Modeling population as an exponential function

WEBVTT
Kind: captions
Language: en

00:00:00.515 --> 00:00:01.976
-[Voiceover] Let's
think a little bit about

00:00:01.976 --> 00:00:06.961
modeling population and
what I have pictures here

00:00:06.961 --> 00:00:09.013
are some of the most known,

00:00:09.013 --> 00:00:10.785
actually this gentleman right over here

00:00:10.785 --> 00:00:14.527
might be the most known
person when people think about

00:00:14.527 --> 00:00:17.540
population and the limits
to grow the population.

00:00:17.657 --> 00:00:19.439
This is Thomas Malthus.

00:00:19.439 --> 00:00:24.439
He was a British cleric
and writer and scholar

00:00:25.065 --> 00:00:28.772
at the end of the 1700s, at
the end of the 18th century,

00:00:29.357 --> 00:00:31.002
early 19th century.

00:00:31.002 --> 00:00:32.585
He really challenged the notion

00:00:32.585 --> 00:00:34.673
that population could grow indefinitely

00:00:35.465 --> 00:00:37.479
and that we would always
through technology

00:00:37.479 --> 00:00:38.901
be able to feed ourselves

00:00:38.901 --> 00:00:40.972
and that really that the
environment would eventually

00:00:40.972 --> 00:00:44.285
put some caps on how much

00:00:44.285 --> 00:00:46.307
or where the population could grow to.

00:00:47.055 --> 00:00:49.191
P.F. Verhulst and I'm sure

00:00:49.686 --> 00:00:51.489
I'm mispronouncing his name here.

00:00:51.489 --> 00:00:56.489
He was a Belgian mathematician
who read Malthusâ€™ work

00:00:57.032 --> 00:00:59.038
and tried to model the behavior

00:00:59.632 --> 00:01:00.730
that Malthus was talking about

00:01:01.612 --> 00:01:04.193
that, okay when there aren't
environmental constraints,

00:01:04.193 --> 00:01:06.922
maybe population does grow
somewhat exponentially

00:01:06.922 --> 00:01:09.018
but then as it approaches
kind of the limits

00:01:09.882 --> 00:01:11.386
set by the environment

00:01:11.386 --> 00:01:13.143
it's going to essentially asymptote

00:01:13.143 --> 00:01:16.518
towards some type of population.

00:01:16.518 --> 00:01:18.403
Malthus in particular,
he actually doesn't think

00:01:18.403 --> 00:01:19.981
it's just going to be
a nice clean asymptote.

00:01:19.981 --> 00:01:21.288
He actually thinks that the population

00:01:21.288 --> 00:01:22.494
would kind of go above the limit

00:01:22.494 --> 00:01:23.829
and you'd have these catastrophes

00:01:23.829 --> 00:01:25.556
and then they would go
crashing below the limit

00:01:25.556 --> 00:01:28.632
and you kind of oscillate
right around the limit

00:01:28.632 --> 00:01:30.757
through these catastrophes.

00:01:30.757 --> 00:01:34.536
You can tell Malthus was
a fairly optimistic guy

00:01:34.536 --> 00:01:36.704
but let's go through a
little bit of the math

00:01:36.704 --> 00:01:38.936
and a little bit of the
differential equations

00:01:38.936 --> 00:01:41.822
although it's not too,
these aren't overly hairy

00:01:41.822 --> 00:01:43.800
differential equations to
think about population.

00:01:44.492 --> 00:01:46.525
The first way to think about population

00:01:46.525 --> 00:01:48.479
and I'll express it as
a differential equation.

00:01:48.479 --> 00:01:50.973
Actually let me just
set some variables here.

00:01:50.973 --> 00:01:53.241
Let's say that N is our population,

00:01:53.241 --> 00:01:55.445
so that's our population

00:01:56.691 --> 00:02:00.139
and we are going to assume
that N is a function of T.

00:02:01.000 --> 00:02:03.113
N as a function of T

00:02:03.113 --> 00:02:05.185
is what we're going to
be thinking about in this

00:02:05.185 --> 00:02:07.659
and frankly the next series of videos.

00:02:08.643 --> 00:02:10.889
One way to think about how to model this

00:02:10.089 --> 00:02:13.045
is just so what is the rate of change

00:02:13.846 --> 00:02:15.590
of population with respect to time?

00:02:15.059 --> 00:02:16.553
How does that relate to things?

00:02:17.096 --> 00:02:19.851
We could say, okay, well
what is the rate of change

00:02:20.715 --> 00:02:22.739
of population with respect to time?

00:02:23.908 --> 00:02:25.903
D, capital N, DT.

00:02:25.903 --> 00:02:27.570
Well one way to think about it

00:02:27.057 --> 00:02:29.568
is it's going to be
proportional to the population.

00:02:30.081 --> 00:02:32.633
You could say well maybe
this is going to be

00:02:33.125 --> 00:02:34.992
some proportionality constant

00:02:34.992 --> 00:02:36.700
times the population,

00:02:36.007 --> 00:02:38.328
times the population itself.

00:02:39.021 --> 00:02:43.772
This makes sense if the
population is smaller

00:02:43.772 --> 00:02:47.085
then you're not going to have as much

00:02:47.085 --> 00:02:50.976
change per unit time as if
the population is larger.

00:02:50.976 --> 00:02:54.159
The larger the population the more

00:02:54.159 --> 00:02:56.850
it's going to grow in a
particular unit of time.

00:02:56.085 --> 00:02:59.332
Actually this is actually
a fairly simple to solve

00:03:00.097 --> 00:03:02.085
differential equation, you
might have done it before.

00:03:02.085 --> 00:03:03.428
I encourage you to pause this video

00:03:03.428 --> 00:03:04.964
if you feel inspired to do so

00:03:04.964 --> 00:03:06.226
but I'll solve it right here

00:03:06.226 --> 00:03:11.187
and you'll see that we get
an exponential function here

00:03:11.187 --> 00:03:12.922
for N, so let's do that.

00:03:12.922 --> 00:03:14.011
Let's solve this and we'll essentially

00:03:14.812 --> 00:03:18.034
separate the variables,
separate the N from the Ts

00:03:18.034 --> 00:03:19.975
although we only see DT here

00:03:19.975 --> 00:03:21.043
but I'll do that in a second.

00:03:21.043 --> 00:03:23.680
If I divide both sides by N.

00:03:23.068 --> 00:03:28.068
I get one over N and if I
multiply both sides by DT

00:03:29.413 --> 00:03:31.186
if you kind of think about the DT

00:03:31.186 --> 00:03:32.517
is something that you can multiply.

00:03:33.363 --> 00:03:34.832
I'm going to multiply both sides

00:03:34.832 --> 00:03:35.953
or I'm going to divide both sides by

00:03:35.953 --> 00:03:38.038
and multiply both sides by DT.

00:03:38.038 --> 00:03:39.377
I'm going to get one over N,

00:03:39.377 --> 00:03:41.328
DN on the left-hand side

00:03:41.328 --> 00:03:43.589
and on the right-hand
side I'm going to get

00:03:43.589 --> 00:03:48.589
R times DT.

00:03:50.002 --> 00:03:51.801
Notice I got the DT
onto the right-hand side

00:03:51.819 --> 00:03:53.334
by multiplying both sides of that

00:03:53.349 --> 00:03:55.338
and then I divided both sides by N

00:03:55.338 --> 00:03:57.012
and I got the one over N right over here.

00:03:57.842 --> 00:04:02.842
Now what we can do is we
can take the antiderivative

00:04:03.002 --> 00:04:04.461
of both sides.

00:04:05.072 --> 00:04:08.828
We can take the
antiderivative of both sides

00:04:09.476 --> 00:04:11.357
and what do we get on the left-hand side?

00:04:11.357 --> 00:04:15.692
Well this is just going
to be the natural log

00:04:15.692 --> 00:04:19.442
of the absolute value of our population

00:04:19.442 --> 00:04:21.457
and actually if we assume
that the population

00:04:21.457 --> 00:04:24.605
is always going to be non-zero

00:04:24.605 --> 00:04:26.683
then we can actually take
this absolute value off

00:04:27.385 --> 00:04:28.417
but I'll do that in a second.

00:04:28.705 --> 00:04:33.705
That's going to be equal to R times T

00:04:36.556 --> 00:04:38.107
and then we could have
added the constant here

00:04:38.107 --> 00:04:39.628
but I'm just going to do it on one side.

00:04:39.628 --> 00:04:44.628
It's R times T plus C.

00:04:45.581 --> 00:04:48.192
Now, if we actually want to solve for N,

00:04:48.192 --> 00:04:50.515
we could take ...

00:04:50.515 --> 00:04:52.668
If this is equal to this
then E to this power

00:04:52.668 --> 00:04:55.303
should be the same as E to this power

00:04:55.303 --> 00:04:56.879
or another way of thinking about it,

00:04:56.879 --> 00:04:59.356
E, the natural log of
the absolute value of N

00:04:59.356 --> 00:04:59.868
is equal to this.

00:04:59.868 --> 00:05:01.714
Is another way of saying that E to this

00:05:01.714 --> 00:05:02.991
is going to be equal to that.

00:05:02.991 --> 00:05:04.593
Well actually let me just do it this way.

00:05:04.593 --> 00:05:09.593
Let me just take E to this power

00:05:12.035 --> 00:05:12.908
and to that power.

00:05:12.908 --> 00:05:14.271
If that's equal to that
then E to that power

00:05:14.271 --> 00:05:15.864
should be the same as E to that power.

00:05:16.617 --> 00:05:20.691
We're going to be left
with E to the natural log

00:05:21.357 --> 00:05:22.293
of the absolute value of N.

00:05:22.293 --> 00:05:24.346
That's going to give you
the absolute value of N.

00:05:24.346 --> 00:05:27.541
Let's just assume N positive,

00:05:27.541 --> 00:05:32.541
let's just assume population
is greater than zero.

00:05:34.081 --> 00:05:39.081
Then we can this left-hand
side right over here

00:05:39.135 --> 00:05:41.161
we'll just simplify to N

00:05:41.395 --> 00:05:43.403
and then our right-hand
side it's going to be ...

00:05:44.195 --> 00:05:47.514
Well it could be E to the RT plus C

00:05:47.514 --> 00:05:49.663
or this is the same thing.

00:05:50.155 --> 00:05:52.223
This thing right over
here is the same thing

00:05:52.835 --> 00:05:57.835
as E to the RT, E to the R times T, times,

00:06:04.596 --> 00:06:06.912
actually let me do that
E in that same color.

00:06:06.912 --> 00:06:11.231
E to the RT, times E to the C,

00:06:11.231 --> 00:06:14.264
times E to the C, right?

00:06:14.561 --> 00:06:16.703
I'm just taking E to the
sum of these two exponents.

00:06:16.703 --> 00:06:19.466
That's going to be E to
the RT, times E to the C.

00:06:20.342 --> 00:06:21.436
If we want to, we could
just hey you know what,

00:06:22.282 --> 00:06:23.984
this is still just going to be some

00:06:23.984 --> 00:06:25.649
arbitrary constant here.

00:06:25.649 --> 00:06:27.745
Actually let's just call this C.

00:06:28.609 --> 00:06:30.086
It's E to the RT times C

00:06:30.086 --> 00:06:35.086
or we could say C times E to the RT.

00:06:42.676 --> 00:06:45.181
Notice, so we have solved
the differential equation.

00:06:45.181 --> 00:06:47.443
We haven't gotten to this kind of

00:06:47.443 --> 00:06:49.558
less than optimistic reality of Malthus

00:06:49.558 --> 00:06:50.289
where we're limiting it.

00:06:50.289 --> 00:06:52.424
This is just hey, if we
just assume population

00:06:52.424 --> 00:06:53.325
is going to ...

00:06:53.325 --> 00:06:55.523
The rate of change of
population with respect to time

00:06:55.523 --> 00:06:57.988
is going to be proportional to population.

00:06:57.988 --> 00:06:59.985
When we solve that differential equation,

00:06:59.985 --> 00:07:01.794
we get that population
is a function of time.

00:07:01.794 --> 00:07:03.477
Actually let me make it explicit

00:07:03.477 --> 00:07:05.030
that this is a function of time.

00:07:05.003 --> 00:07:09.401
Let me just move the N over a little bit,

00:07:09.428 --> 00:07:10.864
so let me write it this way.

00:07:10.864 --> 00:07:15.864
N of T is going to be equal to this.

00:07:17.261 --> 00:07:20.797
This was our solution to
this differential equation.

00:07:20.797 --> 00:07:23.122
Once again this is going
to just grow forever

00:07:23.122 --> 00:07:24.499
and if we know the initial conditions

00:07:24.499 --> 00:07:29.499
and so let's say that
we knew that N of zero

00:07:31.054 --> 00:07:32.093
when time is equal to zero.

00:07:32.093 --> 00:07:33.175
Let's just say that's N sub naught

00:07:34.075 --> 00:07:35.346
so what would C be?

00:07:36.021 --> 00:07:40.698
Well N of zero is going to be equal to C.

00:07:41.559 --> 00:07:44.540
C times E to the zero power.

00:07:44.054 --> 00:07:45.356
You know the zero power is just one

00:07:45.842 --> 00:07:47.041
so this is going to be equal to C,

00:07:47.041 --> 00:07:51.239
so C is equal to N sub naught.

