Overview and history of algebra
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Intro to the coordinate plane
This right here is a picture of Rene Descartes. Once again, one of the great minds in both math and philosophy. And I think you're seeing a little bit of a trend here, that the great philosophers 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 70s. Descartes died at what is only 54 years old. And he's 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, at the end of the day, they really 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 him here as we build our 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, and 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 could have something like y is equal to 2x minus 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. And I could just pick random values for x, and then figure out what y is. But I'll pick relatively straightforward values just so that the math doesn't get too complicated. So for example, if x is negative 2, then y is going to be 2 times negative 2 minus 1, which is negative 4 minus 1, which is negative 5. If x is negative 1, then y is going to be 2 times negative 1 minus 1, which is equal to-- this is negative 2 minus 1, which is negative 3. If x is equal to 0, then y is going to be 2 times 0 minus 1. 2 times 0 is 0 minus 1 is just negative 1. I'll do a couple more. And I could have picked any values here. I could have said, well, what happens if x is the negative square root of 2, or what happens if x is negative 5/2, or positive 6/7? But I'm just picking these numbers because it makes the math 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 times 1 minus 1. 2 times 1 is 2 minus 1 is 1. And I'll do one more. I'll do one more in a color that I have not used yet. Let's see, this purple. If x is 2, then y is going to be 2 times 2-- now our x is 2-- minus 1. So that is 4 minus 1 is equal to 3. So fair enough. I just kind of sampled this relationship. I said, OK, this describes the general relationship between a variable y and a variable x. And then I just made it a little bit more concrete. I said, OK, well, then for each of these values of x, what would be the corresponding value of y? And what Descartes realized is is that you could visualize this. One, you could visualize these individual points, but that could also help you, in general, to visualize this relationship. And 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, it's generally viewed that geometry was Euclidean geometry, and that's essentially the geometry that you studied in a geometry class in eighth or ninth grade 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 go into some depth in that in the geometry playlist. But Descartes said, well, I think I can represent this visually the same way that you could with studying these triangles and these circles. He said, well, if we view a piece of paper, if we think about a two-dimensional plane, you could view a piece of paper as kind of a section of a two-dimensional plane. And we call it two dimensions because there's two directions that you could 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 starting to visualize things, so I'll do it in the geometry color. So you have the up/down direction. And you have the left/right direction. That's why it's called a two-dimensional plane. If we're dealing in three dimensions, you would have an in/out dimension. And it's very easy to do two dimensions on the screen because the screen is two dimensional. And he, says, well, you know, there are two variables here, and they have this relationship. So 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 dependent variable-- the way we did it, it depends on what x is-- 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 the convention of using x's and y's, and we'll see later z's, in algebra so extensively as unknown variables or 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 is negative 3. Let's make this negative 2. This is negative 1. This is 0. Now, I'm just numbering the x direction, the left/right direction. Now this is positive 1. This is positive 2. This is positive 3. And we could do the same in the y direction. So let's see, so this could be, let's say this is negative 5, negative 4, negative 3, negative-- actually, let me do it a little bit neater than that. Let me clean this up a little bit. So let me erase this and extend this down a little bit so I can go all the way down to negative 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 negative 1, negative 2. And these are all just conventions. It could have been labeled the other way. We could've decided to put the x there and the y there and make this the positive direction and make this the negative direction. But this is just the convention that people adopted starting with Descartes. Negative 2, negative 3, negative 4, and negative 5. And he says, well, I think I can associate each of these pairs of values with a point in two dimensions. I can take the x-coordinate, I can take the x value right over here, and I say, OK, that's a negative 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 negative 5 in the vertical direction. So I say the y value is negative 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, negative 2 and negative 5, I can associate it with this point in this plane right over here, in this two-dimensional plane. So I'll say that point has the coordinates, tells me where to find that point, negative 2, negative 5. And these coordinates are called Cartesian coordinates, named for Rene Descartes because he's the guy that came up with these. He's associating, all of a sudden, these relationships with points on a coordinate plane. And then he said, well, OK, let's do another one. There's this other relationship, where I have when x is equal to negative 1, y is equal to negative 3. So x is negative 1, y is negative 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. Negative 1, negative 3, that would be that point right over there. And then you have the point when x is 0, y is negative 1. When x is 0 right over here, which means I don't go to the left or the right, y is negative 1, which means I go 1 down. So that's that point right over there, 0, negative 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 it in that same purple color. When x is 2, y is 3, 2 comma 3. And then this one right over here in orange was 1 comma 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 if you just kept sampling x's, if you tried sampling all the x's in between, you would 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 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 it's about x is 1 and 1/2, y is 2. So me write that, 1.5 comma 2. That is a solution to this equation. When x is 1.5, 2 times 1.5 is 3 minus 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 the coordinates that we use to specify these points are called Cartesian coordinates. And as we'll see, the first type of equations we will study are equations of this form over here. And in a traditional algebra curriculum, they're called linear equations. And you might be saying, well, OK, this is an equation. I see that this is equal to that. But what's so linear about them? What makes them look like a line? And to realize why they are linear, you have to make this jump that Rene 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, we'll see that there's other types of equations where you won't get a line, where you'll get a curve or something kind of crazy or funky.