number
a that is greater than 0.

00:00:38.797 --> 00:00:43.046
And if we know that a is less than b,

00:00:43.046 --> 00:00:46.645
then a squared is going
to be less than b squared.

00:00:46.645 --> 00:00:51.127
If one positive number is less
than another positive number,

00:00:51.127 --> 00:00:52.658
then the square of this positive number

00:00:52.658 --> 00:00:54.980
is going to be less than
the square of that number.

00:00:54.980 --> 00:00:58.534
So one thing that we could
do when we are comparing

00:00:58.534 --> 00:01:00.113
all of these irrational
numbers that involve

00:01:00.113 --> 00:01:02.387
square roots of non-perfect squares,

00:01:02.387 --> 00:01:04.083
let's compare their squares.

00:01:04.083 --> 00:01:05.151
Because their squares

00:01:05.151 --> 00:01:07.125
are not going to be irrational numbers.

00:01:07.125 --> 00:01:08.610
It's going to be much easier to compare,

00:01:08.610 --> 00:01:10.166
and then we can order them.

00:01:10.166 --> 00:01:11.536
Because if we order the squares,

00:01:11.536 --> 00:01:13.255
then they'll tell us what happens

00:01:13.255 --> 00:01:15.507
if we order their square roots.

00:01:15.507 --> 00:01:16.575
What am I talking about?

00:01:16.575 --> 00:01:18.525
Well, I'm just gonna square each of these.

00:01:18.525 --> 00:01:21.868
So if I take this to the second power,

00:01:21.868 --> 00:01:25.397
this is going to be four
square roots of two,

00:01:25.397 --> 00:01:29.089
times four square roots of two.

00:01:29.089 --> 00:01:31.016
You can change the
order of multiplication.

00:01:31.016 --> 00:01:32.643
That's four times four

00:01:32.643 --> 00:01:37.504
times the square root of two
times the square root of two.

00:01:37.504 --> 00:01:40.259
Now, four times four is 16.

00:01:40.259 --> 00:01:42.418
Square root of two times
square root of two,

00:01:42.418 --> 00:01:44.531
well, that's just going to be two.

00:01:44.531 --> 00:01:48.733
So it's gonna be 16 times
two which is equal to 32.

00:01:48.733 --> 00:01:50.776
Now what about two square roots of three?

00:01:50.776 --> 00:01:52.904
Well, same idea.

00:01:52.904 --> 00:01:56.384
Let's square it, let's square it.

00:01:56.384 --> 00:01:58.511
And i'll do this one a little bit faster.

00:01:58.511 --> 00:02:01.319
So if we square two square roots of three,

00:02:01.319 --> 00:02:03.178
this is going to be two squared

00:02:03.178 --> 00:02:05.175
times square root of three squared.

00:02:05.175 --> 00:02:06.986
So it's going to be two squared

00:02:06.986 --> 00:02:11.537
times the square root of three squared.

00:02:11.537 --> 00:02:14.207
Well, two squared is going to be four.

00:02:14.207 --> 00:02:18.061
Square root of three squared
is going to be three.

00:02:18.061 --> 00:02:23.061
So this is going to be equal to 12.

00:02:24.679 --> 00:02:27.767
That's this thing squared.

00:02:27.767 --> 00:02:30.832
If this step seems a little bit confusing,

00:02:30.832 --> 00:02:33.734
if you have the product of
two things raised to a power,

00:02:33.734 --> 00:02:34.616
that's the same thing

00:02:34.616 --> 00:02:36.427
as raising each of them to that power,

00:02:36.427 --> 00:02:38.284
and then taking the product.

00:02:38.284 --> 00:02:40.165
And you can actually see,
I worked it out here,

00:02:40.165 --> 00:02:42.511
why that actually makes sense.

00:02:42.511 --> 00:02:45.134
Notice when I just changed
the order of multiplication

00:02:45.134 --> 00:02:47.503
you had four times four, or four squared,

00:02:47.503 --> 00:02:51.845
times square root of two squared,
which is going to be two.

00:02:51.845 --> 00:02:53.935
So let's keep doing that.

00:02:53.935 --> 00:02:56.721
So what is this value squared?

00:02:56.721 --> 00:02:59.531
It's gonna be three
squared, which is nine,

00:02:59.531 --> 00:03:03.060
times square root of two
squared, which is two.

00:03:03.060 --> 00:03:06.589
Nine times two is 18.

00:03:06.589 --> 00:03:09.329
What's the square root of 17 squared?

00:03:09.329 --> 00:03:12.046
That's just going to be seventeen.

00:03:12.046 --> 00:03:13.345
Do that in blue.

00:03:13.345 --> 00:03:17.340
This is just going to be 17.

00:03:17.340 --> 00:03:19.685
What is three square
roots of three squared?

00:03:19.685 --> 00:03:22.563
It's gonna be three
squared, which is nine,

00:03:22.563 --> 00:03:25.598
times square root of three squared.

00:03:25.598 --> 00:03:26.947
The square root of three
times the square root of three

00:03:26.947 --> 00:03:27.902
is three.

00:03:27.902 --> 00:03:31.153
So it's gonna be nine times three, or 27.

00:03:31.153 --> 00:03:33.614
And what is five squared?

00:03:33.614 --> 00:03:35.309
This is pretty straightforward.

00:03:35.309 --> 00:03:37.909
That's going to be 25.

00:03:37.909 --> 00:03:40.116
So let's order them
from least to greatest.

00:03:40.116 --> 00:03:41.695
Which of them, when I square it,

00:03:41.695 --> 00:03:43.622
gives me the smallest value?

00:03:43.622 --> 00:03:48.567
Compare 32 to 12 to 18 to 17 to 27 to 25.

00:03:48.567 --> 00:03:50.897
12 is the smallest value.

00:03:50.907 --> 00:03:52.205
So if their square is the smallest,

00:03:52.205 --> 00:03:53.085
and these are all positive numbers,

00:03:53.103 --> 00:03:54.822
then this is going to
be the smallest value

00:03:54.822 --> 00:03:55.494
out of all of them.

00:03:55.494 --> 00:03:56.958
Let me write that first.

00:03:56.958 --> 00:03:59.791
Two square roots of three.

00:03:59.791 --> 00:04:01.439
So I've covered that one.

00:04:01.439 --> 00:04:02.785
Now what's next?

00:04:02.785 --> 00:04:04.481
Well, now I have this value.

00:04:04.481 --> 00:04:09.481
17 is the next smallest
square, so its square root

00:04:10.842 --> 00:04:14.464
is going to be the next square root.

00:04:14.464 --> 00:04:17.135
So it's going to be two
square roots of three,

00:04:17.135 --> 00:04:20.525
then square root of 17.

00:04:20.525 --> 00:04:22.846
That is this one here.

00:04:22.846 --> 00:04:24.496
Then we go to 18.

00:04:24.496 --> 00:04:26.370
So if we look at its square root,

00:04:26.370 --> 00:04:28.370
with the numbers we were
originally trying to sort,

00:04:28.370 --> 00:04:33.370
that would be three square roots of two,

00:04:33.825 --> 00:04:36.199
three square roots of two,

00:04:36.199 --> 00:04:37.354
We got that one covered.

00:04:37.354 --> 00:04:39.376
Then the next one is gonna be 25,

00:04:39.376 --> 00:04:40.629
when we look at the squares.

00:04:40.629 --> 00:04:43.276
So the next value out of our original set,

00:04:43.276 --> 00:04:46.315
the next largest one, is going to be five.

00:04:46.315 --> 00:04:51.064
So then we get to five,
we've covered that one.

00:04:51.064 --> 00:04:52.249
Then the next one, let's see.

00:04:52.249 --> 00:04:54.155
We have 27 and 32 left.

