Elly Jackson With Long Hair

transfinite numbers: The continuing

Summary:

+ natural numbers
+ Still Aleph-zero
+ The historical context
- The Continuous
Line of Euclid
The 'numbers
The transcendental number
real numbers
The "Continuous"
power of the continuum
The proof of Cantor
+ the continuum hypothesis

The line of Euclid

Un punto è ciò che è privo di parti

Credo che sia la più celebre definizione della geometria: è la prima fra quelle elencate da Euclide nei suoi "Elementi". Le successive tre invece sono:

Una linea è una lunghezza senza larghezza
Le estremità di una linea sono punti
La retta è quella linea che giace sui suoi punti in modo uniforme

Se fossi uno che non ne sa niente, non so se da queste definizioni, soprattutto la quarta, saprei immaginare cos’è una linea retta... meno male che tutti noi abbiamo almeno una volta tracciato una linea con righello e lapis! Però mi sembrano interessanti un paio di considerazioni:

- A line not only "has no width, but is made by points" that have no parts ": these points are so small, tiny, infinitesimal let's say, if I could draw an imaginary line, it would be absolutely invisible!

- The fact that the points are free to share, leads to form a continuous line of these points must be joined together with an extreme density, appear to see the punishment of "gaps". We shall see later reappear the concepts of density and continuity. ▲

The 'numbers

Working with lines and numbers, Richard Dedekind (1831-1916) has constructed a rigorous theory that we will describe in an intuitive way: a straight line can be used to represent all the existing numbers . In practice, it takes a straight line (of infinite length), draw a point that will be the source to assign the value zero, then you establish a unitary segment, and return to the right on how many times he wants, we get to the right of the zero positive natural numbers, negative numbers on the left.



course, the only points identified by the integers do not result in a line continues, but only a series of points spaced well apart. Then we add the rational numbers: dividing each unit segment into n equal parts, we obtain the representation of fractions of denominator equal to n, repeat this step for each n, we get infinite points on the line, representing rational numbers.



These points are distributed in a "dense", meaning that between any two rational points there are infinitely many other points of this type, because given two points P and Q any one can always find a new point X between P and Q: Just take the average: X = (P + Q) / 2. This can be repeated as many times as you want, so do not make sense to ask how many there are in the range of rational points they marked.

However, in dense, rational numbers do not "fill" the line completely. In fact, the point R = square root of two but is represented above the line (just bring on it, from the origin, the diagonal of the square of the segment [0-1]. However, it is not a rational, as we shown in previous chapters.



Using geometric methods can be drawn on the line many irrational numbers, but not all. For example, remained unresolved, the days of ancient Greece, the problem of "doubling the cube" to do this would be necessary to determine a segment corresponding to the cube root of two (it has proved impossible, at least with the help of the elementary means of classical geometry: line and the compass).

One of the myths that surround the problem of duplication of the cube is as follows. The inhabitants of Delos, Apollo's oracle queried on how to break free from the plague, were ordered to build an altar, rectangular, double the volume compared to the existing one. Given the impossibility of determining a segment with ruler and compass proportional to the cube root of two, this altar must not have been ever built, then who knows the people of Delos as they did to get rid of the plague ...

Whether many points can not be drawn with classical methods, we can imagine, however, to identify our line all algebraic numbers, namely those of the type discussed in the previous chapter: are those numbers which are obtained by solving any polynomial equation of any degree. But even as we cover all the straight, in fact ... ▲

The transcendental number

... esiste una classe di numeri detti "trascendenti", che sono gli irrazionali non algebrici. Il caso più noto è quello di pi greco: dall’antichità si cercava di capire che razza di numero fosse, e se si potesse fare la quadratura del cerchio (disegnare, con riga e compasso, un quadrato di area pari a quella di un cerchio dato). Ne parla anche Dante nel canto XXXIII del Paradiso:

Qual è ‘l geometra che tutto s'affige
per misurar lo cerchio, e non ritrova...

Insomma ce n’è voluto del bello e del buono, ma finalmente (solo!) nel 1882 Ferdinand von Lindemann (1852-1939) dimostrò la trascendenza di pi greco chiudendo definitivamente la questione della quadratura del circle. In those years, they found many other issues of this type, though not so easy to prove the transcendence. ▲

real numbers

Finally, we can define a new class of numbers: the so-called real numbers. Is the set of all numbers, natural, rational, irrational and transcendental. As I mentioned, Dedekind has demonstrated the ability to put points on the line-one correspondence with real numbers, since the points on the line are something continuous but the real numbers that identify a set possiamo definire "continuo".

Rimane però la questione se tutti questi numeri trascendenti, o meglio i numeri reali, possano essere ancora "contati" oppure no, ovvero se l’insieme dei numeri reali abbia cardinalità maggiore rispetto all’insieme dei numeri naturali.

Il problema è che non si può fare un elenco di tutti i possibili numeri trascendenti; alcuni sono i risultati del calcolo di funzioni trigonometriche, logaritmiche e altre cose del genere, ma a molti, a moltissimi altri non sapremmo neanche dare un significato matematico. Mi spiego meglio: se è vero che a un numero decimale periodico so sempre assegnare una frazione generatrice, quindi riesco sempre a understand the nature of a real number I can never know which mathematical expression was derived, to do this should know everyone, but all its infinite decimal.

an example: if I see the number 3.14 ... I immediately comes to mind more greek. But to know that this number is really more greek I know all the places, but everyone, would take a different figure to the billionth decimal place to make it something different!

