Introduction to inflation

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Language: en

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When economists refer
to inflation today,

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they are referring
to a general increase

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in the level of prices
of goods and services.

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So they're really talking
about price inflation.

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The reason why I stress
that is because sometimes,

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or especially when the
term inflation first

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came into usage, it actually was
referring to monetary inflation

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or an increase in
the money supply.

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So it was referring to an
increase in the money supply.

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These two ideas are
closely related,

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but it's important to
realize that people really

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are measuring inflation.

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When they're talking
about inflation today,

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they're talking about
price inflation.

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Because although these
two things are related,

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they aren't always
exactly the same thing.

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It is generally true that if
the money supply-- and the money

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supply is more than
just the amount

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of dollars that are printed.

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It's the amount of
dollars that are printed.

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It's affected by the amount
of lending that's occurring.

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It's affected by the
number of transactions

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that are occurring
in the economy.

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And if that money supply that's
affected by all of those things

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grows faster than the total real
productivity of the economy,

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then it will generally
increase the level of prices.

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But especially in
the short term,

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there could be other
causes of price inflation.

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You could have things
like supply shocks.

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And a supply shock is
the supply of something

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becomes scarce
all of the sudden.

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And the most typical
example of a supply shock

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is especially in the
oil crises in the 1970s.

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If for whatever
reason oil becomes

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scarce in a country
like the United States,

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then the price of oil
and gas would go up.

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But then these are inputs into
a whole set of things, even

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that banana that you buy
at the grocery store.

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If the price of oil
or gas-- or both

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of them, frankly-- or the
price of oil were to shoot up,

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even the price of your
banana would shoot up.

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Because to get that
banana to your store,

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you need to use some gasoline.

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In fact, a significant fraction
of that banana in that store

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was probably the cost of
the gasoline for that ship

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to take that banana
from wherever

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it was grown to your grocery
store and then on a railroad

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and then on a truck or whatever.

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So this would affect the
general prices, not just

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the prices of oil or gas.

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So these two things
are related, but it's

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important to realize
that people are

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referring to price inflation.

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And the general consensus
is a little bit of it

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is a good thing.

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And I want to stress,
little is good.

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And we're talking 1, 2.

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Maybe 3% per year.

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But anything larger than
that gets a little scary

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because it can kind
of snowball on itself.

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And we'll talk about
that in future videos

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when we talk about
hyperinflation.

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And economists are also
afraid if inflation

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were to ever get negative.

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That leads to deflation, and
we'll talk in future videos

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why in many circles that
is viewed as a scary thing.

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Now in the United
States, inflation

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is measured with the
Consumer Price Index, CPI.

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And you'll always hear
this reported in the news,

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especially if you watch some
of the business programming.

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And there are actually multiple
consumer price indices.

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The one that people
report whenever

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they say the CPI went
up 2%, they are actually

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referring to the
CPI-U. And the U here

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stands for Urban consumers.

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And the reason why this
is the headline CPI,

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or the one that people
actually report,

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is because most of the
country in the United States,

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they are urban consumers.

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So this is the CPI that
affects the largest

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number of people's pocketbooks.

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And the way that it's calculated
is, it's like the deflator.

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It's a price index.

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And it's measuring a general
increase or a general change

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in the level of prices.

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But they actually are calculated
in slightly different ways,

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although they should
come into agreement

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or they should be
close to each other

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if they really are
indices from measuring

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the general level of prices.

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But the way that
the CPI works is

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that they take a basket of
goods for this type of consumer

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in a base year.

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So they'll pick up base year.

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And let's take a super simple
example, a ridiculously simple

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example.

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Let's say in our little
country, the urban consumers--

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we'll focus on CPI-U--
only consumes two things.

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In the next video, we'll
see that in reality, we

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consume many more
than two things.

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But the two things--
and they spend

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60% of their money
on apples, and they

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spent 40% of their
money on bananas.

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And in that base year, we just
set that base price of apples

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at 100, and bananas are 100.

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Now we're not saying
that apples and bananas

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cost the same thing.

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We're saying that we're
spending 60% of our money

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on apples, 40% on bananas
in that base year,

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and that this is just that
base year level of prices.

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What will matter is
how much this grew.

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What will this index change
as we go to whatever year

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we want to calculate
the inflation

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in relative to this base year.

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So let's say in
our current year--

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could be the very next year--
so let's say in our current year

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and we're going to assume
these the same rates,

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that we're still spending 60%
on apples and 40% on bananas.

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In our current year,
the apple index--

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let's say that it has
grown 50% percent to 150.

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So it is plus 50%.

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And let's say that the banana
index has grown to 180.

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So bananas have gotten even
more expensive-- plus 80%.

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So how would we measure?

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How much would we say
the CPI-U has grown?

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Well, we would take a weighted
average of these indices,

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or you could say a weighted
average of the growth.

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And you could do it either way.

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So let's do it either way
to get you the same result.

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So in this year, our base index
is 0.6 times 100 plus 0.4 times

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100, and this will
just come out to 100.

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This is 60 plus 40.

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This is equal to
100 as it should.

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That is our base year.

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That is our base for our index.

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Now over here in
our current year,

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so this is what we're
transitioning to,

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there's a couple
of ways to do it.

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You could say, look, we're
spending 60% on something

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that has gone up to 150 now.

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So we would say 0.6 times 150.

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And then we'll say
plus 0.4 times 180.

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And that gets us to-- let
me get my calculator out--

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so this gets us 0.6 times
150 plus 0.4 times 180.

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So that gets us to 162.

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So our general-- if you
look at this basket and this

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is an overly simplified
basket-- we've

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increased from 100 to 162.

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Or you could say
this is plus 62%.

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And you would have
gotten the same result

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if you took the weighted
average of the percentages.

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If you took 0.6 times 50%
plus 0.4 times 80%-- in fact

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we could do that in our head.

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0.6 times 50% is
going to be 30%.

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And then 0.4 times 80%
is going to be 32%.

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to 30 plus 32
gives us 62% growth

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for this basket of goods,
which we are assuming

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is I guess for this
urban consumer,

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from our base year
to the current year.

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Now in the next
video, we'll actually

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look at what the basket
of goods actually

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looks like in the United States
for an actual urban consumer.


Example Calculating Real GDP with a Deflator

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Let's say the 2011 nominal GDP is: 15,294.3 billion dollars.

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And I didn't just make this number up.

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This is actually the advanced estimate of what 2011 GDP was in the fourth quarter.

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This is just a fourth quarter number.

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They took the fourth quarter number and they that analysed this to get to this 15 000 billion, which is esentially 15,294.3 trillion dollars of GDP.

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Let's say the GDP deflator, relative to 2010, you always have to know what you're taking your deflator relative to,

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is a 102.5 and this is, once again, the 2011 GDP deflator.

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And one way to interprate this is if the base year is 2010, that means that prices in 2010 could be viewed as being at 100

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and that now, we're in 2011, we're in 102.5.

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Another way to think about is that generall level of prices, that we have talked about it, this is not an easy thing to measure

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But if attempted to that the generall level of prices has gone up by 2.5 %, when form 100 to 102.5.

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Now that is the way that we know the nominal GDP is, the GDP measured in 2011 dollars,ofen called the current dollar GDP.

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We know the deflator is, can we figure out the real GDP in 2011

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and that will be the real GDP in 2010 dollars, when we have the deflator relative to 2010.

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So to do that we just have to remember that the ratio between our nominal GDP and our real GDP is going to be the ratio,

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you can viewed as current dollar vs. 2010 dollars.

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Or another way of viewing it, is the ratio between our deflator, which is 102.5

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and 100, which is esentially you could use it as kind of deflator in 2010

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or just setting that level of prices to be 100, a prices now are 102.5 .

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So taking it in that way, our nominal current dollar GDP is 15 294.3 billion dollars

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we can viewed that it's 15.2943 trillion, eiter way.

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Our real GDP is what we want to figure out.

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We do not know what it is.

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But we know the deflator, we know that things are gotten 102.5 more expensive

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or that deflator is a 102.5.

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And then we can just solve for the real GDP.

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And the 102.5 over 100, you might be able do this in your head,

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this whole expression right over here just becomes 1.025.

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And so you have this 15 trillions divided by 1.025 or you can divide both sides by 1.025

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and multiply both sides by the real GDP.

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So let's do that. Multiply both sides by the real GDP, divide both sides by 1.025.

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Now, of course, this cancels with that, that why we have multiplay both sides by the real,

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And then this cancels with that and we get our real GDP.

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Our real GDP is equal to our current dollar GDP.

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15 294.3 billion dollars divided by essentially the ratio between our deflator and the 100, divided by 1.025.

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If I were in charge of naming macroeconomic concepts I would actually made this the deflator

00:04:04.916 --> 00:04:12.200
I would set this at 1 and I would call this 1.205, because then you wouldn't had all this sillines multiplaing and dividing by 100.

00:04:12.200 --> 00:04:19.916
You just say: Hey, let's take our current dollar GDP, divide it by the deflator, I guess you could say world - deflated, to get the real GDP.

00:04:19.916 --> 00:04:21.669
That's just make a lot of sense to me.

00:04:21.669 --> 00:04:26.604
But either way, that's essencially what simplify too, this is our current dollar GDP, our nominal GDP

00:04:26.604 --> 00:04:32.800
dividing it by, what I would prefer to call the deflator, but you could use it as the deflator divided by 100

00:04:32.800 --> 00:04:34.810
and that gave us a real GDP.

00:04:34.810 --> 00:04:40.084
We just deflating the current dollar one, and how many the calculate to figure this one out.

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So let's get the calculator out, so 15 294.3, this is in billions, divided by 1.025, gives us 14 921.3

00:04:58.604 --> 00:05:04.469
So let me take this to the screen that I can remember that says. I have a bad memory.

00:05:04.469 --> 00:05:27.855
This is our real GDP, our real GDP is equal to 14 921.3 billion dollars.

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You can also write 14.9213 trillion dollars.


GDP deflator

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In the last video, we studied
a super simplified economy

00:00:04.070 --> 00:00:06.380
that only sold one
good or service.

00:00:06.380 --> 00:00:09.030
But now let's think about things
a little bit more generally,

00:00:09.030 --> 00:00:11.910
or a little bit more
complex economies.

00:00:11.910 --> 00:00:16.170
And let's say that in
year one economists

00:00:16.170 --> 00:00:20.530
have determined that the
level of prices of the goods

00:00:20.530 --> 00:00:24.130
and services produced
in that economy is 100.

00:00:24.130 --> 00:00:26.766
So they've essentially
just multiplied

00:00:26.766 --> 00:00:28.140
and divided by
the right numbers,

00:00:28.140 --> 00:00:30.330
so that their index
that they generate just

00:00:30.330 --> 00:00:32.414
says that that is 100.

00:00:32.414 --> 00:00:34.580
And they do this so that
they can measure the prices

00:00:34.580 --> 00:00:37.380
in other years
relative to year one.

00:00:37.380 --> 00:00:43.930
So let's say in year
two, using their index,

00:00:43.930 --> 00:00:47.990
they realize that
prices are now 110.

00:00:47.990 --> 00:00:49.694
Now, this is not a
simple thing to do.

00:00:49.694 --> 00:00:51.360
This would have been
a very simple thing

00:00:51.360 --> 00:00:54.740
to do if there was only one
good or service in the economy,

00:00:54.740 --> 00:00:56.214
like in our last
example, apples.

00:00:56.214 --> 00:00:58.130
You could have just taken
the price of apples.

00:00:58.130 --> 00:01:01.000
It went from $0.50 to $0.55.

00:01:01.000 --> 00:01:03.410
In the real world, this is
not a simple thing to do.

00:01:03.410 --> 00:01:05.830
You have a gazillion
goods and services.

00:01:05.830 --> 00:01:07.070
Some prices go up.

00:01:07.070 --> 00:01:08.810
Some prices to go down.