00:04:54.155 --> 00:04:58.654
27 is the next largest square,
so the next largest number

00:04:58.654 --> 00:04:59.493
out of the ones we care about,

00:04:59.493 --> 00:05:01.861
is three square roots of three.

00:05:01.861 --> 00:05:06.667
So three square roots of
three, we covered that one.

00:05:06.667 --> 00:05:08.038
And then we finish with--

00:05:08.038 --> 00:05:11.660
This is the largest value,
four square roots of two.

00:05:11.660 --> 00:05:14.446
Four square roots of two.

00:05:14.446 --> 00:05:15.258
And we're done!

00:05:15.258 --> 00:05:15.909
That was pretty neat.

00:05:15.909 --> 00:05:17.075
Without a calculator, we were able

00:05:17.075 --> 00:05:21.821
to sort these irrational numbers,

00:05:21.821 --> 00:05:23.502
well, not all of them are irrational,

00:05:23.502 --> 00:05:24.894
but ones that involve the square root

00:05:24.894 --> 00:05:28.075
of something that is not a perfect square.


How to classify numbers

WEBVTT
Kind: captions
Language: en

00:00:00.505 --> 00:00:02.861
- [Voiceover] So we have a
bunch of numbers listed up here,

00:00:02.861 --> 00:00:05.729
and my goal, in this video,
is to see if we can classify

00:00:05.729 --> 00:00:08.562
them into different types
of number categories,

00:00:08.562 --> 00:00:10.468
and let me draw the categories.

00:00:10.468 --> 00:00:13.694
So this circle, over here,

00:00:13.694 --> 00:00:16.828
this represents all of the numbers

00:00:16.828 --> 00:00:18.895
that can be represented as the fraction

00:00:18.895 --> 00:00:21.066
of two integers, and, of
course, the denominator

00:00:21.066 --> 00:00:23.295
can't be equal to zero,
because we don't know

00:00:23.295 --> 00:00:25.977
what it means to put a
zero in the denominator.

00:00:25.977 --> 00:00:28.659
So, let's call these, or the standard way

00:00:28.659 --> 00:00:29.611
of calling these things.

00:00:29.611 --> 00:00:31.759
These things can be
represented as a fraction

00:00:31.759 --> 00:00:36.588
of two integers, we call
these rational numbers.

00:00:36.588 --> 00:00:37.938
Rational numbers.

00:00:37.938 --> 00:00:42.193
And if something cannot be
represented as a fraction

00:00:42.193 --> 00:00:45.485
of two integers, we
call irrational numbers.

00:00:45.485 --> 00:00:48.283
Irrational numbers.

00:00:48.283 --> 00:00:49.930
Irrational numbers.

00:00:49.930 --> 00:00:52.230
And the size of these circles don't show

00:00:52.230 --> 00:00:53.855
how large these sets are.

00:00:53.855 --> 00:00:56.328
There's actually an
infinite number of rational

00:00:56.328 --> 00:00:58.639
and an infinite number
of irrational numbers.

00:00:58.639 --> 00:01:01.100
So, these are the irrational numbers.

00:01:01.100 --> 00:01:02.923
Irrational.

00:01:02.923 --> 00:01:06.219
So, these cannot be
represented as a fraction

00:01:06.219 --> 00:01:07.753
of two integers.

00:01:07.753 --> 00:01:09.157
And then, within rational numbers,

00:01:09.157 --> 00:01:11.677
you have integers themselves.

00:01:11.677 --> 00:01:15.496
So, I'll do that in, let me
do that in this blue color.

00:01:15.496 --> 00:01:16.954
Integers.

00:01:16.954 --> 00:01:19.288
So, integers are numbers that don't have

00:01:19.288 --> 00:01:23.607
to be represented as a
fraction or a decimal.

00:01:23.607 --> 00:01:28.328
So, these are integers, right over here.

00:01:28.328 --> 00:01:29.876
Integers.

00:01:29.876 --> 00:01:32.802
And then a subset of
integers are whole numbers.

00:01:32.802 --> 00:01:35.281
So, if you essentially say
the non-negative integers,

00:01:35.281 --> 00:01:37.678
you're then talking about whole numbers.

00:01:37.678 --> 00:01:41.753
So let me do that subset, right over here.

00:01:41.753 --> 00:01:45.166
So, these are going to
be the whole numbers.

00:01:45.166 --> 00:01:49.253
So, whole numbers.

00:01:49.253 --> 00:01:52.101
Whole numbers, right over here.

00:01:52.101 --> 00:01:53.669
And, actually, let me just label it all.

00:01:53.669 --> 00:01:54.954
These are rational...

00:01:54.954 --> 00:01:56.637
Let me do that in the same color.

00:01:56.637 --> 00:02:00.344
Rational numbers.

00:02:00.344 --> 00:02:03.104
And, of course, irrational numbers.

00:02:03.104 --> 00:02:05.965
Irrational numbers.

00:02:07.295 --> 00:02:08.746
Irrational numbers.

00:02:08.746 --> 00:02:09.466
An integer.

00:02:09.466 --> 00:02:11.393
Well, if I could say,
"Look, that is an integer.

00:02:11.393 --> 00:02:12.589
"Let's think about the integers."

00:02:12.589 --> 00:02:14.184
But I wouldn't say, "Let's
just think about the rational."

00:02:14.184 --> 00:02:16.026
I'd say, "Let's think about
the rational numbers."

00:02:16.026 --> 00:02:17.523
All right, now that we have
these categories in place,

00:02:17.523 --> 00:02:18.394
let's categorize them.

00:02:18.394 --> 00:02:19.836
Like always, pause the video.

00:02:19.836 --> 00:02:21.893
See if you can figure out what category

00:02:21.893 --> 00:02:23.793
these numbers fall into.

00:02:23.793 --> 00:02:27.427
Where would you put them on this diagram?

00:02:27.427 --> 00:02:29.191
So, let's start off with three.

00:02:29.191 --> 00:02:30.852
This is positive three.

00:02:30.852 --> 00:02:33.661
It can be definitely
represented as a fraction.

00:02:33.661 --> 00:02:36.065
You can represent it as three over one.

00:02:36.065 --> 00:02:38.468
But, it doesn't have to be
represented as a fraction.

00:02:38.468 --> 00:02:41.440
It, literally, could be just
a three, right over there,

00:02:41.440 --> 00:02:42.891
but it's also non-negative.

00:02:42.891 --> 00:02:45.910
So three is a whole number.

00:02:45.910 --> 00:02:49.857
So three, and maybe I'll do it
in the color of the category.

00:02:49.857 --> 00:02:54.559
So, three is a whole number.

00:02:54.559 --> 00:02:56.045
So, it's a member of that set.

00:02:56.045 --> 00:02:57.659
But if you're a whole number,
you're also an integer,

00:02:57.659 --> 00:02:59.401
and you're also a rational number.

00:02:59.401 --> 00:03:01.142
So, three is a whole
number, it's an integer,

00:03:01.142 --> 00:03:03.452
and it's a rational number.

00:03:03.452 --> 00:03:06.087
Now, let's think about negative five.

00:03:06.087 --> 00:03:08.073
Now, negative five, once
again, it can be represented

00:03:08.073 --> 00:03:09.928
as a fraction, but it doesn't have to be,

00:03:09.928 --> 00:03:11.045
but it is negative.

00:03:11.045 --> 00:03:12.621
So, it's not gonna be a whole number.

00:03:12.621 --> 00:03:14.900
So, negative five is going
to sit right over here.

00:03:14.900 --> 00:03:16.967
It's an integer, and if you're an integer,

00:03:16.967 --> 00:03:19.486
you're definitely going
to be a rational number,

00:03:19.486 --> 00:03:22.656
but it's not a whole number
because it is negative.

00:03:24.476 --> 00:03:28.765
Now we have 0.25.

