WEBVTT 00:00:01.062 --> 00:00:03.636 This right here is a picture of René Descartes 00:00:03.636 --> 00:00:05.698 Once again one of the great minds, 00:00:05.698 --> 00:00:07.554 in both math and philosophy. 00:00:07.554 --> 00:00:09.923 And i think you'll be seeing bit of a little trend here 00:00:09.923 --> 00:00:13.190 that the great philosphers were also great mathematicians 00:00:13.190 --> 00:00:15.200 and vice versa 00:00:15.200 --> 00:00:17.021 and he was somewhat of a contemporary of Galileo 00:00:17.021 --> 00:00:18.733 he was 32 years younger. 00:00:18.733 --> 00:00:21.706 although he died shortly after Galileo died. 00:00:21.706 --> 00:00:23.467 This guy died at a much younger age, 00:00:23.467 --> 00:00:25.400 Galileo was well into his 70's 00:00:25.400 --> 00:00:28.067 Descartes died at what, this is only at 54 years old. 00:00:28.067 --> 00:00:30.867 And he is probably most known in popular culture, 00:00:30.867 --> 00:00:32.733 for this quote right over here. 00:00:32.733 --> 00:00:33.800 a very philosophical quote. 00:00:33.800 --> 00:00:35.867 "I think therefore I am" 00:00:35.867 --> 00:00:37.467 but i also wanted to throw in, 00:00:37.467 --> 00:00:38.867 and this isn't that related to algebra, 00:00:38.867 --> 00:00:40.733 but i just thought it was a really neat quote. 00:00:40.733 --> 00:00:42.800 Probably his least famous quote. 00:00:42.800 --> 00:00:44.467 This one right over here. 00:00:44.467 --> 00:00:46.800 And i like it just because it's very practical 00:00:46.800 --> 00:00:48.852 and it makes you realize that these great minds NOTE Paragraph 00:00:48.852 --> 00:00:51.113 these pillars of philosophy and mathematics 00:00:51.113 --> 00:00:52.282 that at the end of the day, 00:00:52.282 --> 00:00:54.467 they were just human beings. 00:00:54.467 --> 00:00:56.498 and he said, "You just keep pushing. 00:00:56.498 --> 00:00:58.133 You just keep pushing. 00:00:58.133 --> 00:01:00.015 I made every mistake that could be made. 00:01:00.015 --> 00:01:02.031 But I just kept pushing." 00:01:02.031 --> 00:01:05.267 Which i think is very very good life advice. 00:01:05.267 --> 00:01:07.733 Now he did many things 00:01:07.733 --> 00:01:09.077 in philosophy and mathematics 00:01:09.077 --> 00:01:11.062 but the reason why I'm including here 00:01:11.062 --> 00:01:12.933 as we build foundations of algebra 00:01:12.933 --> 00:01:15.600 is that he is the individual 00:01:15.600 --> 00:01:18.800 most responsible for a very strong connection 00:01:18.800 --> 00:01:21.425 between algebra and geometry. 00:01:21.425 --> 00:01:22.898 so on the left over here 00:01:22.898 --> 00:01:24.752 you have the world of algebra. 00:01:24.752 --> 00:01:26.415 We've discussed it a little bit. 00:01:26.415 --> 00:01:28.477 You have equations that deal with symbols 00:01:28.477 --> 00:01:30.236 and these symbols are essentially 00:01:30.236 --> 00:01:31.933 they can take on values 00:01:31.933 --> 00:01:32.800 so you can have something like 00:01:32.800 --> 00:01:37.677 y = 2x - 1 00:01:37.677 --> 00:01:39.267 this gives us a relationship 00:01:39.267 --> 00:01:40.733 between whatever x is 00:01:40.733 --> 00:01:42.133 and whatever y is. 00:01:42.133 --> 00:01:44.333 and we can even set up a table here. 00:01:44.333 --> 00:01:46.733 and pick values for x 00:01:46.733 --> 00:01:48.292 and see what the values of y would be. 00:01:48.292 --> 00:01:51.652 I can just pick random values for x 00:01:51.652 --> 00:01:53.133 and then figure out what y is. 00:01:53.133 --> 00:01:55.000 but i'll pick relatively straightforward values 00:01:55.000 --> 00:01:57.662 and so that the maths doesn't get too complicated. 