Basic geometry: language and labels

WEBVTT
Kind: captions
Language: en

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- [Instructor] What I wanna
do in this video is give

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an introduction to the language
or some of the characters

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that we use when we talk about geometry.

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And I guess the best place to start

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is to even think about
what geometry means,

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'cause you might recognize
the first part of geometry.

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Right over here you
have the root word geo,

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the same word that you see in things

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like geography and geology,

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and this refers to the earth.

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This refers, my E look
like a C right over there,

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this refers to the earth.

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And then you see this metry part.

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And you see metry in things
like trigonometry as well.

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And metry, or the metric system,

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and this comes from measurement.

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This comes from measurement,
or measure, measurement.

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So when someone's talking about geometry,

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the word itself comes
from earth measurement

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and that's kind of not so bad of a name,

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because it is such a general subject.

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Geometry really is the study
and trying to understand

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how shapes and space

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and things that we see
relate to each other.

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So, you know, when you start
learning about geometry,

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you learn about lines
and triangles and circles

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and you learn about angles

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and we'll define all of these
things more and more precisely

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as we go further and further on,

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but it also encapsulate
things like patterns

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and three-dimensional shapes,
so it's almost everything

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that we see, all of the
visually mathematical things

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that we understand can in some way

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be categorized in geometry.

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Now, with that out of the way,

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let's just start from the basics,

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a basic starting point from geometry,

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and then we can just grow from there.

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So if we just start at a dot.

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That dot right over there is just a point,

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it's just that little point on
that screen right over there.

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We literally call that a point.

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And I'll call that a definition.

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And the fun thing about mathematics

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is that you can make definitions.

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We could have called this an armadillo,

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but we decided to call this a point,

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which I think makes sense because
it's what we would call it

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in just everyday language as well.

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That is a point.

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Now, what's interesting about a point

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is that it is just a position

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that you can't move on a point.

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If you were on this point and you moved

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in any direction at all,
you would no longer be

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at that point, so you
cannot move on a point.

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Now, there are differences between points,

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for example, that's one point there.

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Maybe I have another point over here

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and then I have another point over here

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and then another point over there.

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And you want to be able to
refer to the different points,

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and not everyone has the luxury of a nice,

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colored pen like I do,

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otherwise they could
refer to the green point

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or the blue point or the pink point,

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and so in geometry, to refer to points,

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we tend to give them labels,

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and the labels tend to have letters.

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So, for example, this could be point A,

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this could be point B,
this would be point C,

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and this right over here could be point D.

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So if someone says, "Hey, circle point C,"

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you know which one to circle.

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You know that you would have to circle

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that point right over there.

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Well, that so far, it's
kind of interesting.

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You have these things called points.

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You really can't move around on a point,

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all they do is specify a position.

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What if we wanna move
around a little bit more?

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What if we wanna get from
one point to another?

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So what if we started at one point

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and we wanted all of the
points, including that point

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that connect that point and another point?

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So all of these points right over here.

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So what would we call this thing,

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all of the points that connect
A and B along a straight,

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and I'll use everyday language here,

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along kind of a straight line like this?

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Well, we'll call this a line segment.

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In everyday language,
you might call it a line,

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but we'll call it a line segment,

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'cause we'll see when we
talk in mathematical terms,

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a line means something slightly different.

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So this is a line segment.

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And if we were to connect D and C,

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this would also be another
line segment, a line segment.

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And once again, because
we always don't have

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the luxury of colors,

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this one is clearly the
orange line segment.

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This is clearly the yellow line segment.

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We want to have labels
for these line segments.

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And the best way to
label the line segments

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are with its endpoints, and
that's another word here.

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So a point is just literally A or B,

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but A and B are also the
endpoints of these line segments,

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'cause it starts and ends at A and B.

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So let me write this A and B.

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A and B are endpoints, another
definition right over here.

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Once again, we could have
called them aardvarks

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or end armadillos, but
we, as mathematicians,

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decided to call them endpoints,

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because that seems to
be a good name for it.

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And once again, we need a way
to label these line segments

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that have the endpoints,

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and what's a better way
to label a line segment

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than with its actual endpoints?

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So we would refer to this
line segment, over here,

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we would put its endpoints there.

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And to show that it's a line segment,

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we would draw a line
over it, just like that.

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This line segment down here,
we would write it like this.

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And we could have just as
easily written it like this,

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CD with a line over it would have referred

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to that same line segment.

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BA, BA with a line over it would refer

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to that same line segment.

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And now you might be saying,

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"Well, I'm not satisfied just
traveling in between A and B."

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And this is actually
another interesting idea.

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When you were just on A,
when you were just on a point

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and you couldn't travel at all,

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you couldn't travel at
all in any direction

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while staying on that point,

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that means you have zero
options to travel in.

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You can't go up or down, left or right,

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in or out of the page and
still be on that point.

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And so that's why we say a
point has zero dimensions,

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zero dimensions.

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Now all of the sudden we have this thing,

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this line segment here,
and this line segment,

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we can at least go to the left

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and the right along this line segment.

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We can go towards A or towards B.

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So we can go back or
forward in one dimension.

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So the line segment is a one-dimensional,

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it is a one-dimensional idea almost,

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or a one-dimensional object,

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although these are more
kind of abstract ideas.

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There is no such thing as
a perfect line segment,

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because a line segment,
you can't move up or down

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on this line segment while being on it,

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while in reality, anything that
we think is a line segment,

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even you know, a stick of some type,

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a very straight stick,
or a string that is taut,

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that still will have some width,

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but the geometrical pure
line segment has no width,

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it only has a length here.

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So you can only move along the line,

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and that's why we it's one-dimensional.

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A point, you can't move at all.

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A line segment, you can only
move in that back and forth

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along that same direction.

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Now, I just hinted that it
can actually have a length.

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How do you refer to that?

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Well, you refer to that by
not writing that line on it.

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So if I write AB with a
line on top of it like that,

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that means I'm referring
to the actual line segment.

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If I say that, let me
do this in a new color,

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if I say that AB is equal to five units,

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it might be centimeters,
or meters, or whatever,

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just the abstract units five,

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that means that the distance
between A and B is five,

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that the length of line
segment AB is actually five.

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Now, let's keep on extending it.

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Let's say we wanna just
keep going in one direction.

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So let's say that I start at A,

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let me do this in a new color,

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let's say I start at
A and I wanna go to D,

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but I want the option of keep
on, I wanna keep on going,

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so I can't go further
in A's direction than A,

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but I can go further in D's direction.

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So this little, this
idea that I just showed,

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essentially, it's like a like segment,

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but I can keep on going
past this endpoint,

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we call this a ray.

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And the starting point for
a ray is called the vertex,

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not a term that you'll see too often.

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You'll see vertex later
on in other contexts,

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but it's good to know, this
is the vertex of the ray.

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It's not the vertex of this line segment,

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so maybe I shouldn't
label it just like that.

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And what's interesting about a ray,

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it's once again a one-dimensional figure,

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but you could keep on going
in one of the (murmurs),

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you can keep on going to or
past one of the endpoints.

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And the way that we would
specify a ray is we would say,

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we would call it AD and we
would put this little arrow

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over on top of it to show that is a ray.

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And in this case, it matters the order

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that we put the letters in.

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If I put DA as a ray, this
would mean a different ray.

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That would mean that we're stating at D

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and then we're going past A,

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so this is not ray DA, this is ray AD.

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Now, the last idea that I'm
sure you're thinking about is,

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well, what if I could keep
on going in both directions?

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So let's say I can keep going in, let me,

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my diagram is getting messy.

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So let me introduce some more points.

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So let's say I have point E

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and then I have point F right over here.

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And let's say that I have this object

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that goes through both E and F,

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but just keeps on going
in both directions.

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This is, when we talk in geometry terms,

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this is what we call a line.

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Now, notice, a line never ends.

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You can keep going in either direction.

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A line segment does end, it
has endpoints, a line does not.

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And actually, a line segment
can sometimes be called

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just a segment.

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And so you would specify line EF,

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you would specify line EF

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with these arrows just like that.

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Now, the thing that you're
gonna see most typically

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when we're studying geometry
are these right over here,

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because we're gonna be
concerned with sides of shapes,

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distances between points.

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And when you're talking
about any of those things,

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things that have finite length,

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things that have an actual length,

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things that don't go off forever
in one or two directions,

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then you are talking about
a segment or a line segment.

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Now, if we go back to a line segment,

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just to kind of keep
talking about new words

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that you might confront in geometry.

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If we go back talking about a line,

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that time I was drawing a ray,

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so let's say I have point X and point Y.

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And so this is line segment XY,

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so I could denote it just like that.

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If I have another point,
let's say I have another point

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right over here, let's call that point Z,

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and I'll introduce another word,

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X, Y, and Z are on the same,

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they all lie on the same line

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if you would imagine that
a line could keep going on

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and on forever and ever.

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So we can say that X,
Y, and Z are colinear.

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So those three points are
co, they are colinear.

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They all sit on the same line

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and they also all sit on line segment XY.

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Now, let's say we know, we're told that XZ

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is equal to ZY and they are all colinear.

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So that means this is
telling us that the distance

00:10:54.530 --> 00:10:59.510
between X and Z is the same as
the distance between Z and Y.

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So sometimes we can mark it like that.

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This distance is the same
as that distance over there.

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So that tells us that Z is
exactly halfway between X and Y.

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So in this situation we
would call Z the midpoint,

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the midpoint of line segment XY,

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'cause it's exactly halfway between.

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Now, to finish up, we've
talked about things

00:11:24.260 --> 00:11:26.140
that have zero dimensions, points.

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We've talked about things that
have one dimension, a line,

00:11:29.240 --> 00:11:30.940
a line segment, or a ray.

00:11:30.940 --> 00:11:33.060
You might say, well,
what has two dimensions?

00:11:33.060 --> 00:11:34.360
Well, in order to have two dimensions,

00:11:34.360 --> 00:11:36.120
that means I can go backwards and forwards

00:11:36.120 --> 00:11:37.640
in two different directions.

00:11:37.640 --> 00:11:40.970
So this page right here, or this video,

00:11:40.970 --> 00:11:42.180
or this screen that you're looking at

00:11:42.180 --> 00:11:43.580
is a two-dimensional object.

00:11:44.500 --> 00:11:46.960
I can go right, left,
that is one dimension,

00:11:46.960 --> 00:11:48.970
or I can go up, down.

00:11:48.970 --> 00:11:53.970
And so this surface of the
monitor you're looking at

00:11:54.010 --> 00:11:57.220
is actually two
dimensions, two dimensions.

00:11:57.220 --> 00:12:00.160
You can go backwards or
forwards in two directions.

00:12:00.160 --> 00:12:02.870
And things that are two
dimensions, we call them planar,

00:12:02.870 --> 00:12:04.050
or we call them planes.

00:12:04.050 --> 00:12:09.050
So if you took a piece of
paper that extended forever,

00:12:09.260 --> 00:12:11.970
it just extended in
every direction forever,

00:12:11.970 --> 00:12:14.570
that in a geometrical sense was a plane.

