[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.06,0:00:03.64,Default,,0000,0000,0000,,This right here is a picture of René Descartes
Dialogue: 0,0:00:03.64,0:00:05.70,Default,,0000,0000,0000,,Once again one of the great minds,
Dialogue: 0,0:00:05.70,0:00:07.55,Default,,0000,0000,0000,,in both math and philosophy.
Dialogue: 0,0:00:07.55,0:00:09.92,Default,,0000,0000,0000,,And i think you'll be seeing bit of a little trend here
Dialogue: 0,0:00:09.92,0:00:13.19,Default,,0000,0000,0000,,that the great philosphers were also great mathematicians
Dialogue: 0,0:00:13.19,0:00:15.20,Default,,0000,0000,0000,,and vice versa
Dialogue: 0,0:00:15.20,0:00:17.02,Default,,0000,0000,0000,,and he was somewhat of a contemporary of Galileo
Dialogue: 0,0:00:17.02,0:00:18.73,Default,,0000,0000,0000,,he was 32 years younger.
Dialogue: 0,0:00:18.73,0:00:21.71,Default,,0000,0000,0000,,although he died shortly after Galileo died.
Dialogue: 0,0:00:21.71,0:00:23.47,Default,,0000,0000,0000,,This guy died at a much younger age,
Dialogue: 0,0:00:23.47,0:00:25.40,Default,,0000,0000,0000,,Galileo was well into his 70's
Dialogue: 0,0:00:25.40,0:00:28.07,Default,,0000,0000,0000,,Descartes died at what, this is only at 54 years old.
Dialogue: 0,0:00:28.07,0:00:30.87,Default,,0000,0000,0000,,And he is probably most known in popular culture,
Dialogue: 0,0:00:30.87,0:00:32.73,Default,,0000,0000,0000,,for this quote right over here.
Dialogue: 0,0:00:32.73,0:00:33.80,Default,,0000,0000,0000,,a very philosophical quote.
Dialogue: 0,0:00:33.80,0:00:35.87,Default,,0000,0000,0000,,"I think therefore I am"
Dialogue: 0,0:00:35.87,0:00:37.47,Default,,0000,0000,0000,,but i also wanted to throw in,
Dialogue: 0,0:00:37.47,0:00:38.87,Default,,0000,0000,0000,,and this isn't that related to algebra,
Dialogue: 0,0:00:38.87,0:00:40.73,Default,,0000,0000,0000,,but i just thought it was a really neat quote.
Dialogue: 0,0:00:40.73,0:00:42.80,Default,,0000,0000,0000,,Probably his least famous quote.
Dialogue: 0,0:00:42.80,0:00:44.47,Default,,0000,0000,0000,,This one right over here.
Dialogue: 0,0:00:44.47,0:00:46.80,Default,,0000,0000,0000,,And i like it just because it's very practical
Dialogue: 0,0:00:46.80,0:00:48.85,Default,,0000,0000,0000,,and it makes you realize that these great minds
Dialogue: 0,0:00:48.85,0:00:51.11,Default,,0000,0000,0000,,these pillars of philosophy and mathematics
Dialogue: 0,0:00:51.11,0:00:52.28,Default,,0000,0000,0000,,that at the end of the day,
Dialogue: 0,0:00:52.28,0:00:54.47,Default,,0000,0000,0000,,they were just human beings.
Dialogue: 0,0:00:54.47,0:00:56.50,Default,,0000,0000,0000,,and he said, "You just keep pushing.
Dialogue: 0,0:00:56.50,0:00:58.13,Default,,0000,0000,0000,,You just keep pushing.
Dialogue: 0,0:00:58.13,0:01:00.02,Default,,0000,0000,0000,,I made every mistake that could be made.
Dialogue: 0,0:01:00.02,0:01:02.03,Default,,0000,0000,0000,,But I just kept pushing."
Dialogue: 0,0:01:02.03,0:01:05.27,Default,,0000,0000,0000,,Which i think is very very good life advice.
Dialogue: 0,0:01:05.27,0:01:07.73,Default,,0000,0000,0000,,Now he did many things
Dialogue: 0,0:01:07.73,0:01:09.08,Default,,0000,0000,0000,,in philosophy and mathematics
Dialogue: 0,0:01:09.08,0:01:11.06,Default,,0000,0000,0000,,but the reason why I'm including here
Dialogue: 0,0:01:11.06,0:01:12.93,Default,,0000,0000,0000,,as we build foundations of algebra
Dialogue: 0,0:01:12.93,0:01:15.60,Default,,0000,0000,0000,,is that he is the individual
Dialogue: 0,0:01:15.60,0:01:18.80,Default,,0000,0000,0000,,most responsible for a very strong connection
Dialogue: 0,0:01:18.80,0:01:21.42,Default,,0000,0000,0000,,between algebra and geometry.