So we found a new class of numbers that is impossible to categorize. We will be able to find a mica class of numbers that is no longer "book", or that they can not be put in correspondence with the numbers natural? ▲

The "Continuous"



The answer is finally ... Yes, Georg Cantor (1845-1918) has found a way to show that there are numbers that can not be counted. I provide a detailed demonstration below, I limit myself here to describe it. It is a reductio ad absurdum: imagine that you have drawn up a numbered list of all real numbers between 0 and 1, each with its infinite number of decimal places. Cantor is able to create a new number, different from all others in this way contradicts the hypothesis (that the real number was represented in the collection each initial).

Here we are in point: we finally have a package that can not have cardinality aleph-0 because it has at least one element that escapes the count. The set of all real numbers then assigns the cardinality c (the letter c was chosen because this set is for "continuous"). For now we know that c is certainly greater than Aleph-0 ... but the developments that ensue from this discovery require an entire chapter of this story (the next). ▲

power of the continuum

begin to analyze this c . Meanwhile, try to see if a long segment has more points than a shorter segment: the answer is no, because you can always show the correspondence between the points of two segments of different lengths (each point in segment [0 -1] has its corresponding A segment [0-2]).

# 1


Now let's see if a line of infinite length of a segment with the most points over: once again the answer is no, because even in this case shows lo stesso tipo di corrispondenza. La dimostrazione si fa con il disegno qui sotto: si mettono in corrispondenza i punti di un semicerchio che ha centro in P con i punti della retta sottostante. A ogni punto del semicerchio corrisponde un punto della retta infinita: A - a , B - b , e C - c con c che evidentemente è ben fuori dal disegno, sulla sinistra.

#2


Ecco un risultato interessante: l’insieme dei punti di un segmento unitario (o l’insieme dei numeri reali nel campo [0-1]) ha la stessa potenza dei punti di una retta infinita (o dell’insieme di tutti i numeri reali).

Ma non finisce qui: proviamo a vedere se una superficie finita ha più punti di un segmento finito: ancora la risposta è no! Proviamo a considerare i punti di un quadrato unitario. Ogni suo punto può essere identificato da una coppia di coordinate X e Y comprese fra 0 e 1:

#3


A partire dai numeri che definiscono le coordinate X e Y posso creare un nuovo numero R intercalando le cifre dei decimi, poi quelle dei centesimi, poi dei millesimi e così via. In questo modo avrò un unico numero che esprime la coppia di coordinate, anch’esso compreso fra 0 e 1: Therefore, there is a correspondence between each pair of coordinates and a real number. Let me give an example:



R From the number I can easily get to the two coordinates X and Y : the decimal digits of odd order R will coordinate X; the order digits of the coordinate Y .

repeating the method described by Figure # 2 for the surface instead of segments / lines can be shown that there is also a correspondence between the points of a surface finished with the points in the infinite plane. And using the system shown in Drawing # 3 I can define a new number R that instead of starting from a pair of coordinates X and Y, use three coordinates X Y Z and three-dimensional space. Then

c, the cardinal number that indicates the power of the continuum, can be used to define the power of the sets of all real numbers, points of the segment, the straight line, plane, space ... etc. etc.!

The proof of Cantor

I describe here the "Cantor's diagonal proof, with some simplifications as not to complicate the reading. Chi fosse interessato può leggere questo articolo su wikipedia .

Supponiamo per assurdo che l'intervallo dei numeri compresi fra 0 e 1 sia numerabile. Questo significa che gli elementi dell'intervallo possono essere posti in corrispondenza biunivoca con i numeri naturali dando luogo ad una successione di numeri reali R1, R2, R3, ... che esaurisce tutti i numeri reali compresi tra 0 e 1.

Possiamo rappresentare ciascun numero della successione in forma decimale e visualizzare la successione di numeri reali come una matrice infinita che avrà più o meno quest'aspetto:

R1 = 0 . 5 1 0 5 1 1 0 ...
R2 = 0 . 4 1 3 2 0 4 3 ...
R3 = 0 . 8 2 0 2 4 5 6 ...
R4 = 0. 0 2 3 3 1 2 6 ...
R5 = 0. 4 1 0 7 4 6 2 ...
R6 = 0. 9 9 3 7 8 3 8 ...
R7 = 0. 0 1 0 5 1 3 5 ...
...

In this table I have indicated in bold type the numbers that appear on the diagonal (the first decimal place of the first issue, the second of the second, and so on). We now construct a new real number X that has all figures different from the sequence on the diagonal. Proceed as follows: If a digit appears on the diagonal is 5, we replace it with a 4, all other cases, we replace it with a 5 (the choice of figures 4 and 5 is arbitrary). In the example we get:

X = 0 . 4 5 5 5 5 5 4 ...

All'inizio dell'argomento avevamo supposto che la nostra lista dei numeri enumerasse tutti i numeri reali compresi tra 0 e 1, quindi dovremmo avere uno dei numeri R, diciamo l'ennesimo, per cui Rn = X. A questo punto emerge una contraddizione: per come abbiamo costruito il numero X, l'ennesima cifra di X dovrebbe essere diversa dall'ennesima cifra del numero Rn. Questo è impossibile, e ne segue che l'ipotesi di partenza è falsa e cioè che l'intervallo dei numeri compreso fra 0 e 1 non è numerabile.    ▲   

Prossimo capitolo: L'ipotesi of Continuous

62 And How To Get An Erection

transfinite numbers: the continuum hypothesis

Summary:

+ natural numbers
+ Still Aleph-zero
+ The historical context
+ The Continuous
- The hypothesis of Continuous
c = aleph-1?
Issues ... theological!