00:07:51.239 --> 00:07:52.606
Now we can even write it

00:07:52.606 --> 00:07:55.118
that the solution to this
thing right over here

00:07:55.118 --> 00:08:00.118
is N as a function of T

00:08:00.034 --> 00:08:03.947
is going to be equal to C times ...

00:08:04.253 --> 00:08:08.484
Be careful, N naught,
our initial population,

00:08:08.484 --> 00:08:13.484
times E to the E to the RT.

00:08:15.937 --> 00:08:18.273
Now once again this is an exponential.

00:08:18.273 --> 00:08:20.514
Essentially our population
is going to look like this.

00:08:20.514 --> 00:08:24.625
If I were to graph it, it's going to look

00:08:25.563 --> 00:08:27.116
if that's my time axis,

00:08:27.116 --> 00:08:30.037
that's my N axis right over here.

00:08:30.037 --> 00:08:32.453
I could say it's Y equals N axis,

00:08:32.453 --> 00:08:35.955
however I want to denote it.

00:08:35.955 --> 00:08:37.538
That would be N naught

00:08:37.538 --> 00:08:40.443
and it's going to grow
exponentially from there.

00:08:40.443 --> 00:08:41.534
The rate of this exponential function

00:08:42.354 --> 00:08:45.366
is going to be dictated by
this constant right over there

00:08:45.366 --> 00:08:46.674
but it's going to be
look something like this

00:08:46.674 --> 00:08:48.681
and it's just going to grow
faster and faster and faster

00:08:48.744 --> 00:08:50.273
forever and ever and ever and ever.

00:08:50.273 --> 00:08:52.596
Now, as I mentioned at the
beginning of the video,

00:08:52.596 --> 00:08:55.988
Malthus does not believe

00:08:55.988 --> 00:08:57.141
that this is going to be true.

00:08:57.141 --> 00:08:59.582
He thinks that we're going
to hit some natural limits

00:08:59.582 --> 00:09:02.651
that are going to start to
constrain the population.

00:09:02.651 --> 00:09:04.603
In his mind the more natural

00:09:05.695 --> 00:09:09.173
or more realistic function
to model population

00:09:09.173 --> 00:09:12.134
would look something like this

00:09:12.134 --> 00:09:15.831
or even potentially something
that you keep going crashing

00:09:15.831 --> 00:09:19.288
around that kind of limit.

00:09:19.288 --> 00:09:22.180
What we'll see in the next
video is that P.F. Verhulst

00:09:22.018 --> 00:09:25.202
actually came up with a very
good one differential equation

00:09:25.364 --> 00:09:27.282
and solution to the differential equation

00:09:27.282 --> 00:09:31.753
that does model a lot
better, that better describes

00:09:31.753 --> 00:09:34.524
the reality that Malthus
believed that we were in.


Converting directly from binary to hexadecimal

WEBVTT
Kind: captions
Language: en

00:00:00.521 --> 00:00:02.919
- [Voiceover] What I would
like to do in this video is

00:00:02.919 --> 00:00:06.843
explore the connection between
the binary number system

00:00:06.843 --> 00:00:08.866
which is clearly, or we've already

00:00:08.866 --> 00:00:11.475
talked about this, is base two.

00:00:11.475 --> 00:00:14.838
Explore the quotient between
that and the hexadecimal,

00:00:14.838 --> 00:00:19.838
hexadecimal number
system, which is base 16.

00:00:20.258 --> 00:00:22.251
The reason why this is interesting is

00:00:22.251 --> 00:00:26.408
because 16 is a power of two.

00:00:26.408 --> 00:00:28.163
What we'll see is you could always view

00:00:28.163 --> 00:00:29.651
the hexadecimal number system.

00:00:29.651 --> 00:00:32.318
It's almost condensed representation

00:00:32.318 --> 00:00:33.788
of the binary number system.

00:00:33.788 --> 00:00:35.503
This is actually why you will actually,

00:00:35.503 --> 00:00:37.609
we've already talked
about the binary system

00:00:37.609 --> 00:00:40.191
is used extensively in computer science

00:00:40.191 --> 00:00:41.905
and in even computer engineering.

00:00:41.905 --> 00:00:44.935
It's the underlying
things that are happening

00:00:44.935 --> 00:00:46.969
or it's the representation
used when we talk

00:00:46.969 --> 00:00:49.918
about logic gates and
transistors and things like that.

00:00:49.918 --> 00:00:52.570
But hexadecimal also shows up a lot

00:00:52.570 --> 00:00:56.342
because it's kind of a condensed
representation of base two.

00:00:56.342 --> 00:00:58.181
What do I mean by that?

00:00:58.181 --> 00:01:03.181
Let's write out a arbitrary
number in base two.

00:01:03.500 --> 00:01:07.329
Let's say I have one, zero, one, one,

00:01:07.329 --> 00:01:12.329
zero, one, one, one, zero.

00:01:12.585 --> 00:01:16.100
This right over here is in binary

00:01:16.100 --> 00:01:17.759
and I can even write in parenthesis .

00:01:17.759 --> 00:01:20.980
This is a binary representation.

00:01:20.980 --> 00:01:25.790
I want to convert this to
hexadecimal representation.

00:01:25.790 --> 00:01:28.772
I encourage you to pause the
video and try out in your own.

00:01:28.772 --> 00:01:30.756
I'll give you a clue
on how you could think

00:01:30.756 --> 00:01:34.066
about converting directly
from base two to base 16.

00:01:34.066 --> 00:01:37.437
Think about which one over
here is in the 16s place

00:01:37.437 --> 00:01:41.137
and what is the 256 place over here.

00:01:41.137 --> 00:01:44.689
Then that might help you convert directly.

00:01:44.689 --> 00:01:46.468
Assuming you had a go at it.

00:01:46.468 --> 00:01:49.970
The really fun thing about
between base two and base 16 is

00:01:49.970 --> 00:01:52.971
you don't have to, well for any
bases, you really don't have

00:01:52.971 --> 00:01:55.852
to go through base 10
but these in particular,

00:01:55.852 --> 00:02:00.114
it's especially easy to go
convert between these two bases.

00:02:00.114 --> 00:02:02.258
The realization that you have to make is,

00:02:02.258 --> 00:02:05.923
what are the powers, which
places here are powers of 16?

00:02:05.923 --> 00:02:09.067
This right over here,
that is the ones place.

00:02:09.067 --> 00:02:11.093
One way to think about it
is all of these is going

00:02:11.093 --> 00:02:13.023
to tell you how many ones we have.

00:02:13.023 --> 00:02:16.029
Ones, twos, fours, and eights,

00:02:16.029 --> 00:02:18.302
but another way to think
about it is this is a count

00:02:18.302 --> 00:02:23.302
of ones, all the way up
to a potential of 15 ones.

00:02:25.324 --> 00:02:29.803
This could count, this is
going to be between zero,

00:02:29.803 --> 00:02:32.096
and I'm going to write it down.

00:02:32.096 --> 00:02:34.714
Actually, let me write it down in base 16.

00:02:34.714 --> 00:02:37.019
It's going to be between zero and F.

00:02:37.019 --> 00:02:39.988
It's going to be between zero and 15.

00:02:39.988 --> 00:02:42.462
It's kind of a count between the number

00:02:42.462 --> 00:02:45.474
of ones, I guess you could say.

00:02:45.474 --> 00:02:47.202
Then this is the 16s place.

00:02:47.202 --> 00:02:49.939
I'm going to do that in different color.

00:02:49.939 --> 00:02:53.914
This right over here is the 16s place.

00:02:53.914 --> 00:02:58.479
You could have between zero and 15s, 16s.

00:02:58.479 --> 00:03:01.517
This is also going to
be between zero and F,

00:03:01.517 --> 00:03:04.827
when you look at this
four digit binary numbers.

00:03:04.827 --> 00:03:07.005
Once again, this whole
thing right over here is

00:03:07.005 --> 00:03:09.755
essentially going to tell
you how many 16s you have.

00:03:09.755 --> 00:03:10.707
This whole thing is going to tell you

00:03:10.707 --> 00:03:13.258
how many ones you have.

00:03:13.258 --> 00:03:15.578
Then the next four, we could keep going,

00:03:15.578 --> 00:03:17.079
although there is only one place here.

00:03:17.079 --> 00:03:22.079
We could go, this right
over here is the 256s place.

00:03:22.128 --> 00:03:24.818
This is going to be the next four digits.

00:03:24.818 --> 00:03:26.360
They really have one right over here,

00:03:26.360 --> 00:03:29.128
but one, two, three,
and then the fourth one.

00:03:29.128 --> 00:03:34.128
This is also going to be
between zero and 15, 256s.

00:03:34.399 --> 00:03:35.584
Hopefully, that helps you a little bit.

00:03:35.584 --> 00:03:37.713
Actually, if this was
a clue, I encourage you

00:03:37.713 --> 00:03:39.157
to pause the video again (laughs) and see

00:03:39.157 --> 00:03:42.173
if you can represent this in hexadecimal.

00:03:42.173 --> 00:03:46.321
Let's try to work this thing together.

00:03:46.321 --> 00:03:47.746
How many ones do we have?

00:03:47.746 --> 00:03:48.681
What number is this?

00:03:48.681 --> 00:03:50.458
These four digits right over here.

00:03:50.458 --> 00:03:54.857
This is eight plus four plus two.

00:03:54.857 --> 00:03:59.453
So eight plus four is 12, plus two is 14.

00:03:59.453 --> 00:04:01.916
This right over here is 14.

00:04:01.916 --> 00:04:04.130
How do we represent that in hexadecimal?

00:04:04.130 --> 00:04:08.411
Well, 14 is one less than
15 so it's going to be E.

00:04:08.411 --> 00:04:10.864
This is going to be E.

00:04:10.864 --> 00:04:13.363
This is E.

00:04:13.363 --> 00:04:18.041
E is our hexadecimal
representation of the number 14

00:04:18.041 --> 00:04:22.264
comes right before our
representation of the number 15 F.

00:04:22.264 --> 00:04:25.593
Alright, now, how many 16s do we have?

00:04:25.593 --> 00:04:27.028
Let's see, I have no eights.

00:04:27.028 --> 00:04:28.972
I have a four, and I have a two.

00:04:28.972 --> 00:04:31.902
We're going to have six 16s.

00:04:31.902 --> 00:04:35.397
So we're going to have six 16s.

00:04:35.397 --> 00:04:38.075
Then, how many 256s do I have?

00:04:38.075 --> 00:04:40.959
I only have one 256.

00:04:40.959 --> 00:04:44.268
One 256.

00:04:44.268 --> 00:04:47.695
This number in hexadecimal,
and I could write that.

00:04:47.695 --> 00:04:52.695
This is in hexadecimal right
over here, is one, six, E.

00:04:53.514 --> 00:04:56.111
One, six, E.

00:04:56.111 --> 00:05:01.111
I guess you could call this 256 E,

00:05:01.188 --> 00:05:05.639
16 E.

00:05:05.639 --> 00:05:07.562
I guess 14.

00:05:07.562 --> 00:05:10.883
I (laughs) finally have to
come up with a better way

00:05:10.883 --> 00:05:12.880
of reading this hexadecimal number.

00:05:12.880 --> 00:05:16.293
If you're not curious what
number is this, because you don't

00:05:16.293 --> 00:05:18.356
have to go through decimal
just so you could comprehend it

00:05:18.356 --> 00:05:21.046
in the number system that
you're used to operating in.

00:05:21.046 --> 00:05:23.611
One that's based off from the
number of fingers you have.

00:05:23.611 --> 00:05:26.279
Feel free to do so.


Multiplying in binary

WEBVTT
Kind: captions
Language: en

00:00:00.377 --> 00:00:03.034
- I now want to show you
that the standard algorithm

00:00:03.034 --> 00:00:06.016
for multiplying numbers can also be used,

00:00:06.016 --> 00:00:07.824
it's not just limited to base 10,

00:00:07.824 --> 00:00:10.035
it can also be used, frankly
it can be used in any base,

00:00:10.035 --> 00:00:11.264
but we're going to do it in base two.

00:00:11.264 --> 00:00:13.033
And base two is especially fun,

00:00:13.033 --> 00:00:14.889
because you essentially have only to know

00:00:14.889 --> 00:00:17.531
your multiplication tables through one.

00:00:17.531 --> 00:00:21.177
So you just have to know
that zero times zero is zero,

00:00:21.177 --> 00:00:22.764
one times zero is zero,

00:00:22.764 --> 00:00:24.562
and that one times one is one.

00:00:24.562 --> 00:00:25.534
And then you're ready to go,

00:00:25.534 --> 00:00:27.772
you're ready to multiply in base two.

00:00:27.772 --> 00:00:29.056
So let's do that.

00:00:29.056 --> 00:00:31.605
So let's say we have...

00:00:32.174 --> 00:00:35.517
Let's say we take nine,

00:00:35.517 --> 00:00:39.193
so let's see, nine is
going to be one eight

00:00:39.193 --> 00:00:44.193
zero fours, zero twos and one.

00:00:45.529 --> 00:00:49.753
So this is nine, right over
here, represented in base two.

00:00:49.770 --> 00:00:53.043
And let's say we were to multiply that

00:00:54.920 --> 00:00:57.001
times seven, so we should, obviously,

00:00:57.001 --> 00:00:58.081
if we know our multiplication tables

00:00:58.081 --> 00:01:00.025
we should get 63.