00:01:08.810 --> 00:01:10.880
The quantities of the
goods and services change.

00:01:10.880 --> 00:01:12.860
In fact, there might
be goods and services

00:01:12.860 --> 00:01:13.950
that were offered
in year one that

00:01:13.950 --> 00:01:15.290
don't exist anymore in year two.

00:01:15.290 --> 00:01:17.770
And there are goods and
services in year two

00:01:17.770 --> 00:01:19.560
that didn't exist in year one.

00:01:19.560 --> 00:01:21.460
But for the sake of
this video, let's just

00:01:21.460 --> 00:01:23.610
assume that economists
are able to say this.

00:01:23.610 --> 00:01:26.820
If you call the general level
of prices 100 in year one,

00:01:26.820 --> 00:01:28.360
it's now 110.

00:01:28.360 --> 00:01:30.110
Or another way to think
about it is things

00:01:30.110 --> 00:01:32.870
have gotten 10% more expensive.

00:01:32.870 --> 00:01:36.710
Now, assuming that we know this
relationship-- and once again,

00:01:36.710 --> 00:01:39.020
it's not an easy
thing to figure out,

00:01:39.020 --> 00:01:41.020
and it actually turns out
there's no perfect way

00:01:41.020 --> 00:01:43.550
to do this-- how
can we figure out

00:01:43.550 --> 00:01:47.660
a relationship between
real GDP and nominal GDP?

00:01:47.660 --> 00:01:50.230
And remember, whenever
we talk about real GDP--

00:01:50.230 --> 00:01:56.830
so we're going to talk
about real GDP in year two--

00:01:56.830 --> 00:01:58.650
whenever you talk
about real GDP,

00:01:58.650 --> 00:02:01.400
you're talking
about GDP in terms

00:02:01.400 --> 00:02:04.480
of the prices in some base year.

00:02:04.480 --> 00:02:07.000
So in this example, we'll
think about real GDP

00:02:07.000 --> 00:02:11.170
in year two in terms
of a year one dollars.

00:02:11.170 --> 00:02:13.980
So whatever were the
goods and services

00:02:13.980 --> 00:02:16.680
that were produced in year two,
we're going to think about,

00:02:16.680 --> 00:02:20.080
well, what if they were at the
same prices as in year one?

00:02:20.080 --> 00:02:23.730
And that will give us
the real GDP in year two.

00:02:23.730 --> 00:02:27.690
So one way to think about
it is really just a ratio.

00:02:27.690 --> 00:02:30.110
So let me write nominal GDP.

00:02:30.110 --> 00:02:33.290
So this is GDP in
year two, measured

00:02:33.290 --> 00:02:37.197
in year two dollars,
divided by-- I

00:02:37.197 --> 00:02:38.780
guess we could call
this a proportion,

00:02:38.780 --> 00:02:44.050
really-- divided by the
real GDP in year two.

00:02:44.050 --> 00:02:46.860
And this is measured
in year one dollars.

00:02:53.226 --> 00:02:54.850
Well, that's going
to be the same thing

00:02:54.850 --> 00:02:58.830
as the ratio of the prices
between year two and year one.

00:02:58.830 --> 00:03:02.160
This is going to be the ratio
of-- we use this indicator

00:03:02.160 --> 00:03:08.100
right over here-- 110 to 100.

00:03:08.100 --> 00:03:10.670
And I want you to just sit and
think about this for a second.

00:03:10.670 --> 00:03:14.400
It's just saying, look, these
are measuring the same goods

00:03:14.400 --> 00:03:15.160
and services.

00:03:15.160 --> 00:03:18.900
The real GDP is measuring
them in year one prices.

00:03:18.900 --> 00:03:22.580
The nominal GDP is measuring
them in year two prices.

00:03:22.580 --> 00:03:24.740
So if things got
10% more expensive

00:03:24.740 --> 00:03:27.100
between year one and
year two, the nominal GDP

00:03:27.100 --> 00:03:29.500
should be 10% larger
than real GDP.

00:03:29.500 --> 00:03:32.020
We should have the
exact same ratios.

00:03:32.020 --> 00:03:35.380
And now we can manipulate this
thing using any type of algebra

00:03:35.380 --> 00:03:36.440
that we want.

00:03:36.440 --> 00:03:40.050
For example, we could
say, well, nominal GDP--

00:03:40.050 --> 00:03:41.770
And I'll just write nominal now.

00:03:41.770 --> 00:03:43.200
This is where I
kind of specified

00:03:43.200 --> 00:03:44.575
exactly what we're
talking about.

00:03:44.575 --> 00:03:46.980
This is a nominal
GDP of year two.

00:03:46.980 --> 00:03:51.790
So now we could say
nominal GDP is equal

00:03:51.790 --> 00:03:54.980
to-- we can multiply both
sides times the real GDP--

00:03:54.980 --> 00:04:01.620
is equal to 110 over
100 times the real GDP.

00:04:04.580 --> 00:04:06.880
And remember, this is
nominal GDP in year two.

00:04:06.880 --> 00:04:12.330
This is real GDP in year two,
measured in year one dollars.

00:04:12.330 --> 00:04:17.130
Or we can divide both
sides of this equation

00:04:17.130 --> 00:04:19.220
by this 110 over 100.

00:04:19.220 --> 00:04:25.651
And then we get nominal
GDP in year two divided

00:04:25.651 --> 00:04:39.960
by 110 over 100 is equal
to real GDP in year two.

00:04:39.960 --> 00:04:41.850
This is nominal GDP in year two.

00:04:45.390 --> 00:04:47.150
And writing it this
way kind of feels

00:04:47.150 --> 00:04:50.100
like you're taking your
nominal GDP in year two,

00:04:50.100 --> 00:04:52.660
and there's been
a general increase

00:04:52.660 --> 00:04:53.730
in the level of prices.

00:04:53.730 --> 00:04:55.010
That's called price inflation.

00:04:55.010 --> 00:04:56.630
We see that right over here.

00:04:56.630 --> 00:05:00.270
And now we're deflating
it to get real GDP.

00:05:00.270 --> 00:05:02.694
We're dividing it by
the ratio of the prices.

00:05:02.694 --> 00:05:05.360
We're dividing it essentially by
how much the prices have grown,

00:05:05.360 --> 00:05:07.880
or I guess you could say the
ratio between the year two

00:05:07.880 --> 00:05:09.540
prices and the year one prices.

00:05:09.540 --> 00:05:12.754
So this quantity right
over here is 1.1.

00:05:12.754 --> 00:05:14.420
So another way you
could think about it,

00:05:14.420 --> 00:05:17.550
we're deflating the
nominal GDP in year two

00:05:17.550 --> 00:05:19.270
to get the real GDP in year two.

00:05:19.270 --> 00:05:22.485
We're getting it in, remember,
this is in year one prices.

00:05:25.830 --> 00:05:30.740
And because of that, this
number right over here

00:05:30.740 --> 00:05:33.370
is referred to as a deflator.

00:05:33.370 --> 00:05:38.270
This is our GDP deflator.

00:05:38.270 --> 00:05:41.900
You pick a base here, in
this case, it was year one.

00:05:41.900 --> 00:05:43.820
That base year could
have been 1985.

00:05:43.820 --> 00:05:45.415
It could've been 2006.

00:05:45.415 --> 00:05:46.540
Who knows what it could be.

00:05:46.540 --> 00:05:47.415
It could be anything.

00:05:47.415 --> 00:05:50.505
Your GDP deflator is going to
be relative to that base year.

00:05:50.505 --> 00:05:52.630
It's going to say, well,
if that base here was 100,

00:05:52.630 --> 00:05:55.270
your deflator's going to
say how much things are now

00:05:55.270 --> 00:05:56.110
in this year.

00:05:56.110 --> 00:05:57.730
And you can even go
backwards in time.

00:05:57.730 --> 00:06:00.180
Year zero, the deflator
might have been 85,

00:06:00.180 --> 00:06:02.110
because maybe things
have gotten cheaper.

00:06:02.110 --> 00:06:04.037
Or you could actually
had prices go down.

00:06:04.037 --> 00:06:05.620
You could have
actually had deflation.

00:06:05.620 --> 00:06:08.810
So maybe in year two your
deflator would be at 98.

00:06:08.810 --> 00:06:10.640
But the reason why
it's called a deflator

00:06:10.640 --> 00:06:13.910
is because generally you have
inflation as time goes on,

00:06:13.910 --> 00:06:16.770
and generally you're going to
be deflating your nominal GDP.

00:06:16.770 --> 00:06:19.890
You're going to be dividing it
by a value greater than one.

00:06:19.890 --> 00:06:22.680
It's going to be something
over 100 divided by 100,

00:06:22.680 --> 00:06:26.878
which is your base year,
to get your real GDP.


Real GDP and nominal GDP

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.400
Let's say we're studying a
very small and oversimplified

00:00:03.400 --> 00:00:08.290
country that only
sells apples, and we

00:00:08.290 --> 00:00:12.400
measure the GDP in year one.

00:00:12.400 --> 00:00:16.050
And we measure
that GDP as $1,000.

00:00:16.050 --> 00:00:18.290
And all of that
is due to apples.

00:00:18.290 --> 00:00:22.850
And we also know that the price
of apples in year one was $0.50

00:00:22.850 --> 00:00:23.720
a pound.

00:00:23.720 --> 00:00:29.670
So I'll write it
as $0.50 per pound.

00:00:29.670 --> 00:00:32.354
And let's say that now
year one has gone by

00:00:32.354 --> 00:00:34.020
and even year two has
gone by, and we're

00:00:34.020 --> 00:00:35.970
able to measure the
GDP in year two.

00:00:35.970 --> 00:00:40.950
So the GDP in year
two is $1,200.

00:00:40.950 --> 00:00:46.670
And the price of apples in year
two, let's just say it is $0.55

00:00:46.670 --> 00:00:47.180
a pound.

00:00:53.600 --> 00:00:57.590
So my question to you is, GDP,
the whole point of measuring

00:00:57.590 --> 00:01:00.900
GDP, is measuring the
productivity of a country.

00:01:00.900 --> 00:01:02.760
I mean we are measuring
in terms of dollars,

00:01:02.760 --> 00:01:04.676
but we care more about
just the dollar amount.

00:01:04.676 --> 00:01:07.360
We really care is, was this
country more productive?

00:01:07.360 --> 00:01:09.670
And if it was more productive,
how much more productive

00:01:09.670 --> 00:01:10.570
was it?

00:01:10.570 --> 00:01:13.230
And if we just look at these
GDP numbers right over here,

00:01:13.230 --> 00:01:16.779
this $1,000 versus this
$1,200, it gives you the sense

00:01:16.779 --> 00:01:19.070
that-- well, at least if you
just look at the numbers--

00:01:19.070 --> 00:01:22.490
$1,200 is 20%
larger than $1,000.

00:01:22.490 --> 00:01:24.740
So if you just look at those
numbers right over there,

00:01:24.740 --> 00:01:27.780
it looks like the
GDP grew by 20%.

00:01:27.780 --> 00:01:29.640
But is that an
accurate representation

00:01:29.640 --> 00:01:32.090
of the productivity
of this country?

00:01:32.090 --> 00:01:35.090
Did it actually
produce 20% more goods?

00:01:35.090 --> 00:01:37.480
And a big clue is looking
at this price here.

00:01:37.480 --> 00:01:39.290
Because some of
this GDP actually

00:01:39.290 --> 00:01:41.410
might have increased
just due to price.

00:01:41.410 --> 00:01:43.930
But that doesn't actually make
the country more productive.

00:01:43.930 --> 00:01:46.247
The quantity, the extra
quantity of apples

00:01:46.247 --> 00:01:47.830
that the country
produces, is actually

00:01:47.830 --> 00:01:50.194
what adds to the
total productivity.

00:01:50.194 --> 00:01:51.610
One way to think
about it-- Let me

00:01:51.610 --> 00:01:55.510
draw a little diagram over here.

