Thermodynamics part 3: Kelvin scale and Ideal gas law example

WEBVTT
Kind: captions
Language: en

00:00:01.140 --> 00:00:04.820
We just finished, hopefully,
getting intuition for why my

00:00:04.820 --> 00:00:08.130
initial pressure times my
initial volume divided by my

00:00:08.130 --> 00:00:11.290
initial temperature is going
to equal-- if I change the

00:00:11.290 --> 00:00:13.540
volume, the pressure, the
temperature, or some

00:00:13.540 --> 00:00:15.820
combination of all of them,
it's going to equal my new

00:00:15.820 --> 00:00:19.780
pressure times my new volume
divided by my new temperature.

00:00:19.780 --> 00:00:23.540
And once again, just remember
all of this-- pressure times

00:00:23.540 --> 00:00:27.750
volume is proportional to the
amount of kinetic energy in

00:00:27.750 --> 00:00:33.660
the system, and temperature is
proportional to the amount of

00:00:33.660 --> 00:00:35.500
kinetic energy per molecule.

00:00:35.500 --> 00:00:39.490
If we don't change the number of
molecules, the amount-- and

00:00:39.490 --> 00:00:42.010
since by the conservation of
energy, the amount of kinetic

00:00:42.010 --> 00:00:44.270
energy isn't going to change
unless we do some work, or get

00:00:44.270 --> 00:00:47.120
some potential energy-- these
quantities and this

00:00:47.120 --> 00:00:48.240
relationship won't change.

00:00:48.240 --> 00:00:49.730
Watch the last video, and
hopefully you'll get some

00:00:49.730 --> 00:00:51.830
intuition-- if it's still
confusing, I'll make another

00:00:51.830 --> 00:00:53.040
video for you guys.

00:00:53.040 --> 00:00:58.230
Before I apply this equation--
this is going to get you

00:00:58.230 --> 00:01:00.860
pretty far in thermodynamics
just knowing this, and even

00:01:00.860 --> 00:01:04.700
more just having the intuition
of what it means.

00:01:04.700 --> 00:01:07.630
I want to clarify something
about temperature-- there's a

00:01:07.630 --> 00:01:10.280
lot of different ways to
measure temperature.

00:01:10.280 --> 00:01:13.720
We know that in Fahrenheit,
what's freezing of water?

00:01:13.720 --> 00:01:17.590
It's 32 degrees Fahrenheit
that's freezing, but that's

00:01:17.590 --> 00:01:22.080
also 0 degrees Celsius--
actually, that's how the

00:01:22.080 --> 00:01:23.170
Celsius scale was determined.

00:01:23.170 --> 00:01:26.470
They said, where does water
freeze, and then where does

00:01:26.470 --> 00:01:27.580
water boil?

00:01:27.580 --> 00:01:30.970
100 degrees for Celsius
is boiling, and that's

00:01:30.970 --> 00:01:33.190
how they rated it.

00:01:33.190 --> 00:01:36.130
You could be colder than the
freezing of water, and you'd

00:01:36.130 --> 00:01:38.120
have to go negative in that
situation-- Fahrenheit, I'm

00:01:38.120 --> 00:01:39.520
actually not sure.

00:01:39.520 --> 00:01:41.380
I need to look that up in
Wikipedia, or that might be

00:01:41.380 --> 00:01:45.150
something for you to do, and
tell me how it came out.

00:01:45.150 --> 00:01:47.610
I think the boiling of water in
Fahrenheit is 212 degrees,

00:01:47.610 --> 00:01:48.710
so it's a little arbitrary.

00:01:48.710 --> 00:01:51.790
I think Fahrenheit might be
somehow related to human body

00:01:51.790 --> 00:01:53.910
temperature, but I'm
just guessing.

00:01:53.910 --> 00:01:57.660
You can have different scales
in this situation, and they

00:01:57.660 --> 00:01:59.230
were all kind of
a bit arbitrary

00:01:59.230 --> 00:02:00.690
when they were designed.

00:02:00.690 --> 00:02:03.520
They were just to have some type
of relative to scale--

00:02:03.520 --> 00:02:06.460
you could say when things are
boiling, they're definitely

00:02:06.460 --> 00:02:08.370
hotter because they have a
higher temperature then when

00:02:08.370 --> 00:02:09.620
things are freezing.

00:02:14.440 --> 00:02:17.790
You can't divide 100 by zero,
but if something is 1 degree,

00:02:17.790 --> 00:02:19.990
is it necessarily the case that
something that is 100

00:02:19.990 --> 00:02:24.160
degrees Celsius is a hundred
times hotter, or has a hundred

00:02:24.160 --> 00:02:26.280
times the kinetic energy?

00:02:26.280 --> 00:02:28.040
Actually, what we'll see is that
no, it's actually not the

00:02:28.040 --> 00:02:30.540
case-- you don't have 100 times
the kinetic energy, so

00:02:30.540 --> 00:02:34.440
this is a bit of an
arbitrary scale.

00:02:34.440 --> 00:02:38.690
The actual interval is
arbitrary-- you could pick the

00:02:38.690 --> 00:02:40.900
1 degree as being one hundredth
of the distance

00:02:40.900 --> 00:02:44.160
between zero and 100, but where
you start-- at least in

00:02:44.160 --> 00:02:45.540
the Celsius scale-- is
a bit arbitrary.

00:02:45.540 --> 00:02:47.690
They picked the freezing
of water.

00:02:47.690 --> 00:02:51.340
Later on, people figured out
that there is an absolute

00:02:51.340 --> 00:02:52.390
point to start at.

00:02:52.390 --> 00:02:56.430
And that absolute point to start
at is the temperature at

00:02:56.430 --> 00:02:59.230
which a molecule or an atom has

00:02:59.230 --> 00:03:03.150
absolutely no kinetic energy.

00:03:03.150 --> 00:03:09.910
Because we said temperature is
equal to the average kinetic

00:03:09.910 --> 00:03:15.310
energy of the system, or the
total kinetic energy of the

00:03:15.310 --> 00:03:17.870
system divided by the
number of molecules.

00:03:17.870 --> 00:03:19.840
Or we could also
say the average

00:03:19.840 --> 00:03:21.390
kinetic energy per molecule.

00:03:24.390 --> 00:03:29.940
The only way to really say that
the temperature is zero--

00:03:29.940 --> 00:03:34.230
and this is proportional,
because the temperature scales

00:03:34.230 --> 00:03:39.370
are still a little bit
arbitrary-- the only way to

00:03:39.370 --> 00:03:43.120
get to a temperature of zero
should be when the kinetic

00:03:43.120 --> 00:03:47.070
energy of each and every
molecule is zero, or the

00:03:47.070 --> 00:03:47.635
average kinetic energy.

00:03:47.635 --> 00:03:51.220
So they're not moving, they're
not vibrating, they're not

00:03:51.220 --> 00:03:54.220
even blinking-- these molecules
are stationary.

00:03:54.220 --> 00:03:57.645
The point at which that occurs
is called absolute zero.

00:04:03.090 --> 00:04:06.330
That actually occurs-- absolute
zero, which is also

00:04:06.330 --> 00:04:16.149
called zero Kelvin, and that is
the same thing as minus 273

00:04:16.149 --> 00:04:17.579
degrees Celsius.

00:04:17.579 --> 00:04:21.570
Nowhere in the universe, at
least that I'm aware of, it is

00:04:21.570 --> 00:04:27.020
it colder than minus 273 degrees
Celsius-- at that

00:04:27.020 --> 00:04:31.810
temperature, nothing moves,
even at the atomic scale.

00:04:31.810 --> 00:04:35.940
I'm talking that the electrons
collapse into the nucleus--

00:04:35.940 --> 00:04:40.230
everything is completely
stationary at zero Kelvin.

00:04:40.230 --> 00:04:44.250
It's a theoretical absolute
limit-- maybe we'll do a bunch

00:04:44.250 --> 00:04:46.390
of videos on how you can get
close to that, but in

00:04:46.390 --> 00:04:48.900
laboratory environments or maybe
in deep space, it gets

00:04:48.900 --> 00:04:50.270
really, really close to this.

00:04:53.570 --> 00:04:57.400
I'm pretty sure nowhere in the
universe do we have absolutely

00:04:57.400 --> 00:05:01.280
zero Kelvin, or at least in any
place where we actually

00:05:01.280 --> 00:05:04.560
have particles, but I might be
wrong there-- that's a little

00:05:04.560 --> 00:05:08.910
bit out of the scope of what
we're talking about.

00:05:08.910 --> 00:05:11.840
The true way to measure
temperature is in Kelvin.

00:05:11.840 --> 00:05:14.460
When you're measuring in Kelvin,
if I say-- I have

00:05:14.460 --> 00:05:19.360
something that is 1 kelvin
versus something that is 5

00:05:19.360 --> 00:05:23.560
kelvin, since we nailed down the
bottom at a point at which

00:05:23.560 --> 00:05:27.530
really do not have kinetic
energy, I can make the

00:05:27.530 --> 00:05:32.410
statement that this has five
times the energy of something

00:05:32.410 --> 00:05:35.600
that's at 5 Kelvin
versus 1 Kelvin.

00:05:35.600 --> 00:05:38.300
That whole long explanation
about Kelvin, that was to just

00:05:38.300 --> 00:05:41.140
to make the point that whenever
we use this formula,

00:05:41.140 --> 00:05:43.440
or really any formula in
thermodynamics that involves

00:05:43.440 --> 00:05:47.160
temperature, we should convert
to Kelvin, unless we're just

00:05:47.160 --> 00:05:48.300
doing change in temperature.

00:05:48.300 --> 00:05:51.550
Then you could you could
probably keep it Celsius, but

00:05:51.550 --> 00:05:52.910
when you're doing
proportionality, or you're

00:05:52.910 --> 00:05:55.400
using it for multiplying or
dividing by temperature, you

00:05:55.400 --> 00:05:56.500
have to use Kelvin.

00:05:56.500 --> 00:06:00.360
Hopefully, I made a little
bit clear of why that is.

00:06:00.360 --> 00:06:01.610
Let's do an example.

00:06:06.570 --> 00:06:08.690
You'd be surprised how
far this takes you.

00:06:08.690 --> 00:06:10.880
Really, the main trick is
just to remember to

00:06:10.880 --> 00:06:11.890
convert things to Kelvin.

00:06:11.890 --> 00:06:15.780
That's the number one reason why
people miss questions on

00:06:15.780 --> 00:06:21.610
thermodynamics exams-- is that
they didn't convert to Kelvin.

00:06:21.610 --> 00:06:24.680
This problem is very typical of
most of what you'll see--

00:06:24.680 --> 00:06:29.150
this is from the Barron's AP
physics B on page 226.

00:06:29.150 --> 00:06:35.320
It says a confined gas is a
temperature of 27 degrees, so

00:06:35.320 --> 00:06:41.520
its initial temperature
is 27 degrees Celsius.

00:06:41.520 --> 00:06:51.640
It has a pressure of 1,000
pascals, or newtons per meter

00:06:51.640 --> 00:07:01.270
squared, and the volume
is 30 meters.

00:07:01.270 --> 00:07:03.650
I think in one of the early
videos, I think I said newtons

00:07:03.650 --> 00:07:06.080
per meter cubed, but it's
newtons per meter squared-- I

00:07:06.080 --> 00:07:07.470
just want to make sure I
didn't confuse people

00:07:07.470 --> 00:07:10.420
previously, so that's
the initial volume.

00:07:10.420 --> 00:07:13.760
It says the volume is decreased,
so then we go to

00:07:13.760 --> 00:07:17.650
this date, where my new
volume is going to

00:07:17.650 --> 00:07:21.270
be 20 meters cubed.

00:07:21.270 --> 00:07:25.610
The new temperature is
increased, and so the new

00:07:25.610 --> 00:07:29.380
temperature is now 50
degrees Celsius.

00:07:29.380 --> 00:07:32.555
They want to know what
is the new pressure?

00:07:35.150 --> 00:07:37.500
Before we just substitute into
the equation, and solve for

00:07:37.500 --> 00:07:39.890
the new pressure, remember--
if they gave it to you in

00:07:39.890 --> 00:07:42.390
Celsius, convert to Kelvin.

00:07:42.390 --> 00:07:44.460
If they gave it to you in
Fahrenheit, which they seldom

00:07:44.460 --> 00:07:49.310
do, then convert into Celsius,
and then convert to Kelvin.

00:07:49.310 --> 00:07:52.200
We already know that zero
Kelvin is equal

00:07:52.200 --> 00:07:57.450
to minus 273 Celsius.

00:07:57.450 --> 00:08:06.720
Or another way you could say it
is x Kelvin is equal to--

00:08:06.720 --> 00:08:11.200
essentially, whatever degree you
get in Celsius, you just

00:08:11.200 --> 00:08:13.370
add 273 to it.

00:08:13.370 --> 00:08:14.560
Does that make sense?

00:08:14.560 --> 00:08:18.440
Think of it this way: if
you're at zero degrees

00:08:18.440 --> 00:08:28.480
Celsius, you're already 273
degrees above zero Kelvin.

00:08:28.480 --> 00:08:30.720
Think about that, and hopefully
that makes sense--

00:08:30.720 --> 00:08:32.650
maybe you want to draw a number
line just to make sure.

00:08:32.650 --> 00:08:36.100
Whatever Celsius degree you
have, just add 273 to it, and

00:08:36.100 --> 00:08:37.350
you'll get Kelvin.

00:08:44.000 --> 00:08:49.340
Add 273 to 27 degrees Celsius,
and that's 300 Kelvin, and

00:08:49.340 --> 00:08:53.510
then 50 degrees Celsius
is-- add 273 to it.

00:08:53.510 --> 00:09:01.170
So 50 plus 273 is 323,
so now we can

00:09:01.170 --> 00:09:02.330
substitute into this formula.

00:09:02.330 --> 00:09:12.560
P1, 1,000 pascals times V1 times
30 divided by the first

00:09:12.560 --> 00:09:14.630
temperature-- remember
to do it in Kelvin--

00:09:14.630 --> 00:09:17.260
300, is equal to P2.

00:09:17.260 --> 00:09:18.450
We don't know what that is.

