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Symmetry in Mathematics II


It's interesting to think of the points on the Euclidean line (the line of geometry you knew in elementary school) as numbers. This invention was responsible for a great revolution in seventeenth-century mathematics. It is the famous Cartesian coordinate idea. With it we can think of geometric problems using numbers and thus algebra. The points of the Euclidean line become, from this point of view, elements of an algebraic structure: it is the very important structure of real numbers. In fact, we can study the points of the Euclidean line, for example, as an algebra structure or as a body structure (we will not need in this article the definition of body). In the previous column we mentioned the problem of whether there would be “holes” between two real numbers or, equivalently, if there would be “empty spaces” between two points of the Euclidean line. Every serious student of mathematics is faced with this problem if he is, of course, in contact with good books on mathematical analysis. Good books put important ideas in order and give competent treatment to their clarification and their logical justification. Here we will continue to expose some of these ideas from an intuitive and informal point of view.
The human need to "see" symmetry in ideas and concepts leads to an attempt to "fill in" the possible voids that may exist between two real numbers, say "in horror of the vacuum." Late nineteenth-century mathematicians, such as Dedekind, found a justification for the absence of empty spaces between two real numbers. Let us now imagine the Euclidean line as the real numerical line.
If the points are also numbers, it is inevitable to ask what algebraic relations these numbers satisfy. For example, as we have already noted, Pythagoras faced a solution of the equation x2 - 2 = 0, the square root of 2, and discovered the problem of "how to accommodate" this point on the Euclidean line, although not from Descartes' perspective. a thousand years later, who clearly “assumed” that each point corresponded to a single number and each number corresponded to a single point of the Euclidean line. Pythagoras would not, for example, have the left side of the line because he did not know the negative numbers. But Pythagoras did not find on the Euclidean line a point to represent the hypotenuse measure of a right-angled triangle whose collars measured 1. Algebraically speaking, the problem was which type of number would solve the equation x2 - 2 = 0. With the completion of Numerical straight line obtained by Dedekind, we can then naturally ask which equations are solved by this completion and which do not. We immediately remember the equation x2 + 1 = 0. There is no way to find a point on the Euclidean line for the solutions of this equation. And speaking of your solutions, what are your solutions?
Again history repeats itself, albeit in a spiral, and now we are at a level above that which led us to discover the hidden symmetry between any two real numbers. The hidden symmetry, which now motivates us to search again for its identity, may be justified by the question: why should there not also be "numbers" that solve the equation x2 + 1 = 0? For the sake of “symmetry” there must be a “geometric space” as well as the space of the Euclidean line containing the “points” that are the “numerical solutions” of this equation.
What clues are there for us to pursue this hidden symmetry? The very idea of ​​symmetry is already a big clue. If we believe in the existence of a point space that symmetrically solves the problem of having all polynomial equations have numerical solutions, then let us begin by describing what this space must be like. That is, if he exists, and if he has the properties that we "need" him to have to satisfy our "desire for symmetry," then can we no longer elect some candidates for this position in the world of mathematics? You have already realized that we are heading for an obvious candidate: the Euclidean plan. If the Euclidean line was able to represent the solutions of some polynomial equations, would not the Euclidean plane be the natural continuation of this space, and would not its points be the “numbers” we eagerly hope to solve all polynomial equations?
Historically this was not easy. Many great mathematicians participated in different times of this saga, obviously without having the point of view of the symmetry we suggested above, so that today we can think of this problem in a simple, unified and harmonious way. Where is the square root of -1? It cannot be on the Euclidean line because there the space is complete, no more points. Algebraically speaking, there is no way to square a real number and get -1. The need to see the world through symmetries leads us to forge a new space of points that does not “spoil” the space we have already conquered in the Euclidean line. The Euclidean plane, continuation of the Euclidean line in a “second dimension” is therefore the natural candidate.
If so, then we are forced to solve some problems. What will the addition of these new numbers look like? What will their multiplication be like? We are led to suppose that they have the possibility of addition and multiplication because they have to be an "extension" of the real numbers. It is natural, then, to suppose that they also add and multiply with each other. Moreover, these operations must satisfy the properties that real numbers satisfy. Thus, if we take two pairs of real numbers to represent two points of the Euclidean plane, we will be forced to conclude that their sum and their product are given by those rules you already know as operations between complex numbers. The question remains whether this solves the problem of having every polynomial equation have a solution. The answer to this question is the famous Algebra Fundamental Theorem, Gauss's Doctoral Thesis: "Every complex polynomial has a complex root."
We now have two algebraic structures: the algebra of the real numbers (represented geometrically by the Euclidean line) and the algebra of the complex numbers (represented geometrically by the Euclidean plane).
Can you now pose a very natural next question in this logical order of ideas: "Is it not then that Euclidean three-dimensional space would be hiding even wider symmetry"? Or put it another way: “Isn't Euclid's three-dimensional space, then, a 3-dimensional algebra that extends the properties of complex numbers to 3-coordinate numbers”?
Make your bets. The pessimist might predict: "The answer is negative because, as Gauss has shown, polynomials are already satisfied with complex numbers, and no more numbers are needed to solve them." The optimist might retort: ​​“It is strange that this symmetry ceases in dimension 2; in dimension 3 it must be hidden! ”.
And they might add, "It's interesting for an algebraic structure to satisfy our desire to symmetrically solve all polynomial equations, but what are all these elucubulations for?"

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