00:12:14.570 --> 00:12:16.870
The piece of paper itself,
the thing that's finite,

00:12:16.870 --> 00:12:18.509
and you'll never see this talked about

00:12:18.509 --> 00:12:20.760
in a typical geometry class,

00:12:20.760 --> 00:12:22.390
but I guess if we were
to draw the analogy,

00:12:22.390 --> 00:12:25.260
you could call a piece of
paper maybe a plane segment,

00:12:25.260 --> 00:12:28.110
because it's a segment of an entire plane.

00:12:28.110 --> 00:12:30.900
If you had a third dimension,
then you're talking about

00:12:30.900 --> 00:12:32.630
kind of our three-dimensional space.

00:12:32.630 --> 00:12:33.990
In three-dimensional space,

00:12:33.990 --> 00:12:36.210
not only could you move left
or right along the screen,

00:12:36.210 --> 00:12:38.220
or up and down, you could also move in

00:12:38.220 --> 00:12:39.750
and our of the screen.

00:12:39.750 --> 00:12:42.160
You could also have this
dimension that I'll try to draw.

00:12:42.160 --> 00:12:43.470
You could go into the screen

00:12:43.470 --> 00:12:47.051
or you could go out of
the screen like that.

00:12:47.051 --> 00:12:49.920
And as we go into higher
and higher mathematics,

00:12:49.920 --> 00:12:51.870
although it becomes
very hard to visualize,

00:12:51.870 --> 00:12:53.860
you'll see that we can
even start to study things

00:12:53.860 --> 00:12:56.563
that have more than three dimensions.


What are variables, expressions, and equations?

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.540
When we're dealing
with basic arithmetic,

00:00:02.540 --> 00:00:04.770
we see the concrete
numbers there.

00:00:04.770 --> 00:00:07.262
We'll see 23 plus 5.

00:00:07.262 --> 00:00:09.220
We know what these numbers
are right over here,

00:00:09.220 --> 00:00:10.303
and we can calculate them.

00:00:10.303 --> 00:00:11.610
It's going to be 28.

00:00:11.610 --> 00:00:13.920
We can say 2 times 7.

00:00:13.920 --> 00:00:17.510
We could say 3 divided by 4.

00:00:17.510 --> 00:00:19.940
In all of these cases, we
know exactly what numbers

00:00:19.940 --> 00:00:20.930
we're dealing with.

00:00:20.930 --> 00:00:23.791
As we start entering into
the algebraic world--

00:00:23.791 --> 00:00:26.040
and you probably have seen
this a little bit already--

00:00:26.040 --> 00:00:30.080
we start dealing with
the ideas of variables.

00:00:30.080 --> 00:00:31.820
And variables, there's
a bunch of ways

00:00:31.820 --> 00:00:33.653
you can think about
them, but they're really

00:00:33.653 --> 00:00:36.240
just values in expressions
where they can change.

00:00:36.240 --> 00:00:38.550
The values in those
expressions can change.

00:00:38.550 --> 00:00:44.770
For example, if
I write x plus 5,

00:00:44.770 --> 00:00:46.760
this is an expression
right over here.

00:00:46.760 --> 00:00:49.110
This can take on
some value depending

00:00:49.110 --> 00:00:51.630
on what the value of x is.

00:00:51.630 --> 00:01:00.450
If x is equal to
1, then x plus 5,

00:01:00.450 --> 00:01:05.870
our expression right over here,
is going to be equal to 1.

00:01:05.870 --> 00:01:07.110
Because now x is 1.

00:01:07.110 --> 00:01:11.050
It'll be 1 plus 5, so x
plus 5 will be equal to 6.

00:01:11.050 --> 00:01:20.330
If x is equal to, I don't
know, negative 7, then x plus 5

00:01:20.330 --> 00:01:25.020
is going to be equal to--
well, now x is negative 7.

00:01:25.020 --> 00:01:28.730
It's going to be negative 7
plus 5, which is negative 2.

00:01:28.730 --> 00:01:34.970
So notice x here is a
variable, and its value can

00:01:34.970 --> 00:01:37.840
change depending on the context.

00:01:37.840 --> 00:01:40.059
And this is in the
context of an expression.

00:01:40.059 --> 00:01:42.100
You'll also see it in the
context of an equation.

00:01:42.100 --> 00:01:43.660
It's actually
important to realize

00:01:43.660 --> 00:01:46.870
the distinction between an
expression and an equation.

00:01:46.870 --> 00:01:48.890
An expression is
really just a statement

00:01:48.890 --> 00:01:51.840
of value, a statement of
some type of quantity.

00:01:51.840 --> 00:01:54.290
So this is an expression.

00:01:54.290 --> 00:01:56.750
An expression would
be something like what

00:01:56.750 --> 00:01:59.220
we saw over here, x plus 5.

00:01:59.220 --> 00:02:00.700
The value of this
expression will

00:02:00.700 --> 00:02:05.740
change depending on what the
value of this variable is.

00:02:05.740 --> 00:02:09.050
And you could just evaluate
it for different values of x.

00:02:09.050 --> 00:02:11.770
Another expression could be
something like, I don't know,

00:02:11.770 --> 00:02:13.130
y plus z.

00:02:13.130 --> 00:02:14.430
Now everything is a variable.

00:02:14.430 --> 00:02:18.570
If y is 1 and z is 2,
it's going to be 1 plus 2.

00:02:18.570 --> 00:02:21.820
If y is 0 and z is
negative 1, it's

00:02:21.820 --> 00:02:24.476
going to be 0 plus negative 1.

00:02:24.476 --> 00:02:26.600
These can all be evaluated,
and they'll essentially

00:02:26.600 --> 00:02:29.820
give you a value depending
on the values of each

00:02:29.820 --> 00:02:32.226
of these variables that
make up the expression.

00:02:32.226 --> 00:02:34.350
An equation, you're
essentially setting expressions

00:02:34.350 --> 00:02:35.610
to be equal to each other.

00:02:35.610 --> 00:02:38.100
That's why they're
called equations.

00:02:38.100 --> 00:02:40.100
You're equating two things.

00:02:40.100 --> 00:02:43.060
An equation, you'll
see one expression

00:02:43.060 --> 00:02:44.830
being equal to
another expression.

00:02:44.830 --> 00:02:49.900
For example, you could say
something like x plus 3

00:02:49.900 --> 00:02:52.230
is equal to 1.

00:02:52.230 --> 00:02:57.250
And in this situation where you
have one equation with only one

00:02:57.250 --> 00:02:59.400
unknown, you can
actually figure out

00:02:59.400 --> 00:03:01.587
what x needs to be
in this scenario.

00:03:01.587 --> 00:03:03.170
And you might even
do it in your head.

00:03:03.170 --> 00:03:04.939
What plus 3 is equal to 1?

00:03:04.939 --> 00:03:06.480
Well, you could do
that in your head.

00:03:06.480 --> 00:03:09.220
If I have negative 2,
plus 3 is equal to 1.

00:03:09.220 --> 00:03:11.140
In this context, an
equation is starting

00:03:11.140 --> 00:03:14.892
to constrain what value
this variable can take on.

00:03:14.892 --> 00:03:16.850
But it doesn't have to
necessarily constrain it

00:03:16.850 --> 00:03:17.400
as much.

00:03:17.400 --> 00:03:25.750
You could have something like
x plus y plus z is equal to 5.

00:03:25.750 --> 00:03:29.360
Now you have this expression is
equal to this other expression.

00:03:29.360 --> 00:03:31.669
5 is really just an
expression right over here.

00:03:31.669 --> 00:03:32.960
And there are some constraints.

00:03:32.960 --> 00:03:35.220
If someone tells you
what y and z is, then

00:03:35.220 --> 00:03:36.480
you're going to get an x.

00:03:36.480 --> 00:03:38.250
If someone tells
you what x and y is,

00:03:38.250 --> 00:03:39.820
then that constrains what z is.

00:03:39.820 --> 00:03:42.660
But it depends on what
the different things are.

00:03:42.660 --> 00:03:51.610
For example, if we said y is
equal to 3 and z is equal to 2,

00:03:51.610 --> 00:03:53.600
then what would be
x in that situation?

00:03:53.600 --> 00:03:57.880
If y is equal to 3
and z is equal to 2,

00:03:57.880 --> 00:04:00.260
then you're going to have
the left-hand expression

00:04:00.260 --> 00:04:02.330
is going to be x plus 3 plus 2.

00:04:02.330 --> 00:04:05.030
It's going to be x plus 5.

00:04:05.030 --> 00:04:07.560
This part right over here is
just going to be 5. x plus 5

00:04:07.560 --> 00:04:08.920
is equal to 5.

00:04:08.920 --> 00:04:11.200
And so what plus
5 is equal to 5?

00:04:11.200 --> 00:04:14.750
Well, now we're
constraining that x

00:04:14.750 --> 00:04:18.269
would have to be equal to 0.

00:04:18.269 --> 00:04:19.769
Hopefully you realize
the difference

00:04:19.769 --> 00:04:20.899
between expression and equation.

00:04:20.899 --> 00:04:22.315
In an equation,
essentially you're

00:04:22.315 --> 00:04:23.640
equating two expressions.

00:04:23.640 --> 00:04:25.390
The important thing
to take away from here

00:04:25.390 --> 00:04:28.370
is that a variable can
take on different values

00:04:28.370 --> 00:04:31.570
depending on the
context of the problem.

00:04:31.570 --> 00:04:34.440
And to hit the point
home, let's just evaluate

00:04:34.440 --> 00:04:37.060
a bunch of expressions
when the variables have

00:04:37.060 --> 00:04:38.350
different values.

00:04:38.350 --> 00:04:47.890
For example, if we had the
expression x to the y power,

00:04:47.890 --> 00:04:55.930
if x is equal to 5
and y is equal to 2,

00:04:55.930 --> 00:04:59.060
then our expression here
is going to evaluate to,

00:04:59.060 --> 00:05:02.680
well, x is now going to be 5.

00:05:02.680 --> 00:05:04.400
y is going to be 2.

00:05:04.400 --> 00:05:06.630
It's going to be 5
to the second power,

00:05:06.630 --> 00:05:09.830
or it's going to evaluate to 25.

00:05:09.830 --> 00:05:16.250
If we changed the values-- let
me do that in that same color--

00:05:16.250 --> 00:05:24.850
if we said x is equal to
negative 2 and y is equal to 3,

00:05:24.850 --> 00:05:26.650
then this expression
would evaluate

00:05:26.650 --> 00:05:33.359
to-- let me do it in
that-- negative 2.

00:05:33.359 --> 00:05:34.900
That's what we're
going to substitute

00:05:34.900 --> 00:05:36.760
for x now in this context.

00:05:36.760 --> 00:05:42.300
And y is now 3, negative
2 to the third power,

00:05:42.300 --> 00:05:44.860
which is negative 2 times
negative 2 times negative 2,

00:05:44.860 --> 00:05:46.842
which is negative 8.