00:09:18.450 --> 00:09:29.970
P2 times V2 times 20 divided by
our new temperature, 323.

00:09:29.970 --> 00:09:33.770
We can simplify this: we could
take two 0's off of here, take

00:09:33.770 --> 00:09:38.450
two 0's off of here, then you
could take a 3 out of here,

00:09:38.450 --> 00:09:42.620
and then take a 3 out of here,
and we're left with 100.

00:09:42.620 --> 00:09:50.050
This is equal to 100-- that was
30,000 divided by 300, and

00:09:50.050 --> 00:09:52.500
so that's 100 on the
left-hand side.

00:09:52.500 --> 00:10:01.590
So we have 100 is equal to
P times 20 over 323.

00:10:01.590 --> 00:10:03.390
I'm running out of time.

00:10:03.390 --> 00:10:13.060
If I were to solve for it, 323
times 100 divided by 20

00:10:13.060 --> 00:10:18.890
equals-- so my new pressure
is 1,615 pascals.

00:10:18.890 --> 00:10:20.780
I just solved this equation,
and the hard part was

00:10:20.780 --> 00:10:22.050
converting to Kelvin.

00:10:22.050 --> 00:10:23.810
See you in the next video.


Thermodynamics part 2: Ideal gas law

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:01.360
Welcome back.

00:00:01.360 --> 00:00:04.680
In the last video, I told you
that pressure times volume is

00:00:04.680 --> 00:00:05.370
a constant.

00:00:05.370 --> 00:00:09.660
That if you increase the
pressure-- or if you increase

00:00:09.660 --> 00:00:11.380
the volume, you're going to
decrease the pressure.

00:00:11.380 --> 00:00:13.250
Hopefully, you got an
intuitive sense why.

00:00:13.250 --> 00:00:17.150
Or likewise, if you squeezed the
balloon, or the box, and

00:00:17.150 --> 00:00:21.100
there are no openings there,
then the pressure within the

00:00:21.100 --> 00:00:22.540
box would increase.

00:00:22.540 --> 00:00:25.340
With that said, let's see if we
can do a couple of fairly

00:00:25.340 --> 00:00:28.995
typical problems that
you'll see.

00:00:28.995 --> 00:00:32.299
So let's say that I have
a box, or a balloon, or

00:00:32.299 --> 00:00:34.775
something, and it has a volume,
and so let me call

00:00:34.775 --> 00:00:36.025
this the initial volume.

00:00:40.980 --> 00:00:49.540
My initial volume is 50 cubic
meters, and my initial

00:00:49.540 --> 00:00:54.730
pressure is 500 pascals.

00:00:54.730 --> 00:00:56.620
Just so you remember,
what's a pascal?

00:00:56.620 --> 00:01:03.655
That's 500 newtons
per meter cubed.

00:01:08.160 --> 00:01:11.260
I take that box, or balloon, or
whatever, and I compress it

00:01:11.260 --> 00:01:13.770
down to 20 meters cubed.

00:01:13.770 --> 00:01:18.020
So I compress it, so I squeeze
it-- that was the first

00:01:18.020 --> 00:01:20.340
example that I gave last time.

00:01:20.340 --> 00:01:22.620
It was the same container,
and I squeeze it down

00:01:22.620 --> 00:01:25.720
to 20 meters cubed.

00:01:25.720 --> 00:01:28.540
What's going to be
the new pressure?

00:01:28.540 --> 00:01:30.500
You should immediately have an
intuition-- what happens when

00:01:30.500 --> 00:01:31.610
you squeeze a balloon?

00:01:31.610 --> 00:01:33.180
It becomes harder to do it.

00:01:43.970 --> 00:01:45.550
What's going to be
the new pressure?

00:01:45.550 --> 00:01:47.690
It's definitely going to be
higher-- when you decrease the

00:01:47.690 --> 00:01:50.470
volume, the pressure increases
are inversely related.

00:01:50.470 --> 00:01:53.670
The pressure's going to go up,
and let's see if we can

00:01:53.670 --> 00:01:54.930
calculate it.

00:01:54.930 --> 00:02:02.250
We know that P1 times v1 is
equal to some constant, and

00:02:02.250 --> 00:02:07.540
since we have no aggregate
change in energy-- I'm just

00:02:07.540 --> 00:02:10.970
telling you that the box is
squeezed, I'm not telling you

00:02:10.970 --> 00:02:15.010
whether it did any work, or
anything like that-- the same

00:02:15.010 --> 00:02:17.620
constant is going to be equal to
the new pressure times the

00:02:17.620 --> 00:02:20.840
new volume, which is equal
to P2 times V2.

00:02:20.840 --> 00:02:23.620
You could just have the general
relationship: P1 times

00:02:23.620 --> 00:02:28.000
V1 is equal to P2 times V2,
assuming that no work was

00:02:28.000 --> 00:02:30.830
done, and there was no exchange
of energy from

00:02:30.830 --> 00:02:32.240
outside of the system.

00:02:32.240 --> 00:02:34.650
In most of these cases, when you
see this on an exam, that

00:02:34.650 --> 00:02:36.350
is the case.

00:02:36.350 --> 00:02:44.140
The old pressure was 500 pascals
times 50 meters cubed.

00:02:44.140 --> 00:02:51.960
One thing to keep in mind,
because this equivalence is

00:02:51.960 --> 00:02:54.430
not equal, and we're not saying
it has to equal some

00:02:54.430 --> 00:02:57.280
necessary absolute number--
for example, we don't know

00:02:57.280 --> 00:02:59.160
exactly what this K is, although
we could figure it

00:02:59.160 --> 00:03:03.270
out right now-- as long as
you're using one unit for

00:03:03.270 --> 00:03:06.070
pressure on this side, and one
unit for volume on this side,

00:03:06.070 --> 00:03:08.330
you just have to use
the same units.

00:03:08.330 --> 00:03:10.620
We could have done this same
exact problem the exact same

00:03:10.620 --> 00:03:14.460
way, if instead of meters cubed,
they said liters, as

00:03:14.460 --> 00:03:16.430
long as we had liters here.

00:03:16.430 --> 00:03:18.120
You just have to make sure
you're using the same units on

00:03:18.120 --> 00:03:19.090
both sides.

00:03:19.090 --> 00:03:21.860
In this case, we have 500
pascals as the pressure, and

00:03:21.860 --> 00:03:24.950
the volume is 50 meters cubed.

00:03:24.950 --> 00:03:31.260
That's going to be equal to the
new pressure, P2, times

00:03:31.260 --> 00:03:33.840
the new volume, 20
meters cubed.

00:03:33.840 --> 00:03:39.090
Let's see what we can do: we can
divide both sides by 10,

00:03:39.090 --> 00:03:42.880
so we can take the 10 out of
there, and we could divide

00:03:42.880 --> 00:03:48.440
both sides by 2, so that
becomes a 250.

00:03:48.440 --> 00:03:56.520
We we get 250 times 5 is equal
to P2, and so P2 is equal to

00:03:56.520 --> 00:04:00.680
1250 pascals, and if we kept
with the units, you

00:04:00.680 --> 00:04:01.620
would have seen that.

00:04:01.620 --> 00:04:09.590
When I decreased the volume
by roughly 60%, I have the

00:04:09.590 --> 00:04:14.950
pressure actually increased by
2 1/2, so that gels with what

00:04:14.950 --> 00:04:17.519
we talked about before.

00:04:17.519 --> 00:04:23.500
Let's add another variable into
this mix-- let's talk

00:04:23.500 --> 00:04:26.120
about temperature.

00:04:26.120 --> 00:04:29.450
Like pressure, volume, work, and
a lot of concepts that we

00:04:29.450 --> 00:04:31.780
talk about in physics,
temperature is something that

00:04:31.780 --> 00:04:35.545
you probably are at least
reasonably familiar with.

00:04:38.800 --> 00:04:41.260
How do you view temperature?

00:04:41.260 --> 00:04:43.390
A high temperature means
something is hot, and a low

00:04:43.390 --> 00:04:46.590
temperature means something is
cold, and I think that also

00:04:46.590 --> 00:04:50.690
gives you intuition that a
higher temperature object has

00:04:50.690 --> 00:04:51.940
more energy.

00:04:57.580 --> 00:05:04.380
The sun has more energy than an
ice cube-- I think that's

00:05:04.380 --> 00:05:05.350
fair enough.

00:05:05.350 --> 00:05:08.620
I think you also have the sense
that-- what would have

00:05:08.620 --> 00:05:10.830
more energy?

00:05:10.830 --> 00:05:33.890
A 100 degree cup of tea, or a
100 degree barrel of tea.

00:05:33.890 --> 00:05:35.950
I want to make them equivalent
in terms of

00:05:35.950 --> 00:05:37.440
what they're holding.

00:05:37.440 --> 00:05:38.400
I think you have a sense.

00:05:38.400 --> 00:05:40.450
Even though they're the same
temperature, they're both

00:05:40.450 --> 00:05:42.390
pretty warm-- let's say this
is 100 degrees Celsius, so

00:05:42.390 --> 00:05:45.800
they're both boiling-- that the
barrel, because there's

00:05:45.800 --> 00:05:49.070
more of it, is going to
have more energy.

00:05:49.070 --> 00:05:54.090
It's equally hot, and there's
just more molecules there.

00:05:54.090 --> 00:05:55.620
That's what temperature is.

00:05:55.620 --> 00:06:04.200
Temperature, in general, is a
measure roughly equal to some

00:06:04.200 --> 00:06:08.440
constant times the kinetic
energy-- the average kinetic

00:06:08.440 --> 00:06:12.950
energy-- per molecule.

00:06:21.570 --> 00:06:24.720
So the average kinetic energy
of the system divided by the

00:06:24.720 --> 00:06:29.570
total number of molecules we
have. Another way we could

00:06:29.570 --> 00:06:30.690
talk about is, temperature is

00:06:30.690 --> 00:06:33.230
essentially energy per molecule.

00:06:33.230 --> 00:06:35.600
So something that has a lot of
molecules, where N is the

00:06:35.600 --> 00:06:38.160
number of molecules.

00:06:38.160 --> 00:06:40.270
Another way we could view this
is that the kinetic energy of

00:06:40.270 --> 00:06:45.840
the system is going to be
equal to the number of

00:06:45.840 --> 00:06:48.170
molecules times the
temperature.

00:06:48.170 --> 00:06:51.960
This is just a constant-- times
1 over K, but we don't

00:06:51.960 --> 00:06:53.430
even know what this is, so we
could say that's still a

00:06:53.430 --> 00:06:56.910
constant-- so the kinetic energy
of the system is going

00:06:56.910 --> 00:07:03.680
to be equal to some constant
times the number of particles

00:07:03.680 --> 00:07:06.170
times temperature.

00:07:06.170 --> 00:07:07.300
We don't know what this
is, and we're going to

00:07:07.300 --> 00:07:08.710
figure this out later.

00:07:08.710 --> 00:07:10.410
This is another interesting
concept.

00:07:10.410 --> 00:07:17.780
We said that pressure times
volume is proportional to the

00:07:17.780 --> 00:07:22.710
kinetic energy of the system--
the aggregate, if you take all

00:07:22.710 --> 00:07:25.080
of the molecules and combine
their kinetic energies.

00:07:25.080 --> 00:07:27.230
These aren't the same K's-- I
could put another constant

00:07:27.230 --> 00:07:28.960
here and call that K1.

00:07:28.960 --> 00:07:36.280
And we also know that the
kinetic energy of the system

00:07:36.280 --> 00:07:44.980
is equal to some other constant
times the number of

00:07:44.980 --> 00:07:49.700
molecules I have times
the temperature.

00:07:49.700 --> 00:07:53.010
If you think about it, you could
also say that this is

00:07:53.010 --> 00:07:57.480
proportional to this, and this
is proportional to this.

00:07:57.480 --> 00:08:01.900
You could say that pressure
times volume is proportional

00:08:01.900 --> 00:08:04.890
to the number-- and these are
all different proportional

00:08:04.890 --> 00:08:06.540
constants, and we'll figure
out this exact constant

00:08:06.540 --> 00:08:09.630
later-- so we could say that
pressure times volume is

00:08:09.630 --> 00:08:13.460
proportional to molecules we
have, times temperature.

00:08:13.460 --> 00:08:16.080
And we said that we can
view temperature

00:08:16.080 --> 00:08:18.580
as energy per molecule.

00:08:18.580 --> 00:08:20.940
Another way we could say is
that if this constant is

00:08:20.940 --> 00:08:24.230
constant, which is by
definition, and the number of

00:08:24.230 --> 00:08:29.090
molecules is constant-- we
have PV over temperature.

00:08:29.090 --> 00:08:31.380
Pressure times volume over
temperature is going to be

00:08:31.380 --> 00:08:33.620
equal to something times the
number of molecules, so we

00:08:33.620 --> 00:08:36.880
could say that's some other
constant, like k4.

00:08:36.880 --> 00:08:39.370
This is another interesting
thing to think about: we said

00:08:39.370 --> 00:08:41.320
pressure times volume is equal
to pressure times volume, and

00:08:41.320 --> 00:08:42.980
now we added temperature
into the mix.

00:08:49.840 --> 00:08:56.880
We could say P1 times V1
over T1 is equal to P2

00:08:56.880 --> 00:09:00.010
times V2 over T2.

00:09:00.010 --> 00:09:02.050
Does this make sense to you?

00:09:02.050 --> 00:09:11.120
What happens if I have another
box, and I have my particles

00:09:11.120 --> 00:09:13.370
bouncing around like always.

00:09:13.370 --> 00:09:16.080
I have some volume, and some
amount of pressure-- what

00:09:16.080 --> 00:09:19.730
happens when the temperature
goes up?

00:09:19.730 --> 00:09:20.720
What am I saying?

00:09:20.720 --> 00:09:24.990
I'm saying that the average
kinetic energy per molecule is

00:09:24.990 --> 00:09:27.270
going to go up, so they're going
to bounce against the

00:09:27.270 --> 00:09:28.390
walls more.

00:09:28.390 --> 00:09:30.930
If they bounce against the walls
more, the pressure's

00:09:30.930 --> 00:09:35.290
going to go up, assuming
volume stays flat.