00:01:57.662 --> 00:01:59.252 so for example, 00:01:59.252 --> 00:02:00.533 if x is -2 00:02:00.533 --> 00:02:03.600 then y is going to be 2 x -2 - 1 00:02:03.600 --> 00:02:06.513 2 x -2 - 1 00:02:06.513 --> 00:02:10.113 which is -4 - 1 00:02:10.113 --> 00:02:12.267 which is -5 00:02:12.267 --> 00:02:14.785 if x is -1 00:02:14.785 --> 00:02:20.452 then y is going to be 2 x -1 - 1 00:02:20.452 --> 00:02:21.733 which is equal to 00:02:21.733 --> 00:02:24.554 this is -2 - 1 which is -3 00:02:24.554 --> 00:02:28.725 if x = 0 00:02:28.725 --> 00:02:32.590 then y is going to be 2 x 0 - 1 00:02:32.600 --> 00:02:35.667 2 x 0 is 0 - 1 is just -1 00:02:35.667 --> 00:02:37.333 i'll do a couple more. 00:02:37.333 --> 00:02:38.282 if x is 1 00:02:38.282 --> 00:02:39.421 and i could've picked any values here 00:02:39.421 --> 00:02:40.352 I could've said what happens 00:02:40.352 --> 00:02:42.005 if x is the negative square root of 2 00:02:42.005 --> 00:02:45.067 or what happens if x is -5 halves 00:02:45.067 --> 00:02:47.867 or positive six seventh. 00:02:47.867 --> 00:02:49.000 but i'm just picking these numbers 00:02:49.000 --> 00:02:50.600 because it makes the maths a lot easier 00:02:50.600 --> 00:02:52.600 when i try to figure out what y is going to be. 00:02:52.600 --> 00:02:54.133 but when x is 1 00:02:54.133 --> 00:02:57.338 y is going to be 2(1) - 1 00:02:57.338 --> 00:02:59.733 2 x 1 is 2 - 1 is 1 00:02:59.733 --> 00:03:03.052 and i'll do one more. 00:03:03.052 --> 00:03:05.133 in the colour I have not used yet. 00:03:05.133 --> 00:03:06.667 let's see this purple. 00:03:06.667 --> 00:03:08.041 if x is 2 00:03:08.041 --> 00:03:09.333 then y is going to be 00:03:09.333 --> 00:03:14.005 2(2) - 1 (now that x is 2) 00:03:14.005 --> 00:03:16.615 so that is 4 - 1, is equal to 3 00:03:16.615 --> 00:03:17.800 so fair enough, 00:03:17.800 --> 00:03:19.548 I just kind of sampled this relationship. 00:03:19.548 --> 00:03:22.533 But I said okay this describes a general relationship 00:03:22.533 --> 00:03:25.200 between a variable y and a variable x 00:03:25.200 --> 00:03:26.908 and then I just made a little more concrete. 00:03:26.908 --> 00:03:28.000 I said ok well then 00:03:28.000 --> 00:03:29.882 if x is one of these variables. 00:03:29.882 --> 00:03:31.200 for each of these values of x, 00:03:31.200 --> 00:03:33.800 what would be the corresponding value of y? 00:03:33.800 --> 00:03:35.698 and what Descartes realized is 00:03:35.717 --> 00:03:37.467 that you could visualize this. 00:03:37.467 --> 00:03:40.405 what you could visualize is individual points. 00:03:40.405 --> 00:03:42.667 But that could also help you in general 00:03:42.667 --> 00:03:45.800 to visualize this relationship. 00:03:45.800 --> 00:03:47.333 so what he essentially did is 00:03:47.333 --> 00:03:52.329 he bridged the worlds of this kind of very abstract symbolic algebra. 00:03:52.329 --> 00:03:55.200 and that and geometry which was concerned 00:03:55.200 --> 00:03:57.600 with shapes and sizes and angles. 00:03:57.600 --> 00:04:02.933 so over here you have the world of geometry. 00:04:02.933 --> 00:04:04.887 and obviously there are people in history 00:04:04.887 --> 00:04:07.067 maybe many people who history may have forgotten 00:04:07.067 --> 00:04:09.067 who might have dabbled in this. 00:04:09.067 --> 00:04:12.467 But before Descartes is generally viewed. 00:04:12.467 --> 00:04:14.800 that geometry was euclidean geometry. 00:04:14.800 --> 00:04:16.133 and that's essentially the geometry 00:04:16.133 --> 00:04:17.533 that you studied in geometry class 00:04:17.533 --> 00:04:20.333 in 8th or 9th or 10th grade. 00:04:20.333 --> 00:04:22.533 in a traditional high school curriculum. 00:04:22.533 --> 00:04:24.200 and that's the geometry of studying 00:04:24.200 --> 00:04:28.554 the relationships between triangles, and their angles. 