00:05:46.842 --> 00:05:49.300
Negative 2 times negative 2 is
positive 4, times negative 2

00:05:49.300 --> 00:05:51.670
again is equal to negative 8.

00:05:56.400 --> 00:05:58.310
We could do even
more complex things.

00:05:58.310 --> 00:06:03.290
We could have an expression
like the square root of x plus y

00:06:03.290 --> 00:06:06.600
and then minus x, like that.

00:06:09.310 --> 00:06:15.890
Let's say that x is equal
to 1 and y is equal to 8.

00:06:15.890 --> 00:06:18.920
Then this expression
would evaluate to, well,

00:06:18.920 --> 00:06:21.460
every time we see an x,
we want to put a 1 there.

00:06:21.460 --> 00:06:24.800
So we would have a 1 there,
and you'd have a 1 over there.

00:06:24.800 --> 00:06:28.350
And every time you see a y,
you would put an 8 in its place

00:06:28.350 --> 00:06:29.320
in this context.

00:06:29.320 --> 00:06:30.760
We're setting these variables.

00:06:30.760 --> 00:06:32.150
So you'd see an 8.

00:06:32.150 --> 00:06:34.640
Under the radical sign,
you would have a 1 plus 8.

00:06:34.640 --> 00:06:38.130
So you'd have the principal
root of 9, which is 3.

00:06:38.130 --> 00:06:39.970
This whole thing would simplify.

00:06:39.970 --> 00:06:41.850
In this context, when
we set these variables

00:06:41.850 --> 00:06:45.580
to be these things, this whole
thing would simplify to be 3.

00:06:45.580 --> 00:06:46.640
1 plus 8 is 9.

00:06:46.640 --> 00:06:48.740
The principal root of that is 3.

00:06:48.740 --> 00:06:54.490
And then you'd have 3 minus
1, which is equal to 2.


Newton, Leibniz, and Usain Bolt

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.350
This is a picture
of Isaac Newton,

00:00:02.350 --> 00:00:06.040
super famous British
mathematician and physicist.

00:00:06.040 --> 00:00:08.990
This is a picture of
a Gottfried Leibnitz,

00:00:08.990 --> 00:00:11.350
super famous, or
maybe not as famous,

00:00:11.350 --> 00:00:14.350
but maybe should be,
famous German philosopher

00:00:14.350 --> 00:00:17.480
and mathematician, and he was
a contemporary of Isaac Newton.

00:00:17.480 --> 00:00:19.940
These two gentlemen
together were really

00:00:19.940 --> 00:00:22.570
the founding
fathers of calculus.

00:00:22.570 --> 00:00:25.150
And they did some of their--
most of their major work

00:00:25.150 --> 00:00:27.250
in the late 1600s.

00:00:27.250 --> 00:00:31.240
And this right over here is
Usain Bolt, Jamaican sprinter,

00:00:31.240 --> 00:00:35.070
whose continuing to do some
of his best work in 2012.

00:00:35.070 --> 00:00:38.540
And as of early 2012, he's
the fastest human alive,

00:00:38.540 --> 00:00:41.820
and he's probably the fastest
human that has ever lived.

00:00:41.820 --> 00:00:45.231
And you might have not made the
association with these three

00:00:45.231 --> 00:00:45.730
gentleman.

00:00:45.730 --> 00:00:48.100
You might not think that
they have a lot in common.

00:00:48.100 --> 00:00:52.070
But they were all obsessed with
the same fundamental question.

00:00:52.070 --> 00:00:54.470
And this is the same
fundamental question

00:00:54.470 --> 00:00:56.910
that differential
calculus addresses.

00:00:56.910 --> 00:01:00.480
And the question is, what
is the instantaneous rate

00:01:00.480 --> 00:01:01.710
of change of something?

00:01:01.710 --> 00:01:05.570
And in the case of Usain Bolt,
how fast is he going right now?

00:01:05.570 --> 00:01:08.630
Not just what his average
speed was for the last second,

00:01:08.630 --> 00:01:10.850
or his average speed
over the next 10 seconds.

00:01:10.850 --> 00:01:14.090
How fast is he going right now?

00:01:14.090 --> 00:01:17.150
And so this is what differential
calculus is all about.

00:01:17.150 --> 00:01:19.730
Instantaneous rates of change.

00:01:19.730 --> 00:01:20.980
Differential calculus.

00:01:23.800 --> 00:01:26.520
Newton's actual original term
for differential calculus

00:01:26.520 --> 00:01:28.970
was the method of
fluxions, which

00:01:28.970 --> 00:01:31.080
actually sounds a
little bit fancier.

00:01:31.080 --> 00:01:33.890
But it's all about what's
happening in this instant.

00:01:37.000 --> 00:01:40.350
And to think about why that
is not a super easy problem

00:01:40.350 --> 00:01:42.880
to address with
traditional algebra, let's

00:01:42.880 --> 00:01:44.510
draw a little graph here.

00:01:44.510 --> 00:01:49.090
So on this axis
I'll have distance.

00:01:49.090 --> 00:01:51.370
I'll say y is equal to distance.

00:01:51.370 --> 00:01:53.200
I could have said d
is equal to distance,

00:01:53.200 --> 00:01:55.500
but we'll see, especially
later on in calculus,

00:01:55.500 --> 00:01:57.190
d is reserved for
something else.

00:01:57.190 --> 00:01:59.160
We'll say y is
equal to distance.

00:01:59.160 --> 00:02:01.532
And in this axis,
we'll say time.

00:02:01.532 --> 00:02:02.990
And I could say t
is equal to time,

00:02:02.990 --> 00:02:04.531
but I'll just say
x is equal to time.

00:02:07.030 --> 00:02:09.759
And so if we were to plot
Usain Bolt's distance

00:02:09.759 --> 00:02:12.190
as a function of time,
well at time zero

00:02:12.190 --> 00:02:13.570
he hasn't gone anywhere.

00:02:13.570 --> 00:02:16.250
He is right over there.

00:02:16.250 --> 00:02:17.750
And we know that
this gentleman is

00:02:17.750 --> 00:02:23.270
capable of traveling 100
meters in 9.58 seconds.

00:02:23.270 --> 00:02:25.650
So after 9.58
seconds, we'll assume

00:02:25.650 --> 00:02:27.550
that this is in seconds
right over here,

00:02:27.550 --> 00:02:29.420
he's capable of
going 100 meters.

00:02:32.060 --> 00:02:33.530
And so using this
information, we

00:02:33.530 --> 00:02:37.910
can actually figure
out his average speed.

00:02:37.910 --> 00:02:41.880
Let me write it this
way, his average speed

00:02:41.880 --> 00:02:46.700
is just going to be
his change in distance

00:02:46.700 --> 00:02:50.469
over his change in time.

00:02:50.469 --> 00:02:52.260
And using the variables
that are over here,

00:02:52.260 --> 00:02:53.500
we're saying y is distance.

00:02:53.500 --> 00:02:54.958
So this is the same
thing as change

00:02:54.958 --> 00:02:59.740
in y over change in x from
this point to that point.

00:02:59.740 --> 00:03:01.670
And this might look
somewhat familiar to you

00:03:01.670 --> 00:03:02.780
from basic algebra.

00:03:02.780 --> 00:03:05.250
This is the slope
between these two points.

00:03:05.250 --> 00:03:11.180
If I have a line that
connects these two points,

00:03:11.180 --> 00:03:13.070
this is the slope of that line.

00:03:13.070 --> 00:03:15.870
The change in distance
is this right over here.

00:03:15.870 --> 00:03:19.580
Change in y is
equal to 100 meters.

00:03:19.580 --> 00:03:22.990
And our change in time
is this right over here.

00:03:22.990 --> 00:03:27.420
So our change in time is
equal to 9.58 seconds.

00:03:27.420 --> 00:03:30.270
We started at 0, we
go to 9.58 seconds.

00:03:30.270 --> 00:03:33.280
Another way to think about it,
the rise over the run you might

00:03:33.280 --> 00:03:34.710
have heard in your
algebra class.

00:03:34.710 --> 00:03:37.780
It's going to be 100
meters over 9.58 seconds.

00:03:37.780 --> 00:03:43.040
So this is 100 meters
over 9.58 seconds.

00:03:43.040 --> 00:03:45.404
And the slope is essentially
just rate of change,

00:03:45.404 --> 00:03:47.070
or you could view it
as the average rate

00:03:47.070 --> 00:03:49.417
of change between
these two points.

00:03:49.417 --> 00:03:51.500
And you'll see, if you
even just follow the units,

00:03:51.500 --> 00:03:54.440
it gives you units
of speed here.

00:03:54.440 --> 00:03:57.150
It would be velocity if we
also specified the direction.

00:03:57.150 --> 00:03:58.660
And we can figure
out what that is,

00:03:58.660 --> 00:04:02.320
let me get the calculator out.

00:04:02.320 --> 00:04:06.590
So let me get the
calculator on the screen.

00:04:06.590 --> 00:04:13.300
So we're going 100 meters
in the 9.58 seconds.

00:04:13.300 --> 00:04:17.079
So it's 10.4, I'll just write
10.4, I'll round to 10.4.

00:04:17.079 --> 00:04:20.490
So it's approximately
10.4, and then the units

00:04:20.490 --> 00:04:23.920
are meters per second.

00:04:23.920 --> 00:04:26.300
And that is his average speed.

00:04:26.300 --> 00:04:28.130
And what we're going
to see in a second

00:04:28.130 --> 00:04:31.660
is how average
speed is different

00:04:31.660 --> 00:04:32.930
than instantaneous speed.

00:04:32.930 --> 00:04:34.513
How it's different
than what the speed

00:04:34.513 --> 00:04:36.130
he might be going
at any given moment.

00:04:36.130 --> 00:04:38.410
And just to have a concept
of how fast this is,

00:04:38.410 --> 00:04:40.140
let me get the calculator back.

00:04:40.140 --> 00:04:41.360
This is in meters per second.

00:04:41.360 --> 00:04:43.860
If you wanted to know how many
meters he's going in an hour,

00:04:43.860 --> 00:04:48.620
well there's 3,600
seconds in an hour.

00:04:48.620 --> 00:04:51.850
So he'll be able to go this
many meters 3,600 times.

00:04:51.850 --> 00:04:53.460
So that's how many
meters he can,

00:04:53.460 --> 00:04:56.010
if he were able to somehow
keep up that speed in an hour.

00:04:56.010 --> 00:04:58.992
This is how fast he's
going meters per hour.

00:04:58.992 --> 00:05:01.200
And then, if you were to
say how many miles per hour,

00:05:01.200 --> 00:05:03.780
there's roughly 1600-- and I
don't know the exact number,

00:05:03.780 --> 00:05:06.390
but roughly 1600
meters per mile.

00:05:06.390 --> 00:05:08.490
So let's divide it by 1600.