00:09:38.710 --> 00:09:42.420
Another way you could think
about it-- let's say the

00:09:42.420 --> 00:09:50.890
temperature goes up, and the
pressure stays flat.

00:09:50.890 --> 00:09:52.730
So what did I have to do?

00:09:52.730 --> 00:09:55.380
I just said if the temperature
goes up, the average kinetic

00:09:55.380 --> 00:09:58.230
energy of each molecule--
they'll bounce more.

00:09:58.230 --> 00:10:02.390
In order to make them bounce
against the sides of the walls

00:10:02.390 --> 00:10:05.170
as often, I'd have to
increase the volume.

00:10:05.170 --> 00:10:07.520
If you hold pressure constant,
the only way you can do that

00:10:07.520 --> 00:10:09.620
is by increasing the volume
while you increase the

00:10:09.620 --> 00:10:10.840
temperature.

00:10:10.840 --> 00:10:14.230
Let's keep this in mind, and we
will use this to solve some

00:10:14.230 --> 00:10:17.010
pretty typical problems
in the next video.


Thermodynamics part 1: Molecular theory of gases

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:04.310
After all the work we've been
doing with fluids, you

00:00:04.310 --> 00:00:07.670
probably have a pretty good
sense of what pressure is.

00:00:07.670 --> 00:00:10.110
Now let's think a little bit
about what it really means,

00:00:10.110 --> 00:00:11.910
especially when we think
about it in terms

00:00:11.910 --> 00:00:14.040
of a gas in a volume.

00:00:14.040 --> 00:00:15.070
Remember, what was
the difference

00:00:15.070 --> 00:00:17.440
between a gas and a liquid?

00:00:17.440 --> 00:00:19.590
They're both fluids, they both
take the shape of their

00:00:19.590 --> 00:00:24.070
containers, but a gas is
compressible, while a liquid

00:00:24.070 --> 00:00:25.230
is incompressible.

00:00:25.230 --> 00:00:27.680
Let's start focusing on gases.

00:00:27.680 --> 00:00:39.120
Let's say I have a container,
and I have a

00:00:39.120 --> 00:00:40.830
bunch of gas in it.

00:00:40.830 --> 00:00:42.430
What is a gas made of?

00:00:42.430 --> 00:00:45.190
It's just made up of a whole
bunch of the molecules of the

00:00:45.190 --> 00:00:47.860
gas itself, and I'll draw each
of the molecules with a little

00:00:47.860 --> 00:00:51.620
dot-- it's just going to have
a bunch of molecules in it.

00:00:51.620 --> 00:00:54.770
There's many, many, many more
than what I've drawn, but

00:00:54.770 --> 00:00:57.090
that's indicative, and they'll
all be going in random

00:00:57.090 --> 00:01:02.440
directions-- this one might be
going really fast in that

00:01:02.440 --> 00:01:04.349
direction, and that one might
be going a little bit slower

00:01:04.349 --> 00:01:06.220
in that direction.

00:01:06.220 --> 00:01:09.130
They all have their own little
velocity vectors, and they're

00:01:09.130 --> 00:01:11.970
always constantly bumping into
each other, and bumping into

00:01:11.970 --> 00:01:14.510
the sides of the container, and
ricocheting here and there

00:01:14.510 --> 00:01:17.030
and changing velocity.

00:01:17.030 --> 00:01:20.250
In general, especially at this
level of physics, we assume

00:01:20.250 --> 00:01:26.410
that this is an ideal gas, that
all of the bumps that

00:01:26.410 --> 00:01:28.040
occur, there's no
loss of energy.

00:01:28.040 --> 00:01:31.710
Or essentially that they're all
elastic bumps between the

00:01:31.710 --> 00:01:32.540
different molecules.

00:01:32.540 --> 00:01:36.620
There's no loss of momentum.

00:01:36.620 --> 00:01:39.280
Let's keep that in mind, and
everything you're going to see

00:01:39.280 --> 00:01:42.280
in high school and on the AP
test is going to deal with

00:01:42.280 --> 00:01:43.560
ideal gases.

00:01:43.560 --> 00:01:46.400
Let's think about what pressure

00:01:46.400 --> 00:01:47.740
means in this context.

00:01:50.530 --> 00:01:53.350
A lot of what we think about
pressure is something pushing

00:01:53.350 --> 00:01:55.110
on an area.

00:01:55.110 --> 00:01:56.930
If we think about pressure
here-- let's pick

00:01:56.930 --> 00:01:57.840
an arbitrary area.

00:01:57.840 --> 00:02:00.770
Let's take this side.

00:02:00.770 --> 00:02:05.295
Let's take this surface
of its container.

00:02:09.870 --> 00:02:10.940
Where's the pressure
going to be

00:02:10.940 --> 00:02:13.540
generated onto this surface?

00:02:13.540 --> 00:02:17.660
It's going to be generated by
just the millions and billions

00:02:17.660 --> 00:02:20.040
and trillions of little
bumps every time-- let

00:02:20.040 --> 00:02:21.850
me draw a side view.

00:02:21.850 --> 00:02:27.160
If this is the side view of the
container, that same side,

00:02:27.160 --> 00:02:32.590
every second there's always
these little molecules of gas

00:02:32.590 --> 00:02:34.720
moving around.

00:02:34.720 --> 00:02:37.930
If we pick an arbitrary period
of time, they're always

00:02:37.930 --> 00:02:41.380
ricocheting off of the side.

00:02:44.200 --> 00:02:48.000
We're looking at time over a
super-small fraction of time.

00:02:48.000 --> 00:02:51.790
And over that period of time,
this one might end up here,

00:02:51.790 --> 00:02:54.750
this one maybe bumped into it
right after it ricocheted and

00:02:54.750 --> 00:02:59.400
came here, this one changes
momentum and goes like that.

00:02:59.400 --> 00:03:01.140
This one might have already been
going in that direction,

00:03:01.140 --> 00:03:02.640
and that one might ricochet.

00:03:02.640 --> 00:03:05.280
But what's happening is, at
any given moment, since

00:03:05.280 --> 00:03:10.760
there's so many molecules,
there's always going to be

00:03:10.760 --> 00:03:14.190
some molecules that
are bumping into

00:03:14.190 --> 00:03:15.960
the side of the wall.

00:03:15.960 --> 00:03:18.275
When they bump, they have
a change in momentum.

00:03:21.520 --> 00:03:24.900
All force is change in
momentum over time.

00:03:30.960 --> 00:03:34.410
What I'm saying is that in any
interval of time, over any

00:03:34.410 --> 00:03:37.070
period or any change in time,
there's just going to be a

00:03:37.070 --> 00:03:41.910
bunch of particles that are
changing their momentum on the

00:03:41.910 --> 00:03:45.500
side of this wall.

00:03:45.500 --> 00:03:48.250
That is going to generate force,
and so if we think

00:03:48.250 --> 00:03:51.560
about how many on average--
because it's hard to keep

00:03:51.560 --> 00:03:55.720
track of each particle
individually, and when we did

00:03:55.720 --> 00:03:58.670
kinematics and stuff, we'd keep
track of the individual

00:03:58.670 --> 00:03:59.460
object at play.

00:03:59.460 --> 00:04:02.780
But when we're dealing with
gases and things on a macro

00:04:02.780 --> 00:04:06.280
level, you can't keep track of
any individual one, unless you

00:04:06.280 --> 00:04:08.830
have some kind of unbelievable
supercomputer.

00:04:08.830 --> 00:04:14.490
We can say, on average, this
many particles are changing

00:04:14.490 --> 00:04:17.240
momentum on this wall in
this amount of time.

00:04:17.240 --> 00:04:20.890
And so the force exerted on this
wall or this surface is

00:04:20.890 --> 00:04:22.670
going to be x.

00:04:22.670 --> 00:04:25.910
If we know what that force is,
and we you know the area of

00:04:25.910 --> 00:04:30.090
the wall, we can figure out
pressure, because pressure is

00:04:30.090 --> 00:04:36.560
equal to force divided
by area.

00:04:36.560 --> 00:04:42.110
What does this help us with?

00:04:42.110 --> 00:04:44.150
I wanted to give you that
intuition first, and now I'm

00:04:44.150 --> 00:04:49.710
just going to give you the one
formula that you really just

00:04:49.710 --> 00:04:50.970
need to know in thermodynamics.

00:04:50.970 --> 00:04:56.430
And then as we go into the next
few videos, I'll prove to

00:04:56.430 --> 00:04:57.750
you why it works, and
hopefully give

00:04:57.750 --> 00:04:58.700
you more of an intuition.

00:04:58.700 --> 00:05:01.230
Now you understand, hopefully,
what pressure means in the

00:05:01.230 --> 00:05:03.610
context of a gas
in a container.

00:05:03.610 --> 00:05:09.910
With that out of the way, let
me give you a formula.

00:05:09.910 --> 00:05:11.890
I hope by the end of this video
you have the intuition

00:05:11.890 --> 00:05:14.520
for why this formula works.

00:05:14.520 --> 00:05:18.490
In general, if I have an ideal
gas in a container, the

00:05:18.490 --> 00:05:23.630
pressure exerted on the gas-- on
the side of the container,

00:05:23.630 --> 00:05:26.910
or actually even at any point
within the gas, because it

00:05:26.910 --> 00:05:30.000
will all become homogeneous at
some point-- and we'll talk

00:05:30.000 --> 00:05:33.160
about entropy in future videos--
but the pressure in

00:05:33.160 --> 00:05:36.920
the container and on its
surface, times the volume of

00:05:36.920 --> 00:05:41.820
the container, is equal
to some constant.

00:05:41.820 --> 00:05:44.230
We'll see in future videos that
that constant is actually

00:05:44.230 --> 00:05:48.830
proportional to the average
kinetic energy of the

00:05:48.830 --> 00:05:50.100
molecules bouncing around.

00:05:50.100 --> 00:05:51.240
That should make sense to you.

00:05:51.240 --> 00:05:56.495
If the molecules were moving
around a lot faster, then you

00:05:56.495 --> 00:05:58.400
would have more kinetic energy,
and then they would be

00:05:58.400 --> 00:06:00.880
changing momentum on the sides
of the surface a lot more, so

00:06:00.880 --> 00:06:03.930
you would have more pressure.

00:06:03.930 --> 00:06:06.780
Let's see if we can get a little
bit more intuition onto

00:06:06.780 --> 00:06:09.185
why pressure times volume
is a constant.

00:06:18.900 --> 00:06:24.180
Let's say I have a container
now, and it's got a bunch of

00:06:24.180 --> 00:06:25.640
molecules of gas in it.

00:06:28.850 --> 00:06:32.190
Just like I showed you in that
last bit right before I

00:06:32.190 --> 00:06:35.260
erased, these are bouncing
off of the sides

00:06:35.260 --> 00:06:36.510
at a certain rate.

00:06:41.390 --> 00:06:42.950
Each of the molecules might
have a different kinetic

00:06:42.950 --> 00:06:44.160
energy-- it's always changing,
because they're always

00:06:44.160 --> 00:06:45.960
transferring momentum
to each other.

00:06:45.960 --> 00:06:50.030
But on average, they all have
a given kinetic energy, they

00:06:50.030 --> 00:06:52.420
keep bumping at a certain rate
into the wall, and that

00:06:52.420 --> 00:06:53.880
determines the pressure.

00:06:53.880 --> 00:06:58.920
What happens if I were able to
squeeze the box, and if I were

00:06:58.920 --> 00:07:03.860
able to decrease the
volume of the box?

00:07:07.310 --> 00:07:09.910
I just take that same box with
the same number of molecules

00:07:09.910 --> 00:07:11.470
in it, but I squeeze.

00:07:11.470 --> 00:07:13.460
I make the volume of
the box smaller--

00:07:13.460 --> 00:07:15.940
what's going to happen?

00:07:15.940 --> 00:07:22.110
I have the same number of
molecules in there, with the

00:07:22.110 --> 00:07:25.890
same kinetic energy, and on
average, they're moving with

00:07:25.890 --> 00:07:26.830
the same velocities.

00:07:26.830 --> 00:07:29.300
So now what's going to happen?

00:07:29.300 --> 00:07:33.150
They're going to be hitting the
sides more often-- at the

00:07:33.150 --> 00:07:36.370
same time here that this
particle went bam, bam, now it

00:07:36.370 --> 00:07:38.680
could go bam, bam, bam.

00:07:38.680 --> 00:07:41.400
They're going to be hitting
the sides more often, so

00:07:41.400 --> 00:07:44.280
you're going to have more
changes in momentum, and so

00:07:44.280 --> 00:07:49.550
you're actually going to have
each particle exert more force

00:07:49.550 --> 00:07:51.130
on each surface.

00:07:51.130 --> 00:07:53.250
Because it's going to be hitting
them more often in a

00:07:53.250 --> 00:07:54.510
given amount of time.

00:07:54.510 --> 00:07:56.350
The surfaces themselves
are smaller.

00:07:56.350 --> 00:07:59.540
You have more force on a
surface, and on a smaller

00:07:59.540 --> 00:08:02.170
surface, you're going to
have higher pressure.

00:08:02.170 --> 00:08:04.060
Hopefully, that gives you an
intuition that if I had some

00:08:04.060 --> 00:08:06.900
amount of pressure in this
situation-- if I squeeze the

00:08:06.900 --> 00:08:09.190
volume, the pressure
increases.

00:08:09.190 --> 00:08:11.750
Another intuition-- if
I have a balloon,

00:08:11.750 --> 00:08:12.940
what blows up a balloon?

00:08:12.940 --> 00:08:18.830
It's the internal air pressure
of the helium, or your own

00:08:18.830 --> 00:08:21.320
exhales that you put
into the balloon.

00:08:21.320 --> 00:08:24.000
The more and more you try to
squeeze a balloon-- if you

00:08:24.000 --> 00:08:26.580
squeeze it from all directions,
it gets harder and

00:08:26.580 --> 00:08:30.040
harder to do it, and that's
because the pressure within

00:08:30.040 --> 00:08:35.289
the balloon increases as you
decrease the volume.

00:08:35.289 --> 00:08:41.480
If volume goes down, pressure
goes up, and that makes sense.

00:08:41.480 --> 00:08:43.669
That follows that when they
multiply each other, you have

00:08:43.669 --> 00:08:45.130
to have a constant.

