Now, you know, in the same way that we can think of a single number as being a point on the line, we can think of a pair of numbers, an ordered pair of numbers, as being a point in the plane. And this idea of identifying numerical concepts called numbers with geometric concepts called points is a very old device and a device that was used and still is used in the curriculum today under the name of the number line, under the name of graphs, and what we will use as a fundamental building block as our course develops. But notice that in either case, I can in a very natural way define or identify either ratio as a point on the line. And if, on the other hand, the line happens to be only 5 inches long, the resulting point is named 5/3. Oh, to be sure, if this line happens to be 6 units long, this point is named 2. Now, if I can geometrically divide this line into three equal parts, and in plain geometry we learn to do this, then the fact is that I can divide this line segment into three equal parts regardless of how long this line happens to be. Either one can divide a line into three parts of equal length or one can't divide the line. However, as soon as we pick a length as our model, the idea is this. And the reason was is that in terms of visualizing tally marks, it was much easier to see how you divide six tallies into three groups than five tallies into three groups. I remember when I was in grade school that this particular problem always seemed more appealing to me than this problem, that 6 divided by 3 seemed natural, but 5 divided by 3 didn't. Oh, as a case in point, consider the problem of 5 divided by 3 versus 6 divided by 3. And this identification, whereby difficult arithmetic concepts are visualized pictorially, is something that begins not only very early in the history of man, but very early in the development of the mathematics curriculum. And in other words then, we begin to establish the feeling that we can identify the analytic term increasing with the geometric term rising. As the profit increases, the curve rises. The reason that we say profits rise is that when the profits are increasing, if we are plotting profit in terms of time, the resulting graph shows a rising tendency. What profits do is they increase or they decrease. Now, you know, profits don't rise unless the safe blows up or something like this. What we hope to do today is to establish the fact that whereas in the study of calculus when we deal with rate of change we are interested in analytical terms, that more often than not, we prefer to visualize things more intuitively in terms of a graph or other suitable visual aid, and that actually, this is not quite as alien or as profound as it may at first glance seem.Ĭonsider, for example, the businessman who says profits rose. Our lecture for today probably should be entitled it should've been functions, but it's analytic geometry instead, or a picture is worth a thousand words. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. The following content is provided under a Creative Commons license.
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