Dialogue: 0,0:01:21.42,0:01:22.90,Default,,0000,0000,0000,,so on the left over here
Dialogue: 0,0:01:22.90,0:01:24.75,Default,,0000,0000,0000,,you have the world of algebra.
Dialogue: 0,0:01:24.75,0:01:26.42,Default,,0000,0000,0000,,We've discussed it a little bit.
Dialogue: 0,0:01:26.42,0:01:28.48,Default,,0000,0000,0000,,You have equations that deal with symbols
Dialogue: 0,0:01:28.48,0:01:30.24,Default,,0000,0000,0000,,and these symbols are essentially
Dialogue: 0,0:01:30.24,0:01:31.93,Default,,0000,0000,0000,,they can take on values
Dialogue: 0,0:01:31.93,0:01:32.80,Default,,0000,0000,0000,,so you can have something like
Dialogue: 0,0:01:32.80,0:01:37.68,Default,,0000,0000,0000,,y = 2x - 1
Dialogue: 0,0:01:37.68,0:01:39.27,Default,,0000,0000,0000,,this gives us a relationship
Dialogue: 0,0:01:39.27,0:01:40.73,Default,,0000,0000,0000,,between whatever x is
Dialogue: 0,0:01:40.73,0:01:42.13,Default,,0000,0000,0000,,and whatever y is.
Dialogue: 0,0:01:42.13,0:01:44.33,Default,,0000,0000,0000,,and we can even set up a table here.
Dialogue: 0,0:01:44.33,0:01:46.73,Default,,0000,0000,0000,,and pick values for x
Dialogue: 0,0:01:46.73,0:01:48.29,Default,,0000,0000,0000,,and see what the values of y would be.
Dialogue: 0,0:01:48.29,0:01:51.65,Default,,0000,0000,0000,,I can just pick random values for x
Dialogue: 0,0:01:51.65,0:01:53.13,Default,,0000,0000,0000,,and then figure out what y is.
Dialogue: 0,0:01:53.13,0:01:55.00,Default,,0000,0000,0000,,but i'll pick relatively straightforward values
Dialogue: 0,0:01:55.00,0:01:57.66,Default,,0000,0000,0000,,and so that the maths doesn't get too complicated.
Dialogue: 0,0:01:57.66,0:01:59.25,Default,,0000,0000,0000,,so for example,
Dialogue: 0,0:01:59.25,0:02:00.53,Default,,0000,0000,0000,,if x is -2
Dialogue: 0,0:02:00.53,0:02:03.60,Default,,0000,0000,0000,,then y is going to be 2 x -2 - 1
Dialogue: 0,0:02:03.60,0:02:06.51,Default,,0000,0000,0000,,2 x -2 - 1
Dialogue: 0,0:02:06.51,0:02:10.11,Default,,0000,0000,0000,,which is -4 - 1
Dialogue: 0,0:02:10.11,0:02:12.27,Default,,0000,0000,0000,,which is -5
Dialogue: 0,0:02:12.27,0:02:14.78,Default,,0000,0000,0000,,if x is -1
Dialogue: 0,0:02:14.78,0:02:20.45,Default,,0000,0000,0000,,then y is going to be 2 x -1 - 1
Dialogue: 0,0:02:20.45,0:02:21.73,Default,,0000,0000,0000,,which is equal to
Dialogue: 0,0:02:21.73,0:02:24.55,Default,,0000,0000,0000,,this is -2 - 1 which is -3
Dialogue: 0,0:02:24.55,0:02:28.72,Default,,0000,0000,0000,,if x = 0
Dialogue: 0,0:02:28.72,0:02:32.59,Default,,0000,0000,0000,,then y is going to be 2 x 0 - 1
Dialogue: 0,0:02:32.60,0:02:35.67,Default,,0000,0000,0000,,2 x 0 is 0 - 1 is just -1
Dialogue: 0,0:02:35.67,0:02:37.33,Default,,0000,0000,0000,,i'll do a couple more.