00:08:45.130 --> 00:08:47.050
Let's take the same example
again, and what happens if you

00:08:47.050 --> 00:08:49.190
make the volume bigger?

00:08:49.190 --> 00:08:56.340
Let's say I have-- it's huge
like that, and I should have

00:08:56.340 --> 00:08:58.890
done it more proportionally, but
I think you get the idea.

00:08:58.890 --> 00:09:02.090
You have the same number of
particles, and if I had a

00:09:02.090 --> 00:09:04.740
particle here, in some period
of time it could have gone

00:09:04.740 --> 00:09:09.930
bam, bam, bam-- it could have
hit the walls twice.

00:09:09.930 --> 00:09:12.420
Now, in this situation, with
larger walls, it might just go

00:09:12.420 --> 00:09:15.300
bam, and in that same amount
of time, it will maybe get

00:09:15.300 --> 00:09:17.150
here and won't even hit
the other wall.

00:09:17.150 --> 00:09:19.670
The particles, on average, are
going to be colliding with the

00:09:19.670 --> 00:09:22.600
wall less often, and the walls
are going to have a larger

00:09:22.600 --> 00:09:24.390
area, as well.

00:09:24.390 --> 00:09:29.860
So in this case, when our volume
goes up, the average

00:09:29.860 --> 00:09:35.520
pressure or the pressure in
the container goes down.

00:09:35.520 --> 00:09:37.560
Hopefully, that gives you a
little intuition, and so

00:09:37.560 --> 00:09:38.970
you'll never forget
that pressure

00:09:38.970 --> 00:09:41.200
times volume is constant.

00:09:41.200 --> 00:09:44.590
And then we can use that to do
some pretty common problems,

00:09:44.590 --> 00:09:46.110
which I'll do in
the next video.

00:09:46.110 --> 00:09:46.917
I'm about to run out of time.

00:09:46.917 --> 00:09:48.167
See you soon.


Complex Numbers (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.160
We learned in the imaginary
numbers video, that hopefully

00:00:05.160 --> 00:00:08.800
you've watched, that every now
and then in certain equations

00:00:08.800 --> 00:00:11.850
you end up with a square
root of a negative number.

00:00:11.850 --> 00:00:14.490
You know, you end up with
square root of negative 1 or

00:00:14.490 --> 00:00:16.640
square root of negative 9.

00:00:16.640 --> 00:00:19.730
And since any real number, when
you square it, is either 0 or

00:00:19.730 --> 00:00:22.630
positive, this was
undefined for us.

00:00:22.630 --> 00:00:25.760
And then in order to make it
defined in that video, people

00:00:25.760 --> 00:00:27.510
came up with this
thing called i.

00:00:27.510 --> 00:00:29.860
And you can call that
the imaginary unit,

00:00:29.860 --> 00:00:32.020
or just the number i.

00:00:32.020 --> 00:00:36.520
And you define i as saying,
well if i squared-- we'll

00:00:36.520 --> 00:00:37.340
just make a definition.

00:00:37.340 --> 00:00:40.170
This thing called i
when you square it,

00:00:40.170 --> 00:00:41.620
it equals negative 1.

00:00:41.620 --> 00:00:44.960
And then that simplifies
the idea of taking a

00:00:44.960 --> 00:00:46.200
negative square root.

00:00:46.200 --> 00:00:48.310
Because then you could say, oh
well the square root of

00:00:48.310 --> 00:00:53.430
negative 9 is the same thing as
i times the square root

00:00:53.430 --> 00:00:56.725
of 9, which equals 3i.

00:00:56.725 --> 00:00:57.880
And how can we say that?

00:00:57.880 --> 00:01:00.400
Well, what happens when
you square this thing?

00:01:00.400 --> 00:01:02.700
So what's 3i squared?

00:01:02.700 --> 00:01:11.120
3i squared is equal to 3
squared times i squared.

00:01:11.120 --> 00:01:13.470
This is just exponent
properties.

00:01:13.470 --> 00:01:15.890
And that equals 9 times
negative 1, which

00:01:15.890 --> 00:01:17.520
equals negative 9.

00:01:17.520 --> 00:01:22.410
So 3i squared is
equal to negative 9.

00:01:22.410 --> 00:01:25.110
And then if we are kind of
extending the definition of

00:01:25.110 --> 00:01:27.190
square roots to negative
numbers, then we can go the

00:01:27.190 --> 00:01:30.610
other way around and we could
say 3i is equal to the

00:01:30.610 --> 00:01:32.790
square root of negative 9.

00:01:32.790 --> 00:01:36.490
And 3i we can call an
imaginary number.

00:01:36.490 --> 00:01:39.850
And the word is "imaginary,"
but it's as real as anything.

00:01:39.850 --> 00:01:42.670
I mean to some degree, do
negative numbers even exist?

00:01:42.670 --> 00:01:48.560
They're just kind of a way of
us-- when we put this sign in

00:01:48.560 --> 00:01:51.180
front of things, it just tells
us something about how it

00:01:51.180 --> 00:01:53.120
relates to this magnitude.

00:01:53.120 --> 00:01:54.390
Anyway, I don't want
to confuse you.

00:01:54.390 --> 00:01:57.010
But imaginary numbers tend to
get a bad rap because they're

00:01:57.010 --> 00:02:00.370
called imaginary numbers, so
some people think that they

00:02:00.370 --> 00:02:02.540
exist less than other things.

00:02:02.540 --> 00:02:06.920
But with that said, any number
times this imaginary unit

00:02:06.920 --> 00:02:10.110
i is an imaginary number.

00:02:10.110 --> 00:02:13.100
When you do the quadratic
equation you realize that

00:02:13.100 --> 00:02:15.930
sometimes you end up with
a number that has a

00:02:15.930 --> 00:02:16.770
little bit of both.

00:02:16.770 --> 00:02:20.180
It has a real part and
an imaginary part.

00:02:20.180 --> 00:02:23.630
So let me give you
an example of that.

00:02:23.630 --> 00:02:31.310
Let's say I have the
number 5 plus 2i.

00:02:31.310 --> 00:02:33.490
That number you might say, oh,
well maybe I can simplify it.

00:02:33.490 --> 00:02:34.440
But you really can't.

00:02:34.440 --> 00:02:37.460
You can't add a real number
plus an imaginary number.

00:02:37.460 --> 00:02:40.100
You can almost kind of imagine
them-- and I don't want to use

00:02:40.100 --> 00:02:44.040
the word "imagine" too much--
as in different dimensions.

00:02:44.040 --> 00:02:48.800
And so a number that has a real
part, like the 5, and an

00:02:48.800 --> 00:02:53.050
imaginary part, like the 2i,
this is called a

00:02:53.050 --> 00:02:54.070
complex number.

00:02:59.970 --> 00:03:04.550
And you could, if you want,
even graph a complex number.

00:03:04.550 --> 00:03:05.050
Let me see.

00:03:05.050 --> 00:03:08.375
You could make the
vertical axis.

00:03:08.375 --> 00:03:11.370
The vertical axis you could
call it the imaginary axis.

00:03:14.710 --> 00:03:18.010
And you could make the
horizontal axis the real axis.

00:03:22.600 --> 00:03:25.390
This is a symbol for the
set of real numbers.

00:03:25.390 --> 00:03:27.650
And 5 plus 2i, well
it's real part is 5.

00:03:27.650 --> 00:03:32.570
So one, two, three, four, five.

00:03:32.570 --> 00:03:33.970
That's five.

00:03:33.970 --> 00:03:37.160
And it's imaginary part is 2i,
so you go along the imaginary

00:03:37.160 --> 00:03:40.870
axis, or the i-axis you
could even say, 2.

00:03:40.870 --> 00:03:46.720
And then you would have this
number here called 5 plus 2i.

00:03:46.720 --> 00:03:49.990
In future videos I'll actually
do examples where we'll make

00:03:49.990 --> 00:03:52.880
more use of complex numbers.

00:03:52.880 --> 00:03:56.520
But now that we've defined what
a complex number is, let's see

00:03:56.520 --> 00:03:58.860
how we can operate with it.

00:03:58.860 --> 00:04:01.690
So what happens when we
add two complex numbers?

00:04:04.870 --> 00:04:07.130
Clear image.

00:04:07.130 --> 00:04:08.790
So let's say we have
two complex numbers.

00:04:08.790 --> 00:04:12.160
One is a plus bi.

00:04:12.160 --> 00:04:16.730
So the real part is a, the
imaginary part is bi.

00:04:16.730 --> 00:04:19.110
And let's say that I
have another complex

00:04:19.110 --> 00:04:26.430
number, c plus di.

00:04:26.430 --> 00:04:29.230
And in general the symbol that
people tend to use-- you know,

00:04:29.230 --> 00:04:32.330
you always use x for when
you're dealing with equations.

00:04:32.330 --> 00:04:32.640
Right?

00:04:32.640 --> 00:04:35.090
Like x could be any general
real number when you're

00:04:35.090 --> 00:04:35.920
dealing with an equation.

00:04:35.920 --> 00:04:39.670
In complex numbers the
convention is to use

00:04:39.670 --> 00:04:41.570
z, the letter z.

00:04:41.570 --> 00:04:43.630
So, for example, we could
call this the complex

00:04:43.630 --> 00:04:44.990
number z sub 1.

00:04:44.990 --> 00:04:45.810
And it could have been any.

00:04:45.810 --> 00:04:48.630
I mean, this choice
of z is arbitrary.

00:04:48.630 --> 00:04:49.920
This is an i right here.

00:04:49.920 --> 00:04:52.310
This could be z sub 2.

00:04:52.310 --> 00:04:57.840
So what is the complex number
z sub 1 plus z sub 2?

00:04:57.840 --> 00:05:04.410
Well that equals this, a plus
bi, plus-- this is z sub

00:05:04.410 --> 00:05:07.210
2 right here-- c plus di.

00:05:07.210 --> 00:05:07.990
Let me switch colors.

00:05:07.990 --> 00:05:10.510
This is getting monotonous.

00:05:10.510 --> 00:05:13.360
So when you add two complex
numbers, all you do is you add

00:05:13.360 --> 00:05:16.150
the real parts to each other
and you add the complex

00:05:16.150 --> 00:05:17.230
parts to each other.

00:05:17.230 --> 00:05:23.530
So that equals a
plus b-- oh sorry.

00:05:23.530 --> 00:05:24.190
The real parts.

00:05:24.190 --> 00:05:25.310
So it's a and c.

00:05:25.310 --> 00:05:26.130
Those are the real parts.

00:05:26.130 --> 00:05:31.360
So that equals a plus c
plus-- and then you add

00:05:31.360 --> 00:05:33.240
the imaginary parts.

00:05:33.240 --> 00:05:35.190
b plus d times i.

00:05:35.190 --> 00:05:38.130
So you have b i's and d i's.

00:05:38.130 --> 00:05:42.450
So when you add them together
you have b plus d i's.

00:05:42.450 --> 00:05:46.060
b plus d in the imaginary
direction, you can

00:05:46.060 --> 00:05:47.400
almost imagine.

00:05:47.400 --> 00:05:49.100
I'm using the word
imagine too much.

00:05:49.100 --> 00:05:49.740
So it's pretty easy.

00:05:49.740 --> 00:05:52.690
You just add the real and you
add the imaginary parts.

00:05:52.690 --> 00:05:57.060
So what happens if you
multiply two complex numbers?

00:05:57.060 --> 00:05:58.820
And actually, let's
just go in order.

00:05:58.820 --> 00:05:59.930
What happens when
you subtract them?

00:05:59.930 --> 00:06:00.830
Well, it's the same thing.

00:06:00.830 --> 00:06:03.880
Nothing fancy here. z
sub 1 minus z sub 2.

00:06:03.880 --> 00:06:06.800
That's just going to be
equal to-- you subtract

00:06:06.800 --> 00:06:08.070
the real parts.

00:06:08.070 --> 00:06:09.420
a minus c.

00:06:09.420 --> 00:06:11.080
And that's the new real part.

00:06:11.080 --> 00:06:13.580
And then you subtract
the imaginary parts.

00:06:13.580 --> 00:06:15.640
b minus d times i.

00:06:15.640 --> 00:06:17.510
And this will be the
new imaginary part.

00:06:17.510 --> 00:06:19.950
So this will be the
new complex number.

00:06:19.950 --> 00:06:21.770
What happens when I
multiply the two numbers?

00:06:21.770 --> 00:06:26.700
So what is z sub
1 times z sub 2?

00:06:26.700 --> 00:06:33.450
Well that equals a plus
bi times c plus di.

00:06:33.450 --> 00:06:36.910
And we essentially can
just FOIL it out.

00:06:36.910 --> 00:06:39.310
I don't know if you've learned
FOIL in 8th grade algebra,

00:06:39.310 --> 00:06:40.100
9th grade algebra.

00:06:40.100 --> 00:06:40.520
I don't like it.

00:06:40.520 --> 00:06:42.130
I actually just think of
it as the distributive

00:06:42.130 --> 00:06:43.270
property twice.

00:06:43.270 --> 00:06:46.260
So what we could do is we
can take the c plus di

00:06:46.260 --> 00:06:48.890
and distribute it on
each of these terms.

00:06:48.890 --> 00:06:49.230
Right?

00:06:49.230 --> 00:06:56.200
So this should be equal
to a times c plus di.

00:06:56.200 --> 00:06:57.020
Right?

00:06:57.020 --> 00:07:02.540
Plus bi times c plus di.

00:07:02.540 --> 00:07:02.850
Right?

00:07:02.850 --> 00:07:05.860
All I did is I took the c plus
di and I multiplied it by this

00:07:05.860 --> 00:07:09.120
to get this, and multiplied
it by this to get this.

00:07:09.120 --> 00:07:10.720
And then we distribute again.

00:07:10.720 --> 00:07:16.060
We get ac plus adi--
this is an i.

00:07:16.060 --> 00:07:17.920
I know my i's aren't
looking good.

00:07:17.920 --> 00:07:25.300
adi plus-- now
what's bi times c?

00:07:25.300 --> 00:07:25.986
Well that's cbi.

00:07:29.680 --> 00:07:33.950
And then we have bi times di.

