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Skip to 0 minutes and 13 seconds The 17th century was an enormously interesting time for mathematics and physics. And a lot of that was generated by the work of two French mathematicians, Pierre Fermat and Rene Descartes. Descartes wrote a very influential book called La Geometrie in which he basically reconfigured the landscape geometry, introducing the idea of a fixed coordinate system to allow us to use algebraic techniques to deal with the geometry of the ancient Greeks. It was a very powerful technique. And we’re going to be reviewing that basic framework today. So the idea is that we have fixed coordinate axes, and so points and lines can be represented very concretely by algebraic little equations.

Skip to 0 minutes and 59 seconds This gives us a handle on slopes of lines and connects that concept to the proportionality constants that were occurring when we were talking about direct proportions.

Skip to 1 minute and 16 seconds So here is the essence of Descartes’ fabulous revolution. It’s really a very simple thing. It’s really a sheet of graph paper with certain axes marked to record the positions of points and then other geometrical objects. So what we see here is a sheet of graph paper blown up, an x-axis in this direction horizontal, y-axis in this direction vertical, spaced off equally. 1, 2, 3, 4. There is 0. And on the other side, minus 1, minus 2, minus 3. So in the y direction, there’s the 0. There’s 1, 2, 3, minus 1, minus 2, minus 3.

Skip to 1 minute and 57 seconds It’s worth remarking that this geometrical number line that’s appearing here is really a primary use of negative numbers, an interpretation of negative numbers, that the negative numbers really correspond to things which are actually physically observable here. So what can we do with this framework? Well, first of all, you see these other lines here that are parallel to the coordinate axes that allow us to pinpoint exactly where points are. So, for example, this is point A right here, and its name is square bracket 2,1 square bracket. So we’ll call it the point A, which is 2,1. It has x-coordinate 2. That’s the first coordinate. And y-coordinate 1. That’s the second coordinate.

Skip to 2 minutes and 44 seconds Here’s the point B, which is 4 over in this direction, 2 up, so it’s the point 4,2. Over here is the point C, point minus 2, minus 1. So minus 2 in the x direction. Minus 1 in the y direction. So we can specify all kinds of points very easily. So, for example, we can choose one of these points randomly, and that would be the point minus 2, 2. Here’s another one. Say, that one up there. That would be the point 3,4. And say, this one down here will be the point minus 1, minus 2. But of course, we can also talk about points which are not just having integer values, but also, say, half integer values.

Skip to 3 minutes and 35 seconds So that point there would be described as the point 2, minus 1 and 1/2 or minus 3/2. And this point here would then be the point– let’s move it a little bit over— the point, say, minus 1 and 1/4 and comma 2 and 1/2. A very flexible device for pinning down what points are and how we specify them. In fact, there’s a big advantage here to what Euclid did in that now there’s the possibility of actually defining what a point is as an ordered pair like this. Euclid had to be a little bit vague about what a point actually was. Descartes allows us to actually say that a point actually is an ordered pair enclosed in square brackets. That’s points.

Skip to 4 minutes and 28 seconds Very good. What about lines? Well, here is a line. It’s a very special line because it goes through the point 0,0, which is called the origin. So that’s a very special point. We call it the origin. Its coordinates are 0,0. And this is a line which passes through these three points that we’ve already talked about. And Descartes realised that all the points that lie on this line were given by a linear equation. There’s a fixed linear relationship, in fact, a direct proportionality in this case, between the x-coordinates and the y-coordinates of all the points that lie on this line.

Skip to 5 minutes and 7 seconds In fact, this line is given by an equation, y equals 1/2 x– it’s exactly a direct proportionality where the factor of proportionality is 1/2. And 1/2 corresponds to the fact that if you look at the ratio of the vertical displacement to the horizontal displacement, 1 to 2, ratio of 1/2, even if we choose some other point like this one, the ratio would be 2 to 4. So this is a way of embodying a physical representation for our direct proportionality.

The Cartesian plane and the beauty of graph paper

The 17th century was scientifically the most important period of modern times, with the introduction of calculus by Newton and Leibniz, and the discovery of the laws of motion and gravitation by Newton.

These new discoveries rested on a crucial new development in geometry: the introduction of coordinates into the earlier, free-form geometry of the ancient Greeks by French mathematicians René Descartes (1596- 1650) and Pierre Fermat (1601-1665).

Cartesian geometry

The basic idea that emerged from Descartes’ groundbreaking work La Geometrie is that we view the plane as a sheet of graph paper, in which two perpendicular axes, called the x-axis and the y-axis, allow us to position any point \(\normalsize{A}\) by its coordinates in these two directions. The special point where the two axes meet is called the origin and is usually denoted by \(\normalsize{O=[0,0]}\).

Picture of coordinates with the line y=2x-3 and the point [2,1]

In this way it turns out that lines through the origin are encoded by the simple algebraic equation \(\normalsize{y=mx}\) with the constant of proportionality \(\normalsize{m}\) representing the geometrical slope of the line.

More general lines, which do not necessarily pass through the origin, have equations of the form \(\normalsize{y=mx+b}\).

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