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