00:07:33.950 --> 00:07:38.130
So you get b times
d, which is bd.

00:07:38.130 --> 00:07:40.790
And then what's i times i?

00:07:40.790 --> 00:07:42.750
It's i squared, or negative 1.

00:07:42.750 --> 00:07:43.150
Right?

00:07:43.150 --> 00:07:45.630
Times negative 1.

00:07:45.630 --> 00:07:47.600
So what does this simplify to?

00:07:47.600 --> 00:07:52.230
Well this simplifies to--
I'm going to go back here.

00:07:52.230 --> 00:07:54.050
So let's see what
the real parts are.

00:07:54.050 --> 00:07:58.070
We have this term, a
times c, that's real.

00:07:58.070 --> 00:08:00.270
So ac.

00:08:00.270 --> 00:08:02.720
And then this last term,
which was bi times di.

00:08:02.720 --> 00:08:05.320
Because we're multiplying i
times i we get a negative 1,

00:08:05.320 --> 00:08:06.490
but it becomes real again.

00:08:06.490 --> 00:08:07.560
Right?

00:08:07.560 --> 00:08:11.920
So we get negative bd minus bd.

00:08:11.920 --> 00:08:13.600
So that's our new real part.

00:08:13.600 --> 00:08:17.950
So that was this term
and that term are real.

00:08:17.950 --> 00:08:19.320
And now what's our
new imaginary part?

00:08:19.320 --> 00:08:21.700
Well it's going to be these
two terms added together.

00:08:21.700 --> 00:08:28.090
So it's-- I'm running out
of space again-- ad plus

00:08:28.090 --> 00:08:30.850
cb, all of that times i.

00:08:30.850 --> 00:08:32.780
It's a little easier with real
numbers, and maybe in the next

00:08:32.780 --> 00:08:34.170
video I'll use real numbers.

00:08:34.170 --> 00:08:38.250
But the important thing to
realize is essentially you just

00:08:38.250 --> 00:08:40.980
use the distributive property
and realize that you can only

00:08:40.980 --> 00:08:42.640
add real terms to each other.

00:08:42.640 --> 00:08:45.225
You can't add a real
to an imaginary.

00:08:45.225 --> 00:08:47.260
And you can only add imaginary
terms to each other.

00:08:47.260 --> 00:08:49.430
And just remember, when you
have two imaginary numbers

00:08:49.430 --> 00:08:53.010
times each other, the i's, when
multiplied times each other,

00:08:53.010 --> 00:08:55.070
and you get negative 1.

00:08:55.070 --> 00:08:58.010
Now the last operation, when
you divide complex numbers.

00:08:58.010 --> 00:09:00.375
This gets a little bit
interesting, and maybe

00:09:00.375 --> 00:09:02.650
a little unintuitive.

00:09:02.650 --> 00:09:08.350
So what happens if I divide
z sub 1 by z sub 2?

00:09:08.350 --> 00:09:17.240
So once again that equals a
plus bi, divided by c plus di.

00:09:17.240 --> 00:09:19.180
How do I divide this?

00:09:19.180 --> 00:09:22.750
Well, we're going to use a
property that hopefully

00:09:22.750 --> 00:09:24.380
you learned in algebra.

00:09:24.380 --> 00:09:29.040
That if I multiply-- let me
do this in this corner.

00:09:29.040 --> 00:09:33.770
a plus b times a minus b.

00:09:33.770 --> 00:09:34.990
What does that equal?

00:09:34.990 --> 00:09:39.620
That equals a squared
minus b squared.

00:09:39.620 --> 00:09:43.230
And so if I multiply
a complex number.

00:09:43.230 --> 00:09:45.560
Let's say, in theory, I
multiply a complex number c

00:09:45.560 --> 00:09:49.950
plus di times c minus di.

00:09:49.950 --> 00:09:51.200
What do I get?

00:09:51.200 --> 00:09:55.600
I get c squared
minus di squared.

00:09:58.480 --> 00:10:00.060
And what's di squared
going to be?

00:10:00.060 --> 00:10:03.870
It's d squared times negative
1, so it becomes c squared--

00:10:03.870 --> 00:10:04.720
oh, I'm out of time.

00:10:04.720 --> 00:10:07.330
Actually, let me continue the
division in the next video,

00:10:07.330 --> 00:10:08.175
because it can get involved.

00:10:08.175 --> 00:10:09.750
See you soon.


Complex Numbers (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.060
So far we've learned what a
complex number is; we've even

00:00:03.060 --> 00:00:04.060
learned how to graph it.

00:00:04.060 --> 00:00:06.170
And we learned how to add,
subtract, and multiply it.

00:00:06.170 --> 00:00:08.640
Where I left off in the last
video was, how do we divide

00:00:08.640 --> 00:00:10.190
two complex numbers.

00:00:10.190 --> 00:00:13.110
So I said, let's say I have
one complex numbers, z1.

00:00:13.110 --> 00:00:17.550
And that equals a plus b i.

00:00:17.550 --> 00:00:18.970
i.

00:00:18.970 --> 00:00:22.150
And I want to
divide that by z2.

00:00:22.150 --> 00:00:25.980
Which is, c plus d i.

00:00:25.980 --> 00:00:27.510
So let me ask you a question.

00:00:27.510 --> 00:00:29.390
And I touched on this
in the last video.

00:00:29.390 --> 00:00:33.030
And let me do it in a
different color over here.

00:00:33.030 --> 00:00:40.800
We know that a plus b times
a minus b is equal to a

00:00:40.800 --> 00:00:42.090
squared minus b squared.

00:00:42.090 --> 00:00:45.400
And you can multiply it out,
in case you're not sure.

00:00:45.400 --> 00:00:49.760
Remember, it's just a times
a plus b times a minus

00:00:49.760 --> 00:00:57.550
a times b, plus a times
a, and you'll get this.

00:00:57.550 --> 00:00:58.910
But you know how to
do this, anyway.

00:00:58.910 --> 00:01:01.010
There's a review of it,
if you need to do it.

00:01:01.010 --> 00:01:05.690
So, given that,
what is c plus d?

00:01:05.690 --> 00:01:09.160
What happens if we do
something very similar

00:01:09.160 --> 00:01:10.670
with a complex number?

00:01:10.670 --> 00:01:20.450
If we say c plus d i
times c minus d i.

00:01:20.450 --> 00:01:22.730
Well, in this case a is c.

00:01:22.730 --> 00:01:24.210
And b is d i, right?

00:01:24.210 --> 00:01:29.560
So this is just going to
be equal to c squared

00:01:29.560 --> 00:01:33.290
minus d i squared.

00:01:33.290 --> 00:01:36.970
d i squared.

00:01:36.970 --> 00:01:44.210
And that equals c squared
minus d squared i squared.

00:01:44.210 --> 00:01:50.280
And that equals c squared
minus d squared.

00:01:50.280 --> 00:01:52.070
And i squared is
negative 1, right?

00:01:52.070 --> 00:01:53.870
So this is going to be
multiplied by negative 1, so

00:01:53.870 --> 00:01:55.160
it cancels out this negative.

00:01:55.160 --> 00:01:58.150
So you get c squared
plus d squared.

00:01:58.150 --> 00:01:59.510
That's interesting.

00:01:59.510 --> 00:02:03.350
When I multiply a complex
number times this other number,

00:02:03.350 --> 00:02:07.270
which is very similar to it,
but it's kind of the imaginary

00:02:07.270 --> 00:02:09.390
part, goes in the
other direction.

00:02:09.390 --> 00:02:13.520
When I multiply the two, I get
a completely real number.

00:02:13.520 --> 00:02:15.600
All of the i's disappear.

00:02:15.600 --> 00:02:19.520
And, in general, this number --
if we call this -- well, in our

00:02:19.520 --> 00:02:28.350
example this was z2, so if we
say that z2 equals c plus d i,

00:02:28.350 --> 00:02:33.670
the quantity c minus d i
is called its conjugate.

00:02:33.670 --> 00:02:35.330
And that's just good
terminology to know.

00:02:35.330 --> 00:02:40.560
And the sign for conjugate
is that line over the top.

00:02:40.560 --> 00:02:42.850
So the conjugate of
z2 is c minus d i.

00:02:42.850 --> 00:02:47.840
Or you could say, the
conjugate of c minus d i

00:02:47.840 --> 00:02:50.610
is equal to c plus d i.

00:02:50.610 --> 00:02:51.830
Or you could say it
the other way around.

00:02:51.830 --> 00:02:58.030
The conjugate of c plus d i
is equal to c minus d i.

00:02:58.030 --> 00:03:01.260
And notice, we're just
switching the direction in

00:03:01.260 --> 00:03:04.130
the imaginary -- along the
imaginary axis, when we take

00:03:04.130 --> 00:03:05.520
the conjugate of something.

00:03:05.520 --> 00:03:08.710
With that said, let me
erase that and go back

00:03:08.710 --> 00:03:09.700
to our original problem.

00:03:09.700 --> 00:03:12.010
Because the conjugate is
the tool we're going

00:03:12.010 --> 00:03:16.050
to use to divide this.

00:03:16.050 --> 00:03:18.820
So we know when we multiply an
imaginary number times its

00:03:18.820 --> 00:03:22.010
conjugate, we get
a real number.

00:03:22.010 --> 00:03:25.210
And we know, also, if we
multiply -- we can multiply

00:03:25.210 --> 00:03:26.800
anything by 1, and we
get the same number.

00:03:26.800 --> 00:03:31.380
So let's multiply the numerator
and denominator of this

00:03:31.380 --> 00:03:35.630
expression by the conjugate
of the denominator.

00:03:35.630 --> 00:03:38.630
So let me do that.

00:03:38.630 --> 00:03:41.705
So the conjugate of the
denominator is going

00:03:41.705 --> 00:03:46.560
to be c minus d i.

00:03:46.560 --> 00:03:49.260
So c minus d i
over c minus d i.

00:03:49.260 --> 00:03:51.800
So this was c plud d i, so
this is its conjugate.

00:03:51.800 --> 00:03:54.020
And so what do we get?

00:03:54.020 --> 00:03:59.730
So in the numerator, we get a c
-- I don't want to run out of

00:03:59.730 --> 00:04:09.890
space, I always do -- a c, so a
times c, minus a d i, minus a d

00:04:09.890 --> 00:04:15.950
i -- these i's are looking
funny -- this is an i.

00:04:15.950 --> 00:04:23.080
Plus b c i; plus b c i.

00:04:23.080 --> 00:04:26.820
And then the last term, we
have a plus b minus b.

00:04:26.820 --> 00:04:29.450
So it's minus b d i squared.

00:04:29.450 --> 00:04:35.580
Minus b d i squared.

00:04:35.580 --> 00:04:37.150
All of that.

00:04:37.150 --> 00:04:39.890
And this is a plus
b times a minus b.

00:04:39.890 --> 00:04:41.250
So it's equal to a
squared minus b squared.

00:04:41.250 --> 00:04:44.690
So this is going to be equal to
-- and it this will become

00:04:44.690 --> 00:04:46.425
second nature to you after a
while, but you might want

00:04:46.425 --> 00:04:48.110
to just multiply it out.

00:04:48.110 --> 00:04:54.470
This equals c squared
plus d squared.

00:04:54.470 --> 00:04:55.480
And don't take my word for it.

00:04:55.480 --> 00:04:57.300
Actually, algebraically,
multiply this out and just

00:04:57.300 --> 00:04:59.330
realize you can only add real
parts to real parts and

00:04:59.330 --> 00:05:00.820
imaginary parts to
imaginary parts.

00:05:00.820 --> 00:05:02.500
So let me simplify that.

00:05:02.500 --> 00:05:05.400
That equals -- let's
see, the real parts.

00:05:05.400 --> 00:05:08.390
This is real, a c.

00:05:08.390 --> 00:05:11.540
And this is minus
b d i squared.

00:05:11.540 --> 00:05:13.870
So the i squared is minus 1.

00:05:13.870 --> 00:05:16.480
So it switches the sign here,
so it becomes plus b d.

00:05:16.480 --> 00:05:18.050
And we can get rid of d i.

00:05:18.050 --> 00:05:23.050
So the real parts
are, a c plus b d.

00:05:23.050 --> 00:05:26.610
That's that, and that.

00:05:26.610 --> 00:05:29.920
And then the imaginary parts
are plus -- this one's

00:05:29.920 --> 00:05:39.410
positive, so I'll put one first
-- b c minus a d i, all of that

00:05:39.410 --> 00:05:43.680
over c squared plus d squared.

00:05:43.680 --> 00:05:45.770
And that still might not look
like a complex number to you.

00:05:45.770 --> 00:05:48.190
But then we can separate them
out and we could say well,

00:05:48.190 --> 00:05:56.530
that equals a c plus b d over
c squared plus d squared.

00:05:56.530 --> 00:05:58.630
And that's the real part.

00:05:58.630 --> 00:06:07.720
Plus b c minus a d over c
squared plus d squared.

00:06:07.720 --> 00:06:10.360
And that times i, and
that's the imaginary part.

00:06:10.360 --> 00:06:15.430
So you can't merge, when you're
adding and subtracting, the

00:06:15.430 --> 00:06:16.920
real part to the
imaginary part.

00:06:16.920 --> 00:06:20.300
But you can most definitely
scale an imaginary

00:06:20.300 --> 00:06:21.410
number by a real number.

00:06:21.410 --> 00:06:22.380
And that's essentially
what we're doing.

00:06:22.380 --> 00:06:24.240
We're multiplying 1
over c squared plus d

00:06:24.240 --> 00:06:25.900
squared times this.

00:06:25.900 --> 00:06:28.820
So, division might seem a
little complicated when I

00:06:28.820 --> 00:06:29.610
write it all in variables.

00:06:29.610 --> 00:06:32.080
But let me give you an example
and you will hopefully see that

00:06:32.080 --> 00:06:35.570
it -- with real numbers, and
-- not real numbers, with

00:06:35.570 --> 00:06:37.930
actual numbers, I should be
careful with what I say.

00:06:37.930 --> 00:06:43.680
Let's say I have 1 plus 2 i.

00:06:43.680 --> 00:06:48.436
And I want to divide that by, I
don't know, let's divide it by,

00:06:48.436 --> 00:06:51.200
I'm going to pick
a random number.