00:01:00.025 --> 00:01:03.256
So seven is one; actually
I'm going to put a zero,

00:01:03.256 --> 00:01:04.601
seven would be one one one,

00:01:04.601 --> 00:01:05.881
but that's a little bit boring

00:01:05.881 --> 00:01:08.169
so let's instead do

00:01:08.828 --> 00:01:12.009
nine times five, which
we know should be 45.

00:01:12.009 --> 00:01:16.392
So this is one four, no twos, and one one.

00:01:16.392 --> 00:01:18.881
So four plus one is going to be five.

00:01:18.881 --> 00:01:20.893
So this is nine times five

00:01:20.893 --> 00:01:24.257
and we're doing it in base two.

00:01:24.747 --> 00:01:26.837
So once again, same algorithm.

00:01:26.837 --> 00:01:28.138
And actually, before I do it,

00:01:28.138 --> 00:01:29.336
I encourage you to just pause the video,

00:01:29.336 --> 00:01:31.406
and try to use the same algorithm

00:01:31.406 --> 00:01:34.133
that you've used for
base ten multiplication,

00:01:34.133 --> 00:01:36.092
use the same algorithm here

00:01:36.092 --> 00:01:37.633
and let's see if we actually do get

00:01:37.633 --> 00:01:41.157
that nine times five is 45.

00:01:41.157 --> 00:01:42.791
So I'm assuming you've had a go at it,

00:01:42.791 --> 00:01:44.516
let's work through it together.

00:01:44.516 --> 00:01:46.451
So let's start right over here

00:01:46.451 --> 00:01:48.086
in the ones place.

00:01:48.086 --> 00:01:50.073
So we multiply, one times one is one,

00:01:50.073 --> 00:01:54.099
one times zero is zero,
one times zero is zero,

00:01:54.099 --> 00:01:57.063
one times one is one.

00:01:57.063 --> 00:01:59.364
Then, we can go

00:01:59.364 --> 00:02:01.272
to the twos place

00:02:01.272 --> 00:02:04.047
and since we're multiplying
all of this times

00:02:04.047 --> 00:02:07.636
the twos place, we can throw
a zero right over here.

00:02:07.636 --> 00:02:09.246
But, of course, zero times each of these

00:02:09.246 --> 00:02:10.933
is just going to be a bunch of zeros,

00:02:10.933 --> 00:02:13.029
you know, zero times one is zero,

00:02:13.029 --> 00:02:15.065
zero times zero is zero,
zero times zero is zero,

00:02:15.065 --> 00:02:16.884
zero times one is zero.

00:02:16.884 --> 00:02:18.452
So I really didn't have to write that,

00:02:18.452 --> 00:02:20.138
but I'm just doing it so that you see

00:02:20.138 --> 00:02:22.380
that I'm using the standard algorithm.

00:02:22.380 --> 00:02:27.237
And then, we go to the fours place.

00:02:27.237 --> 00:02:28.644
And since we're in the fours place,

00:02:28.644 --> 00:02:30.308
let's put some zeros here.

00:02:30.308 --> 00:02:32.702
So zero, zero.

00:02:33.394 --> 00:02:34.918
We're essentially talking about

00:02:34.918 --> 00:02:36.234
a certain number of fours,

00:02:36.234 --> 00:02:38.043
not a certain number of ones.

00:02:38.043 --> 00:02:40.053
So one times that is just going to be

00:02:40.053 --> 00:02:44.245
one, zero, zero, one.

00:02:44.245 --> 00:02:46.215
And now we're ready to add.

00:02:46.215 --> 00:02:49.379
Now we are ready to add.

00:02:49.379 --> 00:02:52.244
So one plus a bunch of zeros is one,

00:02:52.244 --> 00:02:53.859
a bunch of zeros,

00:02:53.859 --> 00:02:55.668
that's going to be one,

00:02:55.668 --> 00:02:57.827
one plus a bunch of zeros is one,

00:02:57.827 --> 00:03:00.202
zeros, and one.

00:03:00.202 --> 00:03:01.466
And we're done.

00:03:01.466 --> 00:03:04.026
And obviously if we had more than one one

00:03:04.026 --> 00:03:06.073
in any of these places right over here,

00:03:06.073 --> 00:03:08.291
then we might have had to carry a one.

00:03:08.291 --> 00:03:12.884
But we saw that, when we
added numbers in binary.

00:03:12.884 --> 00:03:14.012
Well, let's actually verify

00:03:14.012 --> 00:03:16.059
that this is the number that
we would expect it to be.

00:03:16.059 --> 00:03:17.875
Remember, this right over
here is the ones place-

00:03:17.875 --> 00:03:19.948
so let's see, this is ones,

00:03:19.948 --> 00:03:21.347
fours, let me write it down,

00:03:21.347 --> 00:03:26.347
this is ones, twos, fours,
eights, 16s and 32s.

00:03:33.130 --> 00:03:35.119
So we have one 32,

00:03:35.119 --> 00:03:37.196
so it's going to be 32,

00:03:37.196 --> 00:03:39.987
plus one eight,

00:03:39.987 --> 00:03:41.859
and once again, we only have
to do this for ourselves

00:03:41.859 --> 00:03:43.593
because we're so used
to thinking in base 10.

00:03:43.593 --> 00:03:44.953
If we were thinking in base two,

00:03:44.953 --> 00:03:46.548
you'd say, "Oh, I know
what this number is,"

00:03:46.548 --> 00:03:48.228
and you would have some name for it,

00:03:48.228 --> 00:03:49.909
probably different than 45,

00:03:49.909 --> 00:03:52.100
because 45, the name
essentially is kind of

00:03:52.100 --> 00:03:55.268
a base 10 name and we'd
probably have better names

00:03:55.268 --> 00:03:56.310
in terms of base two.

00:03:56.310 --> 00:03:58.286
Somebody should actually do that,

00:04:01.485 --> 00:04:03.427
that would be a fun project.

00:04:03.427 --> 00:04:05.147
So let's see, we have 32,

00:04:05.147 --> 00:04:07.364
plus eight,

00:04:07.364 --> 00:04:11.652
plus four, plus four,

00:04:11.652 --> 00:04:15.174
plus one, plus one.

00:04:15.174 --> 00:04:18.948
Which is indeed equal to 45.


Adding in binary

WEBVTT
Kind: captions
Language: en

00:00:00.653 --> 00:00:02.033
Let's see what it's like to add

00:00:02.033 --> 00:00:04.500
multi-digit numbers in binary.

00:00:04.500 --> 00:00:09.100
So let's say I had the
number one, zero, one, one.

00:00:09.100 --> 00:00:14.100
And to that I wanted to add
the number one, one, one.

00:00:14.500 --> 00:00:15.500
What is that going to be?

00:00:15.500 --> 00:00:17.300
And I encourage you to pause this video,

00:00:17.300 --> 00:00:19.100
and try to work through it on your own.

00:00:20.084 --> 00:00:22.300
So the key here is this is
just a standard algorithm,

00:00:22.300 --> 00:00:23.433
and we're adding numbers.

00:00:23.433 --> 00:00:24.767
But remind yourself you're only restricted

00:00:24.767 --> 00:00:26.833
to the zero and one digits.

00:00:26.833 --> 00:00:28.433
So let's do that.

00:00:28.433 --> 00:00:30.167
So we have one plus one.

00:00:30.501 --> 00:00:31.700
Well, you might say that's two

00:00:31.700 --> 00:00:32.500
but you can't write a two here.

00:00:32.500 --> 00:00:34.034
We can only write a zero or a one.

00:00:34.034 --> 00:00:36.540
And we have to remind ourselves that two

00:00:36.633 --> 00:00:41.633
in decimal is represented
as one zero in binary.

00:00:43.566 --> 00:00:46.233
It's one two and zero ones.

00:00:46.233 --> 00:00:49.367
So you write the zero,
zero ones, and one two.

00:00:49.367 --> 00:00:51.301
You essentially carry the one.

00:00:51.301 --> 00:00:55.600
And now you have one plus one, plus one.

00:00:55.766 --> 00:00:57.233
Well that's going to be three,

00:00:57.233 --> 00:00:58.566
but you can't write three.

00:00:58.566 --> 00:01:03.566
Once again, three in decimal
is equal to one one in binary.

00:01:05.967 --> 00:01:09.666
It's one two, plus, one one is three.

00:01:09.767 --> 00:01:12.033
And so you just have to realize that.

00:01:12.033 --> 00:01:13.900
So one plus one plus one is three,

00:01:13.900 --> 00:01:15.633
which in binary is one one.

00:01:15.633 --> 00:01:20.233
So you write one in the ones
place and then you carry it.

00:01:20.233 --> 00:01:23.933
And then we want to add what
we're doing in the fours place.

00:01:24.100 --> 00:01:26.300
And over here, I have a one and a one,

00:01:26.300 --> 00:01:27.967
which is going to be two

00:01:27.967 --> 00:01:30.566
which we already know we
represent as one zero.

00:01:30.566 --> 00:01:33.834
So you write zero here and
then you carry the one.

00:01:33.834 --> 00:01:36.566
And then once again, one and one is two.

00:01:36.566 --> 00:01:39.966
Which is one zero in
binary, and we're done.

00:01:39.966 --> 00:01:41.633
We just added these numbers.

00:01:41.633 --> 00:01:42.900
Now, you might be saying,

00:01:42.900 --> 00:01:46.600
"Hey, let me make sure
this actually makes sense."

00:01:46.700 --> 00:01:48.636
And we can verify that
this actually makes sense

00:01:48.636 --> 00:01:50.700
by thinking about what these numbers are.

00:01:50.700 --> 00:01:53.966
Remember, this is going to be
this number right over here.

00:01:53.966 --> 00:01:56.033
If we wanted to think of it in decimal.

00:01:56.033 --> 00:01:58.233
It's going to be one plus two,

00:01:59.356 --> 00:02:01.500
plus not four but plus eight.

00:02:02.915 --> 00:02:06.233
So this is 11 if we were
to write it in decimal.

00:02:06.233 --> 00:02:11.233
And this right over here
is one plus two plus four.

00:02:15.038 --> 00:02:18.900
Which is equal to seven if we
were to write it in decimal.

00:02:18.900 --> 00:02:20.633
And now what's this right over here?

00:02:20.802 --> 00:02:23.300
This is equal to one two.

00:02:23.300 --> 00:02:26.866
This is four, eight, 16.

00:02:27.500 --> 00:02:28.567
So and this right over here.

00:02:28.567 --> 00:02:31.900
So we have one 16 plus one two.

00:02:31.900 --> 00:02:36.900
Is going to be equal to 18
if we write it in decimal.

00:02:38.233 --> 00:02:41.800
And we see that, 11 plus
seven is indeed equal to 18.

00:02:41.900 --> 00:02:46.533
11 plus seven is indeed equal to 18.


Converting from decimal to hexadecimal representation

WEBVTT
Kind: captions
Language: en

00:00:00.397 --> 00:00:02.195
-[Voiceover]Let's now
try to convert a number

00:00:02.195 --> 00:00:07.120
from the decimal system
to the hexadecimal system.

00:00:07.120 --> 00:00:11.290
Let's say we want to
convert the number 2,000.

00:00:11.290 --> 00:00:13.326
This is written in decimal form.

00:00:13.326 --> 00:00:15.291
Let's say we want to write
it in hexadecimal form.

00:00:15.291 --> 00:00:17.835
Like always, I encourage
you to pause this video

00:00:17.835 --> 00:00:21.377
and try to work through it on your own.

00:00:21.377 --> 00:00:25.612
The key here is to break this down into

00:00:25.612 --> 00:00:28.303
multiples of powers of 16.

00:00:28.303 --> 00:00:30.518
Let's just write down
our powers of 16 here.

00:00:30.518 --> 00:00:32.781
16 to the zero power is one.

00:00:32.781 --> 00:00:34.962
16 to the first power is 16.

00:00:34.962 --> 00:00:39.168
16 squared is 256.

00:00:39.168 --> 00:00:43.264
16 to the third power is 4,096.

00:00:43.264 --> 00:00:45.305
We've gotten more than large enough.

00:00:45.305 --> 00:00:47.199
Let's start decomposing.

00:00:47.199 --> 00:00:51.530
What's the largest power of 16 that is

00:00:51.530 --> 00:00:54.119
less than or equal to 2,000?

00:00:54.119 --> 00:00:56.430
It's going to be 256.

00:00:56.430 --> 00:00:59.246
How many times does 256 go into 2,000?

00:00:59.246 --> 00:01:02.379
I'll get a calculator out for that.

00:01:03.298 --> 00:01:08.298
2,000 divided by, I want
to figure out how many

00:01:09.565 --> 00:01:13.118
times 256 goes into it, divided by 256.

00:01:13.118 --> 00:01:16.330
It goes seven times, plus a little bit.

00:01:16.330 --> 00:01:17.911
What's going to be left over?

00:01:17.911 --> 00:01:22.911
2,000 minus seven times 256 is equal to,

00:01:25.542 --> 00:01:27.498
you're going to have 208 left over.

00:01:27.498 --> 00:01:28.867
Let me write that.

00:01:28.867 --> 00:01:31.349
It's going to be seven times 256.

00:01:31.349 --> 00:01:36.349
Seven, times 256, plus 208 left over.

00:01:41.230 --> 00:01:44.558
Now let's see if we can
decompose 208 into powers of 16.