Given the results in the previous chapter, it might seem hard to find infinite larger c or cardinal numbers (transfinite) are larger than c ... do exist, and how! I'm not entirely easy to find, but they were identified, however I can assure you that Cantor has shown that there are infinite numbers "transfinite", the first of which is Aleph-0, then there is Aleph-1 Aleph-2 and so on, creating a set ... infinite cardinal numbers transfinite!

c = aleph-1?

Now is a very interesting question. This c what is? We know it must be greater than Aleph-0: Aleph-1 coincide with mica, which is the number transfinite immediately after aleph-0?

We can ask the question another way. You will never find an infinite set has a power that is strictly between the sets of natural numbers and real numbers? That is, a structure which can not be counted by the real numbers, but can not in turn "counts" the continuous? If the answer is no, then we will automatically c = Aleph-1, or it may be c = Aleph-2 Aleph-3 or ...

Cantor has spent the last years of his life trying to prove that what mathematicians call "continuum hypothesis", ie that there is no insieme infinito compreso fra quelli dei numeri naturali e dei numeri reali, ma senza riuscirci.



Nel 1900 a Parigi, durante il Congresso Matematico Internazionale, David Hilbert (1862-1943) sottopose ai matematici di tutto il mondo un elenco di problemi da risolvere (il più noto forse è la dimostrazione dell’ultimo teorema di Fermat); il primo problema della lista era proprio dimostrare l’ipotesi del continuo.

Il problema si è rivelato davvero difficile da aggredire, tant’è che solo nel 1940 Kurt Gödel ha dimostrato che non è dimostrabile la falsità dell’ipotesi del continuo. Invece nel 1963 Paul Cohen ha dimostrato che è impossibile dimostrare che l’ipotesi del continuo sia vera...

... ecco quindi saltar fuori un’antinomia, un enunciato che porta a conclusioni contraddittorie! Solo che, a differenza del paradosso di Russell di cui ho parlato nei capitoli precedenti, che rivelava un’antinomia in una cosa che sembra (almeno a noi comuni mortali) più una pignoleria che un problema reale, adesso l’antinomia salta fuori nello studio dei numeri, cioè al livello più basilare di tutto ciò che è matematica!

A me questo risultato ha sempre riempito di meraviglia: quando l’ho letto la prima volta ho veramente fatto un salto sulla sedia. Mi sembra magnifico che l’intelletto umano sia riuscito a isolare un’antinomia only in studying the nature of the numbers!

The set of ideas that the characters I mentioned in these pages is able to put together is one of the highest peaks reached by humanity. It does not matter if the end result is an apparent "stalemate" with these contradictions that crop up everywhere. I totally agree with Carl Jacobi, who wrote:

"The only purpose of science is the honor of the human spirit, this title is a matter of numbers as a system in the world" The power of

logical processes initiated by Cantor with his approach to infinite sets has proved very formidable, though many mathematicians of his contemporaries did not immediately appreciated. Even in the midst of bitter disputes on this issue, David Hilbert said:

"No one will dispel the Paradise that Cantor has given us!"

Issues ... theological! About

Finally, we add a few anecdotes regarding Galileo and Cantor, struggling with this concept not only numerical but also philosophical and especially theological.

The concept of infinity is beginning to be exploited in theology by Nicholas of Cusa (1401-1464) that compares a number of occasions in the infinity of God with the finite nature of men, and the intellect (Finished) with the Truth (infinite). In the long run then the infinite has condensed a lot of divine attributes ... becoming a term to use with the springs.

As we have seen Galileo Galilei came across some logical reasoning concerning the infinite, but his lack of courage in reaching their ultimate consequences, perhaps is due to the fear of the Inquisition to other topics ... which had problems with its beautiful! Do not forget that a few years earlier, in 1600, Giordano Bruno was condemned to the stake for saying "the universe is therefore an infinite, while the second property ...", the Inquisition (Cardinal Robert Bellarmine is what is condemned Giordano Bruno, Galileo) and runs the universe is finite, while the earth to stand still. Even

Cantor poses some problems when deciding to disclose his views on the infinite, especially knowing the existence of an infinite number of different sizes. As a good Christian churches use the term if not infinite would disturb the ecclesiastical hierarchy was the end of the nineteenth century, there was no danger of going to the stake, but still wanted to know what he thought of this fact the Catholic Church. He went to the Vatican, brought his work to the Holy Office, which was ruled by a German cardinal, who said: "Eminence I have here works in mathematics I say there are countless more, many in fact infinite. " The cardinal said: "Well I do not know the math ... I let her work, I consider them to my secretary. "

The secretaries were the Dominicans, who took two years, because obviously they had to start studying set theory from scratch. After two years, the cardinal said: "In our opinion there is no problem, there is no danger to the faith." So Cantor was summoned to the Vatican and Cardinal of the Holy Office said: "Look you can talk about these endless, provided you call them infinite, because this actually darebbe una brutta idea teologica, cioè farebbe una connessione con la divinità". Allora Cantor scelse il nome "transfiniti". (Per ironia della sorte, i matematici preferiscono chiamarli con il nome di... numeri Cardinali!)

Il cardinale del Santo Uffizio si era anche fatto l’idea che dopo tutti questi transfiniti, là, alla fine, ci fosse il vero infinito assoluto. Chiese a Cantor cosa ne pensasse: "Per noi matematici quello non c'è. Non esiste un infinito assoluto per i matematici, perché ciò sarebbe contraddittorio". Al che il cardinale disse: "Va bene: quello lì è nostro!".    ▲   

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transfinite numbers: the historical context

Sommario:

+ I numeri Naturali
+ Ancora Aleph-zero
– Il contesto storico
       Rifondare la matematica
       Giuseppe Peano
       Bertrand Russell
Gottlob Frege
Kurt Gödel
Irrationality of whole roots
+ + The Continuous
the continuum hypothesis

reshaping the mathematical

In the nineteenth century mathematics had reached a remarkable level of development, and was in constant evolution. Someone began to have doubts about the correctness of this, because it is true that conditions apply, with correct reasoning, it always comes to conclusions as valid, but we are sure that all the assumptions are valid, and that somewhere is not have made a few mistakes of reasoning?