00:06:51.200 --> 00:06:54.530
2 plus 3i.

00:06:54.530 --> 00:06:55.440
And so what do we do?

00:06:55.440 --> 00:07:00.350
We multiply it times the
conjugate of the denominator.

00:07:00.350 --> 00:07:04.730
2 minus 3i over --
over itself, right?

00:07:04.730 --> 00:07:06.060
Because then we're not
changing the number.

00:07:06.060 --> 00:07:08.300
This is just 1, this
simplifies to 1.

00:07:08.300 --> 00:07:11.540
It equals -- the bottom,
we can multiply it out.

00:07:11.540 --> 00:07:13.080
But hopefully it's
second nature to you.

00:07:13.080 --> 00:07:18.090
It equals 4 plus 9, right?

00:07:18.090 --> 00:07:21.830
Because that's just a
squared plus b squared.

00:07:21.830 --> 00:07:22.490
Right?

00:07:22.490 --> 00:07:24.730
Well, I mean, it's a squared
minus b squared, but then the

00:07:24.730 --> 00:07:27.100
i's, when you multiply, and it
becomes a negative number.

00:07:27.100 --> 00:07:29.010
Try it out if you
don't believe me.

00:07:29.010 --> 00:07:32.140
And then the top, we
get 1 times 2 is 2.

00:07:32.140 --> 00:07:36.500
1 times minus 3i is minus 3i.

00:07:36.500 --> 00:07:42.100
And you have 2i times
2, which is plus 4i.

00:07:42.100 --> 00:07:45.160
And then you have
2i times minus 3i.

00:07:45.160 --> 00:07:48.070
So that's minus 6.

00:07:48.070 --> 00:07:50.700
Minus 6i squared.

00:07:50.700 --> 00:07:51.950
Well, what does i
squared equal?

00:07:51.950 --> 00:07:53.600
That equals negative 1.

00:07:53.600 --> 00:07:56.900
So negative 1 times negative 6.

00:07:56.900 --> 00:07:59.610
Get rid of the i squared and
this becomes a positive.

00:07:59.610 --> 00:08:00.900
So then, what are
our real parts?

00:08:00.900 --> 00:08:03.990
Our real parts are 2 and 6.

00:08:03.990 --> 00:08:07.240
so 2 plus 6 is 8.

00:08:07.240 --> 00:08:08.510
And what are our
imaginary parts?

00:08:08.510 --> 00:08:11.300
Minus 3i plus 4i.

00:08:11.300 --> 00:08:13.180
So that's just plus 1i, right?

00:08:13.180 --> 00:08:15.430
Minus 3 plus 4 is positive 1.

00:08:15.430 --> 00:08:17.710
So it's just plus 1i.

00:08:17.710 --> 00:08:19.800
Over 13.

00:08:19.800 --> 00:08:22.170
Or we could write that as -- if
we wanted to write that in the

00:08:22.170 --> 00:08:30.420
traditional complex form --
is 8/13 plus 1 over 13i.

00:08:30.420 --> 00:08:33.740
So when I divided one complex
number by another, I got

00:08:33.740 --> 00:08:35.080
another complex number.

00:08:35.080 --> 00:08:37.280
And an interesting exercise
for you to do is, pick some

00:08:37.280 --> 00:08:38.840
random complex numbers.

00:08:38.840 --> 00:08:42.490
Plot them out on complex plane,
and see what happens when you

00:08:42.490 --> 00:08:44.570
multiply them, when you divide
them, when you add them,

00:08:44.570 --> 00:08:45.680
when you subtract them.

00:08:45.680 --> 00:08:48.490
And when you scale them.

00:08:48.490 --> 00:08:49.910
Or when you take the conjugate.

00:08:49.910 --> 00:08:51.990
And that'll give you a better
intuition of what's going

00:08:51.990 --> 00:08:53.310
on with these numbers.

00:08:53.310 --> 00:08:57.410
Anyway, I will see you
in the next video.


Interest (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.560
So let's generalize a bit
what we learned in the

00:00:02.560 --> 00:00:03.830
last presentation.

00:00:03.830 --> 00:00:07.280
Let's say I'm
borrowing P dollars.

00:00:07.280 --> 00:00:08.790
P dollars, that's what I
borrowed so that's my

00:00:08.790 --> 00:00:10.740
initial principal.

00:00:10.740 --> 00:00:11.420
So that's principal.

00:00:14.710 --> 00:00:17.070
r is equal to the rate,
the interest rate that

00:00:17.070 --> 00:00:18.310
I'm borrowing at.

00:00:18.310 --> 00:00:22.600
We can also write that
as 100r%, right?

00:00:22.600 --> 00:00:24.370
And I'm going to borrow
it for-- well, I

00:00:24.370 --> 00:00:25.510
don't know-- t years.

00:00:29.190 --> 00:00:32.210
Let's see if we can come up
with equations to figure out

00:00:32.210 --> 00:00:35.960
how much I'm going to owe at
the end of t years using either

00:00:35.960 --> 00:00:38.170
simple or compound interest.

00:00:38.170 --> 00:00:41.450
So let's do simple first
because that's simple.

00:00:41.450 --> 00:00:48.460
So at time 0-- so let's make
this the time axis-- how

00:00:48.460 --> 00:00:49.310
much am I going to owe?

00:00:49.310 --> 00:00:51.950
Well, that's right when I
borrow it, so if I paid

00:00:51.950 --> 00:00:55.220
it back immediately, I
would just owe P, right?

00:00:55.220 --> 00:01:00.730
At time 1, I owe P plus the
interest, plus you can kind of

00:01:00.730 --> 00:01:04.460
view it as the rent on that
money, and that's r times P.

00:01:04.460 --> 00:01:06.390
And that previously, in the
previous example, in the

00:01:06.390 --> 00:01:07.900
previous video, was 10%.

00:01:07.900 --> 00:01:11.043
P was 100, so I had to pay $10
to borrow that money for a

00:01:11.043 --> 00:01:13.265
year, and I had to
pay back $110.

00:01:13.265 --> 00:01:18.610
And this is the same thing
as P times 1 plus r, right?

00:01:18.610 --> 00:01:21.830
Because you could
just use 1P plus rP.

00:01:21.830 --> 00:01:24.080
And then after two years,
how much do we owe?

00:01:24.080 --> 00:01:28.190
Well, every year, we just
pay another rP, right?

00:01:28.190 --> 00:01:30.860
In the previous example,
it was another $10.

00:01:30.860 --> 00:01:34.000
So if this is 10%, every
year we just pay 10% of

00:01:34.000 --> 00:01:35.360
our original principal.

00:01:35.360 --> 00:01:38.730
So in year 2, we owe P plus
rP-- that's what we owed in

00:01:38.730 --> 00:01:42.500
year 1-- and then another
rP, so that equals

00:01:42.500 --> 00:01:45.350
P plus 1 plus 2r.

00:01:45.350 --> 00:01:47.720
And just take the P out,
and you get a 1 plus r

00:01:47.720 --> 00:01:49.840
plus r, so 1 plus 2r.

00:01:49.840 --> 00:01:54.770
And then in year 3, we'd owe
what we owed in year 2.

00:01:54.770 --> 00:02:00.330
So P plus rP plus rP, and then
we just pay another rP, another

00:02:00.330 --> 00:02:03.830
say, you know, if r is 10%, or
50% of our original principal,

00:02:03.830 --> 00:02:10.300
plus rP, and so that
equals P times 1 plus 3r.

00:02:10.300 --> 00:02:15.910
So after t years,
how much do we owe?

00:02:15.910 --> 00:02:18.815
Well, it's our original
principal times 1 plus,

00:02:18.815 --> 00:02:22.330
and it'll be tr.

00:02:22.330 --> 00:02:25.920
So you can distribute this out
because every year we pay Pr,

00:02:25.920 --> 00:02:27.390
and there's going
to be t years.

00:02:27.390 --> 00:02:28.970
And so that's why
it makes sense.

00:02:28.970 --> 00:02:31.940
So if I were to say
I'm borrowing-- let's

00:02:31.940 --> 00:02:33.410
do some numbers.

00:02:33.410 --> 00:02:35.460
You could work it out this way,
and I recommend you do it.

00:02:35.460 --> 00:02:37.100
You shouldn't just
memorize formulas.

00:02:37.100 --> 00:02:45.820
If I were to borrow $50 at 15%
simple interest for 15-- or

00:02:45.820 --> 00:02:50.700
let's say for 20 years, at the
end of the 20 years, I would

00:02:50.700 --> 00:03:04.000
owe $50 times 1 plus the
time 20 times 0.15, right?

00:03:04.000 --> 00:03:08.960
And that's equal to $50 times 1
plus-- what's 20 times 0.15?

00:03:08.960 --> 00:03:11.220
That's 3, right?

00:03:11.220 --> 00:03:12.060
Right.

00:03:12.060 --> 00:03:17.550
So it's 50 times 4, which
is equal to $200 to

00:03:17.550 --> 00:03:18.740
borrow it for 20 years.

00:03:18.740 --> 00:03:22.920
So $50 at 15% for 20
years results in a $200

00:03:22.920 --> 00:03:24.700
payment at the end.

00:03:24.700 --> 00:03:27.010
So this was simple
interest, and this was

00:03:27.010 --> 00:03:28.370
the formula for it.

00:03:28.370 --> 00:03:32.560
Let's see if we can do the same
thing with compound interest.

00:03:32.560 --> 00:03:35.770
Let me erase all this.

00:03:39.130 --> 00:03:42.800
That's not how I
wanted to erase it.

00:03:42.800 --> 00:03:44.340
There we go.

00:03:48.200 --> 00:03:53.430
OK, so with compound interest,
in year 1, it's the same thing,

00:03:53.430 --> 00:03:55.020
really, as simple interest, and
we saw that in the

00:03:55.020 --> 00:03:55.820
previous video.

00:03:55.820 --> 00:04:04.810
I owe P plus, and now the rate
times P, and that equals

00:04:04.810 --> 00:04:08.190
P times 1 plus r.

00:04:08.190 --> 00:04:09.450
Fair enough.

00:04:09.450 --> 00:04:12.810
Now year 2 is where compound
and simple interest diverge.

00:04:12.810 --> 00:04:14.820
In simple interest, we would
just pay another rP, and

00:04:14.820 --> 00:04:17.170
it becomes 1 plus 2r.

00:04:17.170 --> 00:04:19.190
In compound interest,
this becomes the new

00:04:19.190 --> 00:04:22.010
principal, right?

00:04:22.010 --> 00:04:25.050
So if this is the new
principal, we are going to pay

00:04:25.050 --> 00:04:28.370
1 plus r times this, right?

00:04:28.370 --> 00:04:29.820
Our original principal was P.

00:04:29.820 --> 00:04:35.000
After one year, we paid 1 plus
r times the original principal

00:04:35.000 --> 00:04:38.270
times 1 plus r rate.

00:04:38.270 --> 00:04:42.520
So to go into year 2, we're
going to pay what we owed at

00:04:42.520 --> 00:04:47.640
the end of year 1, which is P
times 1 plus r, and then we're

00:04:47.640 --> 00:04:49.640
going to grow that
by r percent.

00:04:49.640 --> 00:04:53.240
So we're going to multiply
that again times 1 plus r.

00:04:58.040 --> 00:05:02.900
And so that equals P
times 1 plus r squared.

00:05:02.900 --> 00:05:04.950
So the way you could think
about it, in simple interest,

00:05:04.950 --> 00:05:09.170
every year we added a Pr.

00:05:09.170 --> 00:05:12.330
In simple interest, we
added plus Pr every year.

00:05:12.330 --> 00:05:16.760
So if this was $50 and this is
15%, every year we're adding

00:05:16.760 --> 00:05:19.840
$3-- we're adding--
what was that?

00:05:19.840 --> 00:05:20.460
50%.

00:05:20.460 --> 00:05:23.520
We're adding $7.50 in interest,
where P is the principal,

00:05:23.520 --> 00:05:24.560
r is the rate.

00:05:24.560 --> 00:05:27.480
In compound interest, every
year we're multiplying the

00:05:27.480 --> 00:05:31.680
principal times 1 plus
the rate, right?

00:05:31.680 --> 00:05:33.930
So if we go to year 3,
we're going to multiply

00:05:33.930 --> 00:05:35.230
this times 1 plus r.

00:05:35.230 --> 00:05:39.090
So year 3 is P times 1
plus r to the third.

00:05:39.090 --> 00:05:42.160
So year t is going to be
principal times 1 plus

00:05:42.160 --> 00:05:45.240
r to the t-th power.

00:05:45.240 --> 00:05:47.980
And so let's see
that same example.

00:05:47.980 --> 00:05:50.870
We owe $200 in this example
with simple interest.

00:05:50.870 --> 00:05:53.190
Let's see what we owe
in compound interest.

00:05:53.190 --> 00:05:54.980
The principal is $50.

00:05:59.190 --> 00:06:00.640
1 plus-- and what's the rate?

00:06:00.640 --> 00:06:02.690
0.15.

00:06:02.690 --> 00:06:06.180
And we're borrowing
it for 20 years.

00:06:06.180 --> 00:06:14.910
So this is equal to 50 times
1.15 to the 20th power.

00:06:14.910 --> 00:06:18.070
I know you can't read that,
but let me see what I can

00:06:18.070 --> 00:06:20.680
do about the 20th power.

00:06:20.680 --> 00:06:23.905
Let me use my Excel and
clear all of this.

00:06:28.196 --> 00:06:31.840
Actually, I should just use my
mouse instead of the pen tool

00:06:31.840 --> 00:06:34.950
to the clear everything.

00:06:34.950 --> 00:06:36.770
OK, so let me just
pick a random point.

00:06:36.770 --> 00:06:42.220
So I just want to-- plus 1.15
to the 20th power, and you

00:06:42.220 --> 00:06:46.940
could use any calculator:
16.37, let's say.

00:06:46.940 --> 00:06:55.460
So this equals 50 times 16.37.

00:06:55.460 --> 00:06:58.170
And what's 50 times that?

00:06:58.170 --> 00:07:08.560
Plus 50 times that: $818.