00:01:44.558 --> 00:01:46.167
What's the largest power of 16 that is

00:01:46.167 --> 00:01:48.169
less than or equal to this?

00:01:48.169 --> 00:01:49.750
Well it's just going to be 16.

00:01:49.750 --> 00:01:53.499
So how many 16s go into 208?

00:01:53.499 --> 00:01:56.977
208, I'll just take that, divided by 16.

00:01:56.977 --> 00:01:59.131
13, exactly 13.

00:01:59.131 --> 00:02:00.496
We lucked out.

00:02:00.496 --> 00:02:03.062
So this is exactly 13 times 16.

00:02:03.062 --> 00:02:04.703
Let's just write that.

00:02:04.703 --> 00:02:09.703
This right over here
is exactly 13 times 16.

00:02:11.079 --> 00:02:14.131
Now, we have broken this
down into powers of...

00:02:14.131 --> 00:02:17.515
We have broken this down into
multiples of powers of 16.

00:02:17.515 --> 00:02:19.799
Now we're ready to
write it in hexadecimal.

00:02:19.799 --> 00:02:23.064
We just have to remind
ourselves about the place value.

00:02:23.064 --> 00:02:27.456
This right over here, this is
going to be the ones place.

00:02:27.456 --> 00:02:31.522
Then we're going to have the 16s place.

00:02:31.522 --> 00:02:36.522
Then we're going to have the 256s place.

00:02:39.612 --> 00:02:41.797
We know how many 256s we have.

00:02:41.797 --> 00:02:44.748
We have seven 256s.

00:02:44.748 --> 00:02:47.833
We have zero ones.

00:02:47.833 --> 00:02:49.746
And how many 16s do we have?

00:02:49.746 --> 00:02:51.555
Well we have 13 16s.

00:02:51.555 --> 00:02:53.142
Well what's the digit for 13?

00:02:53.142 --> 00:02:57.321
We can just remind
ourselves that we obviously

00:02:57.321 --> 00:03:00.543
have zero through nine and then we have

00:03:00.543 --> 00:03:05.543
A is equal to 10, B is equal
to 11, C is equal to 12,

00:03:07.405 --> 00:03:11.479
D is equal to 13 and of
course E is equal to 14

00:03:11.479 --> 00:03:14.033
and F is equal to 15.

00:03:14.033 --> 00:03:15.898
But we care about 13.

00:03:15.898 --> 00:03:18.593
We have 13 16s.

00:03:20.070 --> 00:03:22.528
So we put a D right over here.

00:03:22.528 --> 00:03:25.242
So the number "7D0" in hexadecimals...

00:03:25.242 --> 00:03:26.875
Let me write this down,
this is interesting.

00:03:26.875 --> 00:03:31.875
2,000 in decimal form
is equal to "7D0" when

00:03:36.396 --> 00:03:39.375
it is written in hexadecimal form.

00:03:39.375 --> 00:03:42.679
Hope you found that interesting.


Converting larger number from decimal to binary

WEBVTT
Kind: captions
Language: en

00:00:00.877 --> 00:00:02.720
- [Voiceover] Let's now
see if we can convert

00:00:02.720 --> 00:00:06.221
a larger decimal representation to binary.

00:00:06.221 --> 00:00:10.739
So let's say that we have the number 114

00:00:10.739 --> 00:00:12.673
and this is its decimal representation.

00:00:12.673 --> 00:00:14.106
See if you can pause the video

00:00:14.106 --> 00:00:17.781
and rewrite this in its
binary representation.

00:00:18.555 --> 00:00:21.524
So I'm assuming you have at least tried.

00:00:21.524 --> 00:00:23.731
Now we can work on this together.

00:00:23.731 --> 00:00:26.277
So as always, we just
want to decompose this

00:00:26.277 --> 00:00:28.404
into the sum of powers of two.

00:00:28.404 --> 00:00:31.306
You can always decompose
this and any number

00:00:31.306 --> 00:00:34.495
into a sum of powers of two.

00:00:34.495 --> 00:00:37.262
We can once again just remind
ourselves the powers of two.

00:00:37.262 --> 00:00:41.051
Two to the zero is one, two
to the first power is two,

00:00:41.051 --> 00:00:43.437
two to the third power is eight,

00:00:43.437 --> 00:00:45.207
two to the fourth power is 16,

00:00:45.207 --> 00:00:50.207
two to the fifth power is 32,
two to the sixth power is 64,

00:00:51.318 --> 00:00:54.086
two to the seventh power is 128,

00:00:54.086 --> 00:00:55.456
and that gets us large enough.

00:00:55.456 --> 00:00:57.827
We've already gotten larger
than the number here.

00:00:57.827 --> 00:01:01.398
So let's see, 114 can be rewritten as,

00:01:01.398 --> 00:01:03.129
the largest power of two that is less than

00:01:03.129 --> 00:01:06.745
or equal to that is 64.

00:01:06.899 --> 00:01:11.003
So we can rewrite it as 64 plus,

00:01:11.003 --> 00:01:14.891
what's going to be left over, 64 plus 50,

00:01:14.891 --> 00:01:16.678
now we're going to have to rewrite 50

00:01:16.678 --> 00:01:18.952
as the sum of the powers of two.

00:01:18.952 --> 00:01:23.952
And let's see, 50 can be
rewritten as the largest power

00:01:26.420 --> 00:01:30.338
of two that is less than
or equal to 50 is 32.

00:01:30.338 --> 00:01:35.338
So we can rewrite it as 32 plus,

00:01:36.141 --> 00:01:41.141
32 plus 18 and now we have to rewrite 18

00:01:42.099 --> 00:01:44.727
as the sum of some powers of two.

00:01:44.727 --> 00:01:48.263
Well 18, the largest power of two that is

00:01:48.263 --> 00:01:51.570
less than or equal to 18 is 16.

00:01:52.202 --> 00:01:56.846
So this is going to be
16 and then 16 plus,

00:01:56.846 --> 00:02:00.142
16 plus two, and lucky for
us, two, well I guess not

00:02:00.142 --> 00:02:02.245
that lucky, we had to do this a good bit,

00:02:02.245 --> 00:02:07.245
two is a power of two,
so we can rewrite this,

00:02:08.350 --> 00:02:13.350
114 is equal to, lemme give
myself enough real estate here,

00:02:14.109 --> 00:02:19.109
is equal to 64 plus 32 plus 16 plus 2.

00:02:27.907 --> 00:02:31.871
I've just written 114 as
the sum of powers of two.

00:02:31.871 --> 00:02:34.823
And once again we can read
this as one 64 plus one 32

00:02:34.823 --> 00:02:38.185
plus one 16 plus one two.

00:02:38.185 --> 00:02:41.667
Now we're ready to really
rewrite this in binary.

00:02:41.667 --> 00:02:43.757
Let's just write the
different place values.

00:02:43.757 --> 00:02:46.754
So remember, this is the
ones, this right over here

00:02:46.754 --> 00:02:51.171
is the one's place value or
the one's place, I should say.

00:02:51.171 --> 00:02:53.028
Lemme just do this in a different color.

00:02:53.028 --> 00:02:58.028
So this is going to be the
ones, then we're gonna have

00:02:59.461 --> 00:03:04.461
the twos, then we're gonna have the fours,

00:03:06.612 --> 00:03:10.444
the four's place, then we're gonna have

00:03:10.444 --> 00:03:15.444
the eight's place, then we're
gonna have the 16's place,

00:03:19.924 --> 00:03:22.796
tells you how many 16s are in this number.

00:03:22.796 --> 00:03:27.796
Then we're gonna have the 32's place,

00:03:28.042 --> 00:03:29.991
how many 32s are in this number.

00:03:29.991 --> 00:03:33.689
And then you're going
to have the 64's place.

00:03:35.424 --> 00:03:38.582
So how many ones do we have
here? We have zero ones.

00:03:38.582 --> 00:03:42.109
How many twos do we have?
Well, we have one two,

00:03:42.109 --> 00:03:43.505
you're going to have one
of something or zero,

00:03:43.505 --> 00:03:46.636
there's only two digits if
you're thinking in binary.

00:03:46.636 --> 00:03:50.314
Now we have no fours, no fours here,

00:03:50.314 --> 00:03:52.738
and no eights, no eights.

00:03:52.738 --> 00:03:57.738
We have a 16, we have
a 32 and we have a 64.

00:04:02.323 --> 00:04:04.984
So in binary the number 114 in decimal?

00:04:04.984 --> 00:04:06.754
Would be in binary, would be written as

00:04:06.754 --> 00:04:10.754
one, one, one, zero, zero, one, zero.


Converting from decimal to binary

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.633
Let's see if we can get some experience

00:00:02.633 --> 00:00:06.633
converting from a decimal representation

00:00:07.263 --> 00:00:11.467
to a binary representation.

00:00:12.166 --> 00:00:15.156
Let's start with the fairly
straightforward example

00:00:15.156 --> 00:00:16.361
with a fairly low number.

00:00:16.607 --> 00:00:17.566
Let's see if we can convert

00:00:17.566 --> 00:00:21.800
the number 13 in decimal to binary.

00:00:21.833 --> 00:00:23.367
And I encourage you to pause the video,

00:00:23.367 --> 00:00:25.867
and try to work through
it out on your own.

00:00:26.300 --> 00:00:28.366
So I'm assuming you had a go at it.

00:00:28.366 --> 00:00:30.634
So the key here is to see if you can

00:00:30.634 --> 00:00:34.700
deconstruct the number 13
as the sum of powers of two.

00:00:34.700 --> 00:00:36.168
And then it becomes very straightforward

00:00:36.168 --> 00:00:37.900
to represent it in binary.

00:00:37.900 --> 00:00:39.500
Because in binary, you're
essentially saying,

00:00:39.500 --> 00:00:43.400
well what powers of two do you
need to make up this number.

00:00:43.992 --> 00:00:45.900
So let's just write
the powers of two here,

00:00:45.900 --> 00:00:47.633
just to remind ourselves.

00:00:47.633 --> 00:00:50.300
And I'll go until we go right above 13.

00:00:50.300 --> 00:00:52.500
So two to the zero is equal to one,

00:00:53.033 --> 00:00:54.967
Two to the first is equal to 2.

00:00:55.290 --> 00:00:57.767
Two squared is equal to four.

00:00:57.767 --> 00:01:00.700
Two to the third is equal to eight.

00:01:00.700 --> 00:01:03.233
Two to the fourth is equal to 16.

00:01:03.233 --> 00:01:04.833
So now, I'm above 13,

00:01:04.833 --> 00:01:09.833
so I have all the powers of two
that I need to construct 13.

00:01:11.089 --> 00:01:12.766
So what's the largest power of two,

00:01:13.427 --> 00:01:15.233
that is less than or equal to 13?

00:01:16.063 --> 00:01:18.567
Well, 16's too large,
well it would be eight.

00:01:18.567 --> 00:01:22.400
So I could rewrite 13 as eight plus five.

00:01:23.567 --> 00:01:25.433
Now five is not a power of two,

00:01:25.433 --> 00:01:27.433
so I have to keep deconstructing that.

00:01:27.433 --> 00:01:29.101
What's the largest power of two,

00:01:29.101 --> 00:01:31.933
that is less than or equal to five?

00:01:32.500 --> 00:01:34.367
We see it right over here, it's four.

00:01:34.367 --> 00:01:35.900
So let me rewrite that,

00:01:35.900 --> 00:01:38.500
it's eight plus, instead of writing five,

00:01:38.500 --> 00:01:41.933
I'll write four plus one,

00:01:42.100 --> 00:01:44.833
and then the good thing is
at one as a power of two

00:01:44.833 --> 00:01:46.033
we already see is the largest power of two

00:01:46.033 --> 00:01:47.700
that is less than or equal to it is one.

00:01:47.700 --> 00:01:49.367
This already is a power of two.

00:01:49.873 --> 00:01:51.576
So I now have rewritten this

00:01:51.576 --> 00:01:55.600
as the sum of powers of two.

00:01:57.233 --> 00:02:00.034
Notice this is two to the third power.

00:02:03.279 --> 00:02:08.279
This is two squared, and this
is two to the zero power.

00:02:10.330 --> 00:02:11.900
Or I could write it like this.

00:02:11.900 --> 00:02:16.800
I have 1 eight, clearly.

00:02:16.966 --> 00:02:21.966
I have 1 four, and I have 1 one.

00:02:24.501 --> 00:02:28.434
So I can add these two, I
can add these three together.

00:02:28.434 --> 00:02:32.166
13 could be considered 1
eight plus 1 four plus 1 one.

00:02:32.433 --> 00:02:34.100
Why is that helpful?

00:02:34.100 --> 00:02:37.100
Well now, let's go in to binary mode.

00:02:37.100 --> 00:02:41.228
And think about what each of
the place values represent.

00:02:41.288 --> 00:02:46.233
So, this is the ones place,
that's the ones place.

00:02:46.233 --> 00:02:49.833
And then we can go to the twos place.

00:02:51.417 --> 00:02:52.433
Every time we go to the left,

00:02:52.433 --> 00:02:54.700
each place we multiply by two.

00:02:54.700 --> 00:02:56.433
It's the next power of two.

00:02:56.766 --> 00:03:01.766
Then we go to the fours
place, fours place.

00:03:02.766 --> 00:03:06.866
And then we go the eights place, eights.