00:03:28.765 --> 00:03:32.790
Well, this, for sure, can be
represented as a fraction.

00:03:32.790 --> 00:03:35.568
This is 25-hundredths, right over here.

00:03:35.568 --> 00:03:37.461
So, we can represent that as a fraction

00:03:37.461 --> 00:03:39.748
of two integers, I should say.

00:03:39.748 --> 00:03:41.431
It's 25-hundredths.

00:03:41.431 --> 00:03:42.592
But there's no way to represent this

00:03:42.592 --> 00:03:46.131
except using a fraction of two integers.

00:03:46.131 --> 00:03:50.104
So, 0.25 is a rational number,

00:03:50.104 --> 00:03:53.506
but it's not an integer
and not a whole number.

00:03:53.506 --> 00:03:56.559
Now what about 22 over seven.

00:03:56.559 --> 00:03:58.289
Well, here it's clearly
represented, already,

00:03:58.289 --> 00:04:01.108
as a fraction of two integers,

00:04:01.108 --> 00:04:04.316
but I don't think I can
represent this any other,

00:04:04.316 --> 00:04:06.010
except as a fraction of two integers.

00:04:06.010 --> 00:04:08.840
I can't somehow make this

00:04:08.840 --> 00:04:11.675
without using a fraction
or some type of decimal

00:04:11.675 --> 00:04:14.565
that might repeat.

00:04:14.565 --> 00:04:16.679
So, this, right over here, this would

00:04:16.679 --> 00:04:19.570
also be a rational number,

00:04:19.570 --> 00:04:23.297
but it's not an integer,
not a whole number.

00:04:23.297 --> 00:04:25.688
Now this over here.

00:04:28.098 --> 00:04:29.898
0.2713.

00:04:29.898 --> 00:04:30.820
Now the 13 repeats.

00:04:30.820 --> 00:04:35.349
This is the same thing as 0.27131313,

00:04:35.349 --> 00:04:38.007
that's what line up there represents.

00:04:38.007 --> 00:04:40.038
Now, you might not realize it yet,

00:04:40.038 --> 00:04:42.802
but any number that repeats eventually,

00:04:42.802 --> 00:04:43.997
this one does repeat eventually,

00:04:43.997 --> 00:04:45.895
you have the .1313,

00:04:45.895 --> 00:04:49.018
or you have the 0.27131313,

00:04:49.018 --> 00:04:53.364
any number like this can be
represented as a fraction.

00:04:53.364 --> 00:04:54.858
For example, and I'm
not going to do it here,

00:04:54.858 --> 00:04:55.972
just for the sake of time,

00:04:55.972 --> 00:04:58.689
but, for example, 0.3, repeating,

00:04:58.689 --> 00:04:59.978
that's the same thing as one-third.

00:04:59.978 --> 00:05:01.382
And later on, we're gonna see techniques

00:05:01.382 --> 00:05:04.902
of how do you convert this to
a fraction of two integers.

00:05:04.902 --> 00:05:06.698
But, for our sake, we just know

00:05:06.698 --> 00:05:09.115
that this can be represented
as a fraction of two integers

00:05:09.115 --> 00:05:11.808
just the way that 0.3, repeating, can be.

00:05:11.808 --> 00:05:14.629
And so, we would put this
under rational numbers.

00:05:14.629 --> 00:05:18.763
0.2713, repeating.

00:05:18.763 --> 00:05:22.291
But you have to represent
it either as a decimal

00:05:22.291 --> 00:05:24.323
or a fraction of integers.

00:05:24.323 --> 00:05:26.706
If you didn't have to, then
it could have been an integer,

00:05:26.706 --> 00:05:30.187
but we'll throw it up
there in rational numbers.

00:05:30.187 --> 00:05:32.288
Now the square root of ten.

00:05:32.288 --> 00:05:33.358
Square root of ten.

00:05:33.358 --> 00:05:34.743
This is interesting.

00:05:34.743 --> 00:05:38.215
So, any square root of
a non-perfect square

00:05:38.215 --> 00:05:40.432
is going to be irrational.

00:05:40.432 --> 00:05:42.044
So, this is gonna be irrational.

00:05:42.044 --> 00:05:43.985
I'm not proving it to you here,

00:05:43.985 --> 00:05:47.154
but you cannot represent this as the ratio

00:05:47.154 --> 00:05:50.835
of two integers, or a
fraction with two integers,

00:05:50.835 --> 00:05:51.984
with an integer in the numerator

00:05:51.984 --> 00:05:53.377
and an integer in the denominator.

00:05:53.377 --> 00:05:55.618
This will be, if you were to represent it

00:05:55.618 --> 00:05:58.648
as a decimal, it will not repeat.

00:05:58.648 --> 00:06:01.644
It'll just keep being new and new digits.

00:06:01.644 --> 00:06:03.873
It will not repeat over time.

00:06:03.873 --> 00:06:06.311
So, this, right over here,
is an irrational number.

00:06:06.311 --> 00:06:07.366
It's not rational.

00:06:07.366 --> 00:06:11.748
It cannot be represented as
the ratio of two integers.

00:06:11.748 --> 00:06:13.834
All right, 14 over seven.

00:06:13.834 --> 00:06:15.123
This is the ratio of two integers.

00:06:15.123 --> 00:06:17.584
So, this, for sure, is rational.

00:06:17.584 --> 00:06:19.277
But if you think about it, 14 over seven,

00:06:19.277 --> 00:06:20.963
that's another way of saying,

00:06:20.963 --> 00:06:24.480
14 over seven is the same thing as two.

00:06:24.480 --> 00:06:26.547
These two things are equivalent.

00:06:26.547 --> 00:06:29.519
So, 14 over seven is
the same thing as two.

00:06:29.519 --> 00:06:31.238
So, this is actually a whole number.

00:06:31.238 --> 00:06:32.328
It doesn't look like a whole number,

00:06:32.328 --> 00:06:33.466
but, remember, a whole number

00:06:33.466 --> 00:06:36.369
is a non-negative number that doesn't need

00:06:36.369 --> 00:06:39.109
to be represented as the
ratio of two integers.

00:06:39.109 --> 00:06:40.676
And this one, even though
we did represent it

00:06:40.676 --> 00:06:42.313
as the ratio of two
integers, it doesn't need

00:06:42.313 --> 00:06:44.090
to be represented as the
ratio of two integers.

00:06:44.090 --> 00:06:46.190
You could have represent this as just two.

00:06:46.190 --> 00:06:48.440
So, that's going to be a whole number.

00:06:48.440 --> 00:06:51.288
14 over seven, which is
the same thing as two,

00:06:51.288 --> 00:06:54.016
that is a whole number.

00:06:54.016 --> 00:06:55.735
Now, two-pi.

00:06:55.735 --> 00:06:58.927
Now pi is an irrational...

00:06:58.927 --> 00:07:01.400
Pi is an irrational number.

00:07:01.400 --> 00:07:04.697
So if we just take a multiple of pi,

00:07:04.697 --> 00:07:07.786
if we just take a integer multiple of pi,

00:07:07.786 --> 00:07:10.433
like that, this is also going
to be an irrational number.

00:07:10.433 --> 00:07:12.710
If you looked at its
decimal representation,

00:07:12.710 --> 00:07:14.891
it will never repeat.

00:07:14.891 --> 00:07:17.201
So that's two-pi, right over there.

00:07:17.201 --> 00:07:18.188
Now what about...

00:07:18.188 --> 00:07:20.174
Let me do that same, since
I've been consistent,

00:07:20.174 --> 00:07:22.705
relatively consistent, with the colors.

00:07:22.705 --> 00:07:25.665
So, this is two-pi right over there.

00:07:25.665 --> 00:07:28.927
Now, what about the
negative square root of 25.

00:07:28.927 --> 00:07:30.832
Well, 25's a perfect square.