00:01:55.510 --> 00:01:57.210
On this axis, I'll do quantity.

00:01:57.210 --> 00:01:59.950
On this axis, I will do price.

00:01:59.950 --> 00:02:03.870
And P1, so if I want to figure
out the GDP in year one,

00:02:03.870 --> 00:02:06.760
I would have the price
of apples in year one--

00:02:06.760 --> 00:02:09.289
that's the only good or service,
just to simplify things--

00:02:09.289 --> 00:02:12.860
times the quantity of
apples in year one.

00:02:12.860 --> 00:02:14.640
And then this right
over here, the area

00:02:14.640 --> 00:02:17.845
of this green rectangle,
would GDP in year one.

00:02:20.730 --> 00:02:23.310
And then GDP in year two would
be the price in year two.

00:02:23.310 --> 00:02:25.960
So we're going to go
from $0.50 to $0.55.

00:02:25.960 --> 00:02:29.070
The price in year two times
the quantity in year two--

00:02:29.070 --> 00:02:31.560
we'll assume some growth as
occurred-- times the quantity

00:02:31.560 --> 00:02:32.560
in year two.

00:02:32.560 --> 00:02:36.070
And so GDP in year
two would be the area

00:02:36.070 --> 00:02:38.290
of this entire rectangle.

00:02:38.290 --> 00:02:40.600
And if we want to find
the difference between GDP

00:02:40.600 --> 00:02:42.080
in year two and
GDP in year one, it

00:02:42.080 --> 00:02:43.520
would be the difference in area.

00:02:43.520 --> 00:02:46.930
So it would be what I am shading
in, in blue right over here.

00:02:46.930 --> 00:02:49.870
And based on the numbers that
we went over right over here,

00:02:49.870 --> 00:02:52.450
the area that I'm
shading in, in blue--

00:02:52.450 --> 00:02:55.400
so the difference between GDP
in year two and GDP in year one,

00:02:55.400 --> 00:02:57.860
the area I'm shading
in blue-- would

00:02:57.860 --> 00:03:01.290
be this 200, the 200 increment.

00:03:01.290 --> 00:03:05.394
So this area right over
here would be that 200.

00:03:05.394 --> 00:03:06.810
Now when you look
at it over here,

00:03:06.810 --> 00:03:08.780
you see that that
200, some of it

00:03:08.780 --> 00:03:10.465
is due to an
increase in quantity.

00:03:13.760 --> 00:03:17.460
But a lot of it is also due
to an increase in price.

00:03:17.460 --> 00:03:21.320
So if we really wanted to figure
out how much more productive

00:03:21.320 --> 00:03:25.370
the country got, and we still
want to measure GDP in dollars,

00:03:25.370 --> 00:03:27.300
maybe we can take
a measure of GDP

00:03:27.300 --> 00:03:29.440
that measures year two's
GDP, but it does it

00:03:29.440 --> 00:03:31.800
in year one's prices.

00:03:31.800 --> 00:03:33.710
So if we could
somehow multiply--

00:03:33.710 --> 00:03:39.040
if we could multiply year two's
quantity by year one's prices,

00:03:39.040 --> 00:03:41.110
then we would get this
rectangle right over here.

00:03:44.190 --> 00:03:46.570
And then the difference
between that and year one,

00:03:46.570 --> 00:03:52.770
would give us the incremental
GDP in year one prices due

00:03:52.770 --> 00:03:53.450
to quantity.

00:03:53.450 --> 00:03:54.699
And that's what we care about.

00:03:54.699 --> 00:03:56.434
We care about
total productivity.

00:03:56.434 --> 00:03:57.850
When we're thinking
about GDP one,

00:03:57.850 --> 00:04:00.890
we say how much more
productive did the country get?

00:04:00.890 --> 00:04:05.240
So let's try to do it with
these numbers right over here.

00:04:05.240 --> 00:04:06.977
So we can figure
out quantity two,

00:04:06.977 --> 00:04:08.810
we could figure out the
quantity in year two

00:04:08.810 --> 00:04:11.260
just by dividing the
GDP by the price.

00:04:11.260 --> 00:04:14.880
Just by dividing this area
of the entire blue rectangle

00:04:14.880 --> 00:04:17.740
and dividing it by the price,
that will give us the quantity.

00:04:17.740 --> 00:04:21.675
So if we divide 1,200
divided by $0.55-- let

00:04:21.675 --> 00:04:24.770
me get my calculator out.

00:04:24.770 --> 00:04:31.640
So if I do 1,200
divided by $0.55,

00:04:31.640 --> 00:04:35.370
this is my quantity of apples
and in pounds in year two.

00:04:35.370 --> 00:04:38.490
And I'll just round it, 2,182.

00:04:38.490 --> 00:04:40.140
So this is 2,182.

00:04:40.140 --> 00:04:48.290
So the quantity in year
two is 2,182 pounds.

00:04:48.290 --> 00:04:49.720
So this is equal to that.

00:04:49.720 --> 00:04:51.790
And then I could multiply
this times the price.

00:04:51.790 --> 00:04:52.831
So this is this quantity.

00:04:52.831 --> 00:04:54.650
It's 2,182 pounds.

00:04:54.650 --> 00:04:57.550
And then I could multiply it
times the price in year one

00:04:57.550 --> 00:04:58.930
at year one's price.

00:04:58.930 --> 00:05:00.500
So I'm going to
multiply it times--

00:05:00.500 --> 00:05:07.310
P1 is equal to $0.50 a
pound, $0.50 per pound.

00:05:07.310 --> 00:05:10.190
And this will give me-- so let
me just get my calculator out.

00:05:10.190 --> 00:05:11.981
I should be able to do
that one in my head.

00:05:11.981 --> 00:05:13.890
But let's see 0.5.

00:05:13.890 --> 00:05:14.890
And I get 1,090.

00:05:14.890 --> 00:05:17.190
Obviously, I'll
round it to 1,091.

00:05:17.190 --> 00:05:21.470
So this is equal to 1,091.

00:05:21.470 --> 00:05:23.210
And this is an
interesting number.

00:05:23.210 --> 00:05:29.210
So this is-- you could view
this as year two's GDP.

00:05:33.550 --> 00:05:41.700
In year-- or adjusted for-- I'll
write it, adjusted for prices,

00:05:41.700 --> 00:05:43.400
or adjusted for price increases.

00:05:43.400 --> 00:05:45.870
Or you could say
in year one prices.

00:05:50.810 --> 00:05:53.640
And what's useful about
this is, this says, look,

00:05:53.640 --> 00:05:55.960
if prices had remained
constant, this

00:05:55.960 --> 00:05:58.410
is what our GDP
would have gotten to.

00:05:58.410 --> 00:06:00.650
If prices did not
increase, our GDP

00:06:00.650 --> 00:06:01.983
would have gotten to this 1,091.

00:06:01.983 --> 00:06:04.970
1,091 is this area that
I drew in pink here.

00:06:04.970 --> 00:06:07.720
And so now, you could say if
prices were held constant,

00:06:07.720 --> 00:06:14.070
the growth in GDP would
have been $91 not $200.

00:06:14.070 --> 00:06:16.440
So this area right over
here that I'm-- actually,

00:06:16.440 --> 00:06:17.810
let me do it in a color.

00:06:17.810 --> 00:06:19.060
Let me do it in orange, maybe.

00:06:19.060 --> 00:06:22.590
This area right over
here, the actual growth,

00:06:22.590 --> 00:06:25.810
if prices were held constant,
would have been $91.

00:06:25.810 --> 00:06:30.250
We would have gone from
$1,000 of GDP to $1,091.

00:06:30.250 --> 00:06:31.910
So this right over
here, that area,

00:06:31.910 --> 00:06:37.370
is $91 of-- and we could
even call it real growth.

00:06:37.370 --> 00:06:39.850
It really measures
the productivity.

00:06:39.850 --> 00:06:41.840
Now this gives us an
interesting, I guess,

00:06:41.840 --> 00:06:43.360
set of ideas.

00:06:43.360 --> 00:06:47.120
One idea is to just measure
your GDP in the current year's

00:06:47.120 --> 00:06:48.100
dollars.

00:06:48.100 --> 00:06:50.510
So this was GDP measured
in year two's dollars.

00:06:50.510 --> 00:06:53.130
It was year two GDP measured
in year two dollars,

00:06:53.130 --> 00:06:54.610
year two prices.

00:06:54.610 --> 00:07:00.340
So we could call that
year two's nominal GDP.

00:07:00.340 --> 00:07:02.100
Nominal, in name.

00:07:02.100 --> 00:07:04.930
So it's GDP in name,
in that year's prices.

00:07:04.930 --> 00:07:07.120
But this right over
here, where we measured

00:07:07.120 --> 00:07:10.490
year two's GDP, in some
base year's prices--

00:07:10.490 --> 00:07:13.750
so it allows a real
comparison of how much did

00:07:13.750 --> 00:07:15.240
our productivity
actually increase.

00:07:15.240 --> 00:07:17.560
Our productivity
actually increased by 9%.

00:07:17.560 --> 00:07:20.390
We produced 9% more apples.

00:07:20.390 --> 00:07:22.180
This, we call real GDP.

00:07:25.960 --> 00:07:28.410
Because it gives you a
measure of real productivity.

00:07:28.410 --> 00:07:30.450
It tries to take
out price increases.

00:07:30.450 --> 00:07:32.860
What we'll see in the
future, or we might not

00:07:32.860 --> 00:07:35.030
do it in an introductory
course, but in practice,

00:07:35.030 --> 00:07:37.000
it's kind of hard
to really measure

00:07:37.000 --> 00:07:40.790
what the absolute-- this was a
simple economy, where we only

00:07:40.790 --> 00:07:42.950
had one product.

00:07:42.950 --> 00:07:45.170
But if you have many, many,
many products-- actually

00:07:45.170 --> 00:07:47.762
gazillions of products in a
real economy and the prices

00:07:47.762 --> 00:07:49.720
are adjusting and the
quantities are adjusting,

00:07:49.720 --> 00:07:53.710
it's not so easy to figure
out how to adjust for price.

00:07:53.710 --> 00:07:57.560
But the folks running the
national income accounts

00:07:57.560 --> 00:07:58.860
do try to do this.

00:07:58.860 --> 00:08:03.790
So they get a sense of how much
was the actual real growth.