00:03:09.966 --> 00:03:12.234
In binary I only have
two digits, zero and one.

00:03:12.234 --> 00:03:15.566
So I either have zero of a
place or I have one of it.

00:03:15.904 --> 00:03:17.235
So let's go through it.

00:03:17.235 --> 00:03:18.967
How many ones do I have?

00:03:18.967 --> 00:03:21.766
Well I have 1 one, so I write that there.

00:03:21.766 --> 00:03:23.567
How many twos do I have?

00:03:23.567 --> 00:03:25.766
Well in this representation
I don't have any twos.

00:03:26.058 --> 00:03:28.100
I have an eight, I have a four and a one.

00:03:28.100 --> 00:03:29.766
So I'm gonna put I have 0 twos.

00:03:30.627 --> 00:03:32.351
How many fours do I have?

00:03:32.351 --> 00:03:33.766
Well I have 1 four.

00:03:34.658 --> 00:03:36.500
And how many eights do I have?

00:03:36.500 --> 00:03:38.666
Well I have 1 eight.

00:03:38.766 --> 00:03:43.766
So 13, or 13 which is a decimal number.

00:03:45.033 --> 00:03:50.033
If I were to write it in
binary, is one one zero one.


Hexadecimal number system

WEBVTT
Kind: captions
Language: en

00:00:00.649 --> 00:00:02.848
- [Voiceover] We're all
familiar with the base 10

00:00:02.848 --> 00:00:05.332
number system, were
often called the decimal

00:00:05.332 --> 00:00:09.541
number system, where we have 10 digits

00:00:09.541 --> 00:00:13.308
Zero, one, two, three, four,
five, six, seven, eight, nine.

00:00:13.308 --> 00:00:14.585
Now, we started to see that we can

00:00:14.585 --> 00:00:16.304
have alternate number system.

00:00:16.304 --> 00:00:18.072
We can have a base two number system,

00:00:18.072 --> 00:00:19.616
or it's the binary number system,

00:00:19.616 --> 00:00:23.385
where instead of 10 digits
you only have two digits.

00:00:23.385 --> 00:00:25.131
Each place, instead of being a power

00:00:25.131 --> 00:00:28.099
of ten is going to be a power of two.

00:00:28.099 --> 00:00:30.402
Now you can imagine that
we can keep extending this.

00:00:30.402 --> 00:00:33.861
We can extend to base three,
four, five, six, seven,

00:00:33.861 --> 00:00:37.909
eight, nine, or we could even go above 10.

00:00:37.909 --> 00:00:42.909
What I want to show you
in this video is a fairly,

00:00:43.932 --> 00:00:46.569
heavily used number system
that is larger than,

00:00:46.569 --> 00:00:51.358
or that has more digits than
base 10, and that base is 16.

00:00:51.358 --> 00:00:56.358
Base 16, often called the hexadecimal.

00:00:57.045 --> 00:01:01.581
Hexadecimal number system.

00:01:01.581 --> 00:01:04.703
As you can imagine, instead of only

00:01:04.703 --> 00:01:08.858
having 10 digits, it is going to have 16.

00:01:08.858 --> 00:01:11.104
What are those digits going to be?

00:01:11.104 --> 00:01:13.082
As we'll see, instead of
the place is being powers

00:01:13.082 --> 00:01:16.731
of two or powers of ten,
there will be powers of 16.

00:01:16.731 --> 00:01:19.890
Let's see, we can reuse
the existing 10 digits

00:01:19.890 --> 00:01:23.133
from the decimal number system.

00:01:23.133 --> 00:01:27.100
We can reuse zero, one, two, three,

00:01:27.100 --> 00:01:32.100
four, five, six, seven, eight, nine,

00:01:32.783 --> 00:01:37.148
but then we're going to need
to have six more digits.

00:01:37.148 --> 00:01:41.885
The convention is to use
the first six letters.

00:01:41.885 --> 00:01:46.885
A, B, C, D, E, and F.

00:01:47.467 --> 00:01:48.400
You might say this is crazy.

00:01:48.400 --> 00:01:52.016
These are letters, not numbers,
but remember these are just

00:01:52.016 --> 00:01:55.457
arbitrary squiggles of
ink on a piece of paper.

00:01:55.457 --> 00:01:57.464
These are just arbitrary symbols

00:01:57.464 --> 00:01:59.989
that we're grown to associate with things.

00:01:59.989 --> 00:02:03.696
You're grown to associate
this symbol right over here

00:02:03.696 --> 00:02:06.474
with eight thing, with
the word eight which you

00:02:06.474 --> 00:02:09.905
associate with when you
see that many objects.

00:02:09.905 --> 00:02:13.315
If you're thinking in hexadecimal,
this isn't the letter A

00:02:13.315 --> 00:02:15.590
that makes you want to
say "ah", or the letter B

00:02:15.590 --> 00:02:17.632
that makes you want to say "bababababa".

00:02:17.632 --> 00:02:20.386
This is, literally, this represents

00:02:20.386 --> 00:02:23.149
if you had 10 things laying around.

00:02:23.149 --> 00:02:25.674
You would say, "I have
A things over there."

00:02:25.674 --> 00:02:28.420
If you have 11, you'd say,
"I have B things over there."

00:02:28.420 --> 00:02:30.616
12, C things.

00:02:30.616 --> 00:02:32.091
13, instead of saying,
"I have 13 things there",

00:02:32.091 --> 00:02:33.840
"I have D things there."

00:02:33.840 --> 00:02:35.450
Instead of saying, "I have 14",

00:02:35.450 --> 00:02:37.572
you could say, "I can
have E things there."

00:02:37.572 --> 00:02:38.880
Instead of saying, "I have 15",

00:02:38.880 --> 00:02:41.598
you could say, "I have F things there."

00:02:41.598 --> 00:02:43.173
Now, how does that help?

00:02:43.173 --> 00:02:44.607
Well, let's see if we can represent

00:02:44.607 --> 00:02:48.585
the same number 231, or 231 in decimal.

00:02:48.585 --> 00:02:53.585
If we can represent that
same number in hexadecimal.

00:02:54.627 --> 00:02:56.499
What I'll do is I'll give
you what the number is,

00:02:56.499 --> 00:02:58.065
and then I'll show you how we convert it.

00:02:58.065 --> 00:02:59.409
I'll show you the place value,

00:02:59.409 --> 00:03:01.580
and I'll show you how we convert it.

00:03:01.580 --> 00:03:04.750
231 in hexadecimal.

00:03:04.750 --> 00:03:09.750
231 in hexadecimal is the number E seven.

00:03:12.835 --> 00:03:13.997
E seven.

00:03:13.997 --> 00:03:15.588
Once again, you're
like, "This looks crazy.

00:03:15.588 --> 00:03:17.575
"This is like I'm playing
like battleship or something."

00:03:17.575 --> 00:03:19.257
What's E seven?

00:03:19.257 --> 00:03:21.333
This is a number and I would say yes.

00:03:21.333 --> 00:03:22.679
This is a number.

00:03:22.679 --> 00:03:24.095
Now remember, base 16.

00:03:24.095 --> 00:03:26.781
What are these place values represent?

00:03:26.781 --> 00:03:30.701
This first place represents
16 to the zero power

00:03:30.701 --> 00:03:33.504
or still represents the ones place.

00:03:33.504 --> 00:03:34.925
This is the ones place.

00:03:34.925 --> 00:03:36.651
This is seven ones.

00:03:36.651 --> 00:03:39.351
Now, what is this place here represents?

00:03:39.351 --> 00:03:42.426
Well, in base 10, that
was 10 to the first power.

00:03:42.426 --> 00:03:44.470
In base two, that was
two to the first power.

00:03:44.470 --> 00:03:47.103
On base 16, this is going to be,

00:03:47.103 --> 00:03:47.979
I'll leave those there,

00:03:47.979 --> 00:03:52.560
in base 16, this is going
to be 16 to the first power.

00:03:52.560 --> 00:03:55.027
This is literally, well let me write out

00:03:55.027 --> 00:03:58.442
the word, this is literally sixteens.

00:03:58.442 --> 00:04:03.442
This is E sixteens plus seven ones.

00:04:04.140 --> 00:04:05.604
Let me write that down.

00:04:05.604 --> 00:04:10.604
This is E sixteens plus seven ones.

00:04:15.444 --> 00:04:18.426
That's what this number represents.

00:04:18.426 --> 00:04:20.817
Now, if we want to start rewriting this

00:04:20.817 --> 00:04:23.064
or re-conceptualizing it in our decimal

00:04:23.064 --> 00:04:24.882
number system, what is E sixteens?

00:04:24.882 --> 00:04:29.758
Well, the E if we think
in decimal, E is 14.

00:04:29.758 --> 00:04:33.920
E is 14.

00:04:33.920 --> 00:04:36.166
This is really, we can
really think of this

00:04:36.166 --> 00:04:37.941
if you want to think them decimals.

00:04:37.941 --> 00:04:40.748
This is 14 sixteens.

00:04:40.748 --> 00:04:42.270
It's 14 sixteens.

00:04:42.270 --> 00:04:45.530
Well, that's just the
same thing as 14 times 16.

00:04:45.530 --> 00:04:49.077
14 times 16 is equal to 224.

00:04:49.077 --> 00:04:51.503
Maybe I should do that in same color.

00:04:51.503 --> 00:04:56.503
This thing right over
here is going to be 224.

00:04:57.375 --> 00:05:02.375
14 sixteens, 14 times 16
is 224 plus seven ones.

00:05:05.652 --> 00:05:10.652
Well, 224 plus 7 is
going to be give you 231.

00:05:10.674 --> 00:05:11.747
Hopefully, you can appreciate it.

00:05:11.747 --> 00:05:13.957
You can represent the same quantity

00:05:13.957 --> 00:05:17.704
in any of these different number systems.

00:05:17.704 --> 00:05:20.967
In any number that you
can represent in decimal,

00:05:20.967 --> 00:05:24.125
you can also represent
that number in binary,

00:05:24.125 --> 00:05:29.125
or in hexadecimal, or in
base three, or in base 60,

00:05:29.454 --> 00:05:33.557
or in base 31, whatever you want to do.

00:05:33.557 --> 00:05:35.638
You might have noticed the pattern.

00:05:35.638 --> 00:05:38.082
The more symbols that we have,

00:05:38.082 --> 00:05:41.014
so in base 16, you have 16 symbols,

00:05:41.014 --> 00:05:43.025
the less place values we need

00:05:43.025 --> 00:05:44.815
to represent the same quantity.

00:05:44.815 --> 00:05:46.597
One way to think about it is each

00:05:46.597 --> 00:05:49.126
of the places are
containing more information.

00:05:49.126 --> 00:05:51.722
This is one of 16 characters.

00:05:51.722 --> 00:05:53.828
While this over here is
only one of two characters.

00:05:53.828 --> 00:05:55.775
This is one of ten characters.

00:05:55.775 --> 00:05:58.586
The more symbols that
you have, the more digits

00:05:58.586 --> 00:06:01.525
that you could put in each
place, the less places

00:06:01.525 --> 00:06:06.249
that you need to represent
a given quantity.

00:06:06.249 --> 00:06:08.993
Another way to think about it
is when you have a high base,

00:06:08.993 --> 00:06:13.091
like base 16, as you take
powers of 16, the next

00:06:13.091 --> 00:06:18.091
place right over here would be 16 squared,

00:06:18.162 --> 00:06:23.162
which, of course, is two
hundred and, wait a minute, 256.

00:06:23.937 --> 00:06:27.408
You're clearly going to be able
to represent bigger numbers

00:06:27.408 --> 00:06:29.746
faster, I guess you could
say, or with less digits.

00:06:29.746 --> 00:06:32.083
It's just an interesting thing to observe.

00:06:32.083 --> 00:06:34.440
But hopefully, you're
going to kick out of,

00:06:34.440 --> 00:06:36.082
as much of a kick out of base 16

00:06:36.082 --> 00:06:38.291
as I do, and it's actually useful.

00:06:38.291 --> 00:06:42.430
This actually is used if
you look at most web pages.

00:06:42.430 --> 00:06:44.189
If you look at the actual code for there,

00:06:44.189 --> 00:06:47.479
or I guess you could say the
formatting line, the HTML

00:06:47.479 --> 00:06:50.186
for the webpage, when they specify colors,

00:06:50.186 --> 00:06:52.776
they tend to specify in hexadecimal.

00:06:52.776 --> 00:06:55.039
That's because they're specifying
the colors, the intensity

00:06:55.039 --> 00:07:00.039
of the red, the green, or the
blue, between zero and 255.

00:07:00.493 --> 00:07:04.035
Two digits of hexadecimal
are perfect for that,

00:07:04.035 --> 00:07:09.035
because if you think
about it, what is F F?

00:07:11.895 --> 00:07:14.913
What would this be if you
rewrite it in the decimal number

00:07:14.913 --> 00:07:17.948
system, and I encourage you
after this video is done,

00:07:17.948 --> 00:07:21.122
I encourage you to do that to
figure that out on your own.

00:07:21.122 --> 00:07:22.594
If you really want to do something fun,

00:07:22.594 --> 00:07:23.913
let me give you another one.

00:07:23.913 --> 00:07:28.686
Try to figure out what A F three is.

00:07:28.686 --> 00:07:30.205
Again, this isn't very specialized.