Dialogue: 0,0:02:37.33,0:02:38.28,Default,,0000,0000,0000,,if x is 1
Dialogue: 0,0:02:38.28,0:02:39.42,Default,,0000,0000,0000,,and i could've picked any values here
Dialogue: 0,0:02:39.42,0:02:40.35,Default,,0000,0000,0000,,I could've said what happens
Dialogue: 0,0:02:40.35,0:02:42.00,Default,,0000,0000,0000,,if x is the negative square root of 2
Dialogue: 0,0:02:42.00,0:02:45.07,Default,,0000,0000,0000,,or what happens if x is -5 halves
Dialogue: 0,0:02:45.07,0:02:47.87,Default,,0000,0000,0000,,or positive six seventh.
Dialogue: 0,0:02:47.87,0:02:49.00,Default,,0000,0000,0000,,but i'm just picking these numbers
Dialogue: 0,0:02:49.00,0:02:50.60,Default,,0000,0000,0000,,because it makes the maths a lot easier
Dialogue: 0,0:02:50.60,0:02:52.60,Default,,0000,0000,0000,,when i try to figure out what y is going to be.
Dialogue: 0,0:02:52.60,0:02:54.13,Default,,0000,0000,0000,,but when x is 1
Dialogue: 0,0:02:54.13,0:02:57.34,Default,,0000,0000,0000,,y is going to be 2(1) - 1
Dialogue: 0,0:02:57.34,0:02:59.73,Default,,0000,0000,0000,,2 x 1 is 2 - 1 is 1
Dialogue: 0,0:02:59.73,0:03:03.05,Default,,0000,0000,0000,,and i'll do one more.
Dialogue: 0,0:03:03.05,0:03:05.13,Default,,0000,0000,0000,,in the colour I have not used yet.
Dialogue: 0,0:03:05.13,0:03:06.67,Default,,0000,0000,0000,,let's see this purple.
Dialogue: 0,0:03:06.67,0:03:08.04,Default,,0000,0000,0000,,if x is 2
Dialogue: 0,0:03:08.04,0:03:09.33,Default,,0000,0000,0000,,then y is going to be
Dialogue: 0,0:03:09.33,0:03:14.00,Default,,0000,0000,0000,,2(2) - 1 (now that x is 2)
Dialogue: 0,0:03:14.00,0:03:16.62,Default,,0000,0000,0000,,so that is 4 - 1, is equal to 3
Dialogue: 0,0:03:16.62,0:03:17.80,Default,,0000,0000,0000,,so fair enough,
Dialogue: 0,0:03:17.80,0:03:19.55,Default,,0000,0000,0000,,I just kind of sampled this relationship.
Dialogue: 0,0:03:19.55,0:03:22.53,Default,,0000,0000,0000,,But I said okay this describes a general relationship
Dialogue: 0,0:03:22.53,0:03:25.20,Default,,0000,0000,0000,,between a variable y and a variable x
Dialogue: 0,0:03:25.20,0:03:26.91,Default,,0000,0000,0000,,and then I just made a little more concrete.
Dialogue: 0,0:03:26.91,0:03:28.00,Default,,0000,0000,0000,,I said ok well then
Dialogue: 0,0:03:28.00,0:03:29.88,Default,,0000,0000,0000,,if x is one of these variables.
Dialogue: 0,0:03:29.88,0:03:31.20,Default,,0000,0000,0000,,for each of these values of x,
Dialogue: 0,0:03:31.20,0:03:33.80,Default,,0000,0000,0000,,what would be the corresponding value of y?
Dialogue: 0,0:03:33.80,0:03:35.70,Default,,0000,0000,0000,,and what Descartes realized is
Dialogue: 0,0:03:35.72,0:03:37.47,Default,,0000,0000,0000,,that you could visualize this.
Dialogue: 0,0:03:37.47,0:03:40.40,Default,,0000,0000,0000,,what you could visualize is individual points.
Dialogue: 0,0:03:40.40,0:03:42.67,Default,,0000,0000,0000,,But that could also help you in general
Dialogue: 0,0:03:42.67,0:03:45.80,Default,,0000,0000,0000,,to visualize this relationship.
Dialogue: 0,0:03:45.80,0:03:47.33,Default,,0000,0000,0000,,so what he essentially did is
Dialogue: 0,0:03:47.33,0:03:52.33,Default,,0000,0000,0000,,he bridged the worlds of this kind of very abstract symbolic algebra.
Dialogue: 0,0:03:52.33,0:03:55.20,Default,,0000,0000,0000,,and that and geometry which was concerned
Dialogue: 0,0:03:55.20,0:03:57.60,Default,,0000,0000,0000,,with shapes and sizes and angles.
Dialogue: 0,0:03:57.60,0:04:02.93,Default,,0000,0000,0000,,so over here you have the world of geometry.
