3 Answers

  1. In general, if you study higher mathematics at an institute, you will soon realize that the difference between algebra and geometry is very, very conditional. One of the most beautiful, abstract and advanced branches of mathematics is called algebraic geometry.

    There was also geometric algebra. That is, none of her contemporaries called her that, but in general, late mathematics of the Pythagorean school was just such a thing – solving equations that follow from lengths, areas and volumes. It is precisely because the Greeks became too attached to visual science that they never came up with equations higher than the third degree.

    If you look at the root, algebra and geometry in general go back to two different fields of activity.

    Geometry in Greek means “land surveying” and served, of course, quite visual and practical purposes-the production of yumka. Later, it found its application in both architecture and art. Well, it's quite easy to measure those things with a ruler or something else. Although there were always quadratic and cubic equations here-and you will soon have them in the school curriculum in geometry, too.

    Here, by the way, it should be said that in many respects the limitations of Greek mathematics are the fault of Plato, who, despite the famous phrase ” He who does not know geometry will not enter [the Academy]”, was not very friendly with geometry itself. He was more of a mystic, believed that the” right ” instruments are only compasses and a ruler, and imposed this point of view on his students. Although at that time Greek scientists could already draw ellipses, and developed the basics of mathematical analysis. Unfortunately, it was Plato who founded the Academy, not Archimedes, who hobbled European science for two millennia with his authority.

    Algebra, in turn, first of all went from the field of also visual, but in which you can't really measure with a ruler, namely, from astronomy. Like land surveying, this discipline was also closely connected with the production of yam, namely, with the creation of a calendar. And, of course, astronomy up to the Early Modern Period remained inseparable from astrology, which was considered (and the vast majority of people still, unfortunately, are considered) a thing no less practical – everyone wants to know the future. And that's where complex equations were needed to calculate in which constellation when which planet will be. There were, of course, other applications-to calculate, for example, how many rabbits would be bred (Fibonacci numbers), how much grain to leave for sowing, how much interest on debts would run up, how many rice grains to put on the chessboard, and so on.

    Either way, geometry was more concerned with space, while algebra was more concerned with time. It just so happened that geometry was better given to people in the West, and algebra – in the East. No wonder the first of these words is Greek, and the second is Arabic.

    That all began to change when a young French officer, Rene Descartes, was in the service of the Holy Roman Emperor when he fought in what is now the Czech Republic during the Thirty Years ' War. One day, lying on the bed, he looked at the sunbeams on the wall of the room, and suddenly realized that he could describe the position of each of them in two numbers: the distance from the corner of the room and the height from the floor. In other words, points are in a sense also numbers or sequences of them. And sequences of numbers can already be added to each other, multiplied by a number… and in general, a lot of interesting things to do with them. Thus, the point, the fundamental object of geometry, which for a long time remained a thing in itself, began to take on an algebraic meaning, and gradually geometry and algebra began to converge, enriching each other and giving rise to modern mathematics. It was by algebraic methods, in particular, that it was proved impossible to solve the classical problems of geometry that we inherited through the fault of the same Plato, over which people struggled for two millennia: squaring a circle, trisecting an angle, and doubling the volume.

    Currently, it is possible to separate algebra and geometry only in the school curriculum. Just as space and time ceased to be separate with the advent of the theory of relativity, becoming space-time, algebra and geometry have grown together so that it is no longer clear which of them is which. The same point can mean a function – only it will no longer be a point in the two-dimensional space of the wall that Rene Descartes was looking at, but in the infinite-dimensional space of functions. The already mentioned algebraic geometry generally turns everything inside out, defining a point, roughly speaking, by the set of all functions that vanish at this point. And in noncommutative geometry – another branch of modern mathematics-despite the name, there are no points at all, only algebra, although there is, for example, the concept of volume.

    Personally, I was better at geometry at school. Now I am engaged in functional analysis, but despite the fact that in our discipline we constantly have to work with infinite-dimensional spaces that cannot be drawn, sometimes it is convenient to present them in the form of something geometric-this sometimes gives us a good intuitive idea of some phenomenon, and then it can be expressed in strict formulas.

  2. I dare to assume that there is a drawing element in geometry, roughly speaking. Rules, theorems, and other things that are superimposed on the shape are easier to remember, and spatial thinking works. Algebra – dry numbers, constants and variables, may mathematicians not be offended by such a comparison. And so it turns out.

  3. Algebra is easier for me, because reading information written down according to generally accepted standards is simple and understandable.

    And the geometry… There are no problems with it either, but I just didn't know how to draw or draw at all, so I couldn't do geometry from the technological side. The question is about the school course, as far as I understand?

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