So here he says there were two kinds of doubts. The first, already mentioned by Kant in "Critique of Pure Reason", was, as the process of induction can be accomplished without "off on a tangent? The second: the foundation upon which rests the whole castle are really solid?

The Mathematics was (and is) a field key, allowing to describe the behavior of physical systems. In this field you play casually with enough zeros, infinite and infinitesimal, become even infinite sums of infinitesimals. Fortunately, the analysis is often the way to deal with the physics, which historically has also happened that was not the physical to be "understood" by mathematics, but mathematics to be "validated" by physics.

course, this approach was not enough to mathematicians, fussy and perfectionist as I am here, therefore, necessary to make a comprehensive argument on the matter, trying to start from the smallest number of axioms, and then get to the more complex issues for small steps , incontrovertibly dimostrabili. (Gli assiomi, anche detti postulati, sono enunciati che, pur non essendo dimostrati, sono considerati veri; vengono usati per fornire i punti di partenza necessari alla delineazione di un quadro teorico, come può essere quello della teoria degli insiemi).    ▲   

Giuseppe Peano



Dovendo partire dalle cose più semplici, la cosa ovvia era di partire dai numeri naturali. I quali, pur essendo intuitivi, richiedevano una definizione precisa. Giuseppe Peano (1858-1932), il primo logico matematico italiano della storia, called them on the basis of these five axioms:

- There is a natural number zero
- Every natural number has a natural number successor
- different numbers have different successors
- Zero is not the successor of any natural number
- Each set of natural numbers containing zero and the successor of each element coincides precisely with the entire set of natural numbers

seems to me a wonderful definition, but did not appeal to everyone. Why, for example, the definition of "successor" was not considered sufficiently precise, you can count by twos or threes in three ... (i logici, coloro che si occupano di logica, riescono ad essere addirittura più puntigliosi e precisi dei matematici!)

I logici sanno che devono stare molto attenti: non è detto che le loro costruzioni logiche non entrino mai in contraddizione! Lo scoprirono già gli antichi greci, grazie a tale Epimenide di Creta (VI secolo a.C.), il quale, cretese appunto, ebbe a dire che "tutti i Cretesi sono bugiardi". Si capisce bene che si tratta di un paradosso: se egli, cretese, stesse dicendo la verità, allora non sarebbe vero che tutti i cretesi sono bugiardi; se invece fosse bugiardo, starebbe affermando una cosa vera! (*) Contraddizioni di questo tipo, o "antinomie" (le situazioni per cui, posta una particular issue, if they can make two statements apparently valid but which are contrary to each other) are always lurking.

(*) The paradox of the liar, as expressed by Epimenides, is incorrect: just think that at least one Cretan is telling the truth. Epimenides is a liar then actually saying that all Cretans are liars, because in reality there is one that is not. However, the modern logicians have been able to create paradoxes so other ... there is no loophole that takes! ▲

Bertrand Russell



So we think the logic a bit 'up, found that the natural numbers are a concept of "inappropriate", and looking for something more powerful for their meditations: invented the concept of set. Now I have tried several times to read some parts of the "Principia Mathematics" by Bertrand Russell (1872-1970), one of the greatest mathematical logicians ever existed, I say that I've just tried, for accuracy, the endless distinctions, I would say the pedantry are brought to levels ... that are not made for us mere mortals! Then, just to give you an example of what are the arguments of these gentlemen are talking about logic, and risk of saying something not quite right, try to explain how they can be seen the natural numbers.

Sets are collections of one or more distinguishable from one another (there is also the empty set), the number of items from each set is called the power of the whole. The empty set has zero energy and then, those with a single element has a power, and so on. To obtain a full set of natural numbers, just follow these steps:

- the empty set, which has zero elements and thus has power 0, we associate the number zero.

- we create the rule that the successor of a set A B is given by the union of elements of A with the same set A, so I add the element consists of a set A to elements already contained in A, obtaining a set B which has a capacity greater than the whole unit to: Association for the power this new set the corresponding natural number.

We see better how it works: the elements of the empty set (which has none) I add the empty set itself, by getting a set that contains only an empty set, and then have a power, the elements of ' a set (which contains an empty set) add a whole and get the two together (the empty set and will set a), the three will have three elements (The empty set, one and two), and so on, each with power equal to the number of items it contains. Having associated with each of these sets of numbers that express their power, with this we have defined a full set of natural numbers.

In essence, it takes the concepts of set (and the empty set) and successor to do all the work!

Among the nineteenth and twentieth century studies to re-establish the mathematics has been proceeding apace, with and without the use of set theory (not all the mathematicians saw her favorably, but as we have seen, and we'll see still, Georg Cantor used it in a spectacular way to attack the concept of infinity). Bases hours were really solid, and there was widespread belief that he would never have found a contradiction, would never have found a paradox (or antinomy, like that of the liar). ▲

Gottlob Frege



Even Gottlob Frege (1848-1925), considered one of the greatest logicians since Aristotle, had been helping reconstruction of mathematics. He had already published the first volume of his "Principles of Arithmetic" and was about to go to press with the second volume, she receives a letter from Bertrand Russell. Russell faces la seguente questione:

Può un insieme essere elemento di sé stesso, ovvero contenere se stesso?

