Summary:
+ natural numbers
- Still Aleph-zero
Rational Numbers
Irrational Numbers
algebraic
; Negative numbers
irrationality of √ 2
Irrationality of whole roots
The historical context +
+ + The Continuous
the continuum hypothesis
still aleph-zero
In first chapter of this "raid" between infinite sets we found that they are not, so to speak, sets infinitely many "small"! Put in more precise terms: there are no cardinality of infinite sets, or power, the smaller the set of natural numbers, whose cardinality is identified with the letters Aleph-0. Now we want to see if we can find infinite sets larger than Aleph-0.
To find a set of "larger" than that of the natural numbers must be (brace yourselves!) find "a set that has an injective function than that for all natural numbers, but no-one correspondence with it" ! In other words: we must find an infinite set that contains members (also in theory you only need one), that can not be put into correspondence with some element of the natural numbers. It can be said also that the elements of the largest not be "counted" only verify this we will be sure to have found a set of cardinality greater than Aleph-0. ▲
Rational Numbers
We can make a first attempt at examining the rational numbers, which are those obtained from the ratio of two natural numbers ( the term "rational" comes from the Latin "ratio", in his own significance of the report). Every rational number is the result of a division a / b where a is the numerator and denominator b, b must obviously be different from zero, while if we have b = 1 the result is an integer: integers (natural) are a subset of rational numbers.
the interval between each pair of consecutive integers can enter as many rational numbers I want, below I show you a graphical representation of this concept:
between numbers 1 and 2, I added 3 / 2 ( 1.5), 4 / 3 (1.333 ...), 9 / 5 (1.8), obviously there are countless other possibilities (I mean someone with those points questions). By calculating the quotient of numerator and denominator of each of these fractions results in a decimal number with a finite or even infinite number of digits after the decimal point (in the latter case, as we learned in junior high, the decimal point will be periodic).
Here, I can think of at least three reasons why the rational numbers should be in greater quantities than the natural numbers: the fact that between each pair of consecutive natural numbers can I put countless villages and the presence of all those infinite decimals, and finally the fact that every rational number is defined by two natural numbers (numerator and denominator). But ...
# 1
Ecco, in questa griglia ho inserito tutte le frazioni possibili e immaginabili: basta cercare il numeratore sull’asse delle X e il denominatore sull’asse delle Y; ovviamente lo schema può essere ingrandito a piacimento. Ho inserito anche dei pallini colorati e numerati: il numero 1 accanto alla frazione 1/1, poi 2 per 2/1 e 3 per 1/2, poi ancora 4 per 3/1, 5 per 2/2 e 6 per 1/3, e così via. In questo modo sto "contando" per diagonali successive tutte le frazioni possibili, anche quelle non ridotte ai minimi termini.
Eccoci quindi al punto: siccome i numeri naturali sono in grado di "contare" le frazioni, e quindi i numeri razionali (*), vuol dire che i due insiemi (dei numeri naturali e of rational numbers) can be put in correspondence, so they have the same power: always Aleph-0.
(*) Clarification: Some localities that appear in the grid # 1 gives rise to the same natural number (eg 1 / 1 and 2 / 2 = 1) or rational (ie 1 / 2 and 2 / 4 = 0, 5) To have all rational numbers different from each other should be excluded from counting all fractions reduced to lowest terms.
not exclude the villages reduced to a minimum is one thing you can do, even if it is not easy when the numerator and denominator become very large. But it would be a wasted effort:
- the set of natural numbers è un sottoinsieme dell'insieme dei numeri razionali (che comprende tutti i numeri naturali)
— l'insieme dei numeri razionali è un sottoinsieme dell'insieme delle frazioni, in quanto più frazioni danno luogo allo stesso numero razionale
— nella griglia #1 ho mostrato la corrispondenza biunivoca fra gli insiemi dei numeri naturali e delle frazioni, che sono quindi equipotenti e di cardinalità Aleph-0.
— allora anche l'insieme dei numeri razionali, che è apparentemente "compreso", come potenza, fra gli altri due, non può che avere potenza Aleph-0 (ometto la dimostrazione rigorosa). ▲
Irrational Numbers
Let's try again: let's try with the irrational numbers, such as the square root of two. This issue has a long history: the Pythagorean theorem to the square root of two matches the length of the diagonal of a square of unit side:
Pythagoras was an enthusiastic supporter of "comprehensibility" of the universe, in the sense that trying to "measure" the secrets in terms of relationships. For example, had studied the musical notes and found that relations between them have "rational"; had also imagined the system of "celestial spheres" as support for the planets and fixed stars, balls always rational relationship between them, and stirring musical notes with the celestial spheres ... had imagined that today we call the "harmony of the spheres"!