Dialogue: 0,0:04:02.93,0:04:04.89,Default,,0000,0000,0000,,and obviously there are people in history
Dialogue: 0,0:04:04.89,0:04:07.07,Default,,0000,0000,0000,,maybe many people who history may have forgotten
Dialogue: 0,0:04:07.07,0:04:09.07,Default,,0000,0000,0000,,who might have dabbled in this.
Dialogue: 0,0:04:09.07,0:04:12.47,Default,,0000,0000,0000,,But before Descartes is generally viewed.
Dialogue: 0,0:04:12.47,0:04:14.80,Default,,0000,0000,0000,,that geometry was euclidean geometry.
Dialogue: 0,0:04:14.80,0:04:16.13,Default,,0000,0000,0000,,and that's essentially the geometry
Dialogue: 0,0:04:16.13,0:04:17.53,Default,,0000,0000,0000,,that you studied in geometry class
Dialogue: 0,0:04:17.53,0:04:20.33,Default,,0000,0000,0000,,in 8th or 9th or 10th grade.
Dialogue: 0,0:04:20.33,0:04:22.53,Default,,0000,0000,0000,,in a traditional high school curriculum.
Dialogue: 0,0:04:22.53,0:04:24.20,Default,,0000,0000,0000,,and that's the geometry of studying
Dialogue: 0,0:04:24.20,0:04:28.55,Default,,0000,0000,0000,,the relationships between triangles, and their angles.
Dialogue: 0,0:04:28.55,0:04:30.67,Default,,0000,0000,0000,,and the relationships between circles.
Dialogue: 0,0:04:30.67,0:04:33.89,Default,,0000,0000,0000,,and you have radii and then you have triangles
Dialogue: 0,0:04:33.89,0:04:36.20,Default,,0000,0000,0000,,inscribed in circles and all the rest
Dialogue: 0,0:04:36.20,0:04:37.19,Default,,0000,0000,0000,,and we'll go into some depth
Dialogue: 0,0:04:37.19,0:04:39.67,Default,,0000,0000,0000,,in that in the geometry playlist.
Dialogue: 0,0:04:39.67,0:04:42.94,Default,,0000,0000,0000,,But Descarte says, 'well i think i can represent this visually
Dialogue: 0,0:04:42.94,0:04:46.58,Default,,0000,0000,0000,,the same way Euclid was studying these triangles and these circles'
Dialogue: 0,0:04:46.58,0:04:48.30,Default,,0000,0000,0000,,he said 'why don't I ?'
Dialogue: 0,0:04:48.30,0:04:50.58,Default,,0000,0000,0000,,if we view a piece of paper.
Dialogue: 0,0:04:50.58,0:04:52.34,Default,,0000,0000,0000,,if we think about a two-dimensional plane.
Dialogue: 0,0:04:52.34,0:04:53.82,Default,,0000,0000,0000,,you could view a piece of paper
Dialogue: 0,0:04:53.82,0:04:55.92,Default,,0000,0000,0000,,as kind of a section of a two-dimensional plane.
Dialogue: 0,0:04:55.92,0:04:57.82,Default,,0000,0000,0000,,we call it two-dimensions
Dialogue: 0,0:04:57.82,0:04:59.58,Default,,0000,0000,0000,,because there's two directions that you can go in.
Dialogue: 0,0:04:59.58,0:05:01.26,Default,,0000,0000,0000,,there's the up down direction,
Dialogue: 0,0:05:01.26,0:05:02.51,Default,,0000,0000,0000,,that's one direction.
Dialogue: 0,0:05:02.51,0:05:04.82,Default,,0000,0000,0000,,so let me draw that, i'll do it in blue.
Dialogue: 0,0:05:04.84,0:05:06.67,Default,,0000,0000,0000,,because we're trying to visualize things
Dialogue: 0,0:05:06.67,0:05:08.38,Default,,0000,0000,0000,,so i'll do it the geometry colour.
Dialogue: 0,0:05:08.38,0:05:11.83,Default,,0000,0000,0000,,so you have the up down direction
Dialogue: 0,0:05:11.83,0:05:14.14,Default,,0000,0000,0000,,and you have the left right direction.
Dialogue: 0,0:05:14.14,0:05:16.72,Default,,0000,0000,0000,,that's why it's called a two-dimensional plane.
Dialogue: 0,0:05:16.72,0:05:18.16,Default,,0000,0000,0000,,if we're dealing with three-dimensions.
Dialogue: 0,0:05:18.16,0:05:21.34,Default,,0000,0000,0000,,you have an in out dimension.