00:04:28.554 --> 00:04:30.667 and the relationships between circles. 00:04:30.667 --> 00:04:33.887 and you have radii and then you have triangles 00:04:33.887 --> 00:04:36.200 inscribed in circles and all the rest 00:04:36.200 --> 00:04:37.190 and we'll go into some depth 00:04:37.190 --> 00:04:39.667 in that in the geometry playlist. 00:04:39.667 --> 00:04:42.938 But Descarte says, 'well i think i can represent this visually 00:04:42.938 --> 00:04:46.581 the same way Euclid was studying these triangles and these circles' 00:04:46.581 --> 00:04:48.299 he said 'why don't I ?' 00:04:48.299 --> 00:04:50.575 if we view a piece of paper. 00:04:50.575 --> 00:04:52.339 if we think about a two-dimensional plane. 00:04:52.339 --> 00:04:53.825 you could view a piece of paper 00:04:53.825 --> 00:04:55.915 as kind of a section of a two-dimensional plane. 00:04:55.915 --> 00:04:57.819 we call it two-dimensions 00:04:57.819 --> 00:04:59.584 because there's two directions that you can go in. 00:04:59.584 --> 00:05:01.256 there's the up down direction, 00:05:01.256 --> 00:05:02.510 that's one direction. 00:05:02.510 --> 00:05:04.825 so let me draw that, i'll do it in blue. 00:05:04.841 --> 00:05:06.666 because we're trying to visualize things 00:05:06.666 --> 00:05:08.384 so i'll do it the geometry colour. 00:05:08.384 --> 00:05:11.827 so you have the up down direction 00:05:11.827 --> 00:05:14.139 and you have the left right direction. 00:05:14.139 --> 00:05:16.720 that's why it's called a two-dimensional plane. 00:05:16.720 --> 00:05:18.160 if we're dealing with three-dimensions. 00:05:18.160 --> 00:05:21.339 you have an in out dimension. 00:05:21.339 --> 00:05:23.200 and it's very easy to do two-dimensions on the screen 00:05:23.200 --> 00:05:25.425 because the screen is two-dimensional. 00:05:25.425 --> 00:05:27.071 and he says 'Well, you know 00:05:27.071 --> 00:05:29.744 there are two variables here and they have this relationship. 00:05:29.744 --> 00:05:32.548 But why don't I associate each of these variables 00:05:32.548 --> 00:05:34.600 with one of these dimensions over here?' 00:05:34.600 --> 00:05:38.010 and by convention let's make the y variable 00:05:38.010 --> 00:05:39.421 which is really the dependant variable, 00:05:39.421 --> 00:05:40.456 The way we did it, 00:05:40.456 --> 00:05:41.815 it depends on what x is. 00:05:41.815 --> 00:05:43.605 So let's put that on the vertical axis. 00:05:43.605 --> 00:05:45.333 and let's put our independent variable, 00:05:45.333 --> 00:05:46.800 the one where I just randomly picked values for it 00:05:46.800 --> 00:05:48.348 to see what y would become, 00:05:48.348 --> 00:05:50.867 let's put that on the horizontal axis. 00:05:50.867 --> 00:05:52.533 and it actually was Descartes 00:05:52.533 --> 00:05:55.600 who came up with a convention of using x's and y's 00:05:55.600 --> 00:05:58.600 and we'll see later z's in algebra, so extensively 00:05:58.600 --> 00:06:02.098 as unknown variables with the variables that you're manipulating. 00:06:02.098 --> 00:06:03.867 But he says 'Well if we think about it this way 00:06:03.867 --> 00:06:07.452 if we number these dimensions' 00:06:07.452 --> 00:06:09.723 so let's say that in the x direction 00:06:09.723 --> 00:06:15.702 let's make this right over here -3 00:06:15.702 --> 00:06:17.805 let's make this -2 00:06:17.805 --> 00:06:19.498 this is -1 00:06:19.498 --> 00:06:21.067 this is 0 00:06:21.067 --> 00:06:23.800 i'm just numbering the x direction 00:06:23.800 --> 00:06:25.333 the left right direction. 00:06:25.333 --> 00:06:26.837 now this is positive 1 00:06:26.837 --> 00:06:28.338 this is positive 2 00:06:28.338 --> 00:06:30.169 and this is positive 3. 