00:05:08.490 --> 00:05:11.900
And so you see that this is
roughly a little over 23,

00:05:11.900 --> 00:05:14.080
about 23 and 1/2 miles per hour.

00:05:14.080 --> 00:05:16.300
So this is
approximately, and I'll

00:05:16.300 --> 00:05:28.800
write it this way-- this
is approximately 23.5 miles

00:05:28.800 --> 00:05:29.850
per hour.

00:05:29.850 --> 00:05:31.670
And relative to a
car, not so fast.

00:05:31.670 --> 00:05:34.500
But relative to
me, extremely fast.

00:05:34.500 --> 00:05:38.560
Now to see how this is different
than instantaneous velocity,

00:05:38.560 --> 00:05:41.010
let's think about a potential
plot of his distance

00:05:41.010 --> 00:05:42.330
relative to time.

00:05:42.330 --> 00:05:44.785
He's not going to just go
this speed immediately.

00:05:44.785 --> 00:05:46.910
He's not just going to go
as soon as the gun fires,

00:05:46.910 --> 00:05:49.890
he's not just going to go 23 and
1/2 miles per hour all the way.

00:05:49.890 --> 00:05:51.490
He's going to accelerate.

00:05:51.490 --> 00:05:54.210
So at first he's going to start
off going a little bit slower.

00:05:54.210 --> 00:05:56.570
So the slope is going to
be a little bit lot lower

00:05:56.570 --> 00:05:57.780
than the average slope.

00:05:57.780 --> 00:05:58.750
He's going to go a
little bit slower,

00:05:58.750 --> 00:06:00.530
then he's going to
start accelerating.

00:06:00.530 --> 00:06:02.930
And so his speed, and
you'll see the slope here

00:06:02.930 --> 00:06:05.040
is getting steeper and
steeper and steeper.

00:06:05.040 --> 00:06:09.400
And then maybe near the end he
starts tiring off a little bit.

00:06:09.400 --> 00:06:11.520
And so his distance
plotted against time

00:06:11.520 --> 00:06:14.022
might be a curve that
looks something like this.

00:06:14.022 --> 00:06:15.480
And what we calculated
here is just

00:06:15.480 --> 00:06:20.039
the average slope across
this change in time.

00:06:20.039 --> 00:06:21.580
What we could see
at any given moment

00:06:21.580 --> 00:06:23.320
the slope is actually different.

00:06:23.320 --> 00:06:26.680
In the beginning,
he has a slower rate

00:06:26.680 --> 00:06:28.860
of change of distance.

00:06:28.860 --> 00:06:30.830
Then over here, then he
accelerates over here,

00:06:30.830 --> 00:06:32.840
it seems like his rate of
change of distance, which

00:06:32.840 --> 00:06:34.464
would be roughly--
or you could view it

00:06:34.464 --> 00:06:36.880
as the slope of the
tangent line at that point,

00:06:36.880 --> 00:06:38.770
it looks higher
than his average.

00:06:38.770 --> 00:06:40.680
And then he starts
to slow down again.

00:06:40.680 --> 00:06:43.840
When you average it out, it gets
to 23 and 1/2 miles per hour.

00:06:43.840 --> 00:06:46.830
And I looked it up, Usain
Bolt's instantaneous velocity,

00:06:46.830 --> 00:06:49.550
his peak instantaneous
velocity, is actually

00:06:49.550 --> 00:06:51.300
closer to 30 miles per hour.

00:06:51.300 --> 00:06:54.800
So the slope over here might
be 23 whatever miles per hour.

00:06:54.800 --> 00:06:57.000
But the instantaneous,
his fastest point

00:06:57.000 --> 00:07:01.220
in this 9.58 seconds is
closer to 30 miles per hour.

00:07:01.220 --> 00:07:03.550
But you see it's not
a trivial thing to do.

00:07:03.550 --> 00:07:06.520
You could say, OK, let me try
to approximate the slope right

00:07:06.520 --> 00:07:07.690
over here.

00:07:07.690 --> 00:07:09.440
And you could do that
by saying, OK, well,

00:07:09.440 --> 00:07:13.200
what is the change in y over the
change of x right around this?

00:07:13.200 --> 00:07:18.230
So you could say, well, let
me take some change of x,

00:07:18.230 --> 00:07:20.810
and figure out what the
change of y is around it,

00:07:20.810 --> 00:07:23.280
or as we go past that.

00:07:23.280 --> 00:07:23.990
So you get that.

00:07:23.990 --> 00:07:25.310
But that would just
be an approximation,

00:07:25.310 --> 00:07:27.143
because you see that
the slope of this curve

00:07:27.143 --> 00:07:28.590
is constantly changing.

00:07:28.590 --> 00:07:30.240
So what you want
to do is see what

00:07:30.240 --> 00:07:32.620
happens as your change
of x gets smaller

00:07:32.620 --> 00:07:33.620
and smaller and smaller.

00:07:33.620 --> 00:07:35.830
As your change of x get smaller
and smaller and smaller,

00:07:35.830 --> 00:07:37.660
you're going to get a better
and better approximation.

00:07:37.660 --> 00:07:39.390
Your change of y is
going to get smaller

00:07:39.390 --> 00:07:40.630
and smaller and smaller.

00:07:40.630 --> 00:07:41.700
So what you want
to do, and we're

00:07:41.700 --> 00:07:43.366
going to go into depth
into all of this,

00:07:43.366 --> 00:07:44.950
and study it more
rigorously, is you

00:07:44.950 --> 00:07:49.270
want to take the limit
as delta x approaches

00:07:49.270 --> 00:07:57.540
0 of your change in y
over your change in x.

00:07:57.540 --> 00:08:00.300
And when you do
that, you're going

00:08:00.300 --> 00:08:02.967
to approach that
instantaneous rate of change.

00:08:02.967 --> 00:08:04.800
You could view it as
the instantaneous slope

00:08:04.800 --> 00:08:06.010
at that point in the curve.

00:08:06.010 --> 00:08:08.690
Or the slope of the tangent
line at that point in the curve.

00:08:08.690 --> 00:08:10.280
Or if we use
calculus terminology,

00:08:10.280 --> 00:08:12.730
we would view that
as the derivative.

00:08:12.730 --> 00:08:17.080
So the instantaneous
slope is the derivative.

00:08:17.080 --> 00:08:22.290
And the notation we use for
the derivative is a dy over dx.

00:08:22.290 --> 00:08:24.300
And that's why I
reserved the letter y.

00:08:24.300 --> 00:08:25.850
And then you say,
well, how does this

00:08:25.850 --> 00:08:28.070
relate to the word differential?

00:08:28.070 --> 00:08:30.350
Well, the word
differential is relating--

00:08:30.350 --> 00:08:33.760
this dy is a differential,
dx is a differential.

00:08:33.760 --> 00:08:35.460
And one way to
conceptualize it, this

00:08:35.460 --> 00:08:37.679
is an infinitely
small change in y

00:08:37.679 --> 00:08:40.299
over an infinitely
small change in x.

00:08:40.299 --> 00:08:42.970
And by getting super,
super small changes in y

00:08:42.970 --> 00:08:46.190
over change in x, you're able
to get your instantaneous slope.

00:08:46.190 --> 00:08:49.730
Or in the case of this example,
the instantaneous speed

00:08:49.730 --> 00:08:53.300
of Usain Bolt right
at that moment.

00:08:53.300 --> 00:08:55.100
And notice, you can't
just put a 0 here.

00:08:55.100 --> 00:08:56.752
If you just put
change in x is zero,

00:08:56.752 --> 00:08:58.710
you're going to get
something that's undefined.

00:08:58.710 --> 00:09:00.140
You can't divide by 0.

00:09:00.140 --> 00:09:02.829
So we take the limit
as it approaches 0.

00:09:02.829 --> 00:09:04.370
And we'll define
that more rigorously

00:09:04.370 --> 00:09:06.500
in the next few videos.


Introduction to the coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:03.620
This right here is a
picture of Rene Descartes.

00:00:03.620 --> 00:00:05.400
Once again, one
of the great minds

00:00:05.400 --> 00:00:07.240
in both math and philosophy.

00:00:07.240 --> 00:00:09.490
And I think you're seeing
a little bit of a trend

00:00:09.490 --> 00:00:11.520
here, that the great
philosophers were also

00:00:11.520 --> 00:00:14.639
great mathematicians
and vice versa.

00:00:14.639 --> 00:00:16.680
And he was somewhat of a
contemporary of Galileo.

00:00:16.680 --> 00:00:20.270
He was 32 years younger,
although he died shortly

00:00:20.270 --> 00:00:21.140
after Galileo died.

00:00:21.140 --> 00:00:22.900
This guy died at a
much younger age.

00:00:22.900 --> 00:00:24.960
Galileo was well into his 70s.

00:00:24.960 --> 00:00:27.900
Descartes died at what
is only 54 years old.

00:00:27.900 --> 00:00:31.349
And he's probably most known in
popular culture for this quote

00:00:31.349 --> 00:00:32.890
right over here, a
very philosophical

00:00:32.890 --> 00:00:35.454
quote. "I think,
therefore I am."

00:00:35.454 --> 00:00:37.870
But I also wanted to throw in,
and this isn't that related

00:00:37.870 --> 00:00:40.570
to algebra, but I just thought
it was a really neat quote,

00:00:40.570 --> 00:00:44.032
probably his least famous
quote, this one right over here.

00:00:44.032 --> 00:00:45.990
And I like it just because
it's very practical,

00:00:45.990 --> 00:00:48.610
and it makes you realize
that these great minds,

00:00:48.610 --> 00:00:51.210
these pillars of
philosophy and mathematics,

00:00:51.210 --> 00:00:54.320
at the end of the day, they
really were just human beings.

00:00:54.320 --> 00:00:56.160
And he said, "You
just keep pushing.

00:00:56.160 --> 00:00:57.410
You just keep pushing.

00:00:57.410 --> 00:00:59.740
I made every mistake
that could be made.

00:00:59.740 --> 00:01:01.490
But I just kept pushing."

00:01:01.490 --> 00:01:05.400
Which I think is very,
very good life advice.

00:01:05.400 --> 00:01:08.431
Now, he did many things in
philosophy and mathematics.

00:01:08.431 --> 00:01:09.930
But the reason why
I'm including him

00:01:09.930 --> 00:01:12.570
here as we build our
foundations of algebra

00:01:12.570 --> 00:01:16.630
is that he is the
individual most responsible

00:01:16.630 --> 00:01:21.420
for a very strong connection
between algebra and geometry.

00:01:21.420 --> 00:01:24.460
So on the left over here, you
have the world of algebra,

00:01:24.460 --> 00:01:26.240
and we've discussed
it a little bit.

00:01:26.240 --> 00:01:28.230
You have equations
that deal with symbols,

00:01:28.230 --> 00:01:29.860
and these symbols
are essentially

00:01:29.860 --> 00:01:31.730
they can take on values.

00:01:31.730 --> 00:01:37.290
So you could have something
like y is equal to 2x minus 1.