Dialogue: 0,0:05:21.34,0:05:23.20,Default,,0000,0000,0000,,and it's very easy to do two-dimensions on the screen
Dialogue: 0,0:05:23.20,0:05:25.42,Default,,0000,0000,0000,,because the screen is two-dimensional.
Dialogue: 0,0:05:25.42,0:05:27.07,Default,,0000,0000,0000,,and he says 'Well, you know
Dialogue: 0,0:05:27.07,0:05:29.74,Default,,0000,0000,0000,,there are two variables here and they have this relationship.
Dialogue: 0,0:05:29.74,0:05:32.55,Default,,0000,0000,0000,,But why don't I associate each of these variables
Dialogue: 0,0:05:32.55,0:05:34.60,Default,,0000,0000,0000,,with one of these dimensions over here?'
Dialogue: 0,0:05:34.60,0:05:38.01,Default,,0000,0000,0000,,and by convention let's make the y variable
Dialogue: 0,0:05:38.01,0:05:39.42,Default,,0000,0000,0000,,which is really the dependant variable,
Dialogue: 0,0:05:39.42,0:05:40.46,Default,,0000,0000,0000,,The way we did it,
Dialogue: 0,0:05:40.46,0:05:41.82,Default,,0000,0000,0000,,it depends on what x is.
Dialogue: 0,0:05:41.82,0:05:43.60,Default,,0000,0000,0000,,So let's put that on the vertical axis.
Dialogue: 0,0:05:43.60,0:05:45.33,Default,,0000,0000,0000,,and let's put our independent variable,
Dialogue: 0,0:05:45.33,0:05:46.80,Default,,0000,0000,0000,,the one where I just randomly picked values for it
Dialogue: 0,0:05:46.80,0:05:48.35,Default,,0000,0000,0000,,to see what y would become,
Dialogue: 0,0:05:48.35,0:05:50.87,Default,,0000,0000,0000,,let's put that on the horizontal axis.
Dialogue: 0,0:05:50.87,0:05:52.53,Default,,0000,0000,0000,,and it actually was Descartes
Dialogue: 0,0:05:52.53,0:05:55.60,Default,,0000,0000,0000,,who came up with a convention of using x's and y's
Dialogue: 0,0:05:55.60,0:05:58.60,Default,,0000,0000,0000,,and we'll see later z's in algebra, so extensively
Dialogue: 0,0:05:58.60,0:06:02.10,Default,,0000,0000,0000,,as unknown variables with the variables that you're manipulating.
Dialogue: 0,0:06:02.10,0:06:03.87,Default,,0000,0000,0000,,But he says 'Well if we think about it this way
Dialogue: 0,0:06:03.87,0:06:07.45,Default,,0000,0000,0000,,if we number these dimensions'
Dialogue: 0,0:06:07.45,0:06:09.72,Default,,0000,0000,0000,,so let's say that in the x direction
Dialogue: 0,0:06:09.72,0:06:15.70,Default,,0000,0000,0000,,let's make this right over here -3
Dialogue: 0,0:06:15.70,0:06:17.80,Default,,0000,0000,0000,,let's make this -2
Dialogue: 0,0:06:17.80,0:06:19.50,Default,,0000,0000,0000,,this is -1
Dialogue: 0,0:06:19.50,0:06:21.07,Default,,0000,0000,0000,,this is 0
Dialogue: 0,0:06:21.07,0:06:23.80,Default,,0000,0000,0000,,i'm just numbering the x direction
Dialogue: 0,0:06:23.80,0:06:25.33,Default,,0000,0000,0000,,the left right direction.
Dialogue: 0,0:06:25.33,0:06:26.84,Default,,0000,0000,0000,,now this is positive 1
Dialogue: 0,0:06:26.84,0:06:28.34,Default,,0000,0000,0000,,this is positive 2
Dialogue: 0,0:06:28.34,0:06:30.17,Default,,0000,0000,0000,,and this is positive 3.
Dialogue: 0,0:06:30.17,0:06:32.33,Default,,0000,0000,0000,,and we could do the same in the y direction
Dialogue: 0,0:06:32.33,0:06:34.40,Default,,0000,0000,0000,,so let's see we go, so this could be
Dialogue: 0,0:06:34.40,0:06:40.40,Default,,0000,0000,0000,,say this is -5, -4 , -3
Dialogue: 0,0:06:40.40,0:06:42.33,Default,,0000,0000,0000,,actually let me do it a bit neater than that
Dialogue: 0,0:06:42.33,0:06:45.07,Default,,0000,0000,0000,,let me clean this up a little bit.