00:07:30.832 --> 00:07:32.817
Square root of that's just gonna be five.

00:07:32.817 --> 00:07:34.674
So, this thing is going to be,

00:07:34.674 --> 00:07:38.181
this thing is going to be
equivalent to negative five.

00:07:38.181 --> 00:07:39.922
So, this is just another representation

00:07:39.922 --> 00:07:41.664
of this, right over here.

00:07:41.664 --> 00:07:43.196
So, it is an integer.

00:07:43.196 --> 00:07:44.775
It's not a whole number
because it's negative,

00:07:44.775 --> 00:07:46.087
but it's an integer.

00:07:46.087 --> 00:07:47.701
Negative square root of 25.

00:07:47.701 --> 00:07:49.255
These two things are actually...

00:07:49.255 --> 00:07:51.811
These two things are
actually the same number,

00:07:51.811 --> 00:07:54.121
just different ways of representing them.

00:07:54.121 --> 00:07:56.269
And then you have, let's
see, you have the square root

00:07:56.269 --> 00:07:59.450
of nine over...

00:07:59.450 --> 00:08:01.679
The square root of nine over seven.

00:08:01.679 --> 00:08:04.117
Well, what's the principal root of nine?

00:08:04.117 --> 00:08:06.358
This thing is gonna be the same thing,

00:08:06.358 --> 00:08:07.670
this thing is the same...

00:08:07.670 --> 00:08:09.486
Let me do this in a different color.

00:08:09.486 --> 00:08:12.859
This is the same thing as,

00:08:12.859 --> 00:08:14.485
square root of nine is three,

00:08:14.485 --> 00:08:17.318
it's the principal root of
nine, so it's three-sevenths.

00:08:17.318 --> 00:08:19.593
So, this is a ratio of two integers.

00:08:19.593 --> 00:08:21.161
This is a rational number.

00:08:21.161 --> 00:08:22.984
Square root of nine over seven

00:08:22.984 --> 00:08:24.783
is the same thing as three-sevenths.

00:08:24.783 --> 00:08:27.802
Now, let me just give you
one more just for the road.

00:08:27.802 --> 00:08:31.703
What about pi over pi?

00:08:31.703 --> 00:08:33.827
What is that going to be?

00:08:33.827 --> 00:08:37.751
Well, pi divided by pi is
going to be equal to one.

00:08:37.751 --> 00:08:39.528
So, this is actually a whole number.

00:08:39.528 --> 00:08:43.037
So I could write pi over
pi, right over there.

00:08:43.037 --> 00:08:47.052
That's just a very
fancy way of saying one.


How to approximate square roots

WEBVTT
Kind: captions
Language: en

00:00:00.351 --> 00:00:01.931
- [Voiceover] What I want
to do in this video is get

00:00:01.931 --> 00:00:05.112
a little bit of experience,
see a few examples

00:00:05.112 --> 00:00:07.434
of trying to roughly
estimate the square root

00:00:07.434 --> 00:00:09.825
of non-perfect squares.

00:00:09.825 --> 00:00:11.845
So let's say that I had,

00:00:11.845 --> 00:00:15.909
if I wanted to estimate
the square root of 32.

00:00:15.909 --> 00:00:18.161
And in particular, I'm just curious,

00:00:18.161 --> 00:00:23.161
between what two integers
will this square root lie?

00:00:23.269 --> 00:00:25.615
Well one way to think about
it is 32 is in between

00:00:25.615 --> 00:00:27.124
what perfect squares?

00:00:27.124 --> 00:00:29.770
We see 32 is, actually let me

00:00:29.770 --> 00:00:31.976
make sure I have some
space for future examples.

00:00:31.976 --> 00:00:35.042
So 32, what's the perfect square below 32?

00:00:35.042 --> 00:00:38.641
So the greatest perfect
square below 32 is 25.

00:00:38.641 --> 00:00:41.497
32 is greater than 25.

00:00:41.497 --> 00:00:43.238
That's five squared.

00:00:43.238 --> 00:00:44.933
So maybe I should write it this way.

00:00:44.933 --> 00:00:48.323
So five squared is less than 32

00:00:48.323 --> 00:00:51.526
and then 32, what's the next
perfect square after 32?

00:00:51.526 --> 00:00:54.267
Well 32 is less than 36.

00:00:54.267 --> 00:00:57.250
So we could say 32 is
less than six squared.

00:00:58.184 --> 00:00:59.856
So if you were to take the square root

00:00:59.856 --> 00:01:01.225
of all of these sides right over here,

00:01:01.225 --> 00:01:03.362
we could say that instead of here

00:01:03.362 --> 00:01:04.824
we have all of the values squared,

00:01:04.824 --> 00:01:06.241
but instead, if we took the square root,

00:01:06.241 --> 00:01:09.282
we could say five is going to be less than

00:01:09.282 --> 00:01:12.742
the square root of 32, which is less than,

00:01:12.742 --> 00:01:14.275
which is less than six.

00:01:14.275 --> 00:01:16.853
Notice, to go from here to here,

00:01:17.647 --> 00:01:19.854
to go from here to here, and here to here,

00:01:19.854 --> 00:01:21.432
all we did is we squared things,

00:01:21.432 --> 00:01:23.196
we raised everything to the second power.

00:01:23.196 --> 00:01:25.913
But the inequality should still hold.

00:01:25.913 --> 00:01:28.259
So the square root of 32 should be between

00:01:28.259 --> 00:01:29.884
five and six.

00:01:29.884 --> 00:01:32.671
It's going to be five point something.

00:01:32.671 --> 00:01:34.226
Let's do another example.

00:01:34.226 --> 00:01:37.152
Let's say we wanted to estimate,

00:01:37.152 --> 00:01:39.217
we want to say between what two integers

00:01:39.217 --> 00:01:43.336
is the square root of 55?

00:01:44.318 --> 00:01:45.733
Well we can do the same idea.

00:01:45.733 --> 00:01:46.814
Let's square it.

00:01:47.656 --> 00:01:49.583
So if we square the square root of 55,

00:01:49.583 --> 00:01:51.825
we're just gonna get to 55.

00:01:51.825 --> 00:01:52.616
We're just going to get,

00:01:52.616 --> 00:01:53.948
let me do that in the same color,

00:01:54.975 --> 00:01:56.066
55.

00:01:56.066 --> 00:02:00.198
So okay, 55 is between
which two perfect squares?

00:02:00.198 --> 00:02:03.497
So the perfect square that is below 55,

00:02:03.497 --> 00:02:05.261
or I could say the greatest perfect square

00:02:05.261 --> 00:02:07.606
that is less than 55.

00:02:07.606 --> 00:02:11.925
Let's see, six squared is
36 and seven squared is 49,

00:02:11.925 --> 00:02:13.735
eight squared is 64.

00:02:13.735 --> 00:02:17.565
So it would be 49.

00:02:17.565 --> 00:02:19.681
I could write that as seven squared.

00:02:19.681 --> 00:02:22.283
Let me write that, that is the
same thing as seven squared.

00:02:22.283 --> 00:02:23.699
And what's the next
perfect square above it?

00:02:23.699 --> 00:02:24.605
Well we just figured it out.

00:02:24.605 --> 00:02:28.250
Seven squared is 49, eight
squared is larger than 55,

00:02:28.250 --> 00:02:29.400
it's 64.

00:02:30.147 --> 00:02:33.329
So this is going to be less than 64,

00:02:33.329 --> 00:02:35.814
which is eight squared.

00:02:35.814 --> 00:02:38.970
And of course 55, just to
make it clear what's going on.

00:02:38.970 --> 00:02:43.662
55 is the square root of 55 squared.

00:02:43.662 --> 00:02:45.844
That's kind of by
definition, it's going to be

00:02:45.844 --> 00:02:48.445
the square root of 55 squared.

