-
This right here is a picture of René Descartes
-
Once again one of the great minds,
-
in both math and philosophy.
-
And i think you'll be seeing bit of a little trend here
-
that the great philosphers were also great mathematicians
-
and vice versa
-
and he was somewhat of a contemporary of Galileo
-
he was 32 years younger.
-
although he died shortly after Galileo died.
-
This guy died at a much younger age,
-
Galileo was well into his 70's
-
Descartes died at what, this is only at 54 years old.
-
And he is probably most known in popular culture,
-
for this quote right over here.
-
a very philosophical quote.
-
"I think therefore I am"
-
but i also wanted to throw in,
-
and this isn't that related to algebra,
-
but i just thought it was a really neat quote.
-
Probably his least famous quote.
-
This one right over here.
-
And i like it just because it's very practical
-
and it makes you realize that these great minds
-
these pillars of philosophy and mathematics
-
that at the end of the day,
-
they were just human beings.
-
and he said, "You just keep pushing.
-
You just keep pushing.
-
I made every mistake that could be made.
-
But I just kept pushing."
-
Which i think is very very good life advice.
-
Now he did many things
-
in philosophy and mathematics
-
but the reason why I'm including here
-
as we build foundations of algebra
-
is that he is the individual
-
most responsible for a very strong connection
-
between algebra and geometry.
-
so on the left over here
-
you have the world of algebra.
-
We've discussed it a little bit.
-
You have equations that deal with symbols
-
and these symbols are essentially
-
they can take on values
-
so you can have something like
-
y = 2x - 1
-
this gives us a relationship
-
between whatever x is
-
and whatever y is.
-
and we can even set up a table here.
-
and pick values for x
-
and see what the values of y would be.
-
I can just pick random values for x
-
and then figure out what y is.
-
but i'll pick relatively straightforward values
-
and so that the maths doesn't get too complicated.
-
so for example,
-
if x is -2
-
then y is going to be 2 x -2 - 1
-
2 x -2 - 1
-
which is -4 - 1
-
which is -5
-
if x is -1
-
then y is going to be 2 x -1 - 1
-
which is equal to
-
this is -2 - 1 which is -3
-
if x = 0
-
then y is going to be 2 x 0 - 1
-
2 x 0 is 0 - 1 is just -1
-
i'll do a couple more.
-
if x is 1
-
and i could've picked any values here
-
I could've said what happens
-
if x is the negative square root of 2
-
or what happens if x is -5 halves
-
or positive six seventh.
-
but i'm just picking these numbers
-
because it makes the maths a lot easier
-
when i try to figure out what y is going to be.
-
but when x is 1
-
y is going to be 2(1) - 1
-
2 x 1 is 2 - 1 is 1
-
and i'll do one more.
-
in the colour I have not used yet.
-
let's see this purple.
-
if x is 2
-
then y is going to be
-
2(2) - 1 (now that x is 2)
-
so that is 4 - 1, is equal to 3
-
so fair enough,
-
I just kind of sampled this relationship.
-
But I said okay this describes a general relationship
-
between a variable y and a variable x
-
and then I just made a little more concrete.
-
I said ok well then
-
if x is one of these variables.
-
for each of these values of x,
-
what would be the corresponding value of y?
-
and what Descartes realized is
-
that you could visualize this.
-
what you could visualize is individual points.
-
But that could also help you in general
-
to visualize this relationship.
-
so what he essentially did is
-
he bridged the worlds of this kind of very abstract symbolic algebra.
-
and that and geometry which was concerned
-
with shapes and sizes and angles.
-
so over here you have the world of geometry.
-
and obviously there are people in history
-
maybe many people who history may have forgotten
-
who might have dabbled in this.
-
But before Descartes is generally viewed.
-
that geometry was euclidean geometry.
-
and that's essentially the geometry
-
that you studied in geometry class
-
in 8th or 9th or 10th grade.
-
in a traditional high school curriculum.
-
and that's the geometry of studying
-
the relationships between triangles, and their angles.
-
and the relationships between circles.
-
and you have radii and then you have triangles
-
inscribed in circles and all the rest
-
and we'll go into some depth
-
in that in the geometry playlist.
-
But Descarte says, 'well i think i can represent this visually
-
the same way Euclid was studying these triangles and these circles'
-
he said 'why don't I ?'
-
if we view a piece of paper.
-
if we think about a two-dimensional plane.
-
you could view a piece of paper
-
as kind of a section of a two-dimensional plane.
-
we call it two-dimensions
-
because there's two directions that you can go in.
-
there's the up down direction,
-
that's one direction.
-
so let me draw that, i'll do it in blue.
-
because we're trying to visualize things
-
so i'll do it the geometry colour.
-
so you have the up down direction
-
and you have the left right direction.
-
that's why it's called a two-dimensional plane.
-
if we're dealing with three-dimensions.
-
you have an in out dimension.
-
and it's very easy to do two-dimensions on the screen
-
because the screen is two-dimensional.
-
and he says 'Well, you know
-
there are two variables here and they have this relationship.