Dialogue: 0,0:06:45.07,0:06:47.80,Default,,0000,0000,0000,,let me erase this and extend this down a little bit
Dialogue: 0,0:06:47.80,0:06:49.53,Default,,0000,0000,0000,,so I can go all the way down to -5
Dialogue: 0,0:06:49.53,0:06:51.87,Default,,0000,0000,0000,,without making it look too messy.
Dialogue: 0,0:06:51.87,0:06:53.41,Default,,0000,0000,0000,,so let's go all the way down here.
Dialogue: 0,0:06:53.41,0:06:54.87,Default,,0000,0000,0000,,and so we can number it
Dialogue: 0,0:06:54.87,0:06:58.14,Default,,0000,0000,0000,,this is 1, this is 2, this is 3,
Dialogue: 0,0:06:58.14,0:07:00.87,Default,,0000,0000,0000,,and then this could be -1
Dialogue: 0,0:07:00.87,0:07:02.73,Default,,0000,0000,0000,,-2 and these are all just conventions
Dialogue: 0,0:07:02.73,0:07:04.07,Default,,0000,0000,0000,,it could've been labelled the other way.
Dialogue: 0,0:07:04.07,0:07:05.69,Default,,0000,0000,0000,,we could've decided to put the x there
Dialogue: 0,0:07:05.69,0:07:06.73,Default,,0000,0000,0000,,and the y there
Dialogue: 0,0:07:06.73,0:07:07.97,Default,,0000,0000,0000,,and make this the positive direction,
Dialogue: 0,0:07:07.97,0:07:09.28,Default,,0000,0000,0000,,make this the negative direction.
Dialogue: 0,0:07:09.28,0:07:11.33,Default,,0000,0000,0000,,but this is just a convention that people adopted
Dialogue: 0,0:07:11.33,0:07:12.73,Default,,0000,0000,0000,,starting with Descartes.
Dialogue: 0,0:07:12.73,0:07:18.06,Default,,0000,0000,0000,,-2, -3, -4 and -5
Dialogue: 0,0:07:18.06,0:07:20.20,Default,,0000,0000,0000,,and he says 'Well anything i can associate
Dialogue: 0,0:07:20.20,0:07:22.67,Default,,0000,0000,0000,,I can associate each of these pairs of values with
Dialogue: 0,0:07:22.67,0:07:25.33,Default,,0000,0000,0000,,a point in two-dimensions.
Dialogue: 0,0:07:25.33,0:07:28.47,Default,,0000,0000,0000,,I can take the x co-ordinate, I can take the x value
Dialogue: 0,0:07:28.47,0:07:30.33,Default,,0000,0000,0000,,right over here and I say 'Ok that's -2
Dialogue: 0,0:07:30.33,0:07:34.20,Default,,0000,0000,0000,,that would be right over there along the left right direction,
Dialogue: 0,0:07:34.20,0:07:35.83,Default,,0000,0000,0000,,i'm going to the left because it's negative.'
Dialogue: 0,0:07:35.83,0:07:39.40,Default,,0000,0000,0000,,and that's associated with -5 in the vertical direction.
Dialogue: 0,0:07:39.40,0:07:41.67,Default,,0000,0000,0000,,so I say the y value's -5
Dialogue: 0,0:07:41.67,0:07:46.40,Default,,0000,0000,0000,,and so if I go 2 to the left and 5 down.
Dialogue: 0,0:07:46.40,0:07:49.27,Default,,0000,0000,0000,,I get to this point right over there.
Dialogue: 0,0:07:49.27,0:07:53.52,Default,,0000,0000,0000,,so he says 'These two values -2 and -5
Dialogue: 0,0:07:53.52,0:07:55.73,Default,,0000,0000,0000,,I can associate it with this point
Dialogue: 0,0:07:55.73,0:07:59.13,Default,,0000,0000,0000,,in this plane right over here, in this two-dimensional plane.
Dialogue: 0,0:07:59.13,0:08:02.93,Default,,0000,0000,0000,,so I'll say: That point has the co-ordinates,
Dialogue: 0,0:08:02.93,0:08:06.40,Default,,0000,0000,0000,,tells me where do I find that point (-2,-5).
Dialogue: 0,0:08:06.40,0:08:08.96,Default,,0000,0000,0000,,and these coordinates are called 'cartesian coordinates'
Dialogue: 0,0:08:08.96,0:08:12.08,Default,,0000,0000,0000,,named for René Descartes
Dialogue: 0,0:08:12.08,0:08:13.80,Default,,0000,0000,0000,,because he is the guy who came up with these.