00:02:48.445 --> 00:02:51.022
And so the square root of 55
is going to be between what?

00:02:51.022 --> 00:02:53.077
It's going to be between seven and eight.

00:02:54.198 --> 00:02:57.379
So seven is less than
the square root of 55,

00:02:57.379 --> 00:03:00.142
which is less than eight.

00:03:00.142 --> 00:03:01.209
So once again, this is just

00:03:01.209 --> 00:03:03.184
an interesting way to think about,

00:03:03.184 --> 00:03:04.949
what would you, if someone
said the square root of 55 and

00:03:04.949 --> 00:03:06.435
at first you're like, "Oh,
uh, I don't know what that is.

00:03:06.435 --> 00:03:08.407
"I don't have a calculator,"
et cetera et cetera.

00:03:08.407 --> 00:03:09.778
You're like, "Oh wait, wait,
that's going to be between

00:03:09.778 --> 00:03:13.144
"49 and 64, so it's going to
be seven point something."

00:03:13.144 --> 00:03:15.235
It's going to be seven point something.

00:03:15.235 --> 00:03:17.950
And you can even get a rough
estimate of seven point what

00:03:17.950 --> 00:03:21.318
based on how far away
it is from 49 and 64.

00:03:21.318 --> 00:03:23.988
You can begin to approximate things.

00:03:23.988 --> 00:03:25.505
Let's do one more example.

00:03:25.505 --> 00:03:27.549
Let's say we wanted to figure out

00:03:27.549 --> 00:03:31.560
where does the square root of 123 lie?

00:03:32.420 --> 00:03:34.022
And like always, I encourage
you to pause the video

00:03:34.022 --> 00:03:35.160
and try to think about it yourself.

00:03:35.160 --> 00:03:37.500
Between what two integers does this lie?

00:03:38.968 --> 00:03:42.962
Well, if we were to
square it, you get to 123.

00:03:42.962 --> 00:03:46.260
And what's the perfect square that is

00:03:46.260 --> 00:03:48.768
the greatest perfect square less than 123?

00:03:48.768 --> 00:03:50.740
Let's see, 10 squared is 100.

00:03:50.740 --> 00:03:52.552
11 squared is 121.

00:03:52.552 --> 00:03:54.548
12 squared is 144.

00:03:54.548 --> 00:03:56.291
So 11 squared.

00:03:56.291 --> 00:03:58.287
So 123, so we could write

00:03:58.287 --> 00:04:02.183
121 is less than 123, which is

00:04:02.183 --> 00:04:04.901
less than 144,

00:04:04.901 --> 00:04:06.085
that's 12 squared.

00:04:06.085 --> 00:04:08.893
So if we take the square
roots we could write

00:04:08.893 --> 00:04:13.004
that 11 is less than
the square root of 123,

00:04:13.004 --> 00:04:15.789
which is less than 144.

00:04:15.789 --> 00:04:16.859
So once again,

00:04:16.859 --> 00:04:18.088
what's the square root of 123?

00:04:18.088 --> 00:04:20.318
It's going to be 11 point something.

00:04:20.318 --> 00:04:22.199
And in fact, it's going to be closer to 11

00:04:22.199 --> 00:04:23.267
than it's going to be to 12.

00:04:23.267 --> 00:04:26.703
123 is a lot closer to
121 than it is to 144.

00:04:26.703 --> 00:04:30.209
So it might be, I don't know,
11.1, something like that.

00:04:30.209 --> 00:04:31.835
I don't know if that's exactly
right, we would have to

00:04:31.835 --> 00:04:32.973
check that on the calculator.

00:04:32.973 --> 00:04:34.087
But hopefully this gives you,

00:04:34.087 --> 00:04:35.318
oops I,

00:04:35.318 --> 00:04:37.454
that actually will be less than 144.

00:04:37.454 --> 00:04:39.079
But if we want to think about
what consecutive integers

00:04:39.079 --> 00:04:41.982
is that be between, it's going
to be a 12 right over there.

00:04:41.982 --> 00:04:43.398
Almost made a...

00:04:43.398 --> 00:04:44.652
Well anyway, you get the idea.

00:04:44.652 --> 00:04:46.086
Hopefully you enjoyed that.


Dimensions of a cube from its volume

WEBVTT
Kind: captions
Language: en

00:00:00.425 --> 00:00:02.631
- [Voiceover] Let's
say that we had a cube,

00:00:02.631 --> 00:00:04.744
let me draw the cube here,

00:00:04.744 --> 00:00:07.971
So we have a cube, and we know

00:00:07.971 --> 00:00:12.475
that the volume of this cube is equal to

00:00:12.475 --> 00:00:17.119
512 cubic centimeters.

00:00:17.119 --> 00:00:19.488
So my question to you is, what are

00:00:19.488 --> 00:00:21.856
the dimensions of this cube?

00:00:21.856 --> 00:00:25.176
So what is this length gonna be?

00:00:25.176 --> 00:00:27.475
What is this, I guess you could say depth,

00:00:27.475 --> 00:00:29.193
and what is this height going to be?

00:00:29.193 --> 00:00:31.097
And, we know it's a cube, so these are

00:00:31.097 --> 00:00:33.071
all going to be equal.

00:00:33.071 --> 00:00:35.091
And so like always, encourage
you to pause the video

00:00:35.091 --> 00:00:37.602
and try to figure it out.

00:00:37.602 --> 00:00:40.826
Well let's call this length x.

00:00:40.826 --> 00:00:42.614
If that's x then this is going to be x,

00:00:42.614 --> 00:00:44.518
and then this is x as well.

00:00:44.518 --> 00:00:47.954
So if the volume is 512 cubic centimeters,

00:00:47.954 --> 00:00:50.531
that means that

00:00:50.531 --> 00:00:54.268
x times x times x

00:00:54.268 --> 00:00:57.078
is going to be equal to 512.

00:00:57.078 --> 00:00:59.370
Is going to be equal to 512,

00:00:59.370 --> 00:01:01.320
or we could say that x to the third power

00:01:01.320 --> 00:01:03.178
is equal to 512,

00:01:03.178 --> 00:01:04.815
or we could say that x is equal

00:01:04.815 --> 00:01:09.325
to the cube root of 512.

00:01:09.325 --> 00:01:11.456
So what's the cube root of 512?

00:01:11.456 --> 00:01:13.808
And the easiest way I can
think about doing this,

00:01:13.808 --> 00:01:15.038
if I don't have a calculator,

00:01:15.038 --> 00:01:18.777
is to just try to do a prime
factorization of this by hand.

00:01:18.777 --> 00:01:20.819
So that's what I'm going to attempt to do.

00:01:20.819 --> 00:01:23.094
So let's see, does two go into 512?

00:01:23.094 --> 00:01:25.811
Sure, 512s even, so this is going to be

00:01:25.811 --> 00:01:30.017
two times, let's see, 256, yeah,

00:01:30.017 --> 00:01:33.569
two times 256.

00:01:33.569 --> 00:01:36.796
256 that's also divisible by two.

00:01:36.796 --> 00:01:39.351
That's two times 128,

00:01:39.351 --> 00:01:40.817
which is also divisible by two,

00:01:40.817 --> 00:01:44.295
that's two time 64,

00:01:44.295 --> 00:01:45.664
which is also divisible by two.

00:01:45.664 --> 00:01:47.917
That's two times 32,

00:01:47.917 --> 00:01:50.006
let's see, I can keep going,

00:01:50.006 --> 00:01:51.957
that's two times 16.

00:01:51.957 --> 00:01:55.625
Which is two times eight,

00:01:55.625 --> 00:01:57.274
which is two times four,

00:01:57.274 --> 00:01:59.782
which is two times two.