Abstract-ness

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:03.800
You will hear me use the word abstract a lot

00:00:03.800 --> 00:00:07.621
so i thought i would actually give you an attempt at a definition,

00:00:07.621 --> 00:00:09.267
or maybe even more important

00:00:09.267 --> 00:00:11.800
an intuition of what abstract means

00:00:11.800 --> 00:00:14.067
and abstract can be an adjective

00:00:14.067 --> 00:00:16.267
you can have an abstract idea

00:00:16.267 --> 00:00:19.267
you can have abstract art

00:00:19.267 --> 00:00:20.600
or it can be a verb

00:00:20.600 --> 00:00:22.200
you can abstract something

00:00:22.200 --> 00:00:25.067
abstract the idea from some other idea

00:00:25.067 --> 00:00:26.836
and you can even have it as a noun

00:00:26.836 --> 00:00:28.467
you can have an abstract

00:00:28.467 --> 00:00:30.252
and it tends to, if you use it as a noun

00:00:30.252 --> 00:00:31.572
the one I tend to associate

00:00:31.572 --> 00:00:34.200
is the abstract of research paper, which is kind of,

00:00:34.200 --> 00:00:37.133
distills the essence of the research paper,

00:00:37.133 --> 00:00:38.733
which is kind of a summary of that paper

00:00:38.733 --> 00:00:40.200
and the one thing you are going to see

00:00:40.200 --> 00:00:41.400
regardless of how you

00:00:41.400 --> 00:00:44.467
what context you use the word abstract

00:00:44.467 --> 00:00:46.800
there's this kind of notion taking the essence

00:00:46.800 --> 00:00:51.200
of a real world object

00:00:51.200 --> 00:00:54.231
whether you use it as a noun, adjective, or a verb

00:00:54.231 --> 00:00:56.067
so over here we have our real world

00:00:56.067 --> 00:00:58.379
we have our real world

00:00:58.379 --> 00:01:01.410
and then over here you have your world of

00:01:01.410 --> 00:01:05.267
you have ideas and concepts

00:01:05.267 --> 00:01:08.400
and the general idea behind the abstraction

00:01:08.400 --> 00:01:09.733
or abstracting something

00:01:09.733 --> 00:01:10.933
is that you're taking it away

00:01:10.933 --> 00:01:13.467
from the particular concrete real world

00:01:13.467 --> 00:01:17.867
and you're going more into the direction of ideas and concepts

00:01:17.867 --> 00:01:22.267
and probably for me, one of the most tangible ways of

00:01:22.267 --> 00:01:25.733
thinking about abstraction, which is kind of a contradiction in itself

00:01:25.733 --> 00:01:28.400
to think of abstraction in a tangible way

00:01:28.400 --> 00:01:31.713
is things like geometric shapes

00:01:31.713 --> 00:01:34.969
so if i were to tell you to find me some cubes

00:01:34.969 --> 00:01:38.267
you might point to a borg vessel right over there

00:01:38.267 --> 00:01:39.302
a borg vessel

00:01:39.302 --> 00:01:44.800
you might point to a pair of dice

00:01:44.800 --> 00:01:46.867
let me draw a pair of dice

00:01:46.867 --> 00:01:48.952
if you were looking for cubes

00:01:48.952 --> 00:01:51.733
so you might point to a pair of dice

00:01:51.733 --> 00:01:53.933
that looks something like that

00:01:53.933 --> 00:01:55.733
you might point to a Rubik's cube

00:01:55.733 --> 00:01:56.800
anything you might find

00:01:56.800 --> 00:01:58.733
there might be a building that looks like a cube

00:01:58.733 --> 00:02:00.733
it might be a building that looks like a cube

00:02:00.733 --> 00:02:03.467
or maybe there is a box in your house that is a cube

00:02:03.467 --> 00:02:07.667
but in your mind you have a general idea of what a cube is

00:02:07.667 --> 00:02:09.733
like i know a cube when i see one

00:02:09.733 --> 00:02:13.133
and a general idea is distilling the concept

00:02:13.133 --> 00:02:15.600
the idea of what a cube is

00:02:15.600 --> 00:02:17.000
and all of these ideas are very different

00:02:17.000 --> 00:02:19.267
this is some plastic thing i could hold in my hand

00:02:19.267 --> 00:02:20.733
these are these white things

00:02:20.733 --> 00:02:23.200
they aren't even geometrically close to being perfect

00:02:23.200 --> 00:02:26.400
they have these little divets on the side right over there

00:02:26.400 --> 00:02:29.200
this is a large borg vessel that you know

00:02:29.200 --> 00:02:31.067
that doesn't exist yet

00:02:31.067 --> 00:02:32.733
is a fictional thing

00:02:32.733 --> 00:02:35.400
that they all have this cubeness to them

00:02:35.400 --> 00:02:37.800
one of the fun things about geometry

00:02:37.800 --> 00:02:41.733
so really distill the essence of these real world shapes

00:02:41.733 --> 00:02:44.800
and we do have this definition in geometry

00:02:44.800 --> 00:02:49.918
which is an object like this

00:02:49.918 --> 00:02:52.700
where every side has the exact same length

00:02:52.900 --> 00:02:55.108
so this is length one and that would be length one

00:02:55.108 --> 00:02:56.815
that would be length one

00:02:56.815 --> 00:02:58.098
it doesn't have to be

00:02:58.113 --> 00:02:59.621
whatever the length of this side is

00:02:59.621 --> 00:03:00.800
and this dimension

00:03:00.800 --> 00:03:02.200
then this dimension would be that length

00:03:02.200 --> 00:03:04.133
and then that dimension is going to be that length

00:03:04.133 --> 00:03:05.933
and i'm not giving you the rigorous definition

00:03:05.933 --> 00:03:08.267
but i'm just trying to highlight that there is

00:03:08.267 --> 00:03:12.200
pure idea of what a cube is

00:03:12.200 --> 00:03:14.267
of what a cube is

00:03:14.267 --> 00:03:16.733
and the real world, there is nothing that is actually

00:03:16.733 --> 00:03:18.533
a perfect cube

00:03:18.533 --> 00:03:20.333
if you were to get really really close to the die

00:03:20.333 --> 00:03:22.533
if you were to measure exactly their measurements

00:03:22.533 --> 00:03:24.667
they won't be exactly the same measurement

00:03:24.667 --> 00:03:32.267
but the abstract idea is completely the same length as this

00:03:32.267 --> 00:03:35.200
and this and this and all of the edges are going to

00:03:35.200 --> 00:03:37.467
have the exact same length

00:03:37.467 --> 00:03:39.015
so this is going from the concrete

00:03:39.015 --> 00:03:40.933
the specific from the real world

00:03:40.933 --> 00:03:43.933
if you consider the 24th or the 25th century, the real world

00:03:43.933 --> 00:03:49.205
to going to the idea behind it, the general idea

00:03:49.205 --> 00:03:51.748
and you probably also heard the word abstract

00:03:51.748 --> 00:03:54.267
in terms of art, like abstract art

00:03:54.267 --> 00:03:56.467
so this is abstract art

00:03:56.467 --> 00:03:57.933
and it is the same general idea

00:03:57.933 --> 00:03:59.000
so if you to look it up in the dictionary

00:03:59.000 --> 00:04:01.000
you're going to find 20 definitions of the word abstract

00:04:01.000 --> 00:04:04.200
but it's all, essentially trying to say the same thing

00:04:04.200 --> 00:04:09.467
abstract art is art that is not focused on trying to paint

00:04:09.467 --> 00:04:11.667
reality the exact way reality exists

00:04:11.667 --> 00:04:15.000
if you look at kind of lot of Renaissance art

00:04:15.000 --> 00:04:16.998
they are skilled at painting figures

00:04:16.998 --> 00:04:18.933
exactly how they look in the real world

00:04:18.933 --> 00:04:20.333
but the abstract artist

00:04:20.333 --> 00:04:22.060
sometimes they're not even trying

00:04:22.060 --> 00:04:24.267
to represent anything from the real world

00:04:24.267 --> 00:04:26.800
they are trying to represent a raw idea or

00:04:26.800 --> 00:04:31.000
a raw expression of color and form and texture

00:04:31.000 --> 00:04:33.200
and this is a Jackson Pollock painting

00:04:33.200 --> 00:04:35.533
right over here and i printed out so many things wrong

00:04:35.533 --> 00:04:39.267
taken by our own Steven Zucker, our art historian

00:04:39.267 --> 00:04:41.800
and you can see that it's not clear

00:04:41.800 --> 00:04:44.000
Jackson Pollock is not trying to paint a

00:04:44.000 --> 00:04:46.856
you know, a dog or horse or anything like that

00:04:46.856 --> 00:04:49.333
he is painting something that is devoid

00:04:49.333 --> 00:04:51.379
completely independent of anything

00:04:51.379 --> 00:04:55.733
that we actually see in physical reality

00:04:55.733 --> 00:04:58.133
and the word, abstraction, you know,

00:04:58.133 --> 00:05:01.800
it doesn't just apply on just a pure geometry and art

00:05:01.800 --> 00:05:04.800
it applies to almost everything we do on a daily basis

00:05:04.800 --> 00:05:06.467
when we even talk about things

00:05:06.467 --> 00:05:08.815
when we even use words or use symbols

00:05:08.815 --> 00:05:10.867
we are essentially abstracting away

00:05:10.867 --> 00:05:16.471
we're abstracting the essence of something that actually exists in physical reality

00:05:16.471 --> 00:05:19.867
so if i use the word dog

00:05:19.867 --> 00:05:23.671
it is a set of symbols that represent something in our mind

00:05:23.671 --> 00:05:25.533
that we associate with dog

00:05:25.533 --> 00:05:27.467
we have in our minds kind of the quality

00:05:27.467 --> 00:05:29.533
of what a dog actually is

00:05:29.533 --> 00:05:31.667
you know, it has four legs

00:05:31.667 --> 00:05:34.554
and floppy ears and you enjoy petting it

00:05:34.554 --> 00:05:37.458
and they're man's or i guess people's best friend

00:05:37.458 --> 00:05:39.708
you imagine this thing called a dog

00:05:39.708 --> 00:05:41.975
and it has the essence of dog

00:05:41.975 --> 00:05:43.748
and when you actually look at dogs in the real world

00:05:43.748 --> 00:05:46.538
they look very very different type of animals

00:05:46.538 --> 00:05:48.152
like when you look at a great dane

00:05:48.152 --> 00:05:52.385
or kind of a super small poodle

00:05:52.385 --> 00:05:55.429
But we recognize there is an essence of those particulars

00:05:55.429 --> 00:05:58.231
that we can abstract away and say this is a dog

00:05:58.231 --> 00:06:00.871
and we abstract it even more by representing these letter symbols

00:06:00.871 --> 00:06:02.948
that tend to conjure up this image

00:06:02.948 --> 00:06:05.000
even when we write something as simple as a number

00:06:05.000 --> 00:06:09.354
so if i write the number five

00:06:09.354 --> 00:06:11.533
we use it so frequently that to us

00:06:11.533 --> 00:06:14.292
a number 5 seems kind of like a concrete thing

00:06:14.292 --> 00:06:15.687
but it 's so abstract

00:06:15.687 --> 00:06:17.487
it's just a quantity of things

00:06:17.487 --> 00:06:19.000
i can symbolize like that

00:06:19.000 --> 00:06:21.968
i could have symbolized 5 like that

00:06:21.968 --> 00:06:24.887
i could have symbolized 5 in roman numerals like that

00:06:24.887 --> 00:06:26.800
i could symbolize it like that

00:06:26.800 --> 00:06:28.667
and in all of these cases they are

00:06:28.667 --> 00:06:32.333
it is the idea of a quantity of five things

00:06:32.333 --> 00:06:35.933
five, you know, you could say point me to a five

00:06:35.933 --> 00:06:38.400
and someone could draw or point you to something like that

00:06:38.400 --> 00:06:41.477
but they are still pointing you to the symbol of five

00:06:41.477 --> 00:06:45.188
but it's still an very abstract idea

00:06:45.188 --> 00:06:47.267
so hopefully this gives you an appreciation

00:06:47.267 --> 00:06:48.831
for what abstract means

00:06:48.831 --> 00:06:50.200
as you can tell, you know

00:06:50.200 --> 00:06:53.000
it's kind of a, for lack of a better word

00:06:53.000 --> 00:06:56.400
it's kind of an abstract idea

00:06:56.400 --> 00:06:59.108
not to be too cute about that


Origins of algebra

WEBVTT
Kind: captions
Language: en

00:00:00.434 --> 00:00:01.850
What I want to do
in this video is

00:00:01.850 --> 00:00:04.690
think about the
origins of algebra.

00:00:04.690 --> 00:00:08.390
The origins of
algebra, and the word,

00:00:08.390 --> 00:00:10.640
especially in association
with the ideas

00:00:10.640 --> 00:00:15.617
that algebra now represents,
comes from this book,

00:00:15.617 --> 00:00:17.950
or actually this is a page
of the book right over there.

00:00:17.950 --> 00:00:20.430
The English translation
for the title of this book

00:00:20.430 --> 00:00:23.050
is the "Compendious
Book on Calculation

00:00:23.050 --> 00:00:25.280
by Completion and Balancing."

00:00:25.280 --> 00:00:29.120
And it was written by a
Persian mathematician who

00:00:29.120 --> 00:00:32.052
lived in Baghdad
in, I believe, it

00:00:32.052 --> 00:00:33.510
was in the eighth
or ninth century.

