Why is algebra easier for someone, and geometry for someone?

Why is algebra easier for someone, and geometry for someone?

Algebra Basics: What Are Functions? - Math Antics

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answers (3)

Answer 1
March, 2021

In general, if you study higher mathematics at the 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 by the way called: algebraic geometry.

There was also geometric algebra. That is, none of her contemporaries called her that, but in general, the late mathematics of the Pythagorean school was exactly something like that - the solution of equations arising from lengths, areas and volumes. Precisely because the Greeks were too attached to visual science, they never thought of equations above the third degree.

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

Geometry in Greek means "surveying" and served, of course, quite visual and practical purposes - the production of nyamka. Later, it found its application in architecture and art. Well, it's pretty easy to measure those things with a ruler or something else. Although quadratic-cubic equations constantly appeared here - and they will soon appear in your school curriculum in geometry.

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]", he was not very friendly with geometry itself. He was more of a mystic, believed that the "correct" instruments are only a compass 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, the Academy was founded by Plato, and not by Archimedes, who hobbled European science with his authority for two millennia.

Algebra, in turn, first of all came from the field, which is also visual, but in which you can't measure much with a ruler , namely - from astronomy. Like land surveying, this discipline was also closely related to the production of nyamka, namely, to the creation of a calendar. Well, and, of course, astronomy until the Early New Time remained inseparable from astrology, which was considered (and by the overwhelming majority of people even now, unfortunately, considered) a thing no less practical - everyone wants to know the future. And it was there that complex equations were required to calculate in which constellation when which planet would be. Of course, other uses also appeared - to count, for example, how many rabbits (Fibonacci numbers) would be born, how much grain to leave for sowing, how much interest on debts would run, how many grains to put on a chessboard, and so on.

So or otherwise, geometry was more about space, and algebra was more about 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.

Everything began to change when a young French officer Rene Descartes, being in the service of the Emperor of the Holy Roman Empire, fought during the Thirty Years War in modernCzech Republic. Once, lying on the bed, he looked at the sunbeams on the wall of the room, and suddenly realized that he could characterize the position of each of them with two numbers: distance from the corner of the room and height from the floor. That is, points are, in a sense, also numbers or their sequences. And sequences of numbers can already be added to each other, multiplied by a number ... and generally 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 acquire 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 that it was impossible to solve the classical problems of geometry, inherited through the fault of the same Plato, over which people fought for two millennia: squaring a circle, trisecting an angle and doubling the volume.

Currently, divide algebra and geometry are possible only in the school curriculum. Just as space and time, with the advent of the theory of relativity, ceased to be something separate, turning into space-time, algebra and geometry have merged so that it is already not clear where of them what. The same point can mean a function - only it will be a point no longer in the two-dimensional space of the wall, which René Descartes was looking at, but in the infinite-dimensional space of functions. Already mentioned algebraic geometry turns everything inside out in general, setting a point, roughly speaking, by the set of all functions that vanish at this point. And in non-commutative 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'm doing 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 represent them in the form of something geometric - this sometimes gives a good intuitive idea of ​​some phenomenon, and then it can already be expressed in strict formulas.

Answer 2
March, 2021

I dare to assume that there is a drawing element in geometry, roughly speaking. Rules, theorems, etc., imposed on the figure, are easier to remember, spatial thinking works. Algebra is dry numbers, constants and variables, but mathematicians will not take offense at me for such a comparison. And so it goes.

Answer 3
March, 2021

Algebra is easier for me, because it is simple and understandable to read information recorded according to generally accepted standards.

And geometry ... There are no problems with it either, I just absolutely did not know how and do not have to draw and draw, so geometry was not given to me from the technological side. The question is about the school course, as far as I understand?

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