00:01:37.290 --> 00:01:40.030
This gives us a relationship
between whatever x is

00:01:40.030 --> 00:01:41.720
and whatever y is.

00:01:41.720 --> 00:01:45.960
And we can even set up a table
here and pick values for x

00:01:45.960 --> 00:01:48.090
and see what the
values of y would be.

00:01:48.090 --> 00:01:51.211
And I could just pick
random values for x,

00:01:51.211 --> 00:01:52.460
and then figure out what y is.

00:01:52.460 --> 00:01:55.080
But I'll pick relatively
straightforward values

00:01:55.080 --> 00:01:57.680
just so that the math
doesn't get too complicated.

00:01:57.680 --> 00:02:00.230
So for example, if
x is negative 2,

00:02:00.230 --> 00:02:02.110
then y is going to
be 2 times negative

00:02:02.110 --> 00:02:11.770
2 minus 1, which is negative 4
minus 1, which is negative 5.

00:02:11.770 --> 00:02:15.130
If x is negative
1, then y is going

00:02:15.130 --> 00:02:20.440
to be 2 times negative
1 minus 1, which

00:02:20.440 --> 00:02:24.100
is equal to-- this is negative
2 minus 1, which is negative 3.

00:02:24.100 --> 00:02:31.880
If x is equal to 0, then y is
going to be 2 times 0 minus 1.

00:02:31.880 --> 00:02:34.710
2 times 0 is 0 minus
1 is just negative 1.

00:02:34.710 --> 00:02:37.494
I'll do a couple more.

00:02:37.494 --> 00:02:39.160
And I could have
picked any values here.

00:02:39.160 --> 00:02:40.020
I could have said,
well, what happens

00:02:40.020 --> 00:02:41.603
if x is the negative
square root of 2,

00:02:41.603 --> 00:02:47.014
or what happens if x is
negative 5/2, or positive 6/7?

00:02:47.014 --> 00:02:48.430
But I'm just picking
these numbers

00:02:48.430 --> 00:02:50.221
because it makes the
math a lot easier when

00:02:50.221 --> 00:02:52.110
I try to figure out
what y is going to be.

00:02:52.110 --> 00:02:56.610
But when x is 1, y is going
to be 2 times 1 minus 1.

00:02:56.610 --> 00:02:59.115
2 times 1 is 2 minus 1 is 1.

00:02:59.115 --> 00:03:01.030
And I'll do one more.

00:03:01.030 --> 00:03:04.470
I'll do one more in a color
that I have not used yet.

00:03:04.470 --> 00:03:06.180
Let's see, this purple.

00:03:06.180 --> 00:03:10.560
If x is 2, then y is
going to be 2 times 2--

00:03:10.560 --> 00:03:13.730
now our x is 2-- minus 1.

00:03:13.730 --> 00:03:16.200
So that is 4 minus
1 is equal to 3.

00:03:16.200 --> 00:03:16.990
So fair enough.

00:03:16.990 --> 00:03:19.440
I just kind of sampled
this relationship.

00:03:19.440 --> 00:03:22.080
I said, OK, this describes
the general relationship

00:03:22.080 --> 00:03:24.415
between a variable
y and a variable x.

00:03:24.415 --> 00:03:26.540
And then I just made it a
little bit more concrete.

00:03:26.540 --> 00:03:30.150
I said, OK, well, then
for each of these values

00:03:30.150 --> 00:03:33.550
of x, what would be the
corresponding value of y?

00:03:33.550 --> 00:03:35.210
And what Descartes
realized is is

00:03:35.210 --> 00:03:36.820
that you could visualize this.

00:03:36.820 --> 00:03:39.830
One, you could visualize
these individual points,

00:03:39.830 --> 00:03:42.130
but that could also
help you, in general,

00:03:42.130 --> 00:03:45.135
to visualize this relationship.

00:03:45.135 --> 00:03:46.510
And so what he
essentially did is

00:03:46.510 --> 00:03:49.870
he bridged the worlds
of this kind of very

00:03:49.870 --> 00:03:53.740
abstract, symbolic algebra
and that and geometry,

00:03:53.740 --> 00:03:57.290
which was concerned with
shapes and sizes and angles.

00:03:57.290 --> 00:04:02.667
So over here you have
the world of geometry.

00:04:02.667 --> 00:04:05.250
And obviously, there are people
in history, maybe many people,

00:04:05.250 --> 00:04:08.370
who history may have forgotten
who might have dabbled in this.

00:04:08.370 --> 00:04:10.990
But before Descartes,
it's generally

00:04:10.990 --> 00:04:13.644
viewed that geometry
was Euclidean geometry,

00:04:13.644 --> 00:04:15.810
and that's essentially the
geometry that you studied

00:04:15.810 --> 00:04:18.490
in a geometry class in
eighth or ninth grade

00:04:18.490 --> 00:04:21.910
or 10th grade in a traditional
high school curriculum.

00:04:21.910 --> 00:04:23.740
And that's the
geometry of studying

00:04:23.740 --> 00:04:27.940
the relationships between
triangles and their angles,

00:04:27.940 --> 00:04:31.830
and the relationships between
circles and you have radii,

00:04:31.830 --> 00:04:34.822
and then you have triangles
inscribed in circles,

00:04:34.822 --> 00:04:35.530
and all the rest.

00:04:35.530 --> 00:04:38.896
And we go into some depth in
that in the geometry playlist.

00:04:38.896 --> 00:04:40.270
But Descartes
said, well, I think

00:04:40.270 --> 00:04:43.500
I can represent this visually
the same way that you could

00:04:43.500 --> 00:04:46.400
with studying these
triangles and these circles.

00:04:46.400 --> 00:04:50.260
He said, well, if we
view a piece of paper,

00:04:50.260 --> 00:04:52.010
if we think about a
two-dimensional plane,

00:04:52.010 --> 00:04:53.635
you could view a
piece of paper as kind

00:04:53.635 --> 00:04:55.685
of a section of a
two-dimensional plane.

00:04:55.685 --> 00:04:57.560
And we call it two
dimensions because there's

00:04:57.560 --> 00:04:59.270
two directions that
you could go in.

00:04:59.270 --> 00:05:00.780
There's the up/down direction.

00:05:00.780 --> 00:05:02.090
That's one direction.

00:05:02.090 --> 00:05:03.450
So let me draw that.

00:05:03.450 --> 00:05:06.210
I'll do it in blue because we're
starting to visualize things,

00:05:06.210 --> 00:05:08.050
so I'll do it in
the geometry color.

00:05:08.050 --> 00:05:11.630
So you have the
up/down direction.

00:05:11.630 --> 00:05:13.990
And you have the
left/right direction.

00:05:13.990 --> 00:05:16.149
That's why it's called
a two-dimensional plane.

00:05:16.149 --> 00:05:17.690
If we're dealing in
three dimensions,

00:05:17.690 --> 00:05:20.870
you would have an
in/out dimension.

00:05:20.870 --> 00:05:23.140
And it's very easy to do
two dimensions on the screen

00:05:23.140 --> 00:05:25.230
because the screen
is two dimensional.

00:05:25.230 --> 00:05:27.920
And he, says, well, you know,
there are two variables here,

00:05:27.920 --> 00:05:29.520
and they have this relationship.

00:05:29.520 --> 00:05:31.900
So why don't I associate
each of these variables

00:05:31.900 --> 00:05:34.380
with one of these
dimensions over here?

00:05:34.380 --> 00:05:37.975
And by convention, let's make
the y variable, which is really

00:05:37.975 --> 00:05:39.990
the dependent variable--
the way we did it,

00:05:39.990 --> 00:05:43.056
it depends on what x is-- let's
put that on the vertical axis.

00:05:43.056 --> 00:05:44.680
And let's put our
independent variable,

00:05:44.680 --> 00:05:46.490
the one where I just
randomly picked values for it

00:05:46.490 --> 00:05:47.990
to see what y
would become, let's

00:05:47.990 --> 00:05:50.290
put that on the horizontal axis.

00:05:50.290 --> 00:05:52.190
And it actually
was Descartes who

00:05:52.190 --> 00:05:55.650
came up with the convention of
using x's and y's, and we'll

00:05:55.650 --> 00:05:58.320
see later z's, in
algebra so extensively

00:05:58.320 --> 00:06:00.480
as unknown variables
or the variables

00:06:00.480 --> 00:06:01.750
that you're manipulating.

00:06:01.750 --> 00:06:03.791
But he says, well, if we
think about it this way,

00:06:03.791 --> 00:06:07.330
if we number these
dimensions-- so let's

00:06:07.330 --> 00:06:12.270
say that in the x direction,
let's make this right over here

00:06:12.270 --> 00:06:15.450
is negative 3.

00:06:15.450 --> 00:06:17.570
Let's make this negative 2.

00:06:17.570 --> 00:06:18.950
This is negative 1.

00:06:18.950 --> 00:06:20.260
This is 0.

00:06:20.260 --> 00:06:23.290
Now, I'm just numbering
the x direction,

00:06:23.290 --> 00:06:24.810
the left/right direction.

00:06:24.810 --> 00:06:26.450
Now this is positive 1.

00:06:26.450 --> 00:06:27.830
This is positive 2.

00:06:27.830 --> 00:06:29.660
This is positive 3.

00:06:29.660 --> 00:06:31.930
And we could do the
same in the y direction.

00:06:31.930 --> 00:06:36.200
So let's see, so this could be,
let's say this is negative 5,

00:06:36.200 --> 00:06:39.560
negative 4, negative
3, negative-- actually,

00:06:39.560 --> 00:06:42.010
let me do it a little
bit neater than that.

00:06:42.010 --> 00:06:44.070
Let me clean this
up a little bit.

00:06:44.070 --> 00:06:46.369
So let me erase this
and extend this down

00:06:46.369 --> 00:06:48.660
a little bit so I can go all
the way down to negative 5

00:06:48.660 --> 00:06:51.180
without making it
look too messy.

00:06:51.180 --> 00:06:53.240
So let's go all
the way down here.

00:06:53.240 --> 00:06:54.240
And so we can number it.

00:06:54.240 --> 00:06:55.580
This is 1.

00:06:55.580 --> 00:06:56.650
This is 2.

00:06:56.650 --> 00:06:57.770
This is 3.

00:06:57.770 --> 00:07:00.872
And then this could be
negative 1, negative 2.

00:07:00.872 --> 00:07:02.330
And these are all
just conventions.

00:07:02.330 --> 00:07:03.660
It could have been
labeled the other way.

00:07:03.660 --> 00:07:05.909
We could've decided to put
the x there and the y there

00:07:05.909 --> 00:07:07.500
and make this the
positive direction

00:07:07.500 --> 00:07:08.870
and make this the
negative direction.

00:07:08.870 --> 00:07:10.661
But this is just the
convention that people

00:07:10.661 --> 00:07:12.300
adopted starting with Descartes.

00:07:12.300 --> 00:07:17.740
Negative 2, negative 3,
negative 4, and negative 5.