00:00:33.510 --> 00:00:37.740
I believe it was actually 820
AD when he wrote this book.

00:00:37.740 --> 00:00:38.620
AD.

00:00:38.620 --> 00:00:42.730
And algebra is the Arabic word,
that here is the actual title

00:00:42.730 --> 00:00:45.070
that he gave to it, which
is the Arabic title.

00:00:45.070 --> 00:00:47.456
Algebra means restoration
or completion.

00:00:50.600 --> 00:00:55.110
Restoration or completion.

00:00:55.110 --> 00:00:58.380
And he associated it in his book
with a very specific operation,

00:00:58.380 --> 00:01:01.490
really taking something
from one side of an equation

00:01:01.490 --> 00:01:03.659
to another side of an equation.

00:01:03.659 --> 00:01:05.450
But we can actually
see it right over here,

00:01:05.450 --> 00:01:07.075
and I don't know
Arabic, but I actually

00:01:07.075 --> 00:01:08.550
do know some
languages that seems

00:01:08.550 --> 00:01:10.330
to have borrowed a
little bit from Arabic,

00:01:10.330 --> 00:01:12.310
or maybe it went the
other way around.

00:01:12.310 --> 00:01:15.770
But this says Al-kitab,
and I know just enough

00:01:15.770 --> 00:01:18.350
Urdu and Hindi to understand
a good India movie,

00:01:18.350 --> 00:01:20.170
but Al-kitab, kitab means book.

00:01:20.170 --> 00:01:22.550
So this part is book.

00:01:22.550 --> 00:01:23.170
Book.

00:01:23.170 --> 00:01:26.816
Al-mukhtasar, well, I think
that means compendious,

00:01:26.816 --> 00:01:28.690
because I don't know
the word for compendious

00:01:28.690 --> 00:01:29.930
and that seems like that.

00:01:29.930 --> 00:01:33.170
Fihisab, hisab means
calculation in Hindi or Urdu,

00:01:33.170 --> 00:01:35.210
so this is calculation.

00:01:35.210 --> 00:01:36.300
Calculation.

00:01:36.300 --> 00:01:38.500
Al-gabr, this is the root.

00:01:38.500 --> 00:01:41.110
This is the famous algebra,
this is where it shows up.

00:01:41.110 --> 00:01:45.520
So this is for completion, you
could view that as completion.

00:01:45.520 --> 00:01:46.450
Completion.

00:01:46.450 --> 00:01:52.050
And then wa'l-muqabala, and that
means essentially balancing.

00:01:52.050 --> 00:01:52.660
Balancing.

00:01:52.660 --> 00:01:55.305
Completion and balancing.

00:01:55.305 --> 00:01:56.680
So if we wanted
to translate it--

00:01:56.680 --> 00:01:58.890
I know this isn't a video
on translating Arabic,

00:01:58.890 --> 00:02:04.100
but the book, I guess
this is saying compendious

00:02:04.100 --> 00:02:10.720
on calculation by
completion and balancing

00:02:10.720 --> 00:02:12.580
is the rough translation
right over there.

00:02:12.580 --> 00:02:15.470
But that is the source
of the word algebra,

00:02:15.470 --> 00:02:17.940
and this is a very, very,
very important book.

00:02:17.940 --> 00:02:20.700
Not just because it was the
first use of the word algebra,

00:02:20.700 --> 00:02:24.720
but many people viewed
this book as the first time

00:02:24.720 --> 00:02:29.800
that algebra took a
lot of its modern--

00:02:29.800 --> 00:02:31.940
took on many of
its modern ideas.

00:02:31.940 --> 00:02:34.240
Ideas of balancing an equation.

00:02:34.240 --> 00:02:36.320
The abstract problem
itself, not trying

00:02:36.320 --> 00:02:38.540
to do one off problems
here or there.

00:02:38.540 --> 00:02:42.160
But al-Khwarizmi was
not the first person,

00:02:42.160 --> 00:02:44.640
and just to get an idea of
where all this is happening.

00:02:44.640 --> 00:02:47.122
So he was hanging
out in Baghdad,

00:02:47.122 --> 00:02:48.580
and this part of
the world shows up

00:02:48.580 --> 00:02:50.130
a lot in the history of algebra.

00:02:50.130 --> 00:02:51.650
But he was hanging
out right there

00:02:51.650 --> 00:02:54.050
in around the eighth
or ninth century.

00:02:54.050 --> 00:02:56.150
So let me draw a
time line here, just

00:02:56.150 --> 00:02:58.230
so we can appreciate everything.

00:02:58.230 --> 00:03:03.590
So that is timeline,
and then whether or not

00:03:03.590 --> 00:03:06.710
you are religious, most
of our modern dates

00:03:06.710 --> 00:03:11.720
are dependent on the birth of
Jesus, so that is right there.

00:03:11.720 --> 00:03:13.905
Maybe I'll put a cross
over there to signify that.

00:03:13.905 --> 00:03:15.280
When we want to
be non-religious,

00:03:15.280 --> 00:03:17.110
we say the common era.

00:03:17.110 --> 00:03:19.750
Before the common era, when
we want to be religious

00:03:19.750 --> 00:03:22.530
we say AD, which means
in the year of our lord.

00:03:22.530 --> 00:03:25.120
I don't know the Latin,
Anno Domini, I believe,

00:03:25.120 --> 00:03:26.220
year of our lord.

00:03:26.220 --> 00:03:28.970
And then when we want--
in the religious context,

00:03:28.970 --> 00:03:32.180
instead of saying before common
era, we say before Christ, BC.

00:03:32.180 --> 00:03:36.790
But either way, so this
is 1000 in the common era.

00:03:36.790 --> 00:03:39.656
This is 2000 in the common era.

00:03:39.656 --> 00:03:41.530
And obviously, we are
sitting-- at least when

00:03:41.530 --> 00:03:44.400
I'm making this video, I'm
sitting right about there.

00:03:44.400 --> 00:03:48.560
And then this is 1000
before the common era,

00:03:48.560 --> 00:03:52.210
and this is 2000
before the common era.

00:03:52.210 --> 00:03:54.565
So the first traces--
and I'm skipping out,

00:03:54.565 --> 00:03:56.190
and really, it's just
what we can find.

00:03:56.190 --> 00:03:57.420
I'm sure if we were
able to dig more,

00:03:57.420 --> 00:03:59.044
we might be able to
find other evidence

00:03:59.044 --> 00:04:02.080
of different civilizations
and different people stumbling

00:04:02.080 --> 00:04:04.720
on many of the ideas in algebra.

00:04:04.720 --> 00:04:07.430
But our first records of
people really exploring

00:04:07.430 --> 00:04:09.690
the ideas that are
hit upon in algebra

00:04:09.690 --> 00:04:13.590
come from ancient
Babylon around 2000 years

00:04:13.590 --> 00:04:16.160
before the common
era, before Christ.

00:04:16.160 --> 00:04:22.562
So right around there
there are stone tablets

00:04:22.562 --> 00:04:24.270
where it looks like
people were exploring

00:04:24.270 --> 00:04:26.080
some of the fundamental
ideas of algebra.

00:04:26.080 --> 00:04:27.880
They weren't using
the same symbols.

00:04:27.880 --> 00:04:30.780
They weren't using the same ways
of representing the numbers,

00:04:30.780 --> 00:04:33.090
but it was algebra that
they were working on.

00:04:33.090 --> 00:04:35.430
And that was, once again,
in this part of the world.

00:04:35.430 --> 00:04:38.760
Babylon was right about there.

00:04:38.760 --> 00:04:42.720
And Babylon, it's kind of
kept the tradition of Sumeria.

00:04:42.720 --> 00:04:44.920
This whole region was
called Mesopotamia,

00:04:44.920 --> 00:04:47.290
Greek for between two rivers.

00:04:47.290 --> 00:04:49.230
But that's the first
traces of people

00:04:49.230 --> 00:04:51.270
that we know of that
where people were starting

00:04:51.270 --> 00:04:54.550
to do what we would
call real, real algebra.

00:04:54.550 --> 00:04:55.892
And then you fast forward.

00:04:55.892 --> 00:04:57.600
And I'm sure we're
missing-- and I'm sure

00:04:57.600 --> 00:05:01.960
even our historians don't know
all of the different instances

00:05:01.960 --> 00:05:08.300
of people using algebra, but the
major contributions to algebra,

00:05:08.300 --> 00:05:11.450
we saw it here in
Babylon 2000 years ago.

00:05:11.450 --> 00:05:13.990
And then if we fast
forward to about 200 to 300

00:05:13.990 --> 00:05:17.310
AD, so right over there,
you have a Greek gentleman

00:05:17.310 --> 00:05:18.870
who lived in Alexandria.

00:05:18.870 --> 00:05:21.020
So this is Greece
right over here,

00:05:21.020 --> 00:05:23.210
but he lived in Alexandria,
which at the time

00:05:23.210 --> 00:05:25.240
was part of the Roman Empire.

00:05:25.240 --> 00:05:27.890
So Alexandria is
right over here,

00:05:27.890 --> 00:05:30.630
and he was a gentleman by
the name of Diophantus,

00:05:30.630 --> 00:05:31.890
or Diophantus.

00:05:31.890 --> 00:05:36.300
I don't know how to
pronounce it, Diophantus.

00:05:36.300 --> 00:05:38.550
And he is sometimes
credited with being

00:05:38.550 --> 00:05:41.330
the father of algebra,
and it's debatable

00:05:41.330 --> 00:05:44.200
whether it's Diophantus
or al-Khwarizmi.

00:05:44.200 --> 00:05:45.770
al-Khwarizmi, who
kind of started

00:05:45.770 --> 00:05:48.320
using these terms of
balancing equations

00:05:48.320 --> 00:05:50.050
and talking about
math in a purer way,

00:05:50.050 --> 00:05:53.370
while Diophantus was more
focused on particular problems.

00:05:53.370 --> 00:05:55.840
And both of them were
kind of beat to the punch

00:05:55.840 --> 00:05:57.780
by the Babylonians,
although they all

00:05:57.780 --> 00:05:59.170
did contribute in their own way.

00:05:59.170 --> 00:06:00.140
It's not like they
were just copying

00:06:00.140 --> 00:06:01.370
what the Babylonians did.

00:06:01.370 --> 00:06:03.230
They had their own
unique contributions

00:06:03.230 --> 00:06:06.360
to what we now consider algebra.

00:06:06.360 --> 00:06:08.170
But many, especially
Western historians,

00:06:08.170 --> 00:06:11.480
associate Diophantus as
the father of algebra.

00:06:11.480 --> 00:06:13.580
And now, al-Khwarizmi
is sometimes

00:06:13.580 --> 00:06:16.390
what other people would argue
as the father of algebra,

00:06:16.390 --> 00:06:18.290
so he made significant
contributions.

00:06:18.290 --> 00:06:22.200
And if you go to 600 AD-- so
if you go to about 600 AD,

00:06:22.200 --> 00:06:26.140
another famous mathematician
in the history of algebra

00:06:26.140 --> 00:06:29.200
was Brahmagupta, in India.

00:06:29.200 --> 00:06:32.580
Brahmagupta, in India.

00:06:32.580 --> 00:06:34.180
So obviously, and
actually, I don't

00:06:34.180 --> 00:06:35.490
know where in India he lived.

00:06:35.490 --> 00:06:37.900
I should look that
up, but roughly

00:06:37.900 --> 00:06:39.440
in that part of the world.

00:06:39.440 --> 00:06:41.665
And he also made
significant contributions.

00:06:44.290 --> 00:06:45.790
And then you have
al-Khwarizmi, who

00:06:45.790 --> 00:06:50.530
shows up right
there, al-Khwarizmi.