La risposta è sì. Ad esempio, l'insieme di tutti i libri di una biblioteca non è elemento di sé stesso. Invece, l'insieme di tutti gli insiemi con più di 20 elementi è elemento di sé stesso. Allora vediamo quest’altra questione:

Che tipo di insieme salta fuori se ne creo uno che contenga tutti gli insiemi che non contengono se stessi?

Vediamo per tentativi, provando a considerare o meno questo insieme come elemento di se stesso:

— se dico che questo insieme non contiene se stesso, fa parte del gruppo degli insiemi che non contengono se stessi, e quindi dovrebbe be part of it (but the assumption was that it was part)

- if I say that this set contains itself, is not part of the group of sets that do not contain themselves, and therefore should not be part of it (but the ' hypothesis was that it was part)

fatal is the paradox: the creation of all the sets that do not contain themselves follows the appearance of a contradiction, and it is enough to dismantle the illusion of a logical system is complete and consistent . The existence of a contradiction as this is the crack that destroys the castle.

Frege took note of the destructive consequences for the system he had built and he resigned himself to write an addendum to its principles, in which he confessed the failure of his work. The contradictions highlighted by Russell's paradox is insoluble within the set theory, mathematicians and logicians have had a hard enough time learning how to handle this situation. ▲

Kurt Gödel



The illusion of a castle built entirely consistent mathematics was finally broken in 1931 when Kurt Gödel (1906-1978) proved his first incompleteness theorem. This says

In any mathematical theory T ... esiste una formula F tale che, se T è coerente, allora né F né la sua negazione sono dimostrabili in T .

Questo teorema (semplificando) afferma che in un sistema assiomatico salterà sempre fuori un enunciato non dimostrabile a partire dagli assiomi di partenza, ovvero un caso indecidibile del quale non si può dire se sia vero oppure falso: e qui torniamo al paradosso del mentitore... che i greci avessero già capito tutto?

Il Secondo Teorema di Incompletezza recita invece:

Nessun sistema coerente può essere utilizzato per dimostrare la sua stessa coerenza.

In pratica, se voglio costruire un sistema matematico starting from some axiom of departure, I will need some external axiom to the theory in order to verify the validity ... which I do not know if it makes much sense!

But be careful, at this point we do not say that mathematics is "all wrong" means the risk of throwing out the baby with the bathwater! It was discovered that the only logical consistency, when you reach certain limit reasoning, is not fully accessible. But when you pay the bill dell'Ortolano, rest assured that in the calculation of the rest do not present any contradiction! ▲

Next Chapter: The Continuous

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transfinite numbers: Aleph-zero

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 ²

But, for the Pythagorean theorem:

[AC] ² = 2 ּ [AB] ²

And then:

m² = 2 ּ n ²

It 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 ּ
ּ q 2 = n ² n ² 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. ▲

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

How To Connect Ballast With Hps

transfinite numbers

Start here an explanation that begins with some consideration on the natural numbers, their "quantity" e la ricerca di vari tipi di insiemi infiniti. Il tutto con il racconto dei personaggi, delle conquiste e, purtroppo, dei clamorosi fallimenti: è un racconto a tratti davvero avvincente.

Sommario:

– I numeri Naturali
       Insiemi e "corrispondenza biunivoca"
       Numeri infiniti
       Aleph-zero
Infiniti "smaller"?
transfinite numbers
Intermezzo: the hotel with infinite rooms
+ More Aleph-zero
+ The historical context
+ + The Continuous
the continuum hypothesis

Collections and "Match Bijective "

Set theory I was taught in school only when it became" fashionable "I began to wonder what it was. All those endless definitions such as" unions "," intersection "function" injected "and" surjective "... it just seems a way to harass the poor students. So I have not seen the practical value until I came across in the study of infinity, for which studies are really just two only concepts: collection, in fact, and the "correspondence".

A set is a collection of objects of various kinds, all different (or at least distinguishable) one other. For example, I consider that my left hand is all five of his fingers, his right hand and the other set of five fingers.

Question: How do I determine if these two sets (the hands) contain the same number of elements (fingers)? I can count:



Or I can correlate each finger of his left hand with their finger right:



In the latter case I have implemented the concept of "correspondence "to each element (finger) of the first set (left hand) is the one and only one element (finger) of the second set (right hand), and vice versa. Having joined the fingers of both hands as shown in the photo, I can say without a shadow of doubt that the "power", or "cardinality", or simply the number of elements contained by each of the two sets is the same. There is no need to count them, there is no need at all to know how many, the question was: have the two sets the same number of elements? The answer is undoubtedly: Yes.

If the left hand missing two fingers



the correspondence there would be no more. The left hand fingers will find their correspondence in their right hand fingers, ma qualche dito della mano destra non lo troverebbe più nella mano sinistra: ecco quindi che, anche non sapendo quante dita (elementi) contiene ciascuna mano (insieme), posso affermare che la mano destra ha una "potenza", o "cardinalità", o un numero di elementi, superiore alla mano sinistra.

Tutto quanto detto finora è intuitivo, direi quasi banale, perché ci siamo occupati di insiemi non solo finiti, ma insiemi di cui è facile contare il numero di elementi. Facciamo invece un esempio con insiemi più grandi (ma sempre finiti). Ammettiamo di radunare in un una piazza una quantità molto grande di persone, e di voler stabilire se ci sono più maschi o più femmine. Siccome le persone sono tante...