-
But why don't I associate each of these variables
-
with one of these dimensions over here?'
-
and by convention let's make the y variable
-
which is really the dependant variable,
-
The way we did it,
-
it depends on what x is.
-
So let's put that on the vertical axis.
-
and let's put our independent variable,
-
the one where I just randomly picked values for it
-
to see what y would become,
-
let's put that on the horizontal axis.
-
and it actually was Descartes
-
who came up with a convention of using x's and y's
-
and we'll see later z's in algebra, so extensively
-
as unknown variables with the variables that you're manipulating.
-
But he says 'Well if we think about it this way
-
if we number these dimensions'
-
so let's say that in the x direction
-
let's make this right over here -3
-
let's make this -2
-
this is -1
-
this is 0
-
i'm just numbering the x direction
-
the left right direction.
-
now this is positive 1
-
this is positive 2
-
and this is positive 3.
-
and we could do the same in the y direction
-
so let's see we go, so this could be
-
say this is -5, -4 , -3
-
actually let me do it a bit neater than that
-
let me clean this up a little bit.
-
let me erase this and extend this down a little bit
-
so I can go all the way down to -5
-
without making it look too messy.
-
so let's go all the way down here.
-
and so we can number it
-
this is 1, this is 2, this is 3,
-
and then this could be -1
-
-2 and these are all just conventions
-
it could've been labelled the other way.
-
we could've decided to put the x there
-
and the y there
-
and make this the positive direction,
-
make this the negative direction.
-
but this is just a convention that people adopted
-
starting with Descartes.
-
-2, -3, -4 and -5
-
and he says 'Well anything i can associate
-
I can associate each of these pairs of values with
-
a point in two-dimensions.
-
I can take the x co-ordinate, I can take the x value
-
right over here and I say 'Ok that's -2
-
that would be right over there along the left right direction,
-
i'm going to the left because it's negative.'
-
and that's associated with -5 in the vertical direction.
-
so I say the y value's -5
-
and so if I go 2 to the left and 5 down.
-
I get to this point right over there.
-
so he says 'These two values -2 and -5
-
I can associate it with this point
-
in this plane right over here, in this two-dimensional plane.
-
so I'll say: That point has the co-ordinates,
-
tells me where do I find that point (-2,-5).
-
and these coordinates are called 'cartesian coordinates'
-
named for René Descartes
-
because he is the guy who came up with these.
-
He's associating all of a sudden these relationships
-
with points on a co-ordinate plane.
-
and then he says 'well ok, lets do another one'
-
there's this other relationship,
-
when x is equal to -1, y = -3
-
so x is -1, y is -3.
-
that's that point right over there.
-
and the convention is once again.
-
'When you list the co-ordinates,
-
you list the x co-ordinate, then the y co-ordinate
-
and that's just what people decided to do.
-
-1, -3 that would be that point right over there
-
and then you have the point when x is 0, y is -1
-
when x is 0 right over here,
-
which means I don't go the left or the right.
-
y is -1, which means I go 1 down.
-
so that's that point right over there. (0,-1)
-
right over there
-
and I could keep doing this.
-
when x is 1, y is 1
-
when x is 2, y is 3
-
actually let me do that in the same purple colour
-
when x is 2, y is 3
-
2,3 and then this one right over here in orange was 1,1
-
and this is neat by itself,
-
I essentially just sampled possible x's.
-
but what he realized is
-
not only do you sample these possible x's,
-
but it you kept sampling x's,
-
if you tried sampling all of the x's in between,
-
you'd actually end up plotting out a line.
-
So if you were to do every possible x
-
you would end up getting a line
-
that looks something like that... right over there.
-
and any... any relation, if you pick any x
-
and find any y it really represents a point on this line,
-
or another way to think about it
-
any point on this line represents
-
a solution to this equation right over here.
-
so if you have this point right over here.
-
which looks like about x is 1 and a half.
-
y is 2. So let me write that
-
1.5,2
-
that is a solution to this equation.
-
when x is 1.5. 2 x 1.5 is 3 - 1 is 2
-
that is right over there.
-
so all of a sudden he was able to bridge
-
this gap or this relationship between algebra and geometry.
-
we can now visualize all of the x and y pairs
-
that satisfy this equation right over here.
-
and so he is responsible for making this bridge
-
and that's why that co-ordinates
-
that we use to specify these points are called 'cartesian coordinates'
-
and as we'll see and first type of equations
-
we will study our equations of this form over here
-
and in a traditional algebra curriculum.
-
they're called linear equations...
-
linear equations.
-
and you might be saying: well you know, this is an equation,
-
I'll see that this is equal to that on its own.
-
but what's so linear about them?
-
what makes them look like a line?
-
to realize why they're linear,
-
you have to make this jump René Descartes made.
-
because if you were to plot this,
-
using cartesian coordinates.
-
on a Euclidean plane. You will get a line.
-
and in the future you'll see that
-
there's other types of equations where you won't get a line.
-
you get a curve, or something kind of crazy or funky.
