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This right here is a picture of René Descartes

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Once again one of the great minds,

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in both math and philosophy.

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And i think you'll be seeing bit of a little trend here

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that the great philosphers were also great mathematicians

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and vice versa

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and he was somewhat of a contemporary of Galileo

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he was 32 years younger.

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although he died shortly after Galileo died.

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This guy died at a much younger age,

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Galileo was well into his 70's

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Descartes died at what, this is only at 54 years old.

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And he is probably most known in popular culture,

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for this quote right over here.

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a very philosophical quote.

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"I think therefore I am"

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but i also wanted to throw in,

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and this isn't that related to algebra,

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but i just thought it was a really neat quote.

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Probably his least famous quote.

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This one right over here.

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And i like it just because it's very practical

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and it makes you realize that these great minds

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these pillars of philosophy and mathematics

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that at the end of the day,

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they were just human beings.

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and he said, "You just keep pushing.

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You just keep pushing.

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I made every mistake that could be made.

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But I just kept pushing."

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Which i think is very very good life advice.

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Now he did many things

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in philosophy and mathematics

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but the reason why I'm including here

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as we build foundations of algebra

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is that he is the individual

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most responsible for a very strong connection

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between algebra and geometry.

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so on the left over here

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you have the world of algebra.

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We've discussed it a little bit.

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You have equations that deal with symbols

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and these symbols are essentially

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they can take on values

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so you can have something like

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y = 2x - 1

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this gives us a relationship

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between whatever x is

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and whatever y is.

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and we can even set up a table here.

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and pick values for x

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and see what the values of y would be.

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I can just pick random values for x

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and then figure out what y is.

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but i'll pick relatively straightforward values

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and so that the maths doesn't get too complicated.

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so for example,

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if x is -2

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then y is going to be 2 x -2 - 1

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2 x -2 - 1

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which is -4 - 1

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which is -5

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if x is -1

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then y is going to be 2 x -1 - 1

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which is equal to

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this is -2 - 1 which is -3

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if x = 0

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then y is going to be 2 x 0 - 1

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2 x 0 is 0 - 1 is just -1

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i'll do a couple more.

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if x is 1

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and i could've picked any values here

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I could've said what happens

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if x is the negative square root of 2

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or what happens if x is -5 halves

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or positive six seventh.

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but i'm just picking these numbers

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because it makes the maths a lot easier

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when i try to figure out what y is going to be.

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but when x is 1

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y is going to be 2(1) - 1

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2 x 1 is 2 - 1 is 1

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and i'll do one more.

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in the colour I have not used yet.

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let's see this purple.

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if x is 2

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then y is going to be

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2(2) - 1 (now that x is 2)

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so that is 4 - 1, is equal to 3

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so fair enough,

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I just kind of sampled this relationship.

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But I said okay this describes a general relationship

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between a variable y and a variable x

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and then I just made a little more concrete.

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I said ok well then

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if x is one of these variables.

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for each of these values of x,

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what would be the corresponding value of y?

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and what Descartes realized is

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that you could visualize this.

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what you could visualize is individual points.

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But that could also help you in general

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to visualize this relationship.

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so what he essentially did is

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he bridged the worlds of this kind of very abstract symbolic algebra.

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and that and geometry which was concerned

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with shapes and sizes and angles.

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so over here you have the world of geometry.

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and obviously there are people in history

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maybe many people who history may have forgotten

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who might have dabbled in this.

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But before Descartes is generally viewed.

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that geometry was euclidean geometry.

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and that's essentially the geometry

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that you studied in geometry class

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in 8th or 9th or 10th grade.

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in a traditional high school curriculum.

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and that's the geometry of studying

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the relationships between triangles, and their angles.

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and the relationships between circles.

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and you have radii and then you have triangles

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inscribed in circles and all the rest

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and we'll go into some depth

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in that in the geometry playlist.

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But Descarte says, 'well i think i can represent this visually

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the same way Euclid was studying these triangles and these circles'

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he said 'why don't I ?'

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if we view a piece of paper.

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if we think about a two-dimensional plane.

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you could view a piece of paper

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as kind of a section of a two-dimensional plane.

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we call it two-dimensions

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because there's two directions that you can go in.

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there's the up down direction,

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that's one direction.

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so let me draw that, i'll do it in blue.

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because we're trying to visualize things

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so i'll do it the geometry colour.

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so you have the up down direction

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and you have the left right direction.

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that's why it's called a two-dimensional plane.

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if we're dealing with three-dimensions.

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you have an in out dimension.

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and it's very easy to do two-dimensions on the screen

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because the screen is two-dimensional.

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and he says 'Well, you know

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there are two variables here and they have this relationship.

