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Symmetry, Anti-Symmetry, and Symmetry Breaking I


In their interpretation of universe patterns, living things need counting. It is not unique to Homo sapiens sapiens (Hoss), but this species is the only one known to realize that there are infinite possibilities for the simplest of things.

Natural numbers N are the model for the simplest of counts:

N = {0, 1, 2, 3,…}.

The simple counting process involves two operations with the natural numbers, namely the addition of "+" and the multiplication of "×" That is, the basic counting system consists of two systems. The additive system has the distinguished element “0” (zero) and the multiplicative system has the distinguished element “1” (one).

We say that áN, +, 0ñ and áN, ×, 1ñ are monoid, ie both have a neutral element for their operation.

The + 0 = The = 0 + The,

The ×1 = The = 1× The

for any element The in N.

The addition operation comes naturally from the counts performed by the Hoss species, but the multiplication operation is much less natural. To simplify the addition of repeated terms, that is, as a measure of saving thought, time, space, etc., the Hoss specimens invented multiplication. Thus we see that there is an asymmetry between addition and multiplication at its very origin. However, the additive and multiplicative monoids áN, +, 0ñ and áN, ×, 1ñ have some similarities.

For example, both are semigroups, that is, both satisfy the associative property:

The + (B + ç) = (The + B) + ç,

The × (B × ç) = (The × B) × ç

for any elements The, B and ç from N.

In passing, we observe that the addition and multiplication operations coexist well, that is, multiplication naturally distributes in relation to multiplication:

The × (B + ç) = The × B + The × ç

It is natural, therefore, to ask whether these two semigroups could not come together to form a single and larger semigroup.

Another question that immediately arises is: are there any other natural semigroups in the universe?

If any specimen of Hoss wanted to do this research, then what could he do?

It is difficult to find a way of investigation just abstractly. Let us therefore follow one of the possible paths which is the historical path.

Hoss mathematicians could not solve the equation x + The = 0 in semigroup áN, +, 0ñ. They found that the problem was the lack of symmetry in this semigroup. It can be observed by geometrically representing the natural numbers on a line. The points are all on one side only from zero. In other words, a semi-straight line can contain all natural numbers and thus a straight line has a superfluous half.

An idea sprang up unexpectedly to continue investigating the existence of other semigroups in the universe. One can try to solve equations and geometrically represent the new sets of numbers obtained as a way to discover more semigroups in the universe.

This is how the so-called mathematical specimens of Hoss created the negatives of natural numbers. The geometric representation of natural numbers became symmetrical and every equation of the type x + The = 0 could be resolved by negative naturals.

All naturalThe”Now has a negative“ - The”And therefore we have:

x + The = 0 Þ (x + The) + (-The) = 0 + (-The) Þ x + (The + (-The)) = -The Þ x + 0 = -The Þ x = -The.

The new semigroup is z, +, 0ñ, where Z = {…, -3, -2, -1, 0, 1, 2, 3,…} was called a set of integers. The letter Z was borrowed from the German word number "zahlen".

The mind of the Hoss species functions nonlinearly and extremely dynamically. It is immediate to ask if the same phenomenon does not occur with the multiplicative semigroup áN, ×, 1st. That is, the analogous equation x×The = 1 also has no solution except in case The = 1. As immediately as the question arises several natural and interesting problems.

To extend the multiplicative semigroup áN, ×, 1ñ to AZ, ×The mathematicians needed to solve the following interesting problems.

(a) How to multiply negative integers? Would multiplication continue based on the economy of thought, time and space? For example, would a negative summed five times with itself produce a negative sum, so a positive number times a negative would produce a negative product?

(b) How could the two semigroups coexist in a single structure, say, AZ, +, 0, ×, 1ñ?

(c) Would it still be worth associative property?

The creativity of the Hoss species has invented a simple solution: just like the semigroups áN, +, 0ñ and áN, ×, 1ñ could naturally coexist through distributive property, the new structure AZ, +, 0, ×1ñ would have no problem admitting the same rule, and multiplication would continue to save thought, time, and space.

However, there was a price to pay. For example, consider equality (1 - 1) × (1 - 1) = 0. If it is for the associative and distributive properties, then we have:

(1 - 1) × (1 - 1) = 0 Þ (1)×(1) + (1)×(-1) + (-1) ×(1) + (-1)×(-1) = 0 Þ 1 - 1 - 1 + (-1)×(-1) = 0

Þ - 1 + (-1)×(-1) = 0.

We see that a price to pay to realize the desire to expand natural semigroups and to solve equations is to admit that (-1)×(-1) must be an integer exactly opposite -1.

Now then (-1)×(-1) must be 1. The remainder of the debt to be paid is the admission that

(-The)×(-B) = The×B,

as can be seen from the same reasoning above. Therefore, "negative times negative has to be positive".

Every newborn being deserves a name. The new structure z, +, 0, ×, 1, distributivañ, which accommodates in the same environment the additive and multiplicative natural semigroups, received the name of ring. Although much more symmetrical than the natural semigroup, it is not sufficient to solve equations. x×The = B.

Hoss's infinite curiosity then invented the inverse integers: every integer Theexcept zero has multiplicative inverse The-1. Thus, a wider ring emerged: áQ, +, 0, ×, 1, distributivañ, the ring of fractions.

The letter Q came from the word quotient, because the expression B×The-1 was interpreted as “B divided by The”, Ie as the fraction B/The.

So every equation x×The = B, with The ¹ 0, now has solution in ring áQ, +, 0, ×, 1, distributive:

x×The = B Þ (x×The)×The-1 = Þ x×(The×The-1) = B×The-1 Þ x×1 = B×The-1 Þ x = B×The-1.

Mathematical specimens often say that the semigroup az, +, 0ñ is a group because all The has inverse additive -The, or opposite of The. Similarly, the semigroup áQ, +, 0ñ is also a group.

As for the semigroup áQ, ×, 1ñ, the same cannot be said because 0 has no multiplicative inverse. Equation 0×The = 1 has no solution in universes where 0 ¹ 1, because 0×The = 0 for all The.

This asymmetry, therefore, cannot be fixed because 0 cannot have a multiplicative inverse, although its additive inverse, ie its opposite, is itself.

The question now, of course, is: what is the capacity of the ring of fractions q, +, 0, ×, 1, distributivañ solve equations, since their geometric representation is symmetrical and fills the line much better?

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