So Pythagoras was so happy to have found ... just the Pythagorean theorem! But there was very badly when he discovered that the square root of two is not a rational number. Put yourself in his shoes: the world wants to understand the universe, discovered a remarkable theorem which may help to understand better, and the first result that is outside its policy on what is rational and what is not! (For the proof of non-rationality root of the number of two, see below). In other words, he can prove that the root of two can not be defined by any fraction a / b, where a and b are integers.
In more modern times it has been shown a much more general theorem (see more below), which says that every root of any degree, any natural number, or may result in an integer or an irrational number, rational numbers, ever.
So we are finding a huge amount of numbers that are not rational. The failure to be so, it means that their decimal representation does not present any character of periodicity, having an endless stream of decimal places in succession, we say, chaotic. This thing is an easy explanation: if you remember the mathematics of averages, we were taught to find the "breaking-generating" given any recurring decimal, with or without antiperiodo. So if the irrational numbers, such as the square root of two, were recurring decimal, they would have their beautiful village generating ... and then it would be rational and not irrational!
These irrational numbers are good candidates to see if they can give rise to an infinite set larger, or of cardinality larger than that for all natural numbers. But now as you might expect ... is not so! In fact I can use the same trick used to i numeri razionali: invece di mettere nel grafico tutte le frazioni possibili, metto tutte le radici possibili. Quindi avrò la riga delle "radici prime" (di fatto, la riga dei numeri naturali); poi la riga delle radici quadrate, delle radici cubiche, poi delle radici quarte, quinte eccetera. E tutte queste radici le potrò numerare per diagonali, come avevo fatto con le frazioni: quindi neanche in questo caso ho ottenuto il mio scopo!
#2
Per le radici di numeri che danno luogo a numeri interi, come radice quadrata di 4 o radice cubica di 27, vale lo stesso discorso fatto qui sopra per i numeri razionali. ▲
algebraic
There is a small problem: the new "count" since it leaves out over the rational numbers, for example, the fraction 2 / 3 is not included. How to put together the two classes of numbers? Well, just do a double counting. The table # 1 we had counted the rational numbers: if we replace the numbers in the table below root # 2 with their corresponding fractions, we get a table that shows all the roots (the roots of any grade) of all the rational numbers, so:
- the roots of First Instance of the unit fractions with the denominator danno luogo ai numeri naturali;
— le radici di primo grado dei numeri razionali danno luogo ai razionali stessi;
— tutte le radici di secondo, terzo grado e oltre, danno luogo a tutte le radici possibili, dei numeri naturali come dei numeri razionali.
#3
Possiamo immaginare a questo punto di complicare le cose quante volte si vuole: troveremo sempre il modo di "contare" espressioni algebriche sempre più complicate, senza mai trovare un insieme infinito di cardinalità superiore ad Aleph-0!
Nella griglia #3 sto mettendo in corrispondenza i numeri naturali con espressioni del tipo radice ennesima a / b. Even here, many expressions can give rise to the same number, integer, rational or irrational that it is, is still the same argument already above. ▲
Negative Numbers
But now that I can think of, how about zero and negative numbers? At this point is simple: just do a conversion like this:
1-0
2-1
3 - -1
4-2
5 - 6
-2 - , 3
7 - -3
...
Basically I'm putting in correspondence the set of natural numbers (numbers left) with zero and whole numbers of both signs (right). Of course we can replace any number right (unsigned) the respective fraction, or root, root or a fraction, or any other algebraic expression we want!
From what we saw in this chapter and the last sets of natural numbers, sets seemingly "smaller" as the only even numbers, or square or factorial numbers, and sets seemingly "bigger" as the rational numbers, irrational and algebraic, even based on the sign ... all these sets are equipotent and have cardinality aleph-0! ▲
irrationality of √ 2
This demonstration is listed in the "Elements" of Euclid, and is based on reasoning by contradiction. Both
[AB] side and [AC], the diagonal of a square, and suppose that the two segments are to each other as the fraction m / n, reduced to a minimum. Then:
[AC] ² [AB] ² = m² n ²But, for the Pythagorean theorem:
[AC] ² = 2 ּ [AB] ²And then:
m² = 2 ּ n ²It follows which m², and therefore m is even. It must therefore be odd number n. Let m = 2 then
ּThen q = 4 ּ
² ² q = 2 n ² ּa result is even. It is therefore also equal to n, which have proved to be odd. The result is an incompatibility that proves the claim. ▲
ּ q 2 = n ² n ² ²
Irrationality of whole roots
"Let a and n two integers. If the root of n-but does not exist in the integers, it does not even exist in the field of rational numbers. "
This proves by contradiction. Let x = p / q a rational number such that x ª = n, p eq prime. Then also p ª q ª and are relatively prime.
Having to be n = x ª = (p / q) th, is p = n ּ ª q ª q ª and thus is a divisor of p ª. But p and q ª ª are relatively prime, and this can only occur if q = 1 ª. It would then x = p / 1, integer, unlike the case. ▲
Next Chapter: The historical context
0 comments:
Post a Comment