... I can not count them without making mistakes, especially if, as seems likely, there will stand still. Then I ask them to in pairs: each male will have to find a female with whom to hold hands. At this point just to see if the "advanced" men (which would then be a greater number of females) or female (they would be in greater number of males), and if there are leftovers, it means that the number of the two groups is exactly the same . Again, I do not know how many elements of each set (male / female) but I decided what the whole is greater, or if they have the same power and are equipotent.    ▲   

Numeri Infiniti

Parliamo ora di numeri. Impariamo a contare dalla più tenera età, e scopriamo che c’è sempre un "numero più grande". Quando riusciamo a contare fino a cento, scopriamo subito che c’è anche il centouno. Fino al mille, e c’è il milleuno! Intuiamo presto che non ci sarà mai fine: magari non sapremo "come si chiama", ma ci sarà sempre un numero più grande di qualunque numero riusciamo ad immaginare. Ecco, abbiamo trovato il più classico esempio di "insieme infinito": l’insieme dei numeri naturali.

E qual è la cardinalità dell’insieme infinito dei numeri naturali? Non posso certo dire quanti numeri contiene, perché sono infiniti; e dire che questa cardinalità è infinito non sarebbe di nessuna utilità, in quanto "infinito" non è un numero.    ▲   

Aleph-zero

Per risolvere questo problema Georg Cantor (1845 – 1918), il padre della teoria degli insiemi, ha deciso di identificare la cardinalità dell’insieme dei numeri naturali con un simbolo costituito dalla prima lettera dell’alfabeto Hebrew alphabet, Aleph, and the index 0:



Unlike the name "infinite," the Aleph-zero value assumes full dignity of numbers, so much so that on this and other numbers like you can do special calculations arithmetic. ▲

Infiniti "smaller"?

Returning to our natural numbers, we can now invent another infinite set, one containing only even numbers. Apparently a group that lacks all the odd numbers should have less power than the set of all natural numbers. But is it true?

Being infinite, it is obvious that I can not "count" the elements of each of these sets. But I can use the concept of correspondence, and thanks to this stratagem to establish if really all just numbers that are both smaller than the other, much like I did with my hands of three and five fingers.

Then write two columns on the numbers of the two sets: the natural numbers to the left, right, even numbers, then each line contains a number (left) and its dual (right), and the dash is a represent the correspondence between the elements of each set:

1-2
2 - 4
3-6
4-8
........

Evidently each of the set of natural numbers is left of its correspondent in his double in the right, and each number is equal right of all its corresponding number in the left half of the whole. Between the two sets is a state of correspondence, so they have the same power!

I can repeat the process with the square numbers, by matching the natural numbers with their squares:

1 to 1 ² x 1 = 1 = 1
2 to 2 ² = 2 x 2 = 4
3 - 3 ² = 3 x 3 = 9
4-4 ² = 4 x 4 = 16
........

The square numbers are even more "sparse" of even numbers ... and yet the whole has the same power of natural numbers. Just came across this argument in Galileo Galilei, who in "Dialogue Concerning the Two Chief World Systems" says "infinite in number, if we could conceive it, we should say, many many to be the square with the numbers" . In practice understands the principle that an infinite set has the same power as a part of it, saying that "the squares are less than the whole" , but does not venture to say that they own in equal numbers: it concludes "attributes equal major and minor do not have the infinite place, but only in how many terms" .

I now want to take another example, I put just to try to explain what can be great ... a big number! Create another set of numbers, not equal, not square, but not cubic or other powers: we create the set of numbers called factorial. Which are the product of all integers between 1 and the number given (the numbers indicate the factorial is the exclamation point):

1 to 1! = 1
2-2! = 1 x 2 = 2
3-3! = 1 x 2 x 3 = 6
4 - 4! = 1 x 2 x 3 x 4 = 24

The numbers go up very quickly. Let some other examples:

5-5! = 1 x 2 x 3 x 4 x 5 = 120
6-6! = 1 x 2 x 3 x 4 x 5 x 6 = 720
......
59 to 59! = 1 x 2 x 3 x 4 x ... x 58 x 59 = about 1 followed by 80 zeros.

Here, the latter number, a 1 followed by 80 zeros ... is approximately equal to the total number of atoms that make up the entire universe. That is, we are putting the number at 59, a simple number, I would use everyday, with one of the leading figures who have some physical meaning in nature! And going forward with larger numbers, practical results impossible to write ... Can you imagine?

60 to 60! = About 8 followed by 81 zeros, 80 times the number of atoms in the universe
61 - 61! = about 5, followed by 83 zeros, 5000 times the number of atoms in the universe
100 to 100! ; = about 9 followed by 157 zeros
1000 to 1000! = about 4 followed by 2567 zeros
....

So are numbers that have nothing to compare with anything real. Note that the left we came across only one thousand, but if we put a billion, this number would quietly into the relevant factor in all the numbers ... though I can not even imagine how many digits will be the factorial of a billion!

Despite this, the sets of natural numbers and the numbers are indeed equipotent factor and therefore have the same cardinality. ▲

Numeri Transfiniti

Con questi discorsi abbiamo intuito che non è possibile trovare un sottoinsieme dei numeri naturali che sia infinito ma che abbia potenza minore dei numeri naturali stessi. In pratica l’insieme dei numeri naturali è il più piccolo insieme infinito esistente, la cui cardinalità, come dicevamo è identificata dal simbolo Aleph-0. Aleph-0 è il primo dei numeri cosiddetti "transfiniti". Procederemo nel prossimo capitolo alla (per il momento) vana ricerca di quelli successivi, relativi a infiniti "più grandi"... ma adesso facciamo un intermezzo!