00:07:17.740 --> 00:07:20.760
And he says, well, I
think I can associate

00:07:20.760 --> 00:07:25.110
each of these pairs of values
with a point in two dimensions.

00:07:25.110 --> 00:07:28.280
I can take the x-coordinate, I
can take the x value right over

00:07:28.280 --> 00:07:30.090
here, and I say, OK,
that's a negative 2.

00:07:30.090 --> 00:07:33.480
That would be right over there
along the left/right direction.

00:07:33.480 --> 00:07:35.370
I'm going to the left
because it's negative.

00:07:35.370 --> 00:07:37.420
And that's associated
with negative 5

00:07:37.420 --> 00:07:39.040
in the vertical direction.

00:07:39.040 --> 00:07:41.550
So I say the y
value is negative 5,

00:07:41.550 --> 00:07:46.350
and so if I go 2 to
the left and 5 down,

00:07:46.350 --> 00:07:48.780
I get to this point
right over there.

00:07:48.780 --> 00:07:53.090
So he says, these two values,
negative 2 and negative 5,

00:07:53.090 --> 00:07:56.270
I can associate it with
this point in this plane

00:07:56.270 --> 00:07:58.630
right over here, in this
two-dimensional plane.

00:07:58.630 --> 00:08:02.640
So I'll say that point has
the coordinates, tells me

00:08:02.640 --> 00:08:05.700
where to find that point,
negative 2, negative 5.

00:08:05.700 --> 00:08:08.360
And these coordinates are
called Cartesian coordinates,

00:08:08.360 --> 00:08:11.677
named for Rene
Descartes because he's

00:08:11.677 --> 00:08:13.010
the guy that came up with these.

00:08:13.010 --> 00:08:15.260
He's associating, all of a
sudden, these relationships

00:08:15.260 --> 00:08:17.060
with points on a
coordinate plane.

00:08:17.060 --> 00:08:19.440
And then he said, well,
OK, let's do another one.

00:08:19.440 --> 00:08:21.050
There's this other
relationship, where

00:08:21.050 --> 00:08:25.385
I have when x is
equal to negative 1,

00:08:25.385 --> 00:08:27.160
y is equal to negative 3.

00:08:27.160 --> 00:08:29.692
So x is negative
1, y is negative 3.

00:08:29.692 --> 00:08:31.150
That's that point
right over there.

00:08:31.150 --> 00:08:32.730
And the convention
is, once again,

00:08:32.730 --> 00:08:34.146
when you list the
coordinates, you

00:08:34.146 --> 00:08:36.169
list the x-coordinate,
then the y-coordinate.

00:08:36.169 --> 00:08:38.440
And that's just what
people decided to do.

00:08:38.440 --> 00:08:39.900
Negative 1, negative
3, that would

00:08:39.900 --> 00:08:42.039
be that point right over there.

00:08:42.039 --> 00:08:45.040
And then you have the point
when x is 0, y is negative 1.

00:08:45.040 --> 00:08:47.630
When x is 0 right
over here, which

00:08:47.630 --> 00:08:50.630
means I don't go to the left
or the right, y is negative 1,

00:08:50.630 --> 00:08:51.847
which means I go 1 down.

00:08:51.847 --> 00:08:53.430
So that's that point
right over there,

00:08:53.430 --> 00:08:57.000
0, negative 1, right over there.

00:08:57.000 --> 00:08:58.620
And I could keep doing this.

00:08:58.620 --> 00:09:00.430
When x is 1, y is 1.

00:09:03.570 --> 00:09:05.595
When x is 2, y is 3.

00:09:09.000 --> 00:09:11.430
Actually, let me do it in
that same purple color.

00:09:11.430 --> 00:09:16.220
When x is 2, y is 3, 2 comma 3.

00:09:16.220 --> 00:09:20.390
And then this one right over
here in orange was 1 comma 1.

00:09:20.390 --> 00:09:21.820
And this is neat by itself.

00:09:21.820 --> 00:09:24.131
I essentially just
sampled possible x's.

00:09:24.131 --> 00:09:25.630
But what he realized
is, not only do

00:09:25.630 --> 00:09:27.920
you sample these possible
x's, but if you just

00:09:27.920 --> 00:09:30.150
kept sampling x's, if you
tried sampling all the x's

00:09:30.150 --> 00:09:33.630
in between, you would actually
end up plotting out a line.

00:09:33.630 --> 00:09:35.840
So if you were to
do every possible x,

00:09:35.840 --> 00:09:38.130
you would end up getting a
line that looks something

00:09:38.130 --> 00:09:44.560
like that right over there.

00:09:44.560 --> 00:09:48.450
And any relation, if you
pick any x and find any y,

00:09:48.450 --> 00:09:50.800
it really represents
a point on this line.

00:09:50.800 --> 00:09:53.380
Or another way to think about
it, any point on this line

00:09:53.380 --> 00:09:56.781
represents a solution to this
equation right over here.

00:09:56.781 --> 00:09:58.530
So if you have this
point right over here,

00:09:58.530 --> 00:10:02.090
which looks like it's about
x is 1 and 1/2, y is 2.

00:10:02.090 --> 00:10:06.400
So me write that, 1.5 comma 2.

00:10:06.400 --> 00:10:08.470
That is a solution
to this equation.

00:10:08.470 --> 00:10:13.360
When x is 1.5, 2 times
1.5 is 3 minus 1 is 2.

00:10:13.360 --> 00:10:14.980
That is right over there.

00:10:14.980 --> 00:10:16.810
So all of a sudden,
he was able to bridge

00:10:16.810 --> 00:10:22.140
this gap or this relationship
between algebra and geometry.

00:10:22.140 --> 00:10:26.690
We can now visualize
all of the x and y pairs

00:10:26.690 --> 00:10:31.140
that satisfy this
equation right over here.

00:10:31.140 --> 00:10:35.587
And so he is responsible
for making this bridge,

00:10:35.587 --> 00:10:37.170
and that's why the
coordinates that we

00:10:37.170 --> 00:10:42.170
use to specify these points are
called Cartesian coordinates.

00:10:42.170 --> 00:10:45.770
And as we'll see, the first
type of equations we will study

00:10:45.770 --> 00:10:48.000
are equations of
this form over here.

00:10:48.000 --> 00:10:50.010
And in a traditional
algebra curriculum,

00:10:50.010 --> 00:10:51.370
they're called linear equations.

00:10:55.000 --> 00:10:57.540
And you might be saying,
well, OK, this is an equation.

00:10:57.540 --> 00:10:59.100
I see that this
is equal to that.

00:10:59.100 --> 00:11:00.605
But what's so linear about them?

00:11:00.605 --> 00:11:02.110
What makes them
look like a line?

00:11:02.110 --> 00:11:03.720
And to realize why
they are linear,

00:11:03.720 --> 00:11:07.070
you have to make this jump
that Rene Descartes made,

00:11:07.070 --> 00:11:10.480
because if you were to plot
this using Cartesian coordinates

00:11:10.480 --> 00:11:13.622
on a Euclidean plane,
you will get a line.

00:11:13.622 --> 00:11:15.330
And in the future,
we'll see that there's

00:11:15.330 --> 00:11:17.746
other types of equations where
you won't get a line, where

00:11:17.746 --> 00:11:21.870
you'll get a curve or something
kind of crazy or funky.


Phillips curve

WEBVTT
Kind: captions
Language: en

00:00:00.478 --> 00:00:01.938
In the late 1950s,

00:00:01.938 --> 00:00:04.440
William Phillips noticed a correlation

00:00:04.440 --> 00:00:07.348
between unemployment and inflation

00:00:07.348 --> 00:00:09.604
and I have a picture of this gentleman right over here,

00:00:09.604 --> 00:00:10.746
Irving Fisher

00:00:10.746 --> 00:00:13.681
because he actually noticed that relationship a few decades before

00:00:13.681 --> 00:00:17.771
but the relationship has taken on Phillips's name really because

00:00:17.771 --> 00:00:21.272
his publication kind of captured people's imagination

00:00:21.272 --> 00:00:24.520
and the relationship is not that surprising

00:00:24.520 --> 00:00:26.271
it seems to fall a little bit out of common sense

00:00:26.271 --> 00:00:30.770
but we'll talk about examples that seem to go against this relationship

00:00:30.770 --> 00:00:35.012
If I were to plot the unemployment percentage right here

00:00:38.796 --> 00:00:40.211
on my horizontal axis

00:00:40.211 --> 00:00:45.349
and I plot inflation on my vertical axis,

00:00:45.349 --> 00:00:51.347
so let me put inflation here

00:00:51.347 --> 00:00:53.937
Then he noticed, (and I'm just gonna pick random data points)

00:00:53.937 --> 00:00:59.081
but in one year where there was high inflation, there was low unemployment

00:00:59.081 --> 00:01:00.346
and then in another year,

00:01:00.346 --> 00:01:03.015
where there was high unemployment, there was low inflation

00:01:03.015 --> 00:01:05.185
and it's not clear which one's causing which

00:01:05.185 --> 00:01:07.210
or maybe they both circle back on each other

00:01:07.210 --> 00:01:12.475
and then in other years, maybe where you had medium inflation

00:01:12.475 --> 00:01:15.208
or relatively low unemployment

00:01:15.208 --> 00:01:19.184
so he looked at a bunch of years and plotted them on an axis like this

00:01:19.184 --> 00:01:20.934
(let me do them all in the same color)

00:01:20.934 --> 00:01:22.099
so you take a bunch of years

00:01:22.099 --> 00:01:22.935
and you plot them here

00:01:22.935 --> 00:01:28.747
so now you have higher inflation, slightly higher unemployment than that year over there

00:01:28.747 --> 00:01:32.209
and I could just keep plotting some points over here

00:01:32.209 --> 00:01:33.934
you could have deflation

00:01:33.934 --> 00:01:36.476
and what he saw is that there's a correlation here

00:01:36.476 --> 00:01:37.850
that there's an inverse relationship

00:01:37.850 --> 00:01:40.681
that if you were trying to fit a curve to these points

00:01:40.681 --> 00:01:44.270
and you could have more points here, I'm just picking them at random,

00:01:44.347 --> 00:01:50.407
you could fit a curve that looks something like this

00:01:50.407 --> 00:01:54.676
generally saying, when you have high inflation,

00:01:54.676 --> 00:01:59.075
so when you're up here, you have high inflation

00:01:59.075 --> 00:02:10.157
and low unemployment

00:02:10.157 --> 00:02:11.943
the low unemployment part is a good thing

00:02:11.943 --> 00:02:22.835
and now here you have low inflation and high unemployment

00:02:22.835 --> 00:02:29.241
and if this curve goes below the horizontal axis you would actually have deflation

00:02:29.241 --> 00:02:32.074
and it makes reasonable sense

00:02:32.074 --> 00:02:35.405
we've kind of talked bout it in common sense terms before

00:02:35.405 --> 00:02:39.405
but you can imagine - once again, it's not clear which one's causing which -