00:07:08.560 --> 00:07:11.780
So you've now realized that if
someone's giving you a loan and

00:07:11.780 --> 00:07:14.320
they say, oh, yeah, I'll lend
you-- you need a 20-year loan?

00:07:14.320 --> 00:07:16.340
I'm going to lend
it to you at 15%.

00:07:16.340 --> 00:07:19.840
It's pretty important to
clarify whether they're going

00:07:19.840 --> 00:07:24.400
to charge you 15% interest at
simple interest or

00:07:24.400 --> 00:07:25.870
compound interest.

00:07:25.870 --> 00:07:28.770
Because with compound interest,
you're going to end up paying--

00:07:28.770 --> 00:07:31.900
I mean, look at this: just to
borrow $50, you're going to

00:07:31.900 --> 00:07:36.180
be paying $618 more than if
this was simple interest.

00:07:36.180 --> 00:07:40.480
Unfortunately, in the real
world, most of it is

00:07:40.480 --> 00:07:41.690
compound interest.

00:07:41.690 --> 00:07:44.250
And not only is it compounding,
but they don't even just

00:07:44.250 --> 00:07:46.170
compound it every year and they
don't even just compound it

00:07:46.170 --> 00:07:48.810
every six months, they actually
compound it continuously.

00:07:48.810 --> 00:07:50.830
And so you should watch the
next several videos on

00:07:50.830 --> 00:07:53.750
continuously compounding
interest, and then you'll

00:07:53.750 --> 00:07:57.190
actually start to learn
about the magic of e.

00:07:57.190 --> 00:07:59.910
Anyway, I'll see you
all in the next video.


Introduction to interest

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.500
Well now you've learned what I
think is quite possibly one of

00:00:04.500 --> 00:00:07.010
the most useful concepts in
life, and you might already be

00:00:07.010 --> 00:00:11.920
familiar with it, but if you're
not this will hopefully keep

00:00:11.920 --> 00:00:16.330
you from one day filing
for bankruptcy.

00:00:16.330 --> 00:00:20.830
So anyway, I will talk about
interest, and then simple

00:00:20.830 --> 00:00:21.865
versus compound interest.

00:00:21.865 --> 00:00:23.770
So what's interest?

00:00:23.770 --> 00:00:24.840
We all have heard of it.

00:00:24.840 --> 00:00:29.030
Interest rates, or interest
on your mortgage, or how

00:00:29.030 --> 00:00:31.240
much interest do I owe
on my credit card.

00:00:31.240 --> 00:00:34.140
So interest-- I don't know what
the actual formal definition,

00:00:34.140 --> 00:00:35.610
maybe I should look it up
on Wikipedia-- but it's

00:00:35.610 --> 00:00:37.850
essentially rent on money.

00:00:37.850 --> 00:00:41.350
So it's money that you pay
in order to keep money

00:00:41.350 --> 00:00:42.520
for some period of time.

00:00:42.520 --> 00:00:45.420
That's probably not the most
obvious definition, but

00:00:45.420 --> 00:00:46.920
let me put it this way.

00:00:46.920 --> 00:00:52.640
Let's say that I want to
borrow $100 from you.

00:00:52.640 --> 00:00:54.760
So this is now.

00:00:54.760 --> 00:00:59.120
And let's say that this
is one year from now.

00:00:59.120 --> 00:01:00.080
One year.

00:01:00.080 --> 00:01:04.830
And this is you,
and this is me.

00:01:04.830 --> 00:01:07.580
So now you give me $100.

00:01:07.580 --> 00:01:09.915
And then I have the $100
and a year goes by,

00:01:09.915 --> 00:01:12.570
and I have $100 here.

00:01:12.570 --> 00:01:15.980
And if I were to just give you
that $100 back, you would

00:01:15.980 --> 00:01:17.510
have collected no rent.

00:01:17.510 --> 00:01:19.470
You would have just
got your money back.

00:01:19.470 --> 00:01:20.880
You would have
collected no interest.

00:01:20.880 --> 00:01:24.470
But if you said, Sal I'm
willing to give you $100 now if

00:01:24.470 --> 00:01:30.860
you give me $110 a year later.

00:01:30.860 --> 00:01:34.620
So in this situation, how
much did I pay you to keep

00:01:34.620 --> 00:01:36.620
that $100 for a year?

00:01:36.620 --> 00:01:38.200
Well I'm paying you
$10 more, right?

00:01:38.200 --> 00:01:45.610
I'm returning the $100, and
I'm returning another $10.

00:01:45.610 --> 00:01:51.510
And so this extra $10 that I'm
returning to you is essentially

00:01:51.510 --> 00:01:54.570
the fee that I paid to be able
to keep that money and do

00:01:54.570 --> 00:01:56.790
whatever I wanted with that
money, and maybe save

00:01:56.790 --> 00:01:59.630
it, maybe invest it, do
whatever for a year.

00:01:59.630 --> 00:02:02.200
And that $10 is
essentially the interest.

00:02:02.200 --> 00:02:05.530
And a way that it's often
calculated is a percentage

00:02:05.530 --> 00:02:07.850
of the original amount
that I borrowed.

00:02:07.850 --> 00:02:11.140
And the original amount that I
borrowed in fancy banker or

00:02:11.140 --> 00:02:12.980
finance terminology is
just called principal.

00:02:19.200 --> 00:02:23.630
So in this case the rent on the
money or the interest was $10.

00:02:23.630 --> 00:02:27.920
And if I wanted to do it as a
percentage, I would say 10 over

00:02:27.920 --> 00:02:34.240
the principal-- over 100--
which is equal to 10%.

00:02:34.240 --> 00:02:39.480
So you might have said, hey Sal
I'm willing to lend you $100 if

00:02:39.480 --> 00:02:41.420
you pay me 10% interest on it.

00:02:41.420 --> 00:02:44.770
So 10% of $100 was $10, so
after a year I pay you

00:02:44.770 --> 00:02:46.810
$100, plus the 10%.

00:02:46.810 --> 00:02:47.560
And likewise.

00:02:47.560 --> 00:02:51.220
So for any amount of money, say
you're willing to lend me any

00:02:51.220 --> 00:02:53.540
amount of money for
a 10% interest.

00:02:53.540 --> 00:02:58.680
Well then if you were to lend
me $1,000, then the interest

00:02:58.680 --> 00:03:00.950
would be 10% of that,
which would be $100.

00:03:00.950 --> 00:03:11.020
So then after a year I would
owe you $1,000 plus 10% times

00:03:11.020 --> 00:03:14.555
$1,000, and that's
equal to $1,100.

00:03:14.555 --> 00:03:17.780
All right, I just added
a zero to everything.

00:03:17.780 --> 00:03:20.090
In this case $100 would
be the interest, but

00:03:20.090 --> 00:03:22.130
it would still be 10%.

00:03:22.130 --> 00:03:25.170
So let me now make a
distinction between simple

00:03:25.170 --> 00:03:27.000
interest and compound interest.

00:03:30.430 --> 00:03:33.220
So we just did a fairly simple
example where you lent money

00:03:33.220 --> 00:03:36.540
for me for a year at
10% percent, right?

00:03:36.540 --> 00:03:42.280
So let's say that someone were
to say that my interest rate

00:03:42.280 --> 00:03:43.930
that they charge-- or the
interest rate they charge to

00:03:43.930 --> 00:03:51.000
other people-- is-- well 10% is
a good number-- 10% per year.

00:03:51.000 --> 00:03:55.700
And let's say the principal
that I'm going to borrow

00:03:55.700 --> 00:04:01.900
from this person is $100.

00:04:01.900 --> 00:04:03.980
So my question to you-- and
maybe you want to pause it

00:04:03.980 --> 00:04:18.570
after I pose it-- is how
much do I owe in 10 years?

00:04:18.570 --> 00:04:21.140
How much do I owe in 10 years?

00:04:21.140 --> 00:04:23.080
So there's really two ways
of thinking about it.

00:04:23.080 --> 00:04:30.350
You could say, OK in years at
times zero-- like if I just

00:04:30.350 --> 00:04:32.430
borrowed the money, I just
paid it back immediately,

00:04:32.430 --> 00:04:33.730
it'd be $100, right?

00:04:33.730 --> 00:04:35.210
I'm not going to do that,
I'm going to keep it

00:04:35.210 --> 00:04:36.570
for at least a year.

00:04:36.570 --> 00:04:40.270
So after a year, just based on
the example that we just did, I

00:04:40.270 --> 00:04:48.870
could add 10% of that amount to
the $100, and I would

00:04:48.870 --> 00:04:51.050
then owe $110.

00:04:51.050 --> 00:04:55.420
And then after two years, I
could add another 10% of the

00:04:55.420 --> 00:04:57.800
original principal, right?

00:04:57.800 --> 00:04:59.610
So every year I'm
just adding $10.

00:04:59.610 --> 00:05:03.775
So in this case it would be
$120, and in year three,

00:05:03.775 --> 00:05:05.310
I would owe $130.

00:05:05.310 --> 00:05:09.770
Essentially my rent per year to
borrow this $100 is $10, right?

00:05:09.770 --> 00:05:12.580
Because I'm always taking
10% of the original amount.

00:05:12.580 --> 00:05:17.090
And after 10 years-- because
each year I would have had to

00:05:17.090 --> 00:05:20.120
pay an extra $10 in interest--
after 10 years I

00:05:20.120 --> 00:05:22.630
would owe $200.

00:05:22.630 --> 00:05:23.200
Right?

00:05:23.200 --> 00:05:33.520
And that $200 is equal to $100
of principal, plus $100 of

00:05:33.520 --> 00:05:36.580
interest, because I paid
$10 a year of interest.

00:05:36.580 --> 00:05:39.260
And this notion which I just
did here, this is actually

00:05:39.260 --> 00:05:43.020
called simple interest.

00:05:43.020 --> 00:05:45.260
Which is essentially you take
the original amount you

00:05:45.260 --> 00:05:48.840
borrowed, the interest rate,
the amount, the fee that you

00:05:48.840 --> 00:05:51.140
pay every year is the interest
rate times that original

00:05:51.140 --> 00:05:53.090
amount, and you just
incrementally pay

00:05:53.090 --> 00:05:54.380
that every year.

00:05:54.380 --> 00:05:55.980
But if you think about it,
you're actually paying a

00:05:55.980 --> 00:05:58.390
smaller and smaller percentage
of what you owe going

00:05:58.390 --> 00:05:59.170
into that year.

00:05:59.170 --> 00:06:00.950
And maybe when I show
you compound interest

00:06:00.950 --> 00:06:01.690
that will make sense.

00:06:01.690 --> 00:06:05.530
So this is one way to interpret
10% interest a year.

00:06:05.530 --> 00:06:10.960
Another way to interpret it is,
OK, so in year zero it's $100

00:06:10.960 --> 00:06:13.840
that you're borrowing, or if
they handed the money, you say

00:06:13.840 --> 00:06:15.230
oh no, no, I don't want it and
you just paid it back,

00:06:15.230 --> 00:06:16.550
you'd owe $100.

00:06:16.550 --> 00:06:21.630
After a year, you would
essentially pay the

00:06:21.630 --> 00:06:27.450
$100 plus 10% of $100,
right, which is $110.

00:06:27.450 --> 00:06:32.830
So that's $100,
plus 10% of $100.

00:06:32.830 --> 00:06:35.180
Let me switch colors,
because it's monotonous.

00:06:35.180 --> 00:06:36.970
Right, but I think this
make sense to you.

00:06:36.970 --> 00:06:39.030
And this is where simple
and compound interest

00:06:39.030 --> 00:06:40.220
starts to diverge.

00:06:40.220 --> 00:06:42.930
In the last situation we
just kept adding 10%

00:06:42.930 --> 00:06:44.480
of the original $100.

00:06:44.480 --> 00:06:49.310
In compound interest now,
we don't take 10% of

00:06:49.310 --> 00:06:50.310
the original amount.

00:06:50.310 --> 00:06:52.310
We now take 10% of this amount.

00:06:56.340 --> 00:07:02.440
So now we're going
to take $110.

00:07:02.440 --> 00:07:05.470
You can almost view it
as our new principal.

00:07:05.470 --> 00:07:06.840
This is how much we offer
a year, and then we

00:07:06.840 --> 00:07:09.110
would reborrow it.

00:07:09.110 --> 00:07:19.810
So now we're going to owe
$110 plus 10% times 110.

00:07:19.810 --> 00:07:23.220
You could actually undistribute
the 110 out, and that's

00:07:23.220 --> 00:07:32.950
equal to 110 times 110.

00:07:32.950 --> 00:07:34.440
Actually 110 times 1.1.

00:07:39.730 --> 00:07:41.280
And actually I could
rewrite it this way too.

00:07:41.280 --> 00:07:45.850
I could rewrite it as
100 times 1.1 squared,

00:07:45.850 --> 00:07:49.920
and that equals $121.

00:07:49.920 --> 00:07:52.790
And then in year two, this is
my new principal-- this is

00:07:52.790 --> 00:07:55.110
$121-- this is my
new principal.

00:07:55.110 --> 00:07:57.990
And now I have to in year
three-- so this is year two.

00:07:57.990 --> 00:08:01.710
I'm taking more space,
so this is year two.

00:08:01.710 --> 00:08:06.450
And now in year three, I'm
going to have to pay the $121

00:08:06.450 --> 00:08:14.820
that I owed at the end of year
two, plus 10% times the amount

00:08:14.820 --> 00:08:20.450
of money I owed going
into the year, $121.

00:08:20.450 --> 00:08:22.950
And so that's the same thing--
we could put parentheses around

00:08:22.950 --> 00:08:29.270
here-- so that's the same thing
as 1 times 121 plus 0.1 times

00:08:29.270 --> 00:08:35.650
121, so that's the same
thing as 1.1 times 121.

00:08:35.650 --> 00:08:38.800
Or another way of viewing it,
that's equal to our original

00:08:38.800 --> 00:08:44.180
principal times 1.1
to the third power.

00:08:44.180 --> 00:08:46.060
And if you keep doing this--
and I encourage you do it,

00:08:46.060 --> 00:08:48.660
because it'll really give you a
hands-on sense-- at the end of

00:08:48.660 --> 00:08:52.030
10 years, we will owe-- or you,
I forgot who's borrowing from

00:08:52.030 --> 00:08:57.962
whom-- $100 times 1.1
to the 10th power.