Dialogue: 0,0:08:13.80,0:08:15.07,Default,,0000,0000,0000,,He's associating all of a sudden these relationships
Dialogue: 0,0:08:15.07,0:08:17.67,Default,,0000,0000,0000,,with points on a co-ordinate plane.
Dialogue: 0,0:08:17.67,0:08:19.80,Default,,0000,0000,0000,,and then he says 'well ok, lets do another one'
Dialogue: 0,0:08:19.80,0:08:21.60,Default,,0000,0000,0000,,there's this other relationship,
Dialogue: 0,0:08:21.60,0:08:27.45,Default,,0000,0000,0000,,when x is equal to -1, y = -3
Dialogue: 0,0:08:27.45,0:08:30.03,Default,,0000,0000,0000,,so x is -1, y is -3.
Dialogue: 0,0:08:30.03,0:08:31.54,Default,,0000,0000,0000,,that's that point right over there.
Dialogue: 0,0:08:31.54,0:08:33.33,Default,,0000,0000,0000,,and the convention is once again.
Dialogue: 0,0:08:33.33,0:08:34.38,Default,,0000,0000,0000,,'When you list the co-ordinates,
Dialogue: 0,0:08:34.38,0:08:36.60,Default,,0000,0000,0000,,you list the x co-ordinate, then the y co-ordinate
Dialogue: 0,0:08:36.60,0:08:38.40,Default,,0000,0000,0000,,and that's just what people decided to do.
Dialogue: 0,0:08:38.40,0:08:42.07,Default,,0000,0000,0000,,-1, -3 that would be that point right over there
Dialogue: 0,0:08:42.07,0:08:45.93,Default,,0000,0000,0000,,and then you have the point when x is 0, y is -1
Dialogue: 0,0:08:45.93,0:08:48.07,Default,,0000,0000,0000,,when x is 0 right over here,
Dialogue: 0,0:08:48.07,0:08:50.27,Default,,0000,0000,0000,,which means I don't go the left or the right.
Dialogue: 0,0:08:50.27,0:08:52.67,Default,,0000,0000,0000,,y is -1, which means I go 1 down.
Dialogue: 0,0:08:52.67,0:08:56.39,Default,,0000,0000,0000,,so that's that point right over there. (0,-1)
Dialogue: 0,0:08:56.39,0:08:57.36,Default,,0000,0000,0000,,right over there
Dialogue: 0,0:08:57.36,0:08:58.85,Default,,0000,0000,0000,,and I could keep doing this.
Dialogue: 0,0:08:58.85,0:09:03.81,Default,,0000,0000,0000,,when x is 1, y is 1
Dialogue: 0,0:09:03.81,0:09:09.58,Default,,0000,0000,0000,,when x is 2, y is 3
Dialogue: 0,0:09:09.58,0:09:11.73,Default,,0000,0000,0000,,actually let me do that in the same purple colour
Dialogue: 0,0:09:11.73,0:09:15.40,Default,,0000,0000,0000,,when x is 2, y is 3
Dialogue: 0,0:09:15.40,0:09:20.65,Default,,0000,0000,0000,,2,3 and then this one right over here in orange was 1,1
Dialogue: 0,0:09:20.65,0:09:22.20,Default,,0000,0000,0000,,and this is neat by itself,
Dialogue: 0,0:09:22.20,0:09:24.62,Default,,0000,0000,0000,,I essentially just sampled possible x's.
Dialogue: 0,0:09:24.62,0:09:25.87,Default,,0000,0000,0000,,but what he realized is
Dialogue: 0,0:09:25.87,0:09:27.78,Default,,0000,0000,0000,,not only do you sample these possible x's,
Dialogue: 0,0:09:27.78,0:09:29.68,Default,,0000,0000,0000,,but it you kept sampling x's,
Dialogue: 0,0:09:29.68,0:09:31.32,Default,,0000,0000,0000,,if you tried sampling all of the x's in between,
Dialogue: 0,0:09:31.32,0:09:34.00,Default,,0000,0000,0000,,you'd actually end up plotting out a line.
Dialogue: 0,0:09:34.00,0:09:36.07,Default,,0000,0000,0000,,So if you were to do every possible x
Dialogue: 0,0:09:36.07,0:09:38.07,Default,,0000,0000,0000,,you would end up getting a line
Dialogue: 0,0:09:38.07,0:09:44.49,Default,,0000,0000,0000,,that looks something like that... right over there.