00:02:39.405 --> 00:02:41.943
but you can imagine a world, just reasoning through it,

00:02:41.943 --> 00:02:46.624
that if you have low unemployment, or you could view that as high employment,

00:02:48.839 --> 00:02:55.217
one way to view low unemployment is that you have a high utilisation of the labour market

00:02:55.217 --> 00:03:05.482
well, now, in this situation, workers have more leverage

00:03:05.482 --> 00:03:07.274
and if workers have more leverage

00:03:08.766 --> 00:03:12.738
then employers will have to increase wages

00:03:12.738 --> 00:03:15.276
to attract and retain employees

00:03:15.276 --> 00:03:18.034
they have more leverage and they have more options

00:03:18.034 --> 00:03:19.905
there are a lot of people looking to hire them

00:03:19.905 --> 00:03:36.403
employers raise wages to attract and retain employees

00:03:36.403 --> 00:03:41.570
but of course when you increase wages you're increasing buying power, generally for workers

00:03:41.570 --> 00:03:51.677
so this is increasing workers' buying power

00:03:51.677 --> 00:03:54.402
which would increase their demand

00:03:58.385 --> 00:03:59.815
for goods and services

00:03:59.815 --> 00:04:02.009
and if they're increasing the demand for goods and services,

00:04:02.009 --> 00:04:05.070
they're going to increase the utilisation of all of the factors of production:

00:04:05.070 --> 00:04:08.276
of land, of capital, of entrepreneurship,

00:04:08.276 --> 00:04:14.234
and of labour, so that's going to lower unemployment even more

00:04:14.234 --> 00:04:16.735
and you can imagine if you already have low unemployment,

00:04:16.735 --> 00:04:19.481
and this is just one of the factors of production,

00:04:19.481 --> 00:04:23.029
but if you think about all factors of productions, they are probably all highly utilised

00:04:23.029 --> 00:04:25.568
the factories are running at close to full capacity

00:04:25.568 --> 00:04:28.342
the labour market is running at full capacity

00:04:28.342 --> 00:04:33.215
if you increase buying power, if you increase demand in that context,

00:04:33.215 --> 00:04:35.068
so demand is going up

00:04:36.668 --> 00:04:41.276
lower unemployment, labour utilisation is going down

00:04:41.276 --> 00:04:43.067
there's less capacity, more demand

00:04:43.067 --> 00:04:45.816
this is going to cause prices to go up

00:04:45.816 --> 00:04:52.149
so this is going to cause prices generally to go up

00:04:52.149 --> 00:04:57.609
and then if prices go up, now workers have leverage and options of employment

00:04:57.609 --> 00:05:02.942
but they also have higher cost

00:05:02.942 --> 00:05:06.903
higher cost of living

00:05:06.903 --> 00:05:09.149
and this is a very kind of hand, wavy diagram,

00:05:09.149 --> 00:05:11.546
but it's just to make you think about the overall dynamics

00:05:11.546 --> 00:05:14.009
so workers have more options and leverage

00:05:14.009 --> 00:05:15.482
they also have a higher cost of living,

00:05:15.482 --> 00:05:20.148
the whole economy is kind of operating at close to full tilt

00:05:20.148 --> 00:05:23.093
and so they're going to demand higher wages even more

00:05:23.093 --> 00:05:26.076
and then this thing can keep cycling and keep cycling

00:05:26.076 --> 00:05:28.215
now, this seems a bit common sense,

00:05:28.215 --> 00:05:30.481
but there are exceptions to these

00:05:30.481 --> 00:05:34.275
in the 1970s, the US experienced stagflation

00:05:34.275 --> 00:05:39.481
and that's kind of the worst: that's high inflation and high unemployment

00:05:39.481 --> 00:05:41.730
so that is right over here

00:05:41.730 --> 00:05:43.147
and things that could cause it

00:05:43.147 --> 00:05:45.980
and particularly what people point to in the 1970s,

00:05:45.980 --> 00:05:51.761
and it's always important to realise that in economics people don't know for sure what was the exact cause

00:05:51.761 --> 00:05:54.147
and there seldom is only one cause

00:05:54.147 --> 00:05:56.480
but one of the things that's often pointed to

00:05:56.480 --> 00:05:58.753
is that you had a supply shock

00:06:01.091 --> 00:06:04.000
and the supply shock was in oil

00:06:04.000 --> 00:06:07.145
it made the cost of producing everything more expensive

00:06:07.145 --> 00:06:09.343
and so that just drove the prices up

00:06:09.343 --> 00:06:15.145
but really didn't allow the country as a whole to become more productive

00:06:15.145 --> 00:06:18.730
one way to think about it is it drove the prices up here

00:06:18.730 --> 00:06:20.395
causing a higher cost of living right over here

00:06:20.395 --> 00:06:28.076
but this part of this cycle that we associate with low unemployment and high inflation

00:06:28.076 --> 00:06:32.119
was not occurring

00:06:34.687 --> 00:06:37.811
and because those higher prices were essentially going out of the country,

00:06:37.811 --> 00:06:43.895
you could essentially imagine that they were squeezing out people's ability to pay for other things

00:06:43.895 --> 00:06:46.676
so the higher cost of living

00:06:46.676 --> 00:06:48.228
or the higher prices for oil

00:06:48.228 --> 00:06:51.359
you could draw a line from either one and I know it's getting messy now

00:06:51.405 --> 00:06:56.977
you can imagine that it inhibited demand for domestic production

00:06:56.977 --> 00:06:59.565
so I'll draw "negative feedback" right over there

00:06:59.565 --> 00:07:03.310
so it squeezed out people's desire to buy things because they had to pay so much for oil

00:07:03.310 --> 00:07:05.643
and there's other explanations for it

00:07:05.643 --> 00:07:09.977
when people saw the low unemployment, the government wanted to print even more money

00:07:09.977 --> 00:07:15.393
to fuel things, but it did not get this virtual cycle happening

00:07:15.393 --> 00:07:18.245
there were some arguments that there were structural reasons

00:07:18.245 --> 00:07:21.677
why the economy couldn't adapt properly

00:07:21.677 --> 00:07:26.087
so that employees and resources couldn't be allocated efficiently

00:07:26.087 --> 00:07:31.308
but the whole point of bringing up the 1970s is to show you that this isn't a law

00:07:31.308 --> 00:07:33.476
it's not clear what's causing what

00:07:33.476 --> 00:07:37.582
that there was a situation in the 1970s where you had stagflation

00:07:37.582 --> 00:07:42.476
and the opposite of that was what we really saw in the late 90s

00:07:42.476 --> 00:07:48.011
where you had relatively low inflation and you had relatively low unemployment

00:07:48.011 --> 00:07:51.142
so this is a very good situation that we had in the late 90s

00:07:51.142 --> 00:07:54.808
and the argument that many people make why we were able to do that,

00:07:54.808 --> 00:07:57.742
why this cycle didn't keep going on and on

00:07:57.742 --> 00:08:04.226
why the prices didn't go up is that you had this other trend of huge technological improvement

00:08:04.226 --> 00:08:12.141
you had computers, and telecommunications, the Internet, this kind of super productivity curve

00:08:12.141 --> 00:08:18.678
so even though you had the cycle, you had increased buying power, demand was going up,

00:08:18.678 --> 00:08:21.610
the proactivity of the country was increasing so much

00:08:21.610 --> 00:08:24.389
that it did not lead to inflation

00:08:24.389 --> 00:08:27.474
so that's what threw us there in the late 90s

00:08:27.474 --> 00:08:29.550
so, in general, it's a neat thing to think about,

00:08:31.088 --> 00:08:33.056
at least to some degree it seems like common sense,

00:08:33.056 --> 00:08:38.342
but like in all things in economics, it's always a little bit more nuanced and complicated

00:08:38.342 --> 00:08:42.557
than just some little correlation that you might observe

00:08:42.557 --> 00:08:46.557
this could be generally true but there's always going to be exceptions to it


Actual CPI-U Basket of Goods

WEBVTT
Kind: captions
Language: en

00:00:00.332 --> 00:00:01.802
Male voice: What I want
to do in this video

00:00:01.802 --> 00:00:03.868
is explore what the actual basket of goods

00:00:03.868 --> 00:00:06.138
looks like for the consumer price index.

00:00:06.138 --> 00:00:08.735
We had a ridiculously simple
example in the last video.

00:00:08.735 --> 00:00:10.867
Right over here, this is a
table I got, this is from

00:00:10.867 --> 00:00:13.537
a press release issued by the
Bureau of Labor Statistics.

00:00:13.537 --> 00:00:17.474
If you do a search for a CPI or a CPI-U

00:00:17.474 --> 00:00:19.984
in Bureau of Labor
Statistics, you should find

00:00:19.984 --> 00:00:22.447
the press release where
they issue the CPI.

00:00:22.447 --> 00:00:25.028
This is the first table
in that press release.

00:00:25.028 --> 00:00:27.140
They say the Consumer
Price Index for all urban

00:00:27.140 --> 00:00:29.130
consumers, and just
like we talked about in

00:00:29.130 --> 00:00:32.136
the last video, when people
talk about the CPI Index,

00:00:32.136 --> 00:00:33.938
they're really talking about, or they tend

00:00:33.938 --> 00:00:37.169
to be talking about, the
CPI-U. The CPI for urban

00:00:37.169 --> 00:00:40.202
consumers because most consumers
fall under this category.

00:00:40.202 --> 00:00:42.538
U.S. city average by expenditure category

00:00:42.538 --> 00:00:44.069
and commodity and service group.

00:00:44.069 --> 00:00:45.806
Then they define their base year.

00:00:45.806 --> 00:00:48.400
The base year, actually they have a range,

00:00:48.400 --> 00:00:51.281
maybe I guess they take an
average between 1982 and 1984.

00:00:51.281 --> 00:00:55.283
They say that is 100
unless otherwise noted.

00:00:55.283 --> 00:00:56.697
What they do in this first column,

00:00:56.697 --> 00:00:58.611
so these are the different
buckets that people

00:00:58.611 --> 00:01:01.136
spend, that the urban consumer might spend

00:01:01.136 --> 00:01:03.285
some of their money on. This is saying

00:01:03.285 --> 00:01:05.402
the basket of goods
we're giving a weighting.

00:01:05.402 --> 00:01:08.734
About a little under 15% is spent on food

00:01:08.734 --> 00:01:11.557
and beverages. Then they
break down that 15%,

00:01:11.557 --> 00:01:14.470
so 13% is on food and then
they even break it out

00:01:14.470 --> 00:01:17.473
between 7% or a little
under 8% is food at home.

00:01:17.473 --> 00:01:19.805
Then they break it out
between cereals, meats,

00:01:19.805 --> 00:01:21.742
dairy and related products,
fruits and vegetables.

