
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 twodimensional plane.

you could view a piece of paper

as kind of a section of a twodimensional plane.

we call it twodimensions

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 twodimensional plane.

if we're dealing with threedimensions.

you have an in out dimension.

and it's very easy to do twodimensions on the screen

because the screen is twodimensional.

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 twodimensions.

I can take the x coordinate, 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 twodimensional plane.

so I'll say: That point has the coordinates,

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 coordinate 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 coordinates,

you list the x coordinate, then the y coordinate

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 coordinates

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.