Dialogue: 0,0:09:44.49,0:09:47.53,Default,,0000,0000,0000,,and any... any relation, if you pick any x
Dialogue: 0,0:09:47.53,0:09:50.87,Default,,0000,0000,0000,,and find any y it really represents a point on this line,
Dialogue: 0,0:09:50.87,0:09:52.40,Default,,0000,0000,0000,,or another way to think about it
Dialogue: 0,0:09:52.40,0:09:54.17,Default,,0000,0000,0000,,any point on this line represents
Dialogue: 0,0:09:54.17,0:09:57.05,Default,,0000,0000,0000,,a solution to this equation right over here.
Dialogue: 0,0:09:57.05,0:09:58.90,Default,,0000,0000,0000,,so if you have this point right over here.
Dialogue: 0,0:09:58.90,0:10:01.60,Default,,0000,0000,0000,,which looks like about x is 1 and a half.
Dialogue: 0,0:10:01.60,0:10:03.47,Default,,0000,0000,0000,,y is 2. So let me write that
Dialogue: 0,0:10:03.47,0:10:07.13,Default,,0000,0000,0000,,1.5,2
Dialogue: 0,0:10:07.13,0:10:09.13,Default,,0000,0000,0000,,that is a solution to this equation.
Dialogue: 0,0:10:09.13,0:10:13.65,Default,,0000,0000,0000,,when x is 1.5. 2 x 1.5 is 3 - 1 is 2
Dialogue: 0,0:10:13.65,0:10:15.60,Default,,0000,0000,0000,,that is right over there.
Dialogue: 0,0:10:15.60,0:10:17.40,Default,,0000,0000,0000,,so all of a sudden he was able to bridge
Dialogue: 0,0:10:17.40,0:10:22.40,Default,,0000,0000,0000,,this gap or this relationship between algebra and geometry.
Dialogue: 0,0:10:22.40,0:10:27.13,Default,,0000,0000,0000,,we can now visualize all of the x and y pairs
Dialogue: 0,0:10:27.13,0:10:31.50,Default,,0000,0000,0000,,that satisfy this equation right over here.
Dialogue: 0,0:10:31.50,0:10:36.09,Default,,0000,0000,0000,,and so he is responsible for making this bridge
Dialogue: 0,0:10:36.09,0:10:38.07,Default,,0000,0000,0000,,and that's why that co-ordinates
Dialogue: 0,0:10:38.07,0:10:42.68,Default,,0000,0000,0000,,that we use to specify these points are called 'cartesian coordinates'
Dialogue: 0,0:10:42.68,0:10:45.47,Default,,0000,0000,0000,,and as we'll see and first type of equations
Dialogue: 0,0:10:45.47,0:10:48.60,Default,,0000,0000,0000,,we will study our equations of this form over here
Dialogue: 0,0:10:48.60,0:10:50.45,Default,,0000,0000,0000,,and in a traditional algebra curriculum.
Dialogue: 0,0:10:50.45,0:10:52.73,Default,,0000,0000,0000,,they're called linear equations...
Dialogue: 0,0:10:52.73,0:10:55.73,Default,,0000,0000,0000,,linear equations.
Dialogue: 0,0:10:55.73,0:10:57.74,Default,,0000,0000,0000,,and you might be saying: well you know, this is an equation,
Dialogue: 0,0:10:57.74,0:10:59.53,Default,,0000,0000,0000,,I'll see that this is equal to that on its own.
Dialogue: 0,0:10:59.53,0:11:00.74,Default,,0000,0000,0000,,but what's so linear about them?
Dialogue: 0,0:11:00.74,0:11:02.33,Default,,0000,0000,0000,,what makes them look like a line?
Dialogue: 0,0:11:02.33,0:11:04.38,Default,,0000,0000,0000,,to realize why they're linear,
Dialogue: 0,0:11:04.38,0:11:07.47,Default,,0000,0000,0000,,you have to make this jump René Descartes made.
Dialogue: 0,0:11:07.47,0:11:09.13,Default,,0000,0000,0000,,because if you were to plot this,
Dialogue: 0,0:11:09.13,0:11:10.76,Default,,0000,0000,0000,,using cartesian coordinates.
Dialogue: 0,0:11:10.76,0:11:14.49,Default,,0000,0000,0000,,on a Euclidean plane. You will get a line.
Dialogue: 0,0:11:14.49,0:11:15.85,Default,,0000,0000,0000,,and in the future you'll see that
Dialogue: 0,0:11:15.85,0:11:17.72,Default,,0000,0000,0000,,there's other types of equations where you won't get a line.
Dialogue: 0,0:11:17.72,0:11:21.66,Default,,0000,0000,0000,,you get a curve, or something kind of crazy or funky.