00:01:21.742 --> 00:01:24.121
The average, based on
the way the basket looks,

00:01:24.121 --> 00:01:25.954
it looks like they're
spending about the same amount

00:01:25.954 --> 00:01:28.203
on fruits and vegetables
as they're spending

00:01:28.203 --> 00:01:32.336
on alcohol. They're spending
significantly less on ...

00:01:32.336 --> 00:01:34.808
Well, I don't know if
that's a good trend right

00:01:34.808 --> 00:01:37.624
over there, but that's why
this is interesting to look at,

00:01:37.624 --> 00:01:39.946
because this is viewed as a typical basket

00:01:39.946 --> 00:01:43.100
of goods for your average urban consumer.

00:01:43.100 --> 00:01:45.035
You can see they're
spending a little bit more

00:01:45.035 --> 00:01:46.838
on meats, poultry and fish.
You can see the breakdown.

00:01:46.838 --> 00:01:48.369
Then you could keep breaking it down.

00:01:48.369 --> 00:01:51.210
They're spending 41%, the basket of goods,

00:01:51.210 --> 00:01:54.069
on housing, and they even
break that up in terms

00:01:54.069 --> 00:01:56.871
of some of it is your
primary, some of it is your

00:01:56.871 --> 00:02:00.074
general shelter, then there's stuff like

00:02:00.074 --> 00:02:03.619
fuels and utilities.
That's encapsulated in

00:02:03.619 --> 00:02:05.544
your housing, you're going
to have to heat your home

00:02:05.544 --> 00:02:08.866
and whatever else. Then
your furniture, 4% or 4.4%

00:02:08.866 --> 00:02:11.009
is spent on furniture.
We could keep going down.

00:02:11.009 --> 00:02:12.334
This is pretty interesting to look at.

00:02:12.334 --> 00:02:15.074
The basket of goods, so this
is viewed as the typical

00:02:15.074 --> 00:02:19.030
urban consumer spending 3.6% on apparel,

00:02:19.030 --> 00:02:24.412
17% on transportation,
6.6% on medical care,

00:02:24.412 --> 00:02:26.545
and almost a similar amount on recreation.

00:02:26.545 --> 00:02:28.801
A similar amount on
education. They keep breaking

00:02:28.801 --> 00:02:32.473
it down into all of the
different categories.

00:02:32.473 --> 00:02:35.297
A little under 1% on tobacco
and smoking products.

00:02:35.297 --> 00:02:37.537
You might say,"Wait,
most people these days

00:02:37.537 --> 00:02:39.409
"in the U.S. don't smoke,
but the ones who do

00:02:39.409 --> 00:02:41.483
"spend way more than
this." This is an average

00:02:41.483 --> 00:02:45.666
of all of the people in the United States.

00:02:45.666 --> 00:02:48.882
For example, if one
out of ten people spent

00:02:48.882 --> 00:02:51.402
10% of their income on
tobacco and the other

00:02:51.402 --> 00:02:52.997
people don't smoke at
all, then you might get

00:02:52.997 --> 00:02:57.070
on average, the average
basket of goods, is about 1%.

00:02:57.070 --> 00:03:00.135
They keep breaking it
down into other things.

00:03:00.135 --> 00:03:02.893
All of these weightings
combined, they will

00:03:02.893 --> 00:03:08.721
add to 100. That is the
entire basket of goods.

00:03:08.721 --> 00:03:11.952
Now what they do, this is their weighting.

00:03:11.952 --> 00:03:13.662
This column right over
here essentially gives

00:03:13.662 --> 00:03:16.217
us the weighting as of December 2010.

00:03:16.217 --> 00:03:19.215
That's going to change as
people's habits change,

00:03:19.215 --> 00:03:21.332
or as new goods and services
emerge on the market,

00:03:21.332 --> 00:03:23.983
or frankly even as prices
change, that will change.

00:03:23.983 --> 00:03:25.794
But you have to some weighting in which

00:03:25.794 --> 00:03:28.456
to take a weight of the price changes,

00:03:28.456 --> 00:03:31.928
or to weight the average percent changes.

00:03:31.928 --> 00:03:35.987
They told us that unless otherwise noted,

00:03:35.987 --> 00:03:37.937
our base year is going to be 100.

00:03:37.937 --> 00:03:40.653
Relative to that base
year, they then give us

00:03:40.653 --> 00:03:42.930
the prices, the price indices for each

00:03:42.930 --> 00:03:45.871
of these buckets in
November 2011, and then

00:03:45.871 --> 00:03:48.521
December 2011. Then
they're going to actually

00:03:48.521 --> 00:03:51.270
figure out the unadjusted percent change

00:03:51.270 --> 00:03:55.052
to December 2011 from.
This is year over year,

00:03:55.052 --> 00:03:57.192
from December of the
previous year, and this

00:03:57.192 --> 00:03:59.723
is from the previous month. You can see

00:03:59.723 --> 00:04:01.388
the change from the previous year is much

00:04:01.388 --> 00:04:04.119
larger than the change
from the previous month.

00:04:04.119 --> 00:04:06.457
One way to look at this, this is saying

00:04:06.457 --> 00:04:13.720
in November 2011, food
and beverages, on average,

00:04:13.720 --> 00:04:17.339
were about 2.3 times more
expensive than they were

00:04:17.339 --> 00:04:23.097
between 1982 and 1984.
December 2012, sorry,

00:04:23.097 --> 00:04:27.787
December 2011, they were about 2.31 times

00:04:27.787 --> 00:04:32.020
more expensive than
they were in 1982-1984.

00:04:32.020 --> 00:04:33.910
Just as an extra kind of data point, they

00:04:33.910 --> 00:04:35.719
actually give us this
one little line here.

00:04:35.719 --> 00:04:40.468
If we set 1967 as the
base year, then all items,

00:04:40.468 --> 00:04:45.385
if we use the default
base year of 1982-1984,

00:04:45.385 --> 00:04:49.186
in November 2011 all items
were about 2.26 times

00:04:49.186 --> 00:04:52.316
as expensive as they were in 1982-1984.

00:04:52.316 --> 00:04:54.455
But if we use 1967 as our base year,

00:04:54.455 --> 00:04:59.123
now it's 6.77 times as expensive.

00:04:59.123 --> 00:05:01.126
Remember, the base year is equal to 100.

00:05:01.126 --> 00:05:04.455
This is 6.77 times as
expensive as they were

00:05:04.455 --> 00:05:09.605
in 1967. You could go
down all of the categories

00:05:09.605 --> 00:05:11.616
to essentially see these are all relative

00:05:11.616 --> 00:05:14.651
to 1982-1984, so you
can see how much things

00:05:14.651 --> 00:05:16.352
have gotten more expensive.
It's interesting.

00:05:16.352 --> 00:05:20.016
Things like furniture
have not gotten that much

00:05:20.016 --> 00:05:22.515
more expensive relative to the early '80s.

00:05:22.515 --> 00:05:24.266
In fact, there are some
categories that have even

00:05:24.266 --> 00:05:28.123
gotten cheaper. For example,
new and used motor vehicles.

00:05:28.123 --> 00:05:31.253
It hasn't changed much at
all since, based on at least

00:05:31.253 --> 00:05:33.732
this weighting, and they
do all these adjustments

00:05:33.732 --> 00:05:35.509
based on the quality of
the car. You might say,

00:05:35.509 --> 00:05:37.853
"Wait, I'm spending more on
my car than I did in 1982,"

00:05:37.853 --> 00:05:39.733
but they're making
adjustments based on your car

00:05:39.733 --> 00:05:41.263
being that much better and all of that.

00:05:41.263 --> 00:05:44.433
It's not exactly an apples
to apples comparison.

00:05:44.433 --> 00:05:47.268
But you could see that
based on those adjustments,

00:05:47.268 --> 00:05:48.682
it doesn't look like it's changed much.

00:05:48.682 --> 00:05:50.188
Things like medical care have gotten a lot

00:05:50.188 --> 00:05:53.583
more expensive since the
early '80s; four times

00:05:53.583 --> 00:05:56.052
as expensive, you see right over there.

00:05:56.052 --> 00:05:58.399
Video and audio has gotten cheaper.

00:05:58.399 --> 00:06:01.058
Recreation in general
has not gotten that much

00:06:01.058 --> 00:06:03.350
more expensive.
Information and information

00:06:03.350 --> 00:06:06.517
processing has gotten cheaper. Telephone

00:06:06.517 --> 00:06:08.625
service has not gotten
that much more expensive.

00:06:08.625 --> 00:06:10.186
Communication has gotten cheaper.

00:06:10.186 --> 00:06:12.321
I don't know if you were
around in the early '80s,

00:06:12.321 --> 00:06:14.718
but actually the cost to
call someone long distance

00:06:14.718 --> 00:06:17.682
has gone down dramatically.
Even, see, personal

00:06:17.682 --> 00:06:19.982
computers, they have gotten cheaper.

00:06:19.982 --> 00:06:22.991
Once again, like the cars,
there's an adjustment

00:06:22.991 --> 00:06:25.915
for essentially ... They
don't do it directly

00:06:25.915 --> 00:06:27.604
because obviously computers have gotten

00:06:27.604 --> 00:06:28.920
orders of magnitude more powerful,

00:06:28.920 --> 00:06:32.607
but they have gotten on cheaper average,

00:06:32.607 --> 00:06:36.123
and they have gotten
much, much more powerful.

00:06:36.123 --> 00:06:38.061
This is fun to look at.
You can see, looking

00:06:38.061 --> 00:06:40.590
right over here. Tobacco
and smoking products

00:06:40.590 --> 00:06:42.857
have become dramatically more expensive.

00:06:42.857 --> 00:06:44.786
You have more and more
jurisdictions that are,

00:06:44.786 --> 00:06:47.452
for the most part, taxing it or whatever.

00:06:47.452 --> 00:06:49.119
They're making it harder
and harder to buy.

00:06:49.119 --> 00:06:50.794
You see things like ...
It's just fun to look at.

00:06:50.794 --> 00:06:53.187
Dig around here. Medical, oh, we already

00:06:53.187 --> 00:06:54.769
looked at this. This is
the overview services,

00:06:54.769 --> 00:06:57.826
durables. A lot of
these things are in this

00:06:57.826 --> 00:07:00.057
one to two to three price range.

00:07:00.057 --> 00:07:03.403
The stuff like medicine,
tobacco, much more expensive.

00:07:03.403 --> 00:07:05.689
Things like computers,
communications, much cheaper.

00:07:05.689 --> 00:07:08.599
Then as we mentioned before, this measures

00:07:08.599 --> 00:07:12.672
the percent change to December
2011 from the previous year.

00:07:12.672 --> 00:07:14.387
That's why it gives you
a year over year number.

00:07:14.387 --> 00:07:17.101
This is from the previous month.

00:07:17.101 --> 00:07:19.267
These are seasonally adjusted changes.

00:07:19.267 --> 00:07:21.017
If we have time in another
video, I'll talk about

00:07:21.017 --> 00:07:23.385
how you can calculate, or I'll do a simple

00:07:23.385 --> 00:07:26.867
example, of seasonally adjusting things.