00:01:59.782 --> 00:02:01.477
So 512 that's the same thing as

00:02:01.477 --> 00:02:03.148
two to the, let's see you have,

00:02:03.148 --> 00:02:05.284
one, two, three, four, five, six,

00:02:05.284 --> 00:02:07.049
seven, eight, nine.

00:02:07.049 --> 00:02:08.930
That's two to the ninth power.

00:02:08.930 --> 00:02:09.812
But what we care about is

00:02:09.812 --> 00:02:13.527
what times itself times itself is equal,

00:02:13.527 --> 00:02:15.176
what number, if I have three of 'em

00:02:15.176 --> 00:02:18.380
and I multiply 'em
together, get us to 512?

00:02:18.380 --> 00:02:19.796
And to think about that, we could say,

00:02:19.796 --> 00:02:21.956
"Look, I have nine numbers here.

00:02:21.956 --> 00:02:23.953
"So let me divide into three groups."

00:02:23.953 --> 00:02:26.530
So if this is one group,

00:02:26.530 --> 00:02:29.502
and this is the next group,

00:02:29.502 --> 00:02:31.336
and then this is the next group

00:02:31.336 --> 00:02:33.635
right over here,

00:02:33.635 --> 00:02:35.887
we could say that 512 is the same thing as

00:02:35.887 --> 00:02:38.139
two times two times two, which is eight,

00:02:38.139 --> 00:02:40.647
times two times two times
two, which is eight,

00:02:40.647 --> 00:02:43.294
times two times two times two.

00:02:43.294 --> 00:02:47.636
So 512 is the same thing as
eight to the third power.

00:02:47.636 --> 00:02:50.090
So we could say, that,

00:02:50.090 --> 00:02:52.184
x, I'll do it over here,

00:02:52.184 --> 00:02:55.342
x is equal to the cube root of,

00:02:55.342 --> 00:02:57.478
instead of writing 512,

00:02:57.478 --> 00:02:58.964
instead of writing 512,

00:02:58.964 --> 00:03:01.379
I could write eight to the third power.

00:03:01.379 --> 00:03:03.770
Now, what's the cube root of
something to the third power?

00:03:03.770 --> 00:03:05.558
Well, it's just gonna be this something.

00:03:05.558 --> 00:03:09.668
So x, x is going to be equal to eight.

00:03:09.668 --> 00:03:12.826
So if the volume here is
512 cubic centimeters,

00:03:12.826 --> 00:03:15.705
each dimension is going
to be eight centimeters.

00:03:15.705 --> 00:03:19.327
So x is equal to eight centimeters.

00:03:19.327 --> 00:03:20.999
This is equal to eight centimeters.

00:03:20.999 --> 00:03:22.438
I'm just writing the units now.

00:03:22.438 --> 00:03:24.203
This is equal to eight centimeters,

00:03:24.203 --> 00:03:25.341
and we're done.

00:03:25.341 --> 00:03:26.571
But, if you didn't know offhand

00:03:26.571 --> 00:03:28.220
that eight to the third power is 512,

00:03:28.220 --> 00:03:31.354
this is a reasonable way of
coming to that conclusion.

00:03:31.354 --> 00:03:33.296
Anyway, hopefully that helped.


Square root of decimal (example)

WEBVTT
Kind: captions
Language: en

00:00:00.388 --> 00:00:02.431
- Let's see if we can solve the equation

00:00:02.431 --> 00:00:07.431
P squared is equal to 0.81.

00:00:08.157 --> 00:00:10.109
So how could we think about this?

00:00:10.109 --> 00:00:11.543
Well one thing we could
do is we could say,

00:00:11.543 --> 00:00:14.225
look if P squared is equal to 0.81,

00:00:14.225 --> 00:00:16.430
another way of expressing
this is, that well, that means

00:00:16.430 --> 00:00:20.098
that P is going to be equal
to the positive or negative

00:00:20.098 --> 00:00:23.256
square root of 0.81.

00:00:23.256 --> 00:00:25.472
Remember if we just wrote
the square root symbol here,

00:00:25.472 --> 00:00:26.633
that means the principal root,

00:00:26.633 --> 00:00:28.293
or just the positive square root.

00:00:28.293 --> 00:00:31.509
But here P could be positive
or negative, because if you

00:00:31.509 --> 00:00:33.541
square it, if you square
even a negative number,

00:00:33.541 --> 00:00:35.549
you're still going to
get a positive value.

00:00:35.549 --> 00:00:37.070
So we could write that P is equal

00:00:37.070 --> 00:00:39.443
to the plus or minus square root of 0.81,

00:00:39.443 --> 00:00:41.277
which kind of helps us, it's another way

00:00:41.277 --> 00:00:44.203
of expressing the same,
the same, equation.

00:00:44.203 --> 00:00:46.959
But still, what could P be?

00:00:46.959 --> 00:00:49.989
In your brain, you might
immediately say, well okay,

00:00:49.989 --> 00:00:52.624
you know if this was P
squared is equal to 81,

00:00:52.624 --> 00:00:53.866
I kinda know what's going on.

00:00:53.866 --> 00:00:58.335
Because I know that nine
times nine is equal to 81.

00:00:58.335 --> 00:01:01.878
Or we could write that nine
squared is equal to 81,

00:01:01.878 --> 00:01:03.874
or we could write that nine is equal

00:01:03.874 --> 00:01:05.836
to the principal root of 81.

00:01:05.836 --> 00:01:08.413
These are all, I guess,
saying the same truth

00:01:08.413 --> 00:01:12.109
about the universe, but what about 0.81?

00:01:12.979 --> 00:01:17.174
Well 0.81 has two digits behind,
to the right of the decimal

00:01:17.734 --> 00:01:19.717
and so if I were to
multiply something that has

00:01:19.717 --> 00:01:22.062
one digit to the right of
the decimal times itself,

00:01:22.062 --> 00:01:23.333
I'm gonna have something with two digits

00:01:23.333 --> 00:01:24.747
to the right of the decimal.

00:01:24.747 --> 00:01:27.243
And so what happens if I
take, instead of nine squared,

00:01:27.243 --> 00:01:29.599
what happens if I take 0.9 squared?

00:01:29.599 --> 00:01:30.989
Let me try that out.

00:01:30.989 --> 00:01:32.474
Zero, I'm gonna use a different color.

00:01:33.014 --> 00:01:37.377
So let's say I took 0.9 squared.

00:01:38.017 --> 00:01:41.499
0.9 squared, well that's going to be 0.9

00:01:41.499 --> 00:01:45.787
times 0.9, which is going to be equal to?

00:01:45.787 --> 00:01:50.355
Well nine times nine is 81,
and I have one, two, numbers

00:01:50.355 --> 00:01:52.378
to the right of the decimal,
so I'm gonna have two

00:01:52.378 --> 00:01:54.178
numbers to the right of
the decimal in the product.

00:01:54.178 --> 00:01:58.562
So one, two. So that
indeed is equal to 0.81.

00:01:58.562 --> 00:02:03.407
In fact we could write
0.81 as 0.9 squared.

00:02:03.407 --> 00:02:06.627
So we could write this,
we could write that P

00:02:07.257 --> 00:02:10.484
is equal to the plus or
minus, the square root of,

00:02:10.484 --> 00:02:12.117
instead of writing 0.81,

00:02:12.117 --> 00:02:17.017
I could write that as 0.9 squared.

00:02:17.017 --> 00:02:20.137
In fact I could also write
that as negative 0.9 squared.

00:02:20.137 --> 00:02:22.636
Cause if you put a negative
here and a negative here,

00:02:22.636 --> 00:02:24.482
it's still not going to change the value.

00:02:24.482 --> 00:02:27.218
A negative times a negative
is going to be a positive.