Intermezzo: l'albergo con infinite stanze



The other day I walked into this hotel which claimed to have ... endless rooms. When they told me they were all occupied, I asked if we could not do something we thought about the porter, and then asked all guests to move around the room equal to the number who were occupying a more . So the guest room number one has gone in two, one of two in three and so on, as if by magic, has remained free room number one, which I happily busy having first bestowed upon an appropriate tip for the doorman.

The next day came a 1000 bus with tourists, but the hotel was always full: the keeper repeated the trick, asking a tutti gli ospiti di spostarsi nella stanza di numero uguale a quella già occupata più mille. Mi sono ritrovato nella stanza 1001.

Il giorno dopo ancora è arrivato un altro pullman, questa volta con infiniti turisti a bordo. Impossibile trovare alloggio per tutti? Macché: a ogni ospite dell'albergo è stato chiesto di spostarsi nella stanza di numero uguale al doppio di quella occupata. Si sono così liberate tutte le stanze dispari, in cui hanno trovato posto i nuovi arrivati; per quanto riguarda me, mi sono ritrovato nella stanza 2002 (in quest'albergo non si sta mai tranquilli!).

Il giorno dopo ancora, gli infiniti ospiti del pullman del giorno prima se ne sono andati. A questo punto il portiere (che ho scoperto essere anche il proprietario dell'albergo) si è messo a imprecare, e diceva: "come si fa a mandare avanti un albergo come questo, se metà delle stanze sono vuote?"

Gli ho suggerito io la soluzione: basta che ciascun ospite torni nella stanza metà di quella che occupava (in pratica tornando alla situazione precedente all'arrivo degli infiniti turisti). Paradossi dell'infinito...

Per questa storiella mi sono ispirato al racconto "L'hotel straordinario, o il milleunesimo viaggio di Ion il Tranquillo" di Stanislaw Lem, pubblicato nel libro "Racconti matematici" edito da Einaudi. [Lem è anche l'autore del romanzo di fantascienza "Solaris", dai which was made into the 1972 film of the same name].

Next Chapter: still aleph-zero

Swollen Parotid Glands And Bulimia

curiosity and a series of short subjects, beginning with one of the classic problems of geometry!

Summary:
- Task class
Solution
+ The necklace of Democritus

task in class



Soluzione:



Non era difficile! :-)    ▲   

Hot Water Slow In Price Pfister

Gleanings Gleanings: The necklace of Democritus

Sommario:
+ Compito in classe
– La collana di Democrito
       Una pagliuzza d'oro...
       ... ne farò una collana!
       Quanto sarà lunga?
       Soluzione

Una pagliuzza d'oro...

L'altro giorno passeggiavo lungo il fiume, quando ho visto luccicare qualcosa: era una pagliuzza che sembrava d'oro. L'ho raccolta e, dopo averla fatta esaminare dal mio orafo di fiducia, ho scoperto che si trattava proprio di oro puro, e che aveva un volume pari a esattamente un millimetro cubo. Era una frazione di grammo (circa un cinquantesimo), quindi aveva un valore irrisorio; ma mi sono messo a fantasticare su cosa ne avrei potuto fare.    ▲

... I'll make a necklace!

I thought a necklace for my wife, maybe thin. Elaborating 'I could get a little more so long as more subtle, and I thought that in theory I could make a thread though infinitely thin infinitely long: the necklace invisible, but still a necklace!

But then it occurred to me Democritus, who was the first to argue that matter is made of atoms: tiny parts that can no longer divide. Then my necklace could not be infinitely thin, but would had to stop at a minimum thickness equal to the diameter of an atom. So the question: When my necklace would be a long, if I could put in a single row all the atoms of that speck of gold?

might proceed as follows: dividing my cubic millimeter in eight parts (mid-high two slices, each divided into four squares - more or less as you do with potatoes) I got a series of thickness 0.5 mm long and 4 mm (8 cubes of 0.5 mm each). Repeating the operation I got 64 cubes of 0.25 mm, a length of 16mm, then 64mm, 256mm, 1024mm (which are already more than a meter) ... until you reach a series in which each cube would contain a single atom of gold. ▲



How long will?

I did a little 'calculations and have come to determine the theoretical length of my necklace. For now, I propose three alternatives, try to answer: How long can, at most, a necklace made from a cubic millimeter of gold? ▲

A - B
15 km - 15,000 km
C - 15 million kilometers

Solution

For those curious about how I did the math, I post them here. Who is interested in the result only go to see the last line ...



Note: I did a lot of rounding and simplifications, in particular, I assumed that the interatomic distance is simply equal to the cube root of the volume occupied by each atom. However, what I wanted to show is the order of magnitude, which I really incredible: a cubic millimeter of gold covering 15 million kilometers, approximately forty times the distance from Earth to the Moon. With a quantity of gold is covered only ten times the distance from Earth to the Sun ..

Ultima cosa: la collana è lunghissima, ma il suo spessore? Beh, coincide con la lunghezza di un atomo, quindi circa 0,26 milionesimi di millimetro, ovvero 260 picometri!    ▲   

Pirates Of The Caribbean 2 Jesse Jane

Arithmetic: The Ruler Calculator

Sommario:

+ Contare
+ Sistemi di numerazione
+ Addizione e Sottrazione
+ Moltiplicazione
+ Divisione
+ Radice Quadrata
+ Elevamento a potenza
+ Logaritmi
- the slide rule
a magical tool
Origins
Double logarithmic
the regular "modern"
Advanced Calculations
few curious detail

a magical tool

I always thought that the slide rule was an instrument of calculation rather limited ... until I found my father's library in this book of 1936 so as to reveal all the secrets:



It took me a few days to understand, but now I can say ... that was truly a wondrous instrument, capable of making calculations of unexpected complexity. ▲   

Le origini

Prima di iniziare a parlare del regolo calcolatore, riscrivo la definizione di logaritmo (di cui ho parlato nel capitolo precedente ):

Il logaritmo decimale di un numero è l'esponente a cui elevare la base 10 per ottenere il numero dato.