00:06:30.169 --> 00:06:32.333 and we could do the same in the y direction 00:06:32.333 --> 00:06:34.400 so let's see we go, so this could be 00:06:34.400 --> 00:06:40.400 say this is -5, -4 , -3 00:06:40.400 --> 00:06:42.333 actually let me do it a bit neater than that 00:06:42.333 --> 00:06:45.067 let me clean this up a little bit. 00:06:45.067 --> 00:06:47.800 let me erase this and extend this down a little bit 00:06:47.800 --> 00:06:49.533 so I can go all the way down to -5 00:06:49.533 --> 00:06:51.867 without making it look too messy. 00:06:51.867 --> 00:06:53.410 so let's go all the way down here. 00:06:53.410 --> 00:06:54.867 and so we can number it 00:06:54.867 --> 00:06:58.144 this is 1, this is 2, this is 3, 00:06:58.144 --> 00:07:00.867 and then this could be -1 00:07:00.867 --> 00:07:02.733 -2 and these are all just conventions 00:07:02.733 --> 00:07:04.067 it could've been labelled the other way. 00:07:04.067 --> 00:07:05.692 we could've decided to put the x there 00:07:05.692 --> 00:07:06.733 and the y there 00:07:06.733 --> 00:07:07.969 and make this the positive direction, 00:07:07.969 --> 00:07:09.277 make this the negative direction. 00:07:09.277 --> 00:07:11.333 but this is just a convention that people adopted 00:07:11.333 --> 00:07:12.733 starting with Descartes. 00:07:12.733 --> 00:07:18.062 -2, -3, -4 and -5 00:07:18.062 --> 00:07:20.200 and he says 'Well anything i can associate 00:07:20.200 --> 00:07:22.667 I can associate each of these pairs of values with 00:07:22.667 --> 00:07:25.333 a point in two-dimensions. 00:07:25.333 --> 00:07:28.467 I can take the x co-ordinate, I can take the x value 00:07:28.467 --> 00:07:30.333 right over here and I say 'Ok that's -2 00:07:30.333 --> 00:07:34.195 that would be right over there along the left right direction, 00:07:34.195 --> 00:07:35.831 i'm going to the left because it's negative.' 00:07:35.831 --> 00:07:39.395 and that's associated with -5 in the vertical direction. 00:07:39.395 --> 00:07:41.667 so I say the y value's -5 00:07:41.667 --> 00:07:46.400 and so if I go 2 to the left and 5 down. 00:07:46.400 --> 00:07:49.267 I get to this point right over there. 00:07:49.267 --> 00:07:53.518 so he says 'These two values -2 and -5 00:07:53.518 --> 00:07:55.733 I can associate it with this point 00:07:55.733 --> 00:07:59.133 in this plane right over here, in this two-dimensional plane. 00:07:59.133 --> 00:08:02.933 so I'll say: That point has the co-ordinates, 00:08:02.933 --> 00:08:06.400 tells me where do I find that point (-2,-5). 00:08:06.400 --> 00:08:08.959 and these coordinates are called 'cartesian coordinates' 00:08:08.959 --> 00:08:12.077 named for René Descartes 00:08:12.077 --> 00:08:13.800 because he is the guy who came up with these. 00:08:13.800 --> 00:08:15.067 He's associating all of a sudden these relationships 00:08:15.067 --> 00:08:17.667 with points on a co-ordinate plane. 00:08:17.667 --> 00:08:19.800 and then he says 'well ok, lets do another one' 00:08:19.800 --> 00:08:21.600 there's this other relationship, 00:08:21.600 --> 00:08:27.452 when x is equal to -1, y = -3 00:08:27.452 --> 00:08:30.031 so x is -1, y is -3. 00:08:30.031 --> 00:08:31.544 that's that point right over there. 00:08:31.544 --> 00:08:33.333 and the convention is once again. 00:08:33.333 --> 00:08:34.375 'When you list the co-ordinates, 00:08:34.375 --> 00:08:36.600 you list the x co-ordinate, then the y co-ordinate 00:08:36.600 --> 00:08:38.400 and that's just what people decided to do. 00:08:38.400 --> 00:08:42.067 -1, -3 that would be that point right over there 00:08:42.067 --> 00:08:45.933 and then you have the point when x is 0, y is -1 00:08:45.933 --> 00:08:48.067 when x is 0 right over here, 00:08:48.067 --> 00:08:50.267 which means I don't go the left or the right. 