00:02:27.218 --> 00:02:29.041
I could, actually I would
have put a negative there,

00:02:29.041 --> 00:02:30.039
which would have implied

00:02:30.039 --> 00:02:31.724
a negative here and a negative there.

00:02:31.724 --> 00:02:33.877
So either of those are going to be true.

00:02:33.877 --> 00:02:35.549
But it's going to work out for us

00:02:35.549 --> 00:02:37.888
because we are taking the
positive and negative square root.

00:02:37.888 --> 00:02:39.328
So this is going to be,

00:02:39.328 --> 00:02:44.328
P is going to be equal
to plus or minus 0.9.

00:02:44.973 --> 00:02:48.043
Plus or minus 0.9, or we could write it

00:02:48.043 --> 00:02:51.131
that P is equal to 0.9,

00:02:51.131 --> 00:02:55.883
or P could be equal to negative 0.9.

00:02:55.883 --> 00:02:57.160
And you can verify that,

00:02:57.160 --> 00:03:01.838
you would square either of
these things, you get 0.81.


Introduction to cube roots

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.293
- [Voiceover] We already know a little bit

00:00:01.293 --> 00:00:02.467
about square roots.

00:00:02.467 --> 00:00:03.800
For example, if I were to tell you

00:00:03.800 --> 00:00:07.388
that seven squared is equal to 49,

00:00:07.388 --> 00:00:09.864
that's equivalent to saying that seven

00:00:09.864 --> 00:00:13.668
is equal to the square root of 49.

00:00:13.668 --> 00:00:16.269
The square root essentially unwinds

00:00:16.269 --> 00:00:18.110
taking the square of something.

00:00:18.110 --> 00:00:19.565
In fact, we could write it like this.

00:00:19.565 --> 00:00:22.651
We could write the square root of 49,

00:00:22.651 --> 00:00:24.709
so this is whatever number times itself

00:00:24.709 --> 00:00:26.366
is equal to 49.

00:00:26.366 --> 00:00:29.027
If I multiply that number times itself,

00:00:29.027 --> 00:00:32.281
if I square it, well I'm going to get 49.

00:00:32.281 --> 00:00:34.422
And that's going to be
true for any number,

00:00:34.942 --> 00:00:36.974
not just 49.

00:00:38.334 --> 00:00:41.683
If I write the square root of X

00:00:42.183 --> 00:00:43.981
and if I were to square it,

00:00:43.981 --> 00:00:46.429
that's going to be equal to X

00:00:46.429 --> 00:00:48.781
and that's going to be true for any X

00:00:48.781 --> 00:00:52.002
for which we can evaluate the square root,

00:00:52.002 --> 00:00:54.080
evaluate the principle root.

00:00:54.080 --> 00:00:57.236
Now typically and as you advance in math

00:00:57.236 --> 00:01:00.236
you're going to see that this will change,

00:01:00.236 --> 00:01:01.985
but typically you say,
okay if I'm going to take

00:01:01.985 --> 00:01:05.805
the square root of something,
X has to be non-negative.

00:01:05.805 --> 00:01:09.395
X has to be non-negative.

00:01:09.945 --> 00:01:13.972
This is going to change
once we start thinking about

00:01:13.972 --> 00:01:15.901
imaginary and complex numbers,

00:01:15.901 --> 00:01:18.671
but typically for the
principle square root,

00:01:18.671 --> 00:01:21.464
we assume that whatever's
under the radical,

00:01:21.464 --> 00:01:25.509
whatever's under here, is
going to be non-negative

00:01:25.509 --> 00:01:26.902
because it's hard to square a number

00:01:26.902 --> 00:01:28.740
at least the numbers that we know about,

00:01:28.740 --> 00:01:31.580
it's hard to square them
and get a negative number.

00:01:31.580 --> 00:01:34.974
So for this thing to be
defined, for it to make sense,

00:01:34.974 --> 00:01:37.106
it's typical to say
that, okay we need to put

00:01:37.106 --> 00:01:39.184
a non-negative number in here.

00:01:39.184 --> 00:01:41.225
But anyway, the focus of this video

00:01:41.225 --> 00:01:44.109
is not on the square root, it's
really just to review things

00:01:44.109 --> 00:01:47.863
so we can start thinking
about the cube root.

00:01:47.863 --> 00:01:50.201
And as you can imagine,
where does the whole notion

00:01:50.201 --> 00:01:53.768
of taking a square of something
or a square root come from?

00:01:53.768 --> 00:01:55.147
Well it comes from the notion of finding

00:01:55.147 --> 00:01:56.784
the area of a square.

00:01:56.784 --> 00:02:00.580
If I have a square like this
and if this side is seven,

00:02:00.580 --> 00:02:03.264
well if it's a square, all the
sides are going to be seven.

00:02:03.264 --> 00:02:05.687
And if I wanted to find the area of this,

00:02:05.687 --> 00:02:09.078
it would be seven times
seven or seven squared.

00:02:09.078 --> 00:02:11.944
That would be the area of this.

00:02:11.944 --> 00:02:14.421
Or if I were to say, well what is

00:02:14.421 --> 00:02:17.166
if I have a square, if I have,

00:02:17.166 --> 00:02:18.956
and that doesn't look
like a perfect square,

00:02:18.956 --> 00:02:21.608
but you get the idea, all the
sides are the same length.

00:02:21.608 --> 00:02:25.039
If I have a square with area X.

00:02:25.039 --> 00:02:27.037
If the area here is X,

00:02:27.037 --> 00:02:29.232
what are the lengths of
the sides going to be?

00:02:29.232 --> 00:02:31.425
Well it's going to be square root of X.

00:02:31.425 --> 00:02:33.260
All of the sides are
going to be the square root of X,

00:02:33.260 --> 00:02:35.964
so it's going to be the square root
of X by the square root of X

00:02:35.964 --> 00:02:37.776
and this side is going to
be the square root of X as well

00:02:37.776 --> 00:02:39.261
and that's going to be
the square root of X as well.

00:02:39.261 --> 00:02:41.943
So that's where the term
square root comes from,

00:02:41.943 --> 00:02:44.277
where the square comes from.

00:02:44.277 --> 00:02:46.363
Now what do you think cube root?

00:02:46.363 --> 00:02:47.864
Well same idea.

00:02:47.864 --> 00:02:49.775
If I have a cube.

00:02:50.205 --> 00:02:52.055
If I have a cube.

00:02:52.055 --> 00:02:55.317
Let me do my best attempt at
drawing a cube really fast.

00:02:55.317 --> 00:02:59.314
If I have a cube and a
cube, all of it's dimensions

00:02:59.314 --> 00:03:03.669
have the same length so this
is a two, by two, by two cube,

00:03:03.669 --> 00:03:05.876
what's the volume over here.

00:03:05.876 --> 00:03:10.348
Well the volume is going to
be two, times two, times two,

00:03:10.348 --> 00:03:15.232
which is two to the
third power or two cubed.

00:03:15.232 --> 00:03:17.014
This is two cubed.

00:03:17.014 --> 00:03:18.895
That's why they use the word cubed

00:03:18.895 --> 00:03:20.879
because this would be the volume of a cube

00:03:20.879 --> 00:03:23.379
where each of its sides have length two

00:03:23.379 --> 00:03:26.370
and this of course is
going to be equal to eight.

00:03:26.370 --> 00:03:28.233
But what if we went the other way around?

00:03:28.233 --> 00:03:30.740
What if we started with the cube?

00:03:30.740 --> 00:03:32.979
What if we started with this volume?

00:03:33.349 --> 00:03:35.809
What if we started with a cube's volume

00:03:35.809 --> 00:03:40.450
and let's say the volume
here is eight cubic units,

00:03:40.450 --> 00:03:43.930
so volume is equal to eight

00:03:43.930 --> 00:03:46.857
and we wanted to find
the lengths of the sides.

