Comments

Symmetry, Anti-Symmetry, and Symmetry Breaking II


In the previous column we found an inevitable break in symmetry: 0 cannot have a multiplicative inverse, although its additive inverse, that is, its opposite, is itself. Then we come across a fundamental, natural and logical question: 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?

From an important geometric point of view, the fractions symmetrically occupy the Euclidean line, but as Pythagoras realized, certain hypotenuses are of length that should be carried with the compass to a point on the Euclidean line.

Pythagoras had to solve the equation x2 = 2! A rectangle triangle of collars measuring 1 has hypotenuse ¸ which is not a fraction, that is, there are no integers. P and what such that P/what = ¸. With the mental compass, Pythagoras probably imagined that some point on the Euclidean line would be situated at a distance ¸ from an O point.

Thus an infinite number of irrational numbers, that is to say, which are not fractions, but can still be accommodated on the Euclidean line, come into play. Georg Cantor showed us, around 1800, that the infinite quantity of the irrationals is much greater than the infinite quantity of the fractions. For those interested in this, we have already studied in previous columns the cause of the existence of infinite types of infinity.

Getting back to the real numbers, how are we going to conceive them? Through the infinite decimal representation we make the following distinction: those whose decimal expansion is not periodic (the irrational) and those whose decimal expansion is periodic (the fractions or the rational). The new creature is called the "set of real numbers," or simply R.

For example, 1 = 1,00000… 0000… We just have to take special care: we can also write 1 = 0.999999… 9999…! The explanation is simple: the number on the right is not less than 1 because it has infinites 9's and therefore surpasses any one that is less than 1. On the other hand, it is not larger than 1, of course. Therefore, it can only be equal to 1. To avoid ambiguity in the infinite representation of decimal places, we will agree that infinite consecutive zeros will be replaced by infinites 9's by decreasing a unit in the square before the first zero that repeats infinitely. However, to make additions and multiplications, we may use either representation.

For example, 1.39000… 000… could be exchanged for 1.38999… 999…! This way, every real number has a single infinite decimal representation. Number 2 has the representation 1,999… 999…, number 2,1 has the representation 2,0999… 999…, and so on.

We must not forget that certain logical axioms are necessary for our investigation. For example, we just used the axiom of which The no smaller than B and neither is bigger than B, then The = B.

This elegant characterization of real numbers by infinite decimal places makes us definitely committed to "infinity." There is no way to get rid of it and, for now, why should we? Galileo was frightened (like many others) of mathematical infinity, and advised us to avoid it, but that is past water.

We then reach the mental level of the real numbers: ar, +, 0, ×, 1, distributive. How are we going to add two numbers with infinite decimal places, say ¸ + ¹?

We cannot say that we simply add the corresponding decimal places. There is a problem: there is no last square of ¸ and ¹, for example.

Imagine that, although it does not exist, yet the irrational number does exist. But why? Well, by the way!

We still have to explain how we will add, or multiply, two numbers with infinite decimal places. We already know how to do this with fractions.

For example,

0,4999… 999… + 0,333… 333… = 0,5 + 1/3 = ½ + 1/3 = 3/6 + 2/6 = 5/6 = 0,8333… 333…

However, in the case of ¸ + ¹, the joke is much more serious. Well, let's do this: Suppose that this sum exists, that is, that it is also a real number, and therefore also has its infinite decimal representation. In any calculation involving these numbers, all the properties of the fraction ring will also be hypothesized, and that's it, let's touch the boat!

If someone threatens to be left behind, we can say this: indeed, with great will and patience, we could discover the first decimal places of ¸ + ¹. For example, as ¸ = 1,4… and ¹ = 2,2…, we have ¸ + ¹ = 3,… We cannot yet say which is the first place after the comma, but certainly the decimal place of the whole units of ¸ + ¹ is 3. To find the next decimal place, we would have to know what the second decimal place is after the comma of ¸ and ¹. There would be no problem with that, we would just spend time and energy. So we left no unbelievers behind in this game.

The game we'll continue to play is trying to figure out how far Hoss's mind can go with these hypothetical constructs. The Euclidean line was best filled with the irrational. In fact, it has now been completely filled. The possibility of infinite decimal places for each real number covers any hole in the Euclidean line. From now on, we will always imagine the Euclidean line as something continuous, without holes. In other words, between two real numbers there is a third, and therefore there are infinities, far more than the number of fractions!

Since each rational number ¹ 0 has a multiplicative inverse in the ring of fractions, we say that the ring is a body. The body of fractions q, +, 0, ×, 1, distributivañ extended to the body of the real numbers ÁR, +, 0, ×, 1, distributive. The essential fact was that equations of the form xno = The, Where The is a positive fraction, they came to have irrational solutions the way The1 / n, that is, root no-th of The. As with the square root case, any root no-th of a nonnegative rational number comes into being, often as an irrational number. The infinite decimal places of The1 / n can be patiently discovered by approximations as for ¸ = 1.414…

However, you cannot use this strategy to solve the equation x2 = -1. We will need to leave the Euclidean line to accommodate new numbers that will solve this type of equation. We start with the hypothesis that there is a number i that satisfies this equation. Like this, i2 = -1. Since we want to preserve all properties of the body of real numbers, we have to admit the opposite of i which is the -i, and all the other multiples of i the form there, Where The It is a real number.

Complex numbers come into play as a strategy for solving equations like x2 = -The, being The a positive real number.

Back to columns

<