00:07:30.205 --> 00:07:31.347
I just wanted to give you another

00:07:31.347 --> 00:07:33.531
interesting thing to work on.


Introduction to number systems and binary

WEBVTT
Kind: captions
Language: en

00:00:00.596 --> 00:00:01.813
- [Voiceover] For as long as human

00:00:01.813 --> 00:00:03.693
beings have been around we've

00:00:03.693 --> 00:00:06.003
been counting things, and we've been

00:00:06.003 --> 00:00:08.038
looking for ways to keep track and

00:00:08.038 --> 00:00:10.813
represent those things that we counted.

00:00:10.944 --> 00:00:12.547
So, for example if you were

00:00:12.547 --> 00:00:14.519
an early human and you were

00:00:14.519 --> 00:00:16.046
trying to keep track of the days

00:00:16.046 --> 00:00:17.486
since it last rained you might say

00:00:17.486 --> 00:00:20.096
okay let's see it didn't rain today so

00:00:20.096 --> 00:00:24.846
one day has gone by, and we now use

00:00:24.846 --> 00:00:26.055
the word one, but they might have

00:00:26.055 --> 00:00:27.353
not used it back then.

00:00:27.744 --> 00:00:29.171
Now another day goes by.

00:00:29.544 --> 00:00:30.861
Then another day goes by.

00:00:31.128 --> 00:00:32.635
Then another day goes by.

00:00:33.047 --> 00:00:34.378
Another day goes by.

00:00:34.727 --> 00:00:36.153
Another day goes by.

00:00:36.411 --> 00:00:38.276
Another day goes by, then it rained.

00:00:38.832 --> 00:00:40.271
And so when his friend comes

00:00:40.271 --> 00:00:41.296
he says, "Well, how long has it been

00:00:41.296 --> 00:00:42.603
since we last rained."

00:00:43.015 --> 00:00:45.039
Well you would say, "Well, this is how

00:00:45.039 --> 00:00:46.183
many days it's been."

00:00:46.183 --> 00:00:47.326
And your friend would say, "Okay,

00:00:47.326 --> 00:00:49.219
I think I have a general sense of that."

00:00:49.520 --> 00:00:50.555
And at some point they probably

00:00:50.555 --> 00:00:52.040
realized that it's useful to have names

00:00:52.040 --> 00:00:52.962
for these.

00:00:52.966 --> 00:00:55.679
So they would call this one, two, three,

00:00:55.710 --> 00:00:57.709
four, five, six, seven.

00:00:57.709 --> 00:00:59.023
Obviously every language in the world

00:00:59.023 --> 00:01:00.418
has different names for these.

00:01:00.418 --> 00:01:01.453
I'm sure there are lost languages

00:01:01.453 --> 00:01:03.437
that had other names for them.

00:01:03.711 --> 00:01:05.375
But very quickly you start to

00:01:05.375 --> 00:01:07.246
realize that this is a pretty bulky

00:01:07.246 --> 00:01:10.022
way of representing numbers.

00:01:10.451 --> 00:01:12.516
One it takes a long time to write down.

00:01:13.033 --> 00:01:14.666
It takes up a lot of space,

00:01:14.968 --> 00:01:16.336
and then later if someone wants to

00:01:16.336 --> 00:01:17.977
read the number they have to sit here

00:01:17.977 --> 00:01:18.798
and count.

00:01:18.976 --> 00:01:20.587
It's hard enough with seven,

00:01:20.587 --> 00:01:21.675
but you could imagine if there were

00:01:21.675 --> 00:01:24.932
what we call 27 of it, or 1000 of it.

00:01:24.932 --> 00:01:26.959
Then it would take up, possibly, a whole

00:01:26.959 --> 00:01:28.731
page and even when you counted

00:01:28.731 --> 00:01:29.968
you might make a mistake.

00:01:30.476 --> 00:01:32.944
And to solve this human beings

00:01:33.023 --> 00:01:34.935
have invented number systems.

00:01:35.364 --> 00:01:37.785
And it's something that
we take for granted.

00:01:37.972 --> 00:01:39.261
You might say, "Oh, isn't that just the

00:01:39.261 --> 00:01:40.392
way you've always counted?

00:01:40.392 --> 00:01:41.228
But hopefully over the course

00:01:41.228 --> 00:01:42.787
of this video you'll start to appreciate

00:01:42.787 --> 00:01:44.516
the beauty of a number system

00:01:44.539 --> 00:01:47.363
and to realize our number system isn't

00:01:47.363 --> 00:01:49.186
the only number system that is around.

00:01:49.624 --> 00:01:51.027
The number system that most

00:01:51.027 --> 00:01:54.024
of us are familiar with is the base 10

00:01:54.024 --> 00:01:55.008
number system.

00:01:55.202 --> 00:01:57.776
Often called the decimal, the decimal

00:01:57.776 --> 00:01:58.725
number system.

00:01:59.756 --> 00:02:00.865
And why 10?

00:02:01.115 --> 00:02:03.833
Well probably because we have 10 fingers.

00:02:04.056 --> 00:02:05.670
Or most of us have 10 fingers.

00:02:05.892 --> 00:02:07.944
So, it was very natural to think

00:02:07.944 --> 00:02:11.930
in terms of bundles of 10 or to have 10

00:02:11.930 --> 00:02:12.787
symbols.

00:02:12.885 --> 00:02:13.944
So however many bundles you have

00:02:13.944 --> 00:02:16.340
you can use your fingers and eventually

00:02:16.340 --> 00:02:18.834
your symbols to think
about how many there are.

00:02:19.175 --> 00:02:20.769
And since we needed 10 symbols

00:02:20.769 --> 00:02:25.452
we came up with zero,
one, two, three, four,

00:02:25.452 --> 00:02:29.750
five, six, seven, eight, nine.

00:02:29.980 --> 00:02:31.927
These 10 digits, these are our 10 symbols

00:02:31.977 --> 00:02:34.482
that we use in the base 10 system.

00:02:34.978 --> 00:02:36.609
To just give us a little bit of a reminder

00:02:36.609 --> 00:02:41.609
how we use them imagine the number 231.

00:02:43.426 --> 00:02:48.426
So, 231. 231.

00:02:51.425 --> 00:02:52.673
What does this represent?

00:02:53.170 --> 00:02:54.926
Well, what's neat about number systems

00:02:54.926 --> 00:02:56.611
is we have place value.

00:02:57.211 --> 00:02:58.943
This place all the way to the right,

00:02:59.064 --> 00:03:00.463
this is the ones place.

00:03:00.981 --> 00:03:02.335
This is the ones place.

00:03:02.412 --> 00:03:03.952
This literally means one, one.

00:03:04.156 --> 00:03:05.295
One bundle of one.

00:03:05.372 --> 00:03:07.947
So, this is one, one right over here.

00:03:08.349 --> 00:03:10.882
This right over here,
this is in the 10s place.

00:03:11.681 --> 00:03:13.050
This is in the 10s place.

00:03:13.651 --> 00:03:17.435
This three here,
literally means three 10s.

00:03:18.176 --> 00:03:20.472
So this literally means three 10s.

00:03:20.831 --> 00:03:24.222
And this two here, this two here is in

00:03:24.222 --> 00:03:25.323
the 100s place.

00:03:25.907 --> 00:03:27.296
It's in the 100s place.

00:03:30.442 --> 00:03:32.686
So, this represent two 100s.

00:03:34.853 --> 00:03:36.447
You add them together and once again

00:03:36.447 --> 00:03:38.202
I'm still thinking in base 10, you'd

00:03:38.202 --> 00:03:39.698
get 231.

00:03:39.766 --> 00:03:44.766
This is two 100s plus three 10s plus one.

00:03:47.427 --> 00:03:49.681
In our base 10 system notice every

00:03:49.681 --> 00:03:52.117
time we move to the left we're thinking

00:03:52.117 --> 00:03:54.569
in bundles of 10 of the space

00:03:54.569 --> 00:03:55.472
to the right.

00:03:55.864 --> 00:03:57.828
So, this is the ones place.

00:03:58.401 --> 00:04:00.951
You multiply by 10, you
go to the 10s place.

00:04:01.327 --> 00:04:04.114
You want to go to the next place

00:04:04.114 --> 00:04:05.656
you multiply by 10 again.

00:04:05.951 --> 00:04:07.653
You get the 100s place.

00:04:08.471 --> 00:04:09.814
If you're familiar with exponents,

00:04:10.270 --> 00:04:13.053
one is the same thing
as 10 to the zero power.

00:04:13.384 --> 00:04:18.384
10 is the same thing as
10 to the first power.

00:04:20.248 --> 00:04:22.304
So this is the 10s place. Three tens.

00:04:22.733 --> 00:04:24.829
And 100 is the same thing as 10

00:04:24.829 --> 00:04:26.948
to the second power.

00:04:27.124 --> 00:04:28.680
Obviously we could keep going on and on

00:04:28.680 --> 00:04:29.696
and on and on and on.

00:04:29.696 --> 00:04:31.915
That is the power of the base 10 system.

00:04:32.722 --> 00:04:34.272
So, you might be curious now.

00:04:34.826 --> 00:04:36.387
"Well, what if this wasn't 10 here?

00:04:36.759 --> 00:04:38.372
What if we did, let's just go as simple

00:04:38.372 --> 00:04:39.459
as we can. You can almost view this

00:04:39.459 --> 00:04:40.732
as a base one system.

00:04:40.817 --> 00:04:43.253
You only have one symbol right over here.

00:04:43.736 --> 00:04:46.039
But what if we went to something slightly

00:04:46.039 --> 00:04:48.696
more complex, a base two system.

00:04:49.305 --> 00:04:52.907
You'd be happy to know that not only

00:04:52.907 --> 00:04:55.521
can we do this, but the base two system

00:04:55.521 --> 00:04:57.173
often called the binary system.

00:04:57.348 --> 00:04:58.549
This is called the decimal system.

00:04:58.554 --> 00:05:00.329
The base two system often called

00:05:00.329 --> 00:05:02.360
the binary system is the basis of all

00:05:02.360 --> 00:05:03.624
modern computing.

00:05:03.810 --> 00:05:05.902
It's the underlying mathematics

00:05:06.131 --> 00:05:07.617
and operations that computers perform

00:05:07.617 --> 00:05:09.418
are based on binary.

00:05:10.136 --> 00:05:12.117
And in binary you have two symbols.

00:05:12.565 --> 00:05:14.791
You have zero and you have one.

00:05:15.103 --> 00:05:16.929
The reason why this is
useful for computation

00:05:16.929 --> 00:05:18.873
is because all the hardware that we use

00:05:18.873 --> 00:05:21.112
to make our modern computers, all

00:05:21.112 --> 00:05:22.994
of the transistors and the logic gates

00:05:22.994 --> 00:05:25.839
they either result in
an on or an off state.

00:05:26.160 --> 00:05:28.027
On or an off state.

00:05:28.313 --> 00:05:29.850
And so what we do is when you use

00:05:29.850 --> 00:05:31.657
your calculator or whatever you might

00:05:31.657 --> 00:05:33.522
be operating in base 10, but underlying

00:05:33.522 --> 00:05:35.987
everything it is doing the operations

00:05:35.987 --> 00:05:37.915
in binary.

00:05:38.309 --> 00:05:39.351
But you might say well how do we

00:05:39.351 --> 00:05:41.534
actually think in terms of binary?

00:05:41.889 --> 00:05:45.006
Well, we can construct
similar places here,

00:05:45.040 --> 00:05:47.545
but instead of them being powers of 10

00:05:47.920 --> 00:05:50.217
they're going to be powers of two.

00:05:50.511 --> 00:05:52.107
So, let's set up some places here.

00:05:52.481 --> 00:05:54.445
So, all the way on the right

00:05:55.224 --> 00:05:57.658
two to the zero power is still one.

00:05:57.753 --> 00:05:59.664
So we can still call that the ones place.

00:06:00.273 --> 00:06:02.156
Then we can move to the left of that.

00:06:02.892 --> 00:06:04.309
We can move to the left of that.

00:06:04.503 --> 00:06:05.927
That would be two to the first power.

00:06:05.960 --> 00:06:07.899
So we could call that the twos place,

00:06:08.030 --> 00:06:09.428
and I can even write it out if I want.

00:06:09.835 --> 00:06:12.523
Twos place instead of the 10s place.

00:06:13.105 --> 00:06:14.234
Then I could keep going.

00:06:14.321 --> 00:06:16.901
Instead of this space being the 10 to

00:06:16.901 --> 00:06:18.578
the second or the 100s place, it will be

00:06:18.578 --> 00:06:23.212
the two to the second, or the fours place.

00:06:23.679 --> 00:06:25.058
And I can keep going.

00:06:26.785 --> 00:06:27.793
I encourage you actually to pause

00:06:27.793 --> 00:06:29.151
the video and try to build this out

00:06:29.151 --> 00:06:30.063
for yourself.

00:06:30.473 --> 00:06:31.512
What would this be?

00:06:31.906 --> 00:06:33.585
Well this would be two to the third,

00:06:33.585 --> 00:06:34.875
or the eights place.

00:06:34.937 --> 00:06:36.241
Notice every time we're doing this

00:06:36.241 --> 00:06:37.770
we're multiplying by two.

00:06:37.770 --> 00:06:41.786
Everytime we go to the left,

00:06:41.786 --> 00:06:43.602
just like we multiplied by 10 here.