00:08:57.962 --> 00:08:59.050
And what does that equal?

00:08:59.050 --> 00:09:01.320
Let me get my spreadsheet out.

00:09:01.320 --> 00:09:02.690
Let me just pick a random cell.

00:09:02.690 --> 00:09:10.980
So plus 100 times 1.1
to the 10th power.

00:09:10.980 --> 00:09:14.160
So $259 and some change.

00:09:19.890 --> 00:09:22.730
So it might seem like a very
subtle distinction, but it ends

00:09:22.730 --> 00:09:24.580
up being a very big difference.

00:09:24.580 --> 00:09:30.610
When I compounded it 10% for
10 years using compound

00:09:30.610 --> 00:09:33.100
interest, I owe $259.

00:09:33.100 --> 00:09:37.290
When I did it using simple
interest, I only owe $200.

00:09:37.290 --> 00:09:40.770
So that $59 was kind of the
increment of how much more

00:09:40.770 --> 00:09:43.360
compound interest cost me.

00:09:43.360 --> 00:09:45.610
I'm about to run out of time,
so I'll do a couple more

00:09:45.610 --> 00:09:47.560
examples in the next video,
just you really get a deep

00:09:47.560 --> 00:09:50.460
understanding of how to do
compound interest, how the

00:09:50.460 --> 00:09:53.680
exponents work, and what
really is the difference.

00:09:53.680 --> 00:09:54.060
I'll see you in the next video.


Compound interest and e (part 4)

WEBVTT
Kind: captions
Language: en

00:00:00.880 --> 00:00:04.760
In the last video, I hopefully
showed you that if I borrowed P

00:00:04.760 --> 00:00:10.500
dollars, and I borrow it for a
year, and you were to charge me

00:00:10.500 --> 00:00:16.680
an interest rate of r, or you
could say 10r%, then at

00:00:16.680 --> 00:00:19.310
the end-- and we were to
compound continuously.

00:00:19.310 --> 00:00:22.900
So we compound every zillionth
of a second, but we compound it

00:00:22.900 --> 00:00:26.100
a trillion times, however many
of those intervals are in a

00:00:26.100 --> 00:00:30.610
year, that at the end of a
year, I would owe you P

00:00:30.610 --> 00:00:34.710
times e to the r dollars.

00:00:34.710 --> 00:00:35.780
Fair enough.

00:00:35.780 --> 00:00:38.380
Now, what happens if I
borrow it for two years?

00:00:38.380 --> 00:00:42.070
Well, after one year, we
already said that I would

00:00:42.070 --> 00:00:45.470
owe you P times e to
the r dollars, right?

00:00:45.470 --> 00:00:47.570
And then after two
years, what happens?

00:00:47.570 --> 00:00:49.060
Well, this becomes
the new principal.

00:00:49.060 --> 00:00:52.380
You can kind of view it as like
I borrowed this much, then I

00:00:52.380 --> 00:00:54.900
owed this much after a year,
and so this is the new

00:00:54.900 --> 00:00:57.750
principal, so I can
reborrow this, right?

00:00:57.750 --> 00:01:00.980
So if I reborrow this,
this becomes the new P.

00:01:00.980 --> 00:01:06.770
So that becomes the new P, so I
could write Pe to the r, and

00:01:06.770 --> 00:01:08.780
it's going to-- and that
new principal is going to

00:01:08.780 --> 00:01:11.340
compound for another year.

00:01:11.340 --> 00:01:13.340
So e to the r.

00:01:13.340 --> 00:01:17.880
So that equals Pe to the 2r.

00:01:17.880 --> 00:01:21.330
And similarly, this is
now my new principal.

00:01:21.330 --> 00:01:25.710
If I were to borrow it
for another year, it

00:01:25.710 --> 00:01:28.820
becomes Pe to the 3r.

00:01:28.820 --> 00:01:31.740
So, in general, if I borrow P
dollars, that's my initial

00:01:31.740 --> 00:01:35.870
principal, I borrow it at a
rate of r and I borrow it for

00:01:35.870 --> 00:01:38.740
two years, the amount that I
owe after two years

00:01:38.740 --> 00:01:43.110
is Pe to the rt.

00:01:43.110 --> 00:01:48.990
And once you know this, you are
ready to become your local

00:01:48.990 --> 00:01:51.910
banker and to lend people
money continuously.

00:01:51.910 --> 00:01:54.220
And let me just do a couple of
examples because I think it

00:01:54.220 --> 00:01:56.670
might be a little confusing in
the abstract, but with some

00:01:56.670 --> 00:02:00.310
numbers it might all clear up.

00:02:00.310 --> 00:02:07.700
OK, so let's say
I borrow $1,000.

00:02:07.700 --> 00:02:12.440
Let's say that the
interest rate is 25%.

00:02:12.440 --> 00:02:16.060
That's the annual
interest rate.

00:02:16.060 --> 00:02:21.040
Rate is equal to 25%, which
is the same thing as 0.25.

00:02:21.040 --> 00:02:23.810
And let's say I were to
borrow it for three years.

00:02:23.810 --> 00:02:29.140
So t is equal to three years.

00:02:29.140 --> 00:02:33.580
And we're going to continuously
compound this interest.

00:02:33.580 --> 00:02:36.930
So our formula says that the
amount that I'll owe at the end

00:02:36.930 --> 00:02:45.870
of this is how much I borrowed,
$1,000 times e, to my interest

00:02:45.870 --> 00:02:52.240
rate power, 0.25, times the
number of years, times t.

00:02:52.240 --> 00:02:55.630
And so let's-- oh,
sorry, that's 3, right?

00:02:55.630 --> 00:03:02.610
So that equals 1,000e to
the 0.75 power, and let

00:03:02.610 --> 00:03:04.770
me calculate what
that is in Excel.

00:03:04.770 --> 00:03:06.180
And just so you know, I
don't know if you're

00:03:06.180 --> 00:03:07.250
familiar with Excel.

00:03:07.250 --> 00:03:12.710
In Excel, the e to the power--
so I wrote that it's a 1,000

00:03:12.710 --> 00:03:16.700
times-- e to a power
in Excel is exp.

00:03:16.700 --> 00:03:22.270
So that's e to some power;
in this case, it's 0.75.

00:03:22.270 --> 00:03:26.845
So I get my answer.

00:03:26.845 --> 00:03:27.480
I don't know.

00:03:27.480 --> 00:03:29.360
I think it fell off the
bottom of the screen.

00:03:29.360 --> 00:03:30.650
There it is right here.

00:03:30.650 --> 00:03:32.780
That's my answer.

00:03:32.780 --> 00:03:35.470
Zoom in a little bit because I
think you might have trouble

00:03:35.470 --> 00:03:36.710
reading it because it kind of
shrinks it when I

00:03:36.710 --> 00:03:37.540
go to YouTube.

00:03:37.540 --> 00:03:43.010
$2,117.

00:03:43.010 --> 00:03:50.210
It equals $2,117, and that's
what you would owe me at

00:03:50.210 --> 00:03:51.930
the end of three years.

00:03:51.930 --> 00:03:54.010
This is actually the power
of compounding interest.

00:03:54.010 --> 00:03:56.250
A lot of people, you know, when
you hear 10% interest rate or

00:03:56.250 --> 00:04:00.070
even a 25% interest rate, that
no one really makes a

00:04:00.070 --> 00:04:00.870
big deal about it.

00:04:00.870 --> 00:04:03.120
But when you compound it, and
especially when you compound

00:04:03.120 --> 00:04:07.420
it continuously, it can very
quickly turn into very,

00:04:07.420 --> 00:04:09.430
very large numbers.

00:04:09.430 --> 00:04:13.380
But let's do another example,
and this might be another kind

00:04:13.380 --> 00:04:17.200
of more complicated example,
or something that you might

00:04:17.200 --> 00:04:18.950
actually see in a textbook.

00:04:18.950 --> 00:04:23.700
Let's say that I borrow $50.

00:04:23.700 --> 00:04:27.400
I borrow $50, and let's say
it's continuously compounded

00:04:27.400 --> 00:04:31.520
at some rate r, and let's
say it is continuously

00:04:31.520 --> 00:04:34.470
compounded for 10 years.

00:04:34.470 --> 00:04:41.690
At the end of 10
years, I owe $500.

00:04:41.690 --> 00:04:44.660
What was the rate at
which it was compounded?

00:04:44.660 --> 00:04:47.440
So once again, we can
use the same formula.

00:04:47.440 --> 00:04:51.650
We could say, well, if my
original principle is $50--

00:04:51.650 --> 00:04:56.410
so it's going to be $50
times e to the rate.

00:04:56.410 --> 00:05:00.160
We don't know the rate, but
we know that t is equal

00:05:00.160 --> 00:05:04.170
to 10 years, so its 10r.

00:05:04.170 --> 00:05:08.010
That equals my final payment or
how much I owe once all the

00:05:08.010 --> 00:05:10.800
interest and the principal
has compounded.

00:05:10.800 --> 00:05:12.440
It's equal to $500.

00:05:12.440 --> 00:05:14.000
So we can divide
both sides by 50.

00:05:14.000 --> 00:05:22.010
You get e to the 10r
is equal to 10.

00:05:22.010 --> 00:05:22.980
And then how do we solve that?

00:05:22.980 --> 00:05:25.900
Well, we could take the
log base e of both sides.

00:05:25.900 --> 00:05:29.905
Hopefully, you might want to
review the logarithm, but log

00:05:29.905 --> 00:05:32.910
base e-- e is just a number,
if you ever get confused-- is

00:05:32.910 --> 00:05:36.580
equal to log base e of 10.

00:05:36.580 --> 00:05:41.160
And log base e on your
calculator is often

00:05:41.160 --> 00:05:42.420
written the natural log.

00:05:42.420 --> 00:05:45.410
And they called it the natural
log because I'll show you e in

00:05:45.410 --> 00:05:47.470
a hundred different-- not
a hundred, but in many

00:05:47.470 --> 00:05:48.230
different applications.

00:05:48.230 --> 00:05:51.050
It shows up all over nature,
and I think that's why it's

00:05:51.050 --> 00:05:52.750
called the natural log.

00:05:52.750 --> 00:05:54.660
Anyway, let me see if I can
figure out what Excel's

00:05:54.660 --> 00:05:56.590
natural log function is.

00:05:56.590 --> 00:06:02.680
So I need to figure out the
natural log, log base e of 10.

00:06:02.680 --> 00:06:05.750
Equals LN 10.

00:06:05.750 --> 00:06:08.580
Oh, there we go.

00:06:08.580 --> 00:06:09.470
There it is right there: 2.3.

00:06:15.010 --> 00:06:19.250
So first of all, if I say log
base e of e to the 10r, that's

00:06:19.250 --> 00:06:23.180
like saying e to what power
is equal to e to the 10r?

00:06:23.180 --> 00:06:28.940
So this is the same
thing as just 10r.

00:06:31.700 --> 00:06:32.600
Why is that?

00:06:32.600 --> 00:06:35.350
Because remember, logarithm is
an exponent, so this is saying

00:06:35.350 --> 00:06:38.230
e to the 10r is equal
to e to the 10r.

00:06:38.230 --> 00:06:40.290
Review my logarithm videos if
that's a little confusing.

00:06:40.290 --> 00:06:41.990
I know it's a little
confusing at first.

00:06:41.990 --> 00:06:44.510
And then we just figured out
that log base e, so e to the

00:06:44.510 --> 00:06:47.780
what power is 10, is 2.--
what was the number?

00:06:47.780 --> 00:06:51.850
2.30.

00:06:51.850 --> 00:06:53.970
And now-- oh, this
isn't 10 to the r.

00:06:53.970 --> 00:06:56.030
This is 10r, right?

00:06:56.030 --> 00:06:57.560
And so we want to
figure out what r is.

00:06:57.560 --> 00:06:58.700
We divide both sides by 10.

00:06:58.700 --> 00:07:04.800
We get r is equal
to 0.23, or 23%.

00:07:04.800 --> 00:07:07.920
So essentially, if I
continuously compound at an

00:07:07.920 --> 00:07:12.500
annual rate of 23%, after 10
years, I'll essentially

00:07:12.500 --> 00:07:14.460
owe 10 times the money.

00:07:14.460 --> 00:07:16.310
So that's something
good to keep it mind.

00:07:16.310 --> 00:07:19.660
Anyway, I'll leave you there,
and I really encourage you to

00:07:19.660 --> 00:07:21.440
go back a couple of videos,
rewatch them, play

00:07:21.440 --> 00:07:22.030
with the numbers.

00:07:22.030 --> 00:07:23.960
Prove to yourself that
that limit exists.

00:07:23.960 --> 00:07:27.640
Take that limit that we showed
in the beginning, the limit as

00:07:27.640 --> 00:07:32.570
n approaches infinity of 1
over 1 plus n to the n.

00:07:32.570 --> 00:07:35.070
All you have to do to prove
this is just put in larger

00:07:35.070 --> 00:07:36.010
and larger numbers for n.

00:07:36.010 --> 00:07:37.540
And, of course, whatever
number you put in there,

00:07:37.540 --> 00:07:38.560
you have to put over here.

00:07:38.560 --> 00:07:40.250
You can't put a million
here and a trillion there.

00:07:40.250 --> 00:07:41.650
You have to put a trillion
and a trillion, or a

00:07:41.650 --> 00:07:42.810
million and a million.

00:07:42.810 --> 00:07:44.600
And you'll see that
it converges to e.

00:07:44.600 --> 00:07:46.710
And rewatch the videos and
make sure you get kind of

00:07:46.710 --> 00:07:48.750
an intuitive understanding
of everything we did.

00:07:48.750 --> 00:07:51.770
And then this formula, which
most people, frankly, just

00:07:51.770 --> 00:07:55.090
memorize, this Pe to the rt,
will make a lot of sense to

00:07:55.090 --> 00:07:59.370
you, and you will have a
permanent neuron for it

00:07:59.370 --> 00:08:00.550
the rest of your life.

00:08:00.550 --> 00:08:01.790
Anyway, I'll see you
in the next video.