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Symmetry, Anti-Symmetry, and Symmetry Breaking III


A new number indicated by the letter i with the property that i2 = -1. This number does not fit on the Euclidean line where the real numbers have accommodated leaving no room for any other numbers. Mathematicians fulfilled the desire to geometrically represent the multiples of i choosing a Euclidean plane, with a straight line to represent the real and the perpendicular line that goes through zero to represent the pure imaginary. They named the new axis "imaginary axis" and the number i became the generator of this axis, just as real 1 is the generator of the real axis.

So any real number of the form The.1 now has an imaginary analogue of the form The.i. Immediately, an inevitable question arises: what universe is this where real numbers inhabit? The and imaginary numbers B.i Where B it's real?

It is clear that by the very nature of the invention of the imaginary axis, this universe is the Euclidean plane. That is, each point on the plane represents a new number that we will call a "complex number." The geometrical representation of these numbers in the Euclidean plane can be done by arrows, originating from the intersection point of the chosen axes that can be represented by coordinates (0, 0) or by the expression 0 = 0 + 0i.

We have already noted that we want to preserve all body properties of real numbers. The next inevitable question is: Do complex numbers also have an algebraic body structure? The answer is known to readers and is affirmative. We can say that AC, +, 0, ×, 1, distributivañ is a body where C = {The + Bi: The, B Î R} is the set of complex numbers. The first big difference is that we have lost the order compatible with the addition and multiplication operations. That is, we cannot work with the notion that "one complex number is smaller than another" in a manner compatible with addition and multiplication. Remember that if for two real numbers we have the relation The < B, what if ç is a positive real number so B.C < bc. If we apply this reasoning to the number i, assuming it is positive, we get: 0 < i implies 0.i < i.i, ie 0 <-1, which is absurd. Similarly, it will not work to suppose that i It is negative. How i It is not zero, we see that the order properties of real numbers cannot be extended to complex numbers.

It is an intriguing loss. Does it represent a limitation to nature's phenomena or, on the contrary, allow a greater diversity of her behaviors? This is the biggest and difficult question that motivates us to write these notes.

We must not forget that the invention of i2 = -1 was also “intriguing”. In fact, we better change this adjective to "stimulant." We were blocked in the desire to extend the order of reals to complexes, but on the other hand, we gained a lot in the ability to solve polynomial equations. In fact, not only the equation x2 = -1 now has solutions in the universe of complexes, as well as any polynomial of the form xno + Then-1xn-1 +… + The1x + The0 = 0, where the coefficients are real or complex, have “no solutions ”. The great mathematician C. F. Gauss gave us this knowledge rigorously. We are then facing a major expansion of the numerical universe. In it we can extract no roots no - very z = The + Bi. Except, of course, in the case of 0 that has only one root no - That's right, himself. All other complex numbers z have no roots no - nth symmetrically distributed in a circumference centered on 0 radius equal to the root no - ith distance from z up until 0 indicated

by |z|.

The symmetries present in the algebraic structure of complex numbers are remarkable. We cannot present them here, but we need to present their vector structure. To add two complex numbers, z = The + Bi and w = ç + diwe can imagine two forces applied at the point (0, 0) and their resulting z + w = (The + ç) + (B + d)i. Our reader knows this interpretation as the "parallelogram rule." This interpretation is a great application of vectors to the physics of nature. It is inevitable, then, to ask: what other interpretations of physical phenomena are possible with vectors?

There are countless applications of complex numbers to the phenomena of nature. First of all, the applications of complex numbers to geometry and other areas of mathematics are fantastic. We cannot present them here, not even give an idea. However, we will need the idea of ​​“vector multiplication”. By imposing that complex numbers maintain as many properties as possible of real numbers, we are forced to admit that z.w = (The + Bi).(ç + di) = (B.C - bd) + (ad + bc)i.

But geometrically, it is important to "see" the effect of multiplication. So, using the great mathematician Leonhard Euler's discovery that z = |z| andiwhat and w = |w| andif, Where and is Euler's number, and what and f represent the angles between the vectors z and w with the real axis called “arguments” of complex numbers. So if z = The + Bithen |z| = (The2 + B2)1/2, by the Pythagorean Theorem, |z| waistband what = The, |z| sen what = B, and Euler's discovery is written as andiwhat = waistband what + i sen what .

There is, in Euler's formula, an implicit notion of complex powers that we cannot discuss here, but remember that the rule of multiplying powers of the same base is maintained by holding the base and adding the exponents. By this rule we can write z.w = |z| andiwhat . |w| andif = |z|.|w| andi(what +f). This result clearly shows that in the multiplication of complexes, their distances to the origin are multiplied and their arguments summed. This geometric interpretation of complex multiplication is of great importance and applies to the study of nature phenomena.

Since we are interested in the most basic applications of nature's phenomena, we are interested in mentioning the possibility of multiplying complex plane vectors in two other ways. The first is to represent the work done by the force z = The + Bi assuming she displaces a body of THE up until B. The dots THE and B can also be represented by complex numbers and so we can write B - THE = w = ç + di. Physical theory tells us that only the component of z towards w produces the offset w. It is fortunate to have a simple algebraic formula for calculating the work done on this shift. It is called the scalar product: z · w = (The + Bi) · (ç + di) = B.C + bd! The simplicity of this formula is even more impressive when we remember that physicists defined the work done by force. z with displacement w through the formula z · w = |z|.|w| cosa, Where The is the angle that the force z causes the displacement w. That's because |z|.cosa is the force intensity that moves the body in the intensity shift |w|.

We can demonstrate that z · w = |z|.|w| waistband The = B.C + bd! It's a great formula that immediately leads us to the question: what other simple and interesting formulas exist involving multiplication of vectors that measure nature's phenomena? Surprisingly, we find that there is yet another permissible multiplication for complex plane vectors of important physical significance. The vectors z = The + Bi and w = ç + di can be interpreted geometrically as generating sides of a parallelogram with lengths |z| and |w|. Unsurprisingly for the reader what area of ​​the parallelogram = |z|.|w| sen The. What interests us here is the extraordinary formula “parallelogram area = |z ´ w |", Where z ´ w It is the so-called vector product known also from our reader.

This product measures, for example, the torque produced by the force w applied at a point that determines the lever arm z. More precisely, what we want to explore is the fact that physical quantities, such as torque, are simply represented by a vector multiplication which is one of the possibilities of vector multiplication. It is inevitable to ask: are there still other possible multiplications for vectors with relevant physical significance?

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