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Symmetry in Mathematics III


Study without thought is lost work; Thinking without study is dangerous.

CONFUSION (551-479 BC), Analecta.

… “It's interesting that an algebraic structure satisfies our desire to symmetrically solve all polynomial equations, but what are all these elucubulations for?”.

The trick to solving the equation x2 + 1 = 0 was to imagine a second line of numbers, which can be represented perpendicular to the line of real numbers on a Euclidean plane. This second line contains the imaginary numbers (a name that was chosen in history because for a long time the reason for the existence of these numbers was a mystery)…, -3i, -2i, -i, 0, i, 2i, 3i,… Are the multiples of the imaginary unit i. But these are not the only multiples of i. Any real number can be multiplied by i. For example, p i is a multiple of i, or yet, Ö 2 i, the square root of 2 times i. In this additional line, perpendicular to the line of real numbers in the Euclidean plane, lie all pure imaginary numbers. That is, the numbers invented to solve the equation x2 + a = 0 where The is a positive real number. Since we want this line to bear a great resemblance to the actual line we already knew, it must contain exactly one copy of the previous one. We can understand this copy very easily: imagine that the real line's 1 is replaced by the imaginary unit i. Okay, now just imagine a real number The any and your imaginary copy The i. In algebraic terms, if we multiply The i perB i, we get (The i) (B i) = ab i2 = - ab. As for the sum of two pure imaginary we have, easily, by imitation of the real sum: The i + B i = (The + B) i. Thus we have perfect symmetry between the real line and the imaginary line. Let us note that the word "imaginary" is only a mode of expression because the imaginary line is really nothing imaginary, since it is a simple line perpendicular to the real line that was already familiar to us in Euclidean geometry.

Very well, but what about the other points of the Euclidean plan? For now we use only two lines, perpendicular to each other, from the Euclidean plane. What to do with the other points of this plan? Well, the natural question that is falling ripe to be asked here is: can we do something with them? Why not try to do with them the same things we do with real numbers, such as adding, multiplying, dividing, calculating powers, extracting their square, cubic roots, etc.? We have an excellent guide which is the algebra structure of real numbers, and we can, imitating everything that happens in that structure, try to transplant it to the complex numbers, that is, to the points of the plane. Before we embark on this adventure, let us recall the basic properties of the structure of real numbers. Imitating real numbers is the only way we can succeed with the structure of imaginary or complex numbers.

Real numbers have body structure. What is a body? It is a set of symbols that can be manipulated according to certain rules. There are only two operations: a “+” addition and a “'” multiplication. Symbols can be imagined as the points of a Euclidean line. There are two special symbols: 0 and 1. Actually, 0 works in addition as 1 in multiplication, that is, we have almost perfect symmetry of behavior of these two symbols. We're saying "two" but we don't know that yet. That is, we have not yet discussed why they are different. This is a very interesting fundamental idea of ​​mathematics. Why does 0 have to be different from 1? In fact, there is no reason for that. Otherwise let's see: if 0 equals 1, then 2 = 1 + 1 = 0 + 0 = 0, that is, no new numbers generated by adding unit 1 to itself! This causes the structure generated by 1, which is equal to 0, to collapse forming a set of only a single symbol {0} = {1}. Now what to do with a set of only one symbol? At most it will model a universe of only one object…! Therefore, there is no contradiction in identifying 0 with 1. The only drawback is that the structure generated in this case is not funny. Mathematicians would say:it's an uninteresting structure" Uninteresting because there are no patterns to be discovered and studied. The symmetry is total. When the symmetry of a structure is total, or too perfect, we cannot detect anything in it. It's like imagining a circle and rotating it from a 45-degree angle: what difference is there between the two? None. But if you rotate a square from a 45 degree angle, what difference is there between the two? Well, now we see a diamond (which is still a square in this case, but the visual feel is completely different)…! This is the key to understanding why the "idea of ​​symmetry breaking" has been so successful in physics for the past 50 years. The reality perceived by man seems "a symmetry that has been broken". A big question is, therefore, where is the hidden symmetry? We can consider the "collapsed structure 0 = 1" to be perfect, total symmetry. The main consequence of this is that we cannot think of anything else. There is nothing to be noticed, nothing to be asked, no clues to any interesting patterns. The only pattern comes down to dull operation The + The = 0 = The ' The = 0 = 1 + 1 = 1 = 0 + 0 = 0 ' 1 = 1 ' 1 = 1 = 0 = The. That is, there is no reality, there is nothing but a point 0 = 1, and nothing escapes that point, and nothing interesting happens to that point.

One way to escape this doldrums is to imagine a universe where 1 is not 0. This "innocent hypothesis" alone gives us a whirlwind of possibilities. The first question, of course, is: if 1 is not 0, then how much is 1 + 1? Instantly we fall into a very rich situation where clues of interesting patterns gush, but uncertainties as to the possible structures remain. That is, we need to decide which possibility we will investigate. What clue to find a reality we will follow. Continuing this research requires that we answer the question: Is 1 + 1 different from 1? We can organize our thinking in a simple way by arguing that since only two symbols, 0 and 1, populate our universe for now, it is crucial to decide whether 1 + 1 would be a new entity. Moreover, the same question applies to cases 0 + 1, 0 '1, 1' 1, 0 '0 and 0 + 0: will new ones be generated here too? A body is a structure with two operations, addition and multiplication, with their respective neutral elements 0 and 1, and satisfying certain properties. By conceiving 0 and 1 as neutral elements, we have already solved half of the previous problem. That is, the questions are automatically resolved: 1 '1 = 1, 0 + 1 = 1, 0 + 0 = 0, 1' 0 = 0. Just note that 0 and 1 are the neutral elements, respectively, of the addition and the multiplication. For example, 1 '0 = 0 because 1 is the neutral element of multiplication, so it does not affect the number by which it is multiplying. Symmetrically, 0 + 1 = 1, because 0 is neutral in addition. The problem of whether 1 + 1 is different from 1 is much more engaging. We have discussed in previous columns the cases where 1 +… + 1 can be 0. These are the finite number structures. We now begin the discussion of the case where 1 +… + 1 is never 0, whatever the number of plots in this sum, that is, the case of the real body.

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