00:06:50.530 --> 00:06:53.560
And he is the gentleman
that definitely

00:06:53.560 --> 00:06:56.810
we credit with the name
algebra, comes from Arabic

00:06:56.810 --> 00:07:00.080
for restoration, and some
people also consider him to be,

00:07:00.080 --> 00:07:02.170
if not the father of
algebra, although some people

00:07:02.170 --> 00:07:05.370
say he is the father, he is
one of the fathers of algebra

00:07:05.370 --> 00:07:08.340
because he really started
to think about algebra

00:07:08.340 --> 00:07:11.410
in the abstract sense, devoid
of some specific problems

00:07:11.410 --> 00:07:13.440
and a lot of the way
a modern mathematician

00:07:13.440 --> 00:07:17.110
would start to think
about the field.


The beauty of algebra

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.520
Before we get into
the meat of algebra,

00:00:02.520 --> 00:00:04.728
I wanted to give you a quote
from one of the greatest

00:00:04.728 --> 00:00:08.029
minds in human history, Galileo
Galilei, because I think

00:00:08.029 --> 00:00:11.240
this quote encapsulates
the true point of algebra

00:00:11.240 --> 00:00:13.960
and really mathematics
in general.

00:00:13.960 --> 00:00:17.230
He said, "Philosophy is written
in that great book which ever

00:00:17.230 --> 00:00:20.610
lies before our eyes--
I mean the universe--

00:00:20.610 --> 00:00:23.850
but we cannot understand
it if we do not first learn

00:00:23.850 --> 00:00:27.320
the language and grasp the
symbols in which is written.

00:00:27.320 --> 00:00:30.440
This book is written in
the mathematical language,

00:00:30.440 --> 00:00:35.580
without which one wanders in
vain through a dark labyrinth."

00:00:35.580 --> 00:00:38.030
So very dramatic, but very deep.

00:00:38.030 --> 00:00:40.830
And this really is the
point of mathematics.

00:00:40.830 --> 00:00:43.639
And what we'll see as we start
getting deeper and deeper

00:00:43.639 --> 00:00:46.180
into algebra is that we're going
to start abstracting things,

00:00:46.180 --> 00:00:48.700
and we're going to start
getting to core ideas that

00:00:48.700 --> 00:00:51.640
start explaining really how
the universe is structured.

00:00:51.640 --> 00:00:54.500
Sure, these ideas can be
applied to things like economics

00:00:54.500 --> 00:00:56.850
and finance and
physics and chemistry.

00:00:56.850 --> 00:00:59.080
But at their core,
they're the same idea,

00:00:59.080 --> 00:01:01.890
and so they're even more
fundamental, more pure,

00:01:01.890 --> 00:01:04.800
than any one of
those applications.

00:01:04.800 --> 00:01:08.190
And to see what I mean by
getting down to the root idea,

00:01:08.190 --> 00:01:11.770
let's go with a-- I guess we
started with the very grand,

00:01:11.770 --> 00:01:13.410
the philosophy of
the universe is

00:01:13.410 --> 00:01:15.076
written in mathematics--
but let's start

00:01:15.076 --> 00:01:17.409
with a very concrete,
simple idea.

00:01:17.409 --> 00:01:18.950
But we'll keep
abstracting, and we'll

00:01:18.950 --> 00:01:23.080
see how the same idea
connects across many domains

00:01:23.080 --> 00:01:24.330
in our universe.

00:01:24.330 --> 00:01:26.370
So let's just say
we're at the store,

00:01:26.370 --> 00:01:27.800
and we're going
to buy something.

00:01:27.800 --> 00:01:29.370
And there is a sale.

00:01:29.370 --> 00:01:34.800
The sale says that it
is 30% percent off,

00:01:34.800 --> 00:01:36.080
and I'm interested.

00:01:36.080 --> 00:01:37.802
I don't shop at
too fancy a store.

00:01:37.802 --> 00:01:39.760
So let's say I'm interested
in a pair of pants.

00:01:39.760 --> 00:01:43.150
And the pair of pants before
the sale even is about $20.

00:01:43.150 --> 00:01:45.800
And that is about how
much I spend on my pants.

00:01:45.800 --> 00:01:48.360
So I'm interested in
a $20 pair of pants.

00:01:48.360 --> 00:01:53.770
But it's even better, there's
a 30% off sale on these pants.

00:01:53.770 --> 00:01:55.480
Well, how would I
think about how much

00:01:55.480 --> 00:01:57.656
I'm going to get
off of that $20?

00:01:57.656 --> 00:01:58.780
And this isn't algebra yet.

00:01:58.780 --> 00:02:01.071
This is something that you've
probably had exposure to.

00:02:01.071 --> 00:02:04.590
You would multiply
the 30% times the $20.

00:02:04.590 --> 00:02:09.919
So you would say your
discount is equal

00:02:09.919 --> 00:02:14.100
to-- you could write
it as 30% times $20.

00:02:14.100 --> 00:02:16.147
I'll do the $20 in purple.

00:02:16.147 --> 00:02:18.230
Or you could write it, if
you wanted to write this

00:02:18.230 --> 00:02:23.900
as a decimal, you could
write this as 0.30 times $20.

00:02:23.900 --> 00:02:27.060
And if you were to do the
math, you would get $6.

00:02:27.060 --> 00:02:29.640
So nothing new over there.

00:02:29.640 --> 00:02:31.690
But what if I want to
generalize it a little bit?

00:02:31.690 --> 00:02:35.032
That's the discount on this
particular pair of pants.

00:02:35.032 --> 00:02:37.240
But what if I wanted to know
the discount on anything

00:02:37.240 --> 00:02:38.310
in the store?

00:02:38.310 --> 00:02:39.880
Well, then I could
say, well, let

00:02:39.880 --> 00:02:44.020
x be the price-- let me do
this in a different color.

00:02:44.020 --> 00:02:45.740
So I'm just going
to make a symbol.

00:02:45.740 --> 00:02:49.930
Let x be the price
of the product

00:02:49.930 --> 00:02:57.520
I want to buy, price,
the non-discount price

00:02:57.520 --> 00:03:01.052
of the product in the store.

00:03:01.052 --> 00:03:02.510
So now, all of a
sudden, we can say

00:03:02.510 --> 00:03:11.305
that our discount is
equal to 30% times x.

00:03:15.070 --> 00:03:17.306
Or if we wanted to
write it as a decimal,

00:03:17.306 --> 00:03:18.930
if we wanted to write
30% as a decimal,

00:03:18.930 --> 00:03:25.190
we could write 0.30 times x.

00:03:25.190 --> 00:03:26.590
Now, this is interesting.

00:03:26.590 --> 00:03:29.180
Now you give me the price
of any product in the store,

00:03:29.180 --> 00:03:31.320
and I can substitute
it in for x.

00:03:31.320 --> 00:03:34.390
And then I can essentially
multiply 0.3 times that,

00:03:34.390 --> 00:03:35.620
and I would get the discount.

00:03:35.620 --> 00:03:38.530
So now we're starting
to, very slowly,

00:03:38.530 --> 00:03:41.740
we're starting to get into
the abstraction of algebra.

00:03:41.740 --> 00:03:44.560
And we'll see that these will
get much more nuanced and deep

00:03:44.560 --> 00:03:46.470
and, frankly, more
beautiful as we

00:03:46.470 --> 00:03:52.310
start studying more and more
kind of algebraic ideas.

00:03:52.310 --> 00:03:53.540
But we aren't done here.

00:03:53.540 --> 00:03:55.480
We can abstract this even more.

00:03:55.480 --> 00:03:57.500
Over here, we've said
we've generalized

00:03:57.500 --> 00:03:58.570
this for any product.

00:03:58.570 --> 00:04:00.430
We're not just saying
for this $20 product.

00:04:00.430 --> 00:04:03.950
If there's a $10 product, we
can put that $10 product in here

00:04:03.950 --> 00:04:05.140
for x.

00:04:05.140 --> 00:04:08.280
And then we would
say 0.30 times 10,

00:04:08.280 --> 00:04:10.610
and the discount would be $3.

00:04:10.610 --> 00:04:13.940
It might be $100 product, then
the discount would be $30.

00:04:13.940 --> 00:04:15.630
But let's generalize even more.

00:04:15.630 --> 00:04:18.290
Let's say, well, what is
the discount for any given

00:04:18.290 --> 00:04:20.430
sale when the sale is
a certain percentage?

00:04:20.430 --> 00:04:23.320
So now we can say
that the discount--

00:04:23.320 --> 00:04:25.170
let me define a variable.

00:04:25.170 --> 00:04:32.320
So let's let m equal-- or I'll
say p just so it makes sense.

00:04:32.320 --> 00:04:36.800
p is equal to the
percentage off.

00:04:41.300 --> 00:04:42.590
Now what can we do?

00:04:42.590 --> 00:04:47.040
Well, now we can say
that the discount

00:04:47.040 --> 00:04:49.610
is equal to the percentage off.

00:04:49.610 --> 00:04:52.170
In these other examples,
we were picking 30%.

00:04:52.170 --> 00:04:53.580
But we can say now it's p.

00:04:53.580 --> 00:04:55.250
It's the percentage off.

00:04:55.250 --> 00:04:56.630
It's p.

00:04:56.630 --> 00:05:00.870
That's the percentage off
times the product in question,

00:05:00.870 --> 00:05:02.895
times the price, the
non-discount price

00:05:02.895 --> 00:05:04.020
of the product in question.

00:05:04.020 --> 00:05:05.330
Well, that was x.

00:05:05.330 --> 00:05:08.437
The discount is
equal to p times x.

00:05:08.437 --> 00:05:09.770
Now, this is really interesting.

00:05:09.770 --> 00:05:12.350
Now we have a general
way of calculating

00:05:12.350 --> 00:05:16.420
a discount for any given
percentage off and any given

00:05:16.420 --> 00:05:18.880
product x.

00:05:18.880 --> 00:05:21.720
And we didn't have to use
these words and these letters.

00:05:21.720 --> 00:05:24.295
We could have said let
y equal the discount.

00:05:29.320 --> 00:05:31.989
Then we could have written
the same underlying idea.

00:05:31.989 --> 00:05:33.530
Instead of writing
discount, we could

00:05:33.530 --> 00:05:40.810
have written y is equal to
the percentage off p times

00:05:40.810 --> 00:05:44.801
the non-discount price
of the product, times x.

00:05:44.801 --> 00:05:47.300
And you could have defined these
letters any way you wanted.

00:05:47.300 --> 00:05:48.980
Instead of writing
y there, you could

00:05:48.980 --> 00:05:50.260
have written a
Greek letter, or you

00:05:50.260 --> 00:05:51.530
could have written
any symbol there.

00:05:51.530 --> 00:05:53.030
As long as you can
keep track of it,

00:05:53.030 --> 00:05:56.847
that symbol represents the
actual dollar discount.

00:05:56.847 --> 00:05:58.430
But now things get
really interesting.

00:05:58.430 --> 00:06:02.330
Because we can use this type
of a relationship, which

00:06:02.330 --> 00:06:04.550
is an equation--
you're equating y

00:06:04.550 --> 00:06:07.632
to this right over here, that's
why we call it an equation--

00:06:07.632 --> 00:06:09.590
this can be used for
things that are completely

00:06:09.590 --> 00:06:12.610
unrelated to the price,
the discount price,

00:06:12.610 --> 00:06:15.330
at the store over here.

00:06:15.330 --> 00:06:18.050
So in physics,
you'll see that force

00:06:18.050 --> 00:06:21.180
is equal to mass
times acceleration.

00:06:21.180 --> 00:06:23.590
The letters are different,
but these are fundamentally

00:06:23.590 --> 00:06:25.130
the same idea.

00:06:25.130 --> 00:06:36.360
We could've let y is equal to
force, and mass is equal to p.

00:06:36.360 --> 00:06:39.580
So let me write p
is equal to mass.