00:06:43.602 --> 00:06:45.441
So notice everywhere you see this 10s

00:06:45.509 --> 00:06:46.926
we're now dealing with twos.

00:06:47.308 --> 00:06:48.219
Let's keep going.

00:06:48.505 --> 00:06:49.937
Let's keep going and then we can actually

00:06:49.937 --> 00:06:52.656
represent this number using binary.

00:06:53.168 --> 00:06:54.376
So, let's do that.

00:06:54.879 --> 00:06:56.370
So, this right over here I've already

00:06:56.370 --> 00:06:57.417
used that color.

00:06:58.449 --> 00:07:01.221
This right over here, this is two

00:07:01.221 --> 00:07:01.949
to the fourth.

00:07:01.949 --> 00:07:04.040
We could call that the 16s place.

00:07:04.550 --> 00:07:07.201
Then we could have --

00:07:07.201 --> 00:07:08.329
I'll reuse some of these colors.

00:07:08.342 --> 00:07:09.713
This is two to the fifth.

00:07:10.205 --> 00:07:13.645
We could call this the 32s place.

00:07:14.163 --> 00:07:18.615
Then we can go two to the sixth.

00:07:21.430 --> 00:07:24.133
We can call that, multiply by two again,

00:07:24.138 --> 00:07:25.878
or two to the six is 64.

00:07:26.159 --> 00:07:27.520
So this is the 64s place.

00:07:27.923 --> 00:07:31.587
Tells us how many 64s we
have. Zero or one 64s.

00:07:31.792 --> 00:07:32.975
We'll see that in a second.

00:07:33.573 --> 00:07:35.287
Then we can go over here.

00:07:35.365 --> 00:07:37.831
This would be two to the seventh.

00:07:38.389 --> 00:07:41.343
That would be the 128s place.

00:07:46.548 --> 00:07:47.655
And we can obviously keep going on

00:07:47.655 --> 00:07:49.402
and on and on, but this
should be enough for

00:07:49.402 --> 00:07:51.078
me to represent this number.

00:07:51.128 --> 00:07:52.650
In future videos I will show you how

00:07:52.650 --> 00:07:54.771
to do that, but let's actually represent

00:07:54.771 --> 00:07:55.555
the number.

00:07:55.555 --> 00:07:56.962
It turns out that this number

00:07:56.962 --> 00:07:58.156
in decimal can be represented

00:07:58.156 --> 00:08:03.156
as 11100111 in binary.

00:08:22.578 --> 00:08:23.433
What does this mean?

00:08:23.433 --> 00:08:26.405
This means you have one 128 plus one 64,

00:08:26.500 --> 00:08:30.877
plus one 32, plus no 16s, plus no eights,

00:08:31.043 --> 00:08:36.043
plus one four, plus one two, plus one one.

00:08:38.324 --> 00:08:39.278
So you can see that these are going

00:08:39.278 --> 00:08:40.640
to be the same thing.

00:08:40.880 --> 00:08:43.115
Notice, this is one 128.

00:08:43.239 --> 00:08:48.239
So it's 128, plus 64, plus 32.

00:08:51.228 --> 00:08:54.338
We have zero 16s, zero eights.

00:08:54.603 --> 00:08:55.666
So we're not going to add those.

00:08:55.988 --> 00:08:59.564
Plus four, one four.

00:08:59.973 --> 00:09:01.201
Plus one two.

00:09:04.012 --> 00:09:06.096
Plus one one.

00:09:06.355 --> 00:09:07.653
And add these together,

00:09:07.659 --> 00:09:08.811
and once again when we're doing this,

00:09:08.811 --> 00:09:09.873
when I'm writing it this way I'm

00:09:09.873 --> 00:09:11.204
kind of using the number system

00:09:11.204 --> 00:09:12.458
that we're most familiar with.

00:09:12.473 --> 00:09:14.573
We're most used to
doing the operations in,

00:09:14.614 --> 00:09:16.908
but when you do it you will see that

00:09:16.908 --> 00:09:19.908
this is the exact same number as 231.

00:09:19.940 --> 00:09:21.614
This is just another representation.

00:09:21.614 --> 00:09:23.379
One isn't better than the other.

00:09:23.379 --> 00:09:24.188
The only reason why I converted this

00:09:24.188 --> 00:09:27.358
is this is what I'm used to thinking in.

00:09:27.420 --> 00:09:29.456
It's what I'm used to doing operations in.

00:09:29.685 --> 00:09:31.670
So, hopefully you find
that pretty interesting.

00:09:31.810 --> 00:09:33.987
To me, this kind of opened my mind

00:09:33.987 --> 00:09:36.150
to the power of even our decimal system.

00:09:36.471 --> 00:09:38.027
In future videos we'll explore other

00:09:38.027 --> 00:09:39.047
number systems.

00:09:39.050 --> 00:09:40.736
The most used ones, base 10 is

00:09:40.736 --> 00:09:43.975
used very heavily, binary and there's

00:09:43.975 --> 00:09:45.901
also hexadecimal where you don't have

00:09:45.901 --> 00:09:50.901
two digits or not 10 digits,
but you have 16 digits.

00:09:51.499 --> 00:09:53.579
And we'll explore those in future videos

00:09:53.596 --> 00:09:55.970
and how to convert between or rewrite the

00:09:56.131 --> 00:09:58.844
the different representations
and different bases.


Mean value theorem for integrals

WEBVTT
Kind: captions
Language: en

00:00:00.643 --> 00:00:03.926
- [Voiceover] We have many
videos on the mean value theorem,

00:00:03.926 --> 00:00:07.742
but I'm going to review it
a little bit, so that we can

00:00:07.742 --> 00:00:10.444
see how this connects
the mean value theorem

00:00:10.444 --> 00:00:11.959
that we learned in differential calculus,

00:00:11.959 --> 00:00:13.393
how that connects to what we learned about

00:00:13.393 --> 00:00:17.110
the average value of a function
using definite integrals.

00:00:17.110 --> 00:00:21.310
So the mean value theorem
tells us that if I have

00:00:21.310 --> 00:00:26.310
some function f that is
continuous on the closed interval,

00:00:28.409 --> 00:00:31.277
so it's including the
endpoints, from a to b,

00:00:31.277 --> 00:00:35.693
and it is differentiable,
so the derivative

00:00:35.693 --> 00:00:39.660
is defined on the open
interval, from a to b,

00:00:39.660 --> 00:00:43.242
so it doesn't necessarily
have to be differentiable

00:00:43.242 --> 00:00:45.310
at the boundaries, as long
as it's differentiable

00:00:45.310 --> 00:00:49.943
between the boundaries, then we know that

00:00:49.943 --> 00:00:54.943
there exists some value, or some number c,

00:01:04.927 --> 00:01:09.927
such that c is between the
two endpoints of our interval,

00:01:11.042 --> 00:01:16.042
so such that a < c < b, so
c is in this interval, AND,

00:01:20.038 --> 00:01:23.727
and this is kind of the meat of it,

00:01:23.727 --> 00:01:28.727
is that the derivative of
our function at that point,

00:01:28.943 --> 00:01:31.092
you could use as the
slope of the tangent line

00:01:31.092 --> 00:01:36.092
at that point, is equal
to essentially the average

00:01:37.360 --> 00:01:39.443
rate of change over the
interval, or you could even

00:01:39.443 --> 00:01:42.443
think about it as the slope
between the two endpoints.

00:01:42.443 --> 00:01:44.910
So the slope between the
two endpoints is gonna be

00:01:44.910 --> 00:01:47.277
your change in y, which is going to be

00:01:47.277 --> 00:01:48.977
your change in your function value,

00:01:48.977 --> 00:01:53.977
so f of b, minus f of a, over
b minus a, and once again,

00:01:57.677 --> 00:01:59.893
we go into much more depth in this

00:01:59.893 --> 00:02:02.477
when we covered it the first
time in differential calculus,

00:02:02.477 --> 00:02:04.278
but just to give you
a visualization of it,

00:02:04.278 --> 00:02:06.311
'cause I think it's always handy, the

00:02:06.311 --> 00:02:09.560
mean value theorem that we
learned in differential calculus

00:02:09.560 --> 00:02:14.478
just tells us, hey look,
if this is a, this is b,

00:02:14.478 --> 00:02:18.660
I've got my function doing
something interesting,

00:02:18.660 --> 00:02:23.576
so this is f of a, this is
f of b, so this quantity

00:02:23.576 --> 00:02:26.060
right over here, where
you're taking the change

00:02:26.060 --> 00:02:29.076
in the value of our function,
so this right over here is

00:02:29.076 --> 00:02:33.393
f of b, minus f of a,
is this change in the

00:02:33.393 --> 00:02:35.561
value of our function,
divided by the change

00:02:35.561 --> 00:02:38.645
in our x-axis, so it's a
change in y over change in x,

00:02:38.645 --> 00:02:40.610
that gives us the slope,
this right over here gives us

00:02:40.610 --> 00:02:44.727
the slope of this line,
the slope of the line

00:02:44.727 --> 00:02:48.877
that connects these two
points, that's this quantity,

00:02:48.877 --> 00:02:51.592
and the mean value theorem
tell us that there's some c

00:02:51.592 --> 00:02:54.626
in between a and b where you're
gonna have the same slope,

00:02:54.626 --> 00:02:57.443
so it might be at LEAST
one place, so it might be

00:02:57.443 --> 00:03:00.695
right over there, where you
have the exact same slope,

00:03:00.695 --> 00:03:03.742
there exists a c where the
slope of the tangent line

00:03:03.742 --> 00:03:06.444
at that point is going to be
the same, so this would be

00:03:06.444 --> 00:03:08.076
a c right over there, and
we actually might have

00:03:08.076 --> 00:03:10.542
a couple of c's, that's
another candidate c.

00:03:10.542 --> 00:03:12.228
There's at least one c where the slope

00:03:12.228 --> 00:03:15.160
of the tangent line is the
same as the average slope

00:03:15.160 --> 00:03:18.061
across the interval, and
once again, we have to assume

00:03:18.061 --> 00:03:21.177
that f is continuous,
and f is differentiable.

00:03:21.177 --> 00:03:25.828
Now when you see this, it
might evoke some similarities

00:03:25.828 --> 00:03:28.992
with what we saw when
we saw how we defined,

00:03:28.992 --> 00:03:30.478
I guess you could say, or the formula

00:03:30.478 --> 00:03:32.061
for the average value of a function.

00:03:32.061 --> 00:03:35.444
Remember, what we saw for the
average value of a function,

00:03:35.444 --> 00:03:40.310
we said the average value
of a function is going to be

00:03:40.310 --> 00:03:44.760
equal to 1 over b minus a,
notice, 1 over b minus a,

00:03:44.760 --> 00:03:46.626
you have a b minus a in
the denominator here,

00:03:46.626 --> 00:03:51.626
times the definite integral
from a to b, of f of x dx.

00:03:53.459 --> 00:03:57.478
Now this is interesting, 'cause
here we have a derivative,

00:03:57.478 --> 00:04:01.827
here we have an integral, but
maybe we could connect these.

00:04:01.827 --> 00:04:05.643
Maybe we could connect these two things.

00:04:05.643 --> 00:04:07.242
Well one thing that might jump out at you

00:04:07.242 --> 00:04:11.010
is maybe we could rewrite

00:04:11.010 --> 00:04:16.010
this numerator right over
here in this form somehow.

00:04:16.041 --> 00:04:18.443
And I encourage you to pause
the video and see if you can,

00:04:18.443 --> 00:04:21.477
and I'll give you actually
quite a huge hint,

00:04:21.477 --> 00:04:23.361
instead of it being an f of x
here, what happens if there's

00:04:23.361 --> 00:04:27.542
an f prime of x there, so I
encourage you to try to do that.

00:04:27.542 --> 00:04:29.493
So once again, let me rewrite all of this,

00:04:29.493 --> 00:04:31.360
this is going to be equal to...

00:04:31.360 --> 00:04:33.593
This over here is the exact same thing as

00:04:33.593 --> 00:04:38.593
the definite integral from
a to b, of f prime of x dx.

00:04:40.394 --> 00:04:41.143
Think about it.

00:04:41.143 --> 00:04:43.242
You're gonna take the
anti-derivative of f prime of x,

00:04:43.242 --> 00:04:45.144
which is going to be f
of x, and you're going to

00:04:45.144 --> 00:04:47.660
evaluate it at b, f of
b, and then from that

00:04:47.660 --> 00:04:50.943
you're going to subtract it
evaluated at a, minus f of a.

00:04:50.943 --> 00:04:52.992
These two things are identical.

00:04:52.992 --> 00:04:57.992
And then, you can of
course divide by b minus a.

00:04:59.977 --> 00:05:02.893
Now this is starting to get interesting.

00:05:02.893 --> 00:05:07.893
One way to think about
it is, there must be a c

00:05:09.955 --> 00:05:12.157
that takes on the average value of,

00:05:12.157 --> 00:05:14.639
there must be a c, that when you

00:05:14.639 --> 00:05:17.291
evaluate the derivative at c,

00:05:17.291 --> 00:05:20.924
it takes on the average
value of the derivative.