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But why don't I associate each of these variables

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with one of these dimensions over here?'

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and by convention let's make the y variable

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which is really the dependant variable,

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The way we did it,

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it depends on what x is.

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So let's put that on the vertical axis.

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and let's put our independent variable,

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the one where I just randomly picked values for it

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to see what y would become,

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let's put that on the horizontal axis.

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and it actually was Descartes

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who came up with a convention of using x's and y's

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and we'll see later z's in algebra, so extensively

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as unknown variables with the variables that you're manipulating.

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But he says 'Well if we think about it this way

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if we number these dimensions'

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so let's say that in the x direction

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let's make this right over here -3

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let's make this -2

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this is -1

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this is 0

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i'm just numbering the x direction

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the left right direction.

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now this is positive 1

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this is positive 2

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and this is positive 3.

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and we could do the same in the y direction

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so let's see we go, so this could be

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say this is -5, -4 , -3

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actually let me do it a bit neater than that

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let me clean this up a little bit.

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let me erase this and extend this down a little bit

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so I can go all the way down to -5

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without making it look too messy.

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so let's go all the way down here.

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and so we can number it

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this is 1, this is 2, this is 3,

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and then this could be -1

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-2 and these are all just conventions

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it could've been labelled the other way.

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we could've decided to put the x there

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and the y there

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and make this the positive direction,

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make this the negative direction.

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but this is just a convention that people adopted

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starting with Descartes.

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-2, -3, -4 and -5

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and he says 'Well anything i can associate

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I can associate each of these pairs of values with

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a point in two-dimensions.

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I can take the x co-ordinate, I can take the x value

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right over here and I say 'Ok that's -2

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that would be right over there along the left right direction,

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i'm going to the left because it's negative.'

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and that's associated with -5 in the vertical direction.

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so I say the y value's -5

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and so if I go 2 to the left and 5 down.

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I get to this point right over there.

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so he says 'These two values -2 and -5

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I can associate it with this point

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in this plane right over here, in this two-dimensional plane.

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so I'll say: That point has the co-ordinates,

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tells me where do I find that point (-2,-5).

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and these coordinates are called 'cartesian coordinates'

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named for René Descartes

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because he is the guy who came up with these.

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He's associating all of a sudden these relationships

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with points on a co-ordinate plane.

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and then he says 'well ok, lets do another one'

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there's this other relationship,

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when x is equal to -1, y = -3

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so x is -1, y is -3.

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that's that point right over there.

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and the convention is once again.

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'When you list the co-ordinates,

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you list the x co-ordinate, then the y co-ordinate

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and that's just what people decided to do.

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-1, -3 that would be that point right over there

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and then you have the point when x is 0, y is -1

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when x is 0 right over here,

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which means I don't go the left or the right.

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y is -1, which means I go 1 down.

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so that's that point right over there. (0,-1)

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right over there

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and I could keep doing this.

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when x is 1, y is 1

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when x is 2, y is 3

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actually let me do that in the same purple colour

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when x is 2, y is 3

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2,3 and then this one right over here in orange was 1,1

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and this is neat by itself,

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I essentially just sampled possible x's.

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but what he realized is

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not only do you sample these possible x's,

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but it you kept sampling x's,

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if you tried sampling all of the x's in between,

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you'd actually end up plotting out a line.

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So if you were to do every possible x

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you would end up getting a line

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that looks something like that... right over there.

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and any... any relation, if you pick any x

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and find any y it really represents a point on this line,

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or another way to think about it

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any point on this line represents

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a solution to this equation right over here.

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so if you have this point right over here.

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which looks like about x is 1 and a half.

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y is 2. So let me write that

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1.5,2

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that is a solution to this equation.

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when x is 1.5. 2 x 1.5 is 3 - 1 is 2

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that is right over there.

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so all of a sudden he was able to bridge

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this gap or this relationship between algebra and geometry.

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we can now visualize all of the x and y pairs

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that satisfy this equation right over here.

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and so he is responsible for making this bridge

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and that's why that co-ordinates

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that we use to specify these points are called 'cartesian coordinates'

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and as we'll see and first type of equations

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we will study our equations of this form over here

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and in a traditional algebra curriculum.

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they're called linear equations...

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linear equations.

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and you might be saying: well you know, this is an equation,

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I'll see that this is equal to that on its own.

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but what's so linear about them?

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what makes them look like a line?

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to realize why they're linear,

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you have to make this jump René Descartes made.

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because if you were to plot this,

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using cartesian coordinates.

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on a Euclidean plane. You will get a line.

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and in the future you'll see that

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there's other types of equations where you won't get a line.

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you get a curve, or something kind of crazy or funky.