00:08:50.267 --> 00:08:52.667 y is -1, which means I go 1 down. 00:08:52.667 --> 00:08:56.390 so that's that point right over there. (0,-1) 00:08:56.390 --> 00:08:57.359 right over there 00:08:57.359 --> 00:08:58.852 and I could keep doing this. 00:08:58.852 --> 00:09:03.810 when x is 1, y is 1 00:09:03.810 --> 00:09:09.575 when x is 2, y is 3 00:09:09.575 --> 00:09:11.733 actually let me do that in the same purple colour 00:09:11.733 --> 00:09:15.400 when x is 2, y is 3 00:09:15.400 --> 00:09:20.652 2,3 and then this one right over here in orange was 1,1 00:09:20.652 --> 00:09:22.195 and this is neat by itself, 00:09:22.195 --> 00:09:24.615 I essentially just sampled possible x's. 00:09:24.615 --> 00:09:25.867 but what he realized is 00:09:25.867 --> 00:09:27.775 not only do you sample these possible x's, 00:09:27.775 --> 00:09:29.677 but it you kept sampling x's, 00:09:29.677 --> 00:09:31.318 if you tried sampling all of the x's in between, 00:09:31.318 --> 00:09:34.000 you'd actually end up plotting out a line. 00:09:34.000 --> 00:09:36.067 So if you were to do every possible x 00:09:36.067 --> 00:09:38.067 you would end up getting a line 00:09:38.067 --> 00:09:44.492 that looks something like that... right over there. 00:09:44.492 --> 00:09:47.533 and any... any relation, if you pick any x 00:09:47.533 --> 00:09:50.867 and find any y it really represents a point on this line, 00:09:50.867 --> 00:09:52.400 or another way to think about it 00:09:52.400 --> 00:09:54.171 any point on this line represents 00:09:54.171 --> 00:09:57.051 a solution to this equation right over here. 00:09:57.051 --> 00:09:58.902 so if you have this point right over here. 00:09:58.902 --> 00:10:01.600 which looks like about x is 1 and a half. 00:10:01.600 --> 00:10:03.467 y is 2. So let me write that 00:10:03.467 --> 00:10:07.133 1.5,2 00:10:07.133 --> 00:10:09.133 that is a solution to this equation. 00:10:09.133 --> 00:10:13.652 when x is 1.5. 2 x 1.5 is 3 - 1 is 2 00:10:13.652 --> 00:10:15.600 that is right over there. 00:10:15.600 --> 00:10:17.400 so all of a sudden he was able to bridge 00:10:17.400 --> 00:10:22.400 this gap or this relationship between algebra and geometry. 00:10:22.400 --> 00:10:27.133 we can now visualize all of the x and y pairs 00:10:27.133 --> 00:10:31.498 that satisfy this equation right over here. 00:10:31.498 --> 00:10:36.092 and so he is responsible for making this bridge 00:10:36.092 --> 00:10:38.067 and that's why that co-ordinates 00:10:38.067 --> 00:10:42.677 that we use to specify these points are called 'cartesian coordinates' 00:10:42.677 --> 00:10:45.467 and as we'll see and first type of equations 00:10:45.467 --> 00:10:48.600 we will study our equations of this form over here 00:10:48.600 --> 00:10:50.446 and in a traditional algebra curriculum. 00:10:50.446 --> 00:10:52.733 they're called linear equations... 00:10:52.733 --> 00:10:55.733 linear equations. 00:10:55.733 --> 00:10:57.738 and you might be saying: well you know, this is an equation, 00:10:57.738 --> 00:10:59.533 I'll see that this is equal to that on its own. 00:10:59.533 --> 00:11:00.744 but what's so linear about them? 00:11:00.744 --> 00:11:02.333 what makes them look like a line? 00:11:02.333 --> 00:11:04.379 to realize why they're linear, 00:11:04.379 --> 00:11:07.467 you have to make this jump René Descartes made. 00:11:07.467 --> 00:11:09.133 because if you were to plot this, 00:11:09.133 --> 00:11:10.759 using cartesian coordinates. 00:11:10.759 --> 00:11:14.492 on a Euclidean plane. You will get a line. 00:11:14.492 --> 00:11:15.846 and in the future you'll see that 00:11:15.846 --> 00:11:17.723 there's other types of equations where you won't get a line. 00:11:17.723 --> 00:11:21.656 you get a curve, or something kind of crazy or funky.