00:06:39.580 --> 00:06:41.850
And this wouldn't be an
intuitive way to define it,

00:06:41.850 --> 00:06:42.620
but I want to show
you that this is

00:06:42.620 --> 00:06:44.680
the same idea, the
same relationship,

00:06:44.680 --> 00:06:48.060
but it's being applied to two
completely different things.

00:06:48.060 --> 00:06:50.105
And we could say x is
equal to acceleration.

00:06:57.380 --> 00:07:00.710
Well, then the famous force
is equal to mass times

00:07:00.710 --> 00:07:02.210
acceleration can be rewritten.

00:07:02.210 --> 00:07:04.380
And it's really
the same exact idea

00:07:04.380 --> 00:07:07.310
as y, which we've
defined as force,

00:07:07.310 --> 00:07:08.890
can be equal to
mass, which we're

00:07:08.890 --> 00:07:10.620
going to use the
symbol p, which is

00:07:10.620 --> 00:07:12.930
equal to p times acceleration.

00:07:12.930 --> 00:07:15.690
And we're just going to happen
to use the letter x here,

00:07:15.690 --> 00:07:16.840
times x.

00:07:16.840 --> 00:07:19.560
Well, this is the
exact same equation.

00:07:19.560 --> 00:07:20.995
This is the exact same equation.

00:07:20.995 --> 00:07:23.290
And we could see that we
can take this equation,

00:07:23.290 --> 00:07:27.650
and it can apply to
things in economics,

00:07:27.650 --> 00:07:30.700
or it can apply to
things in finance,

00:07:30.700 --> 00:07:33.890
or it can apply to things in
computer science, or logic,

00:07:33.890 --> 00:07:36.410
or electrical engineering,
or anything, accounting.

00:07:36.410 --> 00:07:38.900
There's an infinite
number of applications

00:07:38.900 --> 00:07:40.150
of this one equation.

00:07:40.150 --> 00:07:42.000
And what's neat
about mathematics

00:07:42.000 --> 00:07:43.890
and what's neat about
algebra in particular

00:07:43.890 --> 00:07:47.240
is we can focus on
this abstraction.

00:07:47.240 --> 00:07:49.480
We can focus on
the abstract here,

00:07:49.480 --> 00:07:51.920
and we can manipulate
the abstract here.

00:07:51.920 --> 00:07:54.160
And what we discover
from these ideas,

00:07:54.160 --> 00:07:56.090
from these
manipulations, can then

00:07:56.090 --> 00:08:00.630
go and be reapplied to all
of these other applications,

00:08:00.630 --> 00:08:01.720
to all of them.

00:08:01.720 --> 00:08:03.560
And even neater, it's
kind of telling us

00:08:03.560 --> 00:08:05.540
the true structure
of the universe

00:08:05.540 --> 00:08:08.420
if you were to strip away all
of these human definitions

00:08:08.420 --> 00:08:10.470
and all of these
human applications.

00:08:10.470 --> 00:08:13.730
So for example, we could say,
look, if y is equal to p times

00:08:13.730 --> 00:08:16.860
x-- so literally, if someone
said, hey, this is y,

00:08:16.860 --> 00:08:19.250
and someone says, on the
other hand, I have p times x,

00:08:19.250 --> 00:08:21.333
I could say, well, you
have the same thing in both

00:08:21.333 --> 00:08:22.320
of your hands.

00:08:22.320 --> 00:08:25.460
And if you were to divide
one of them by a number,

00:08:25.460 --> 00:08:27.420
and if you wanted them
to still be equal,

00:08:27.420 --> 00:08:29.640
you would divide the
other one by that number.

00:08:29.640 --> 00:08:35.770
So for example, we know that
y is equal to p times x.

00:08:35.770 --> 00:08:38.200
Well, what if you wanted
to have them both be equal?

00:08:38.200 --> 00:08:39.789
And you say, well,
what is y divided

00:08:39.789 --> 00:08:41.549
by x going to be equal to?

00:08:41.549 --> 00:08:44.159
Well, y was equal to p
times x, so y divided

00:08:44.159 --> 00:08:45.870
by x is going to be
the same thing as p

00:08:45.870 --> 00:08:48.370
times x divided by x.

00:08:48.370 --> 00:08:49.790
But now this is interesting.

00:08:49.790 --> 00:08:52.290
Because p times x
divided by x-- well,

00:08:52.290 --> 00:08:55.155
if you multiply by something and
then divide by that something,

00:08:55.155 --> 00:08:57.280
it's just you're going to
get your original number.

00:08:57.280 --> 00:08:59.710
If you multiply by
5 and divide by 5,

00:08:59.710 --> 00:09:02.430
you're just going to start with
p or whatever this number is.

00:09:02.430 --> 00:09:04.140
So those would cancel out.

00:09:04.140 --> 00:09:08.750
But we were able to manipulate
the abstraction here and get y

00:09:08.750 --> 00:09:12.060
over x is equal to p-- and
let me make that x green.

00:09:16.050 --> 00:09:20.890
And now this has implications
for every one of these ideas.

00:09:20.890 --> 00:09:23.090
One is telling us
a fundamental truth

00:09:23.090 --> 00:09:25.010
about the universe,
almost devoid

00:09:25.010 --> 00:09:26.470
of any of these applications.

00:09:26.470 --> 00:09:28.784
But now we can go and take
them back to any place

00:09:28.784 --> 00:09:29.450
that we applied.

00:09:29.450 --> 00:09:30.910
And the really
interesting thing is

00:09:30.910 --> 00:09:32.909
we're going to find there
are an infinite number

00:09:32.909 --> 00:09:34.950
of applications,
and we don't even

00:09:34.950 --> 00:09:36.230
know, frankly, most of them.

00:09:36.230 --> 00:09:39.930
We're going to discover new ones
for them in a thousand years.

00:09:39.930 --> 00:09:41.710
And so hopefully this
gives you a sense

00:09:41.710 --> 00:09:45.450
for why Galileo said
what he said about really

00:09:45.450 --> 00:09:48.870
mathematics is really the
language with which we can

00:09:48.870 --> 00:09:51.620
understand the philosophy
of the universe.

00:09:51.620 --> 00:09:54.590
And that's why people tell us
that if a completely alien life

00:09:54.590 --> 00:09:57.370
form were to ever
contact humans,

00:09:57.370 --> 00:10:00.150
mathematics would probably
be our first common ground,

00:10:00.150 --> 00:10:03.910
the place that we can start
to form a basis that we

00:10:03.910 --> 00:10:06.620
can start to communicate from.


How to write expressions with variables

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:01.920
What I want to do
in this video is

00:00:01.920 --> 00:00:06.000
write the algebraic expressions
that represent the same thing

00:00:06.000 --> 00:00:07.930
that these statements
are saying.

00:00:07.930 --> 00:00:11.250
So this first statement, they
say the sum of negative 7

00:00:11.250 --> 00:00:14.140
and the quantity 8 times x.

00:00:14.140 --> 00:00:17.790
So the sum-- so we're going
to have an addition here--

00:00:17.790 --> 00:00:24.590
of negative 7 and the
quantity 8 times x.

00:00:24.590 --> 00:00:28.770
So the quantity 8 times
x, well, that's just 8x.

00:00:28.770 --> 00:00:30.790
So I can just write
8x over there.

00:00:30.790 --> 00:00:33.750
So it is negative 7 plus 8x.

00:00:33.750 --> 00:00:37.060
Or you could view this
as the sum of negative 7

00:00:37.060 --> 00:00:39.950
and the quantity 8 times x.

00:00:39.950 --> 00:00:41.410
Let's do the next one.

00:00:41.410 --> 00:00:49.165
Take the quantity negative
3 times x and then add 1.

00:00:51.750 --> 00:00:54.410
So the quantity
negative 3 times x,

00:00:54.410 --> 00:00:57.340
we can write that
as negative 3x.

00:00:57.340 --> 00:00:59.280
And then we need
to add 1 to that.

00:00:59.280 --> 00:01:03.240
So that's going to be plus 1.

00:01:03.240 --> 00:01:04.239
Now this one.

00:01:04.239 --> 00:01:08.760
Negative 6 plus--
so we can write

00:01:08.760 --> 00:01:13.830
negative 6 plus something-- the
product of negative 1 and x.

00:01:13.830 --> 00:01:17.060
So the product of
negative 1 and x,

00:01:17.060 --> 00:01:20.410
that's just going to
be negative 1x, which

00:01:20.410 --> 00:01:22.710
is the same thing as negative x.

00:01:22.710 --> 00:01:26.360
So we can write this as
negative 6 plus negative x.

00:01:26.360 --> 00:01:28.260
Or we can just
write this-- this is

00:01:28.260 --> 00:01:35.480
the exact same thing
as negative 6 minus x.

00:01:35.480 --> 00:01:37.510
And we are done.


How to write algebraic expressions with parentheses

WEBVTT
Kind: captions
Language: en

00:00:00.780 --> 00:00:03.890
First consider the
expression for negative 5

00:00:03.890 --> 00:00:06.620
plus the quantity of 4 times x.

00:00:06.620 --> 00:00:09.650
Now, take the product of
negative 8 and that expression

00:00:09.650 --> 00:00:11.160
and then add 6.

00:00:11.160 --> 00:00:12.930
So let's do it step by step.

00:00:12.930 --> 00:00:15.340
First, we're going to have
this expression-- negative 5

00:00:15.340 --> 00:00:17.040
plus something.

00:00:17.040 --> 00:00:22.115
So it's going to be negative 5
plus the quantity of 4 times x.

00:00:24.730 --> 00:00:26.730
Well, that's just
going to be 4x.

00:00:26.730 --> 00:00:29.510
So it's going to be
negative 5 plus 4x.

00:00:29.510 --> 00:00:31.520
So that's this
expression up here.

00:00:31.520 --> 00:00:33.770
Now, take the product
of negative 8,

00:00:33.770 --> 00:00:36.120
so were going to
just take negative 8,

00:00:36.120 --> 00:00:38.500
and we're going to multiply
the product of negative 8

00:00:38.500 --> 00:00:39.595
and that expression.

00:00:39.595 --> 00:00:40.970
So we're going to
take negative 8

00:00:40.970 --> 00:00:43.390
and multiply it
so that expression

00:00:43.390 --> 00:00:48.290
is this thing right over here.

00:00:48.290 --> 00:00:52.150
So if we say the product of
negative 8 in that expression

00:00:52.150 --> 00:00:56.520
is going to be negative
8 times that expression,

00:00:56.520 --> 00:01:01.180
that expression is negative 5
plus 4x, so that's negative 8.

00:01:01.180 --> 00:01:02.640
That's that expression.

00:01:02.640 --> 00:01:08.269
The product of the two, so
we could put a multiplication

00:01:08.269 --> 00:01:10.060
sign there, or we could
just leave that out

00:01:10.060 --> 00:01:12.210
and implicitly it would
mean multiplication,

00:01:12.210 --> 00:01:14.720
take the product of negative
8 and that expression

00:01:14.720 --> 00:01:16.170
and then add 6.

00:01:19.960 --> 00:01:23.550
So that would be then
adding 6 right over here.

00:01:23.550 --> 00:01:27.860
So we could write it as negative
8 open parentheses negative 5

00:01:27.860 --> 00:01:31.370
plus 4x and then add 6.

00:01:31.370 --> 00:01:33.360
Let's do one more.

00:01:33.360 --> 00:01:39.920
First, consider the expression
the sum of 7 and-- so

00:01:39.920 --> 00:01:41.710
that's going to be
7 plus something--

00:01:41.710 --> 00:01:44.590
and the product of
negative 2 and x.

00:01:44.590 --> 00:01:49.690
The product of negative
2 and x is negative 2x.

00:01:49.690 --> 00:01:52.020
So it's 7 plus negative 2x.

00:01:52.020 --> 00:01:55.150
We could write
that as 7 minus 2x.

00:01:55.150 --> 00:01:59.300
So this is equal to 7 minus 2x.