00:03:46.857 --> 00:03:51.180
So we wanted to figure out what X is

00:03:51.180 --> 00:03:53.606
cause that's X, that's X, and that's X.

00:03:53.606 --> 00:03:56.810
It's a cube so all the
dimensions have the same length.

00:03:56.810 --> 00:03:58.512
Well there's two ways that
we could express this.

00:03:58.512 --> 00:04:01.280
We could say that X times X times X

00:04:01.280 --> 00:04:05.158
or X to the third power is equal to eight

00:04:05.158 --> 00:04:08.640
or we could use the cube root symbol,

00:04:08.640 --> 00:04:11.837
which is a radical with a
little three in the right place.

00:04:12.677 --> 00:04:15.838
Or we could write that X is equal to,

00:04:15.838 --> 00:04:18.526
it's going to look very
similar to the square root.

00:04:18.526 --> 00:04:20.238
This would be the square root of eight,

00:04:20.238 --> 00:04:21.526
but to make it clear,
they were talking about

00:04:21.526 --> 00:04:23.914
the cube root of eight, we would write

00:04:23.914 --> 00:04:26.054
a little three over there.

00:04:26.054 --> 00:04:27.261
In theory for square root,

00:04:27.261 --> 00:04:28.782
you could put a little two over here,

00:04:28.782 --> 00:04:29.656
but that'd be redundant.

00:04:29.656 --> 00:04:30.825
If there's no number here,

00:04:30.825 --> 00:04:33.715
people just assume that
it's the square root.

00:04:33.715 --> 00:04:36.641
But if you're figuring out the cube root

00:04:36.641 --> 00:04:38.775
or sometimes you say the third root,

00:04:38.775 --> 00:04:39.743
well then you have to say,

00:04:39.743 --> 00:04:41.738
well you have to put this
little three right over here

00:04:41.738 --> 00:04:46.161
in this little notch in the
radical symbol right over here.

00:04:46.161 --> 00:04:48.354
And so this is saying X
is going to be some number

00:04:48.354 --> 00:04:51.617
that if I cube it, I get eight.

00:04:51.617 --> 00:04:52.476
So with that out of the way,

00:04:52.476 --> 00:04:54.544
let's do some examples.

00:04:54.544 --> 00:04:56.583
Let's say that I have...

00:04:56.583 --> 00:04:59.174
Let's say that I want to calculate

00:05:00.934 --> 00:05:03.284
the cube root of 27.

00:05:03.284 --> 00:05:04.752
What's that going to be?

00:05:04.752 --> 00:05:07.540
Well if say that this is
going to be equal to X,

00:05:07.540 --> 00:05:09.902
this is equivalent to
saying that X to the third

00:05:09.902 --> 00:05:14.355
or that 27 is equal to
X to the third power.

00:05:14.355 --> 00:05:16.491
So what is X going to be?

00:05:17.001 --> 00:05:19.653
Well X times X times X is equal to 27,

00:05:19.653 --> 00:05:23.154
well the number I can think of is three,

00:05:23.154 --> 00:05:26.850
so we would say that X, let
me scroll down a little bit,

00:05:26.850 --> 00:05:28.953
X is equal to three.

00:05:28.953 --> 00:05:31.156
Now let me ask you a question.

00:05:32.256 --> 00:05:34.413
Can we write something like...

00:05:34.413 --> 00:05:36.355
Can we pick a new color?

00:05:36.355 --> 00:05:41.355
The cube root of, let
me write negative 64.

00:05:41.993 --> 00:05:43.359
I already talked about
that if we're talking

00:05:43.359 --> 00:05:45.495
the square root, it's fairly typical that

00:05:45.495 --> 00:05:47.097
hey you put a negative number in there

00:05:47.097 --> 00:05:48.850
at least until we learn
about imaginary numbers,

00:05:48.850 --> 00:05:50.121
we don't know what to do with it.

00:05:50.121 --> 00:05:52.605
But can we do something with this?

00:05:53.085 --> 00:05:57.342
Well if I cube something,
can I get a negative number?

00:05:57.342 --> 00:05:58.444
Sure.

00:05:58.914 --> 00:06:01.173
So if I say this is equal to X,

00:06:01.173 --> 00:06:04.445
this is the same thing as
saying that negative 64

00:06:04.445 --> 00:06:07.006
is equal to X to the third power.

00:06:07.006 --> 00:06:09.097
Well what could X be?

00:06:09.097 --> 00:06:11.540
Well what happens if
you take negative four

00:06:11.540 --> 00:06:15.180
times negative four times negative four?

00:06:15.180 --> 00:06:17.523
Negative four times four is positive 16,

00:06:17.523 --> 00:06:21.297
but then times negative
four is negative 64

00:06:21.297 --> 00:06:23.423
is equal to negative 64.

00:06:23.423 --> 00:06:25.103
So what could X be here?

00:06:25.103 --> 00:06:27.722
Well X could be equal to negative four.

00:06:27.722 --> 00:06:31.112
X could be equal to negative four.

00:06:31.112 --> 00:06:32.708
So based on the math that we know so far

00:06:32.708 --> 00:06:37.708
you actually can take the cube
root of a negative number.

00:06:37.838 --> 00:06:39.885
And just so you know, you
don't have to stop there.

00:06:39.885 --> 00:06:40.904
You could take a fourth root

00:06:40.904 --> 00:06:42.471
and in this case you'd have a four here,

00:06:42.471 --> 00:06:43.469
a fifth root, a sixth root,

00:06:43.469 --> 00:06:45.385
a seventh root of numbers

00:06:45.385 --> 00:06:48.368
and we'll talk about that later

00:06:48.368 --> 00:06:49.761
in your mathematical career.

00:06:49.761 --> 00:06:50.771
But most of what you're going to see

00:06:50.771 --> 00:06:51.770
is actually going to be square root

00:06:51.770 --> 00:06:53.627
and every now and then you're
going to see a cube root.

00:06:53.627 --> 00:06:54.962
Now you might be saying, well hey look,

00:06:54.962 --> 00:06:58.511
you know, you just knew that
three to the third power is 27,

00:06:58.511 --> 00:06:59.768
you took the cube root, you get X,

00:06:59.768 --> 00:07:02.046
is there any simple way to do this?

00:07:02.046 --> 00:07:04.609
And like you know if i give
you an arbitrary number.

00:07:04.609 --> 00:07:07.000
If I were to just say, I don't know,

00:07:07.000 --> 00:07:12.000
if I were to say cube root of 125.

00:07:12.416 --> 00:07:13.989
And the simple answer is,

00:07:13.989 --> 00:07:17.542
well the easiest way to
actually figure this out

00:07:17.542 --> 00:07:20.312
is actually just to do a factorization

00:07:20.312 --> 00:07:22.998
and particular prime
factorization of this thing

00:07:22.998 --> 00:07:25.206
right over here and then
you would figure it out.

00:07:25.206 --> 00:07:29.369
So you would say, okay
well 125 is five times 25,

00:07:29.369 --> 00:07:31.624
which is five times five.

00:07:31.624 --> 00:07:36.024
Alright, so this is the same
thing as the cube root of five

00:07:36.024 --> 00:07:37.765
to the third power, which of course,

00:07:37.765 --> 00:07:41.979
is going to be equal to five.

00:07:41.979 --> 00:07:43.302
If you have a much larger number here,

00:07:43.302 --> 00:07:47.905
yes, there's no very simple way to compute

00:07:47.905 --> 00:07:49.798
what a cube root or a fourth root

00:07:49.798 --> 00:07:51.755
or a fifth root might
be and even square root

00:07:51.755 --> 00:07:53.960
can get quite difficult.

00:07:53.960 --> 00:07:56.305
There's no very simple
way to just calculate it

00:07:56.305 --> 00:07:59.248
the way that you might
multiply things or divide it.