00:05:20.924 --> 00:05:23.707
Or another way to think about it,

00:05:23.707 --> 00:05:28.707
if we were to just write g of
x is equal to f prime of x,

00:05:31.458 --> 00:05:33.908
then we get very close to
what we have over here,

00:05:33.908 --> 00:05:37.991
because this right over
here is going to be g of c,

00:05:37.991 --> 00:05:40.659
remember, f prime of c is
the same thing as g of c,

00:05:40.659 --> 00:05:45.659
is equal to 1 over b minus
a, so there exists a c

00:05:50.224 --> 00:05:53.107
where g of c is equal to 1 over b minus a,

00:05:53.107 --> 00:05:58.107
times the definite integral
from a to b, of g of x dx,

00:06:00.506 --> 00:06:02.908
f prime of x is the same thing as g of x.

00:06:02.908 --> 00:06:05.189
So another way of thinking
about it, this is actually

00:06:05.189 --> 00:06:07.074
another form of the mean value theorem,

00:06:07.074 --> 00:06:12.074
it's called the mean value
theorem for integrals.

00:06:14.374 --> 00:06:15.291
I'll just write the acronym,

00:06:15.291 --> 00:06:20.291
mean value theorem for
integrals, or integration,

00:06:22.173 --> 00:06:24.542
which essentially, to
give it in a slightly more

00:06:24.542 --> 00:06:29.542
formal sense, is if you have
some function g, so if g is,

00:06:30.258 --> 00:06:33.991
let me actually go down a
little bit, which tells us that

00:06:33.991 --> 00:06:38.991
if g of x is continuous
on this closed interval,

00:06:42.340 --> 00:06:47.340
going from a to b, then there
exists a c in this interval

00:06:54.959 --> 00:06:57.574
where g of c is equal to, what is this?

00:06:57.574 --> 00:07:00.575
This is the average value of our function.

00:07:00.575 --> 00:07:05.575
There exists a c where g of c is equal to

00:07:07.773 --> 00:07:11.023
the average value of your
function over the interval.

00:07:11.023 --> 00:07:14.257
This was our definition of the
average value of a function.

00:07:14.257 --> 00:07:16.491
So anyway, this is just
another way of saying you might

00:07:16.491 --> 00:07:19.507
see some of the mean value
theorem of integrals,

00:07:19.507 --> 00:07:22.139
and just to show you that
it's really closely tied,

00:07:22.139 --> 00:07:24.390
it's using different
notation, but it's usually,

00:07:24.390 --> 00:07:27.140
it's essentially the
same exact idea as the

00:07:27.140 --> 00:07:30.659
mean value theorem you learned
in differential calculus,

00:07:30.659 --> 00:07:32.325
but now your different
notation, and I guess

00:07:32.325 --> 00:07:34.510
you could have a slightly
different interpretation.

00:07:34.510 --> 00:07:36.939
We were thinking about it
in differential calculus,

00:07:36.939 --> 00:07:39.472
we're thinking about having a point where

00:07:39.472 --> 00:07:41.954
the slope of the tangent line
of the function of that point

00:07:41.954 --> 00:07:44.273
is the same as the average
rate, so that's when

00:07:44.273 --> 00:07:46.287
we had our kind of differential
mode, and we were kind of

00:07:46.287 --> 00:07:49.371
thinking in terms of slopes,
and slopes of tangent lines,

00:07:49.371 --> 00:07:51.573
and now, when we're in
integral mode, we're thinking

00:07:51.573 --> 00:07:54.157
much more in terms of average value,

00:07:54.157 --> 00:07:57.107
average value of the
function, so there's some c,

00:07:57.107 --> 00:07:59.822
where g of c, there's some c,

00:07:59.822 --> 00:08:01.856
where the function
evaluated at that point,

00:08:01.856 --> 00:08:04.989
is equal to the average
value, so another way

00:08:04.989 --> 00:08:09.989
of thinking about it, if
I were to draw g of x,

00:08:17.721 --> 00:08:22.721
that's x, that is my
y-axis, this is the graph of

00:08:23.205 --> 00:08:26.239
y is equal to g of x, which
of course is the same thing

00:08:26.239 --> 00:08:29.773
as f prime of x, but we've
just rewritten it now

00:08:29.773 --> 00:08:32.804
to be more consistent with
our average value formula,

00:08:32.804 --> 00:08:37.338
and we're talking about
the interval from a to b,

00:08:37.338 --> 00:08:42.338
we've already seen how to
calculate the average value,

00:08:42.855 --> 00:08:47.405
so maybe the average value
is that right over there,

00:08:47.405 --> 00:08:52.304
so that is g average, so
our average value is this,

00:08:52.304 --> 00:08:53.988
the mean value theorem for
integrals just tells us

00:08:53.988 --> 00:08:58.787
there's some c where our
function must take on

00:08:58.787 --> 00:09:03.105
that value at c, whereas that c is inside,

00:09:03.105 --> 00:09:06.806
where the c is in that interval.


Average acceleration over interval

WEBVTT
Kind: captions
Language: en

00:00:00.637 --> 00:00:02.378
- [Voiceover] Let's say
that we have a particle

00:00:02.378 --> 00:00:04.824
that's traveling in one
dimension, and its position

00:00:04.824 --> 00:00:07.859
as a function of time is given as t

00:00:07.859 --> 00:00:12.859
to the third power plus
two over t-squared.

00:00:13.384 --> 00:00:17.140
What I would like you to
do is pause this video

00:00:17.140 --> 00:00:22.140
and figure out what the
average acceleration is

00:00:22.281 --> 00:00:26.676
of this particle over the
interval, the closed interval,

00:00:26.676 --> 00:00:30.645
from t is equal to one
to t is equal to two.

00:00:30.645 --> 00:00:32.024
What is this?

00:00:32.024 --> 00:00:34.725
What is this going to be equal to?

00:00:34.725 --> 00:00:37.115
Assuming you've given a
go, and the first thing you

00:00:37.115 --> 00:00:38.996
might have realized is we're
trying to get the average

00:00:38.996 --> 00:00:41.024
value of a function that we
don't know explicitly at.

00:00:41.024 --> 00:00:42.026
We know the position function

00:00:42.026 --> 00:00:43.814
but not the acceleration function.

00:00:43.814 --> 00:00:47.166
But luckily, we also know
that the acceleration function

00:00:47.166 --> 00:00:49.649
is derivative with respective
time of the velocity function

00:00:49.649 --> 00:00:51.337
which is the derivative with respect

00:00:51.337 --> 00:00:53.329
to time of the position function.

00:00:53.329 --> 00:00:56.393
The acceleration function is
the second derivative of this.

00:00:56.393 --> 00:00:59.078
We have to just find its average
value over this interval.

00:00:59.078 --> 00:01:00.356
Let's do that.

00:01:00.356 --> 00:01:02.063
Let's take the derivative of this twice.

00:01:02.063 --> 00:01:03.918
But before we do that, let
me just even rewrite this.

00:01:03.918 --> 00:01:04.604
It's just going to be a little bit

00:01:04.604 --> 00:01:06.932
easier to differentiate it.

00:01:06.932 --> 00:01:09.120
If we just take each of these
two terms in the numerator

00:01:09.120 --> 00:01:10.943
and divide them by t
squared, we're going to get t

00:01:10.943 --> 00:01:13.936
to the third, divided by
t-squared is just the t.

00:01:13.936 --> 00:01:16.016
Then two divided by T
squared, we could write that

00:01:16.016 --> 00:01:19.851
as plus two t to the negative two power.

00:01:19.851 --> 00:01:23.063
Now, let's take the derivative.

00:01:23.063 --> 00:01:26.600
The velocity function, as
the velocity as the function

00:01:26.600 --> 00:01:29.920
of time, just the derivative
with this with respect to time.

00:01:29.920 --> 00:01:33.152
It's going to be derivative
of t with respect to t as one.

00:01:33.152 --> 00:01:35.148
Derivative of two t to the negative two.

00:01:35.148 --> 00:01:39.758
Let's see, negative two times
positive two is negative four.

00:01:39.758 --> 00:01:43.108
T to the, we just decremented
the exponent here,

00:01:43.108 --> 00:01:47.729
t to the negative three power.

00:01:47.729 --> 00:01:50.019
Now, to find acceleration as
a function of time, we just

00:01:50.019 --> 00:01:52.884
find, take the derivative of
this with respect to time.

00:01:52.884 --> 00:01:57.346
Acceleration as a function
of time is equal to,

00:01:57.346 --> 00:01:59.184
actually it's already used
at color for the average

00:01:59.184 --> 00:02:01.929
so let me do a different color now.

00:02:01.929 --> 00:02:05.725
Acceleration has a
function of time is just

00:02:05.725 --> 00:02:07.560
the derivative of this with respect to t.

00:02:07.560 --> 00:02:09.070
Derivative of a constant with respect

00:02:09.070 --> 00:02:11.547
to time was not changing so it's a zero.

00:02:11.547 --> 00:02:13.159
Then over here, negative three times

00:02:13.159 --> 00:02:17.051
negative four is positive
12, times t to the,

00:02:17.051 --> 00:02:21.017
let's decrement that exponent
to the negative four power.

00:02:21.017 --> 00:02:24.360
Now to find the average
value, all we have to do now,

00:02:24.360 --> 00:02:28.708
average value is essentially
take the definitive role

00:02:28.708 --> 00:02:30.989
of this over the interval, and divide that

00:02:30.989 --> 00:02:33.372
by the width of the interval.

00:02:33.372 --> 00:02:34.649
Or we could say, we could take,

00:02:34.649 --> 00:02:36.954
we can divide by the
width of the interval.

00:02:36.954 --> 00:02:38.274
One, over two minus one,

00:02:38.274 --> 00:02:40.026
and this also simplifies to one.

00:02:40.026 --> 00:02:44.672
Times the definitive role of the interval.

00:02:44.672 --> 00:02:47.698
One to two of a of t which is,

00:02:47.698 --> 00:02:52.698
so this could be 12 t to
the negative four power d t.

00:02:54.398 --> 00:02:55.777
What is this simplified to?

00:02:55.777 --> 00:02:57.911
Once again, this is one over one.

00:02:57.911 --> 00:03:00.136
That's just going to be one.

00:03:00.136 --> 00:03:04.941
We take the antiderivative of this.

00:03:04.941 --> 00:03:07.681
Let me just, so this
is going to be equal to

00:03:07.681 --> 00:03:11.225
the antiderivative of this
is, so we go t to the negative

00:03:11.225 --> 00:03:15.188
three power but we
divide by negative three.

00:03:15.188 --> 00:03:17.563
An antiderivative of this
is going to be, if we don't

00:03:17.563 --> 00:03:22.369
take that, an antiderivative
is going to be negative four

00:03:22.369 --> 00:03:25.402
t to the negative three power,
and we saw that over here.

00:03:25.402 --> 00:03:27.987
Obviously, if you were really
just taking an in-definitive

00:03:27.987 --> 00:03:29.623
role, we would have to
put some concept here.

00:03:29.623 --> 00:03:33.160
But in the definitive role,
if we put a concept here,

00:03:33.160 --> 00:03:35.334
Assuming the same concept
that we get canceled out

00:03:35.334 --> 00:03:37.627
when you actually do a
calculation, but the entire

00:03:37.627 --> 00:03:40.398
derivative of this, we
increment the exponent,

00:03:40.398 --> 00:03:42.398
and then we divide by that new exponent.

00:03:42.398 --> 00:03:45.142
Twelve divided by negative
three is negative four.

00:03:45.142 --> 00:03:50.142
We are going to evaluate
that from two and that one.

00:03:50.761 --> 00:03:55.353
This is going to be equal to,
when we evaluated it at two

00:03:55.353 --> 00:03:57.170
at the upper bound of our
intervals, it could be

00:03:57.170 --> 00:04:01.242
negative four times two to
the negative three powers.

00:04:01.242 --> 00:04:04.526
It's negative four times, what is that?

00:04:04.526 --> 00:04:08.260
Two, that's one over to
the third of times 1/8,

00:04:08.260 --> 00:04:09.949
is one way to think about that.

00:04:09.949 --> 00:04:12.693
Then we are going to have
minus this evaluated one.

00:04:12.693 --> 00:04:16.734
Minus negative four times
t to the negative three

00:04:16.734 --> 00:04:18.416
of one to the negative three is just one.

00:04:18.416 --> 00:04:21.583
This is going to be
negative four times one.

00:04:21.583 --> 00:04:22.917
This is going to be equal to,

00:04:22.917 --> 00:04:24.850
or really in the homestretch now,

00:04:24.850 --> 00:04:26.705
this is equal to, this part right

00:04:26.705 --> 00:04:30.550
over here, is negative 1/2.

00:04:30.550 --> 00:04:32.854
This is negative 1/2, and this part

00:04:32.854 --> 00:04:35.990
right over here is positive four.

00:04:35.990 --> 00:04:38.288
Positive four minus 1/2.

00:04:38.288 --> 00:04:40.921
We could either write
that as three and a half,

00:04:40.921 --> 00:04:43.516
or if we wanted to write
that as an improper fraction,

00:04:43.516 --> 00:04:47.112
we could write this as 7/2.

00:04:47.112 --> 00:04:49.172
The average value of our acceleration

00:04:49.172 --> 00:04:52.017
over this interval is 7/2.

00:04:52.017 --> 00:04:55.856
If this position was given a
meters and time was in seconds,

00:04:55.856 --> 00:04:59.613
then this would be 7/2
meters per seconds squared,

00:04:59.613 --> 00:05:02.459
is the average acceleration between time

00:05:02.459 --> 00:05:06.173
in one second and time at two seconds.