00:01:59.300 --> 00:02:01.350
These are the same expression.

00:02:01.350 --> 00:02:04.420
What expression would be 4 plus?

00:02:04.420 --> 00:02:09.830
So now we're saying
4 plus the quantity

00:02:09.830 --> 00:02:11.460
of 2 times that expression?

00:02:11.460 --> 00:02:16.200
So it's going to be
4 plus some quantity.

00:02:16.200 --> 00:02:17.570
I'll put that in parentheses.

00:02:17.570 --> 00:02:20.680
The quantity of 2
times-- I'll do this

00:02:20.680 --> 00:02:24.610
in magenta or in yellow.

00:02:24.610 --> 00:02:29.920
2 times that expression--
let me do this in blue--

00:02:29.920 --> 00:02:32.720
that expression is this
thing right over here.

00:02:32.720 --> 00:02:39.230
So 4 plus the quantity of 2
times that expression, 2 times

00:02:39.230 --> 00:02:42.340
7 minus 2x.

00:02:42.340 --> 00:02:44.408
And we are done.


Examples of accounting for GDP

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:02.680
What I've done here
is listed a bunch

00:00:02.680 --> 00:00:05.307
of events that might
occur in a given period.

00:00:05.307 --> 00:00:07.140
And what I want to think
about in this video

00:00:07.140 --> 00:00:11.560
is how, if at all, they might
be accounted for in GDP,

00:00:11.560 --> 00:00:13.990
especially in this
expenditure view of GDP.

00:00:13.990 --> 00:00:16.320
And I encourage you
to pause this video

00:00:16.320 --> 00:00:17.640
and try it out yourself.

00:00:17.640 --> 00:00:20.690
See how, if each of these
events happen in the period

00:00:20.690 --> 00:00:23.360
for which we are trying
to calculate GDP,

00:00:23.360 --> 00:00:26.000
how would they be accounted
for, according to the buckets

00:00:26.000 --> 00:00:28.760
we thought about, the
composition of the expenditure

00:00:28.760 --> 00:00:30.160
view of GDP.

00:00:30.160 --> 00:00:31.960
So now I'm assuming
that you have unpaused,

00:00:31.960 --> 00:00:34.460
you've tried it yourself, and
so let's try to go through it.

00:00:34.460 --> 00:00:36.760
So Khan Academy is a firm.

00:00:36.760 --> 00:00:37.960
It's a not-for-profit firm.

00:00:37.960 --> 00:00:39.340
No one really owns Khan Academy.

00:00:39.340 --> 00:00:43.250
I guess society owns Khan
Academy, but it is a firm.

00:00:43.250 --> 00:00:45.410
So Khan Academy employs
a software engineer

00:00:45.410 --> 00:00:47.520
and pays them $100,000.

00:00:47.520 --> 00:00:50.960
Well, this is a firm
making the expenditure.

00:00:50.960 --> 00:00:52.636
And arguably and
even conceptually,

00:00:52.636 --> 00:00:53.760
this also is an investment.

00:00:53.760 --> 00:00:57.240
Because this $100,000 is going
to be used to develop code

00:00:57.240 --> 00:00:58.840
that has future benefit.

00:00:58.840 --> 00:01:02.350
So this is going to be
the investment category.

00:01:02.350 --> 00:01:04.390
Let me do it in that same color.

00:01:04.390 --> 00:01:11.150
So I, investment, is going
to get plus $100,000.

00:01:11.150 --> 00:01:15.616
In general, the spending by
firms goes into investment.

00:01:15.616 --> 00:01:17.240
Now, let's look at
the second scenario.

00:01:17.240 --> 00:01:19.720
Accenture, which is
another firm, and this

00:01:19.720 --> 00:01:23.690
is a for-profit firm, earns $10
million-- or maybe I should say

00:01:23.690 --> 00:01:28.570
gets $10 million
in revenue, just

00:01:28.570 --> 00:01:33.900
to be clear what we're
talking about-- by building

00:01:33.900 --> 00:01:35.840
a new IT system for California.

00:01:35.840 --> 00:01:38.090
And the important thing to
think about, you might say,

00:01:38.090 --> 00:01:40.820
oh, OK, wait, this is
OK, Accenture is a firm,

00:01:40.820 --> 00:01:44.650
but California is
clearly the government.

00:01:44.650 --> 00:01:46.210
So how do you account for this?

00:01:46.210 --> 00:01:48.970
And it's the
expenditure view of GDP.

00:01:48.970 --> 00:01:53.650
So in this situation, California
is spending $10 million

00:01:53.650 --> 00:01:57.450
in the period for
a new IT system.

00:01:57.450 --> 00:01:59.760
So this is going
to be government.

00:01:59.760 --> 00:02:03.220
The government category is going
to be increased by $10 million,

00:02:03.220 --> 00:02:06.340
because of this expenditure.

00:02:06.340 --> 00:02:07.160
Now next one.

00:02:07.160 --> 00:02:09.250
My mother sells her
house in New Orleans

00:02:09.250 --> 00:02:11.930
to a Swedish woman for $200,000.

00:02:11.930 --> 00:02:14.780
Once again, a
house is being sold

00:02:14.780 --> 00:02:17.660
from someone in the country
to someone who was foreign,

00:02:17.660 --> 00:02:18.680
what do we do?

00:02:18.680 --> 00:02:20.390
But the important
thing to realize

00:02:20.390 --> 00:02:22.540
is that this is not a new house.

00:02:22.540 --> 00:02:24.690
This is a transfer
of an existing house.

00:02:24.690 --> 00:02:26.510
Nothing was produced here.

00:02:26.510 --> 00:02:30.359
So this has no
contribution to GDP.

00:02:30.359 --> 00:02:32.900
It doesn't matter it's a Swedish
woman or anything like that.

00:02:32.900 --> 00:02:34.110
The house existed before.

00:02:34.110 --> 00:02:35.350
It just changed hands.

00:02:35.350 --> 00:02:37.380
A new house did
not get produced.

00:02:37.380 --> 00:02:40.320
So nothing happens to GDP here.

00:02:40.320 --> 00:02:41.290
Next one.

00:02:41.290 --> 00:02:43.391
I-- and I'm assuming
that I am here,

00:02:43.391 --> 00:02:45.140
sitting here in Mountain
View, California,

00:02:45.140 --> 00:02:47.950
American citizen-- I
buy a Japanese made

00:02:47.950 --> 00:02:50.600
lawn mower for $200.

00:02:50.600 --> 00:02:53.290
Now this one is interesting.

00:02:53.290 --> 00:02:55.320
Because if you think
about it theoretically,

00:02:55.320 --> 00:02:57.080
nothing was produced
in the United States,

00:02:57.080 --> 00:03:00.020
so nothing should be added
to GDP on a net-net basis.

00:03:00.020 --> 00:03:01.940
And we'll see that that
is actually the case.

00:03:01.940 --> 00:03:04.810
But it's going to show up
by adding to consumption

00:03:04.810 --> 00:03:07.040
and then taking away
from net exports.

00:03:07.040 --> 00:03:08.990
So two things are
going to happen here.

00:03:08.990 --> 00:03:12.020
We'll say, OK, Sal is
an American consumer.

00:03:12.020 --> 00:03:14.150
If we just look at how
much more he spent,

00:03:14.150 --> 00:03:17.890
he spent $200 more, so it's
going to be added there.

00:03:17.890 --> 00:03:21.230
But then we're going to
take it out of net exports.

00:03:21.230 --> 00:03:27.160
So net exports-- let me do it
in that same green color-- net

00:03:27.160 --> 00:03:28.230
exports.

00:03:28.230 --> 00:03:30.020
Everything else is neutral.

00:03:30.020 --> 00:03:31.580
So in this thing
right over here,

00:03:31.580 --> 00:03:33.660
there was no foreign purchases.

00:03:33.660 --> 00:03:36.430
But there is me buying
a foreign product.

00:03:36.430 --> 00:03:38.100
And let me subtract that out.

00:03:38.100 --> 00:03:41.640
So I'm going to subtract
out $200 right over there.

00:03:41.640 --> 00:03:45.350
So net exports will
be lower by $200,

00:03:45.350 --> 00:03:47.280
because essentially
this was a $200 import.

00:03:47.280 --> 00:03:49.820
And that completely
cancels out the $200

00:03:49.820 --> 00:03:51.720
increase in consumption.

00:03:51.720 --> 00:03:54.820
And so this will have
zero net effect on GDP.

00:03:54.820 --> 00:03:57.640
These two terms will cancel out.

00:03:57.640 --> 00:04:03.210
Now I buy a new home in
California for $500,000.

00:04:03.210 --> 00:04:05.590
Now household spending
for the most part

00:04:05.590 --> 00:04:10.670
is considered C, except when
you are buying a new home.

00:04:10.670 --> 00:04:14.760
So even though I am
not a firm, because I

00:04:14.760 --> 00:04:19.899
am buying a house, a new house,
this will go into investment.

00:04:19.899 --> 00:04:24.170
So investment will
go up by $500,000.

00:04:24.170 --> 00:04:28.110
And then finally American
Airlines buys a new Airbus jet,

00:04:28.110 --> 00:04:30.690
and Airbus jets
are made in Europe.

00:04:30.690 --> 00:04:32.650
So what's going to happen here?

00:04:32.650 --> 00:04:35.530
So once again,
net-net, nothing was

00:04:35.530 --> 00:04:36.920
produced in the United States.

00:04:36.920 --> 00:04:40.320
So on a net basis, this
should not contribute to GDP.

00:04:40.320 --> 00:04:42.810
And we'll see that on net
basis, it will break out,

00:04:42.810 --> 00:04:46.170
it will be neutral, but it
will be like this situation.

00:04:46.170 --> 00:04:48.169
There's an American firm
that made a purchase--

00:04:48.169 --> 00:04:49.960
and actually, I didn't
put the amount here.

00:04:49.960 --> 00:04:52.134
So let's say it
was $100 million.

00:04:52.134 --> 00:04:54.550
I think that's actually about
what a passenger plane might

00:04:54.550 --> 00:04:57.890
actually cost, for $100 million.

00:04:57.890 --> 00:04:59.760
So the way we would
account for it,

00:04:59.760 --> 00:05:03.480
investment would go
up by $100 million.

00:05:03.480 --> 00:05:08.512
You have an American firm
making a purchase, $100 million.

00:05:08.512 --> 00:05:09.720
Conceptually, it makes sense.

00:05:09.720 --> 00:05:11.720
It's going to provide
future goods and services,

00:05:11.720 --> 00:05:13.750
going to give
transportation to people.

00:05:13.750 --> 00:05:20.200
But it's going to be netted out,
because you have a net import.

00:05:20.200 --> 00:05:23.030
So what this is going
to do to net exports,

00:05:23.030 --> 00:05:25.590
on this side of it, you're
going to have $100 million,

00:05:25.590 --> 00:05:26.716
because this was an import.

00:05:26.716 --> 00:05:28.631
So you're going to have
negative $100 million,

00:05:28.631 --> 00:05:30.780
when you think of it from
an export point of view.

00:05:30.780 --> 00:05:34.430
And you had no corresponding
positive export.

00:05:34.430 --> 00:05:37.920
So you're going to have net
exports-- net exports is

00:05:37.920 --> 00:05:40.590
going to go down $100 million.

00:05:40.590 --> 00:05:42.670
This was a net import
of $100 million,

00:05:42.670 --> 00:05:44.950
so it makes sense that
net exports would go down.

00:05:44.950 --> 00:05:46.900
It would be negative
net exports.

00:05:46.900 --> 00:05:48.820
And these two, once
again, are going

00:05:48.820 --> 00:05:50.880
to cancel out with
each other, so that you

00:05:50.880 --> 00:05:53.240
have no net GDP,
which makes sense,

00:05:53.240 --> 00:05:57.130
because this plane was not
produced in the United States.