Ecco come si esprime questo concetto in formule per due numeri N1 e N2, ma anche per il loro prodotto:



Fra le proprietà delle potenze (di cui invece ho parlato qui ) c'è quella per cui il prodotto di due potenze di pari base è la stessa cosa di una potenza della stessa base con esponente uguale alla somma dei due esponenti di partenza. Quindi:



Dalle formule qui sopra risulta quindi che il logaritmo del prodotto è uguale alla somma dei logaritmi dei fattori. Il "trucco" alla base del funzionamento del regolo calcolatore è proprio il fatto che siamo riusciti a trasformare un prodotto in una somma!

A seguito degli studi di Nepero sui Logaritmi ci fu subito chi pensò di sfruttare l'idea in modo da velocizzare i calcoli, anche a scapito della precisione. Già nel 1623 Edmund Gunter, professore di astronomia al Gresham College di Londra, sviluppa una scala logaritmica sulla quale, con l'aiuto di un compasso, si possono eseguire graficamente moltiplicazioni e divisioni. Ecco... ma cos'è esattamente una  scala logaritmica ?



Si tratta di un righello in cui si riportano tacche a distanze proporzionali ai logaritmi dei numeri da 1 a 10. Nel diagramma sopra specifico che ogni tacca corrisponde al logaritmo del numero, ma la sigla "log" non è assolutamente necessaria:



Notare che a destra si scrive un 1 e non un 10: questa è una pratica utilizzata in tutti i regoli calcolatori. In pratica "si sa" che all'uno di destra corrisponde un 10; inoltre, come we shall see, in some cases the right one is used just as ... 1 and not 10!.

is built on a logarithmic scale! Now suppose you want to multiply by 1.5 to 4 with this scale and a compass, just like the Gunter: just open the compass with an opening corresponding to the logarithm of 1.5 and contain the same opening at 4. We see the process step by step:



The point of the compass is positioned right on the mark of 1.5. The left point to be put on a notch, because remember that this mark is the logarithm of 1, and the logarithm of 1 is 0. In this way the opening of the compass is the difference between the logarithm of 1.5 and a log of the following:

log (1.5) - log (1) = log (1.5) - 0 = log (1.5)

Once you have found the opening of the compass , just passing it to the right:



is that by putting the left point of the compass on the rear of 4 I find myself right on the tip 6 (remember that the logarithm of 1 is 0) :

log (4) + [log (1.5) - log (1)] = log (4) + log (1.5) = log (4 x 1.5) = log (6)

(that is 1.5 x 4 = 6).

With the same exact positions I could have done the inverse calculation: in fact with the same opening corresponding to the number 1.5, I could make the division 6: 4 = 1.5: Whereas the right to put the tip of the compass on the 6, the tip of my left would have given the quotient correctly sought (the difference of the exponents is in fact the division of powers).

log (6) - [log (1.5) - log (1)] = log (6) - log (1.5) = log (6: 1.5) = log (4) ▲

Double logarithmic

In 1630 Edmund Gunter Wingate uses two scales at each other to perform multiplication and division directly, without using the compass. Let the multiplication 1.5 x 3:



The red area in the lower scale, has the same amplitude of the compass that we have seen above. By starting the source (ie the mark of 1) the scale than just the end of the red zone, I see that the right edge of the area falls on the blue 4.5: in fact 1.5 x 3 = 4, 5.

This procedure is also good to calculate numbers with different orders of magnitude: eg. 15 x 300 = 4500; In these cases the zeros more or less, or any movement of the decimal point must be made by hand (in this sense the errors were still there ... it was necessary to be very, very careful and the ideal was more or less understand what was the result before calculating it, and try to adjust only the accuracy of significant digits).

And if you want to calculate 5 x 3? Here comes a problem, because the top three on the ladder is positioned outside the lower scale, so I can not read the result:



In these cases it uses a trick: instead of multiplying by 3 multiply by 0.3:



La zona marcata in azzurro qui sopra indica la differenza fra il logaritmo di 3 e il logaritmo di 10, quindi

log( 3 ) – log( 10 ) = log( 3 : 10 ) = log( 0,3 )

Allora basta posizionare, sul 5 della scala inferiore, non la tacca dell'uno sinistro della scala superiore, ma la tacca dell'uno destro :



In questo modo la moltiplicazione 5 x 0,3 si riesce a fare senza uscire dalle scale, e il risultato si legge sul 1,5 della scala inferiore; ma siccome abbiamo moltiplicato per 0,3 e non per 3, occorre "aggiustare" il risultato multiplying by 10: The final result is 15!

I said that among the properties of powers (and logarithms) is one for which the division of powers is the subtraction of exponents. So I can calculate the Division 6: 4



From the graph above we see that on 5 of lower scale but not the one I put on 4 of the top scale, one of the top scale is then on the left and the red and blue areas are "escape" here is the mark of a higher scale of 1.5 falls on a smaller scale, determining the correct result.

Let us now calculate the "standard" (Which I've actually found this to be just reading the book I mentioned in the caption of the photo): multiplication and division in one fell swoop! In practice the calculation of a proportion: a number multiplied by a fraction of which the numerator and denominator data, such as the calculation of the three means of five (5 x 3 / 2).



aligning two of the top scale with the scale of less than 5 of which have the mark of a left upper scale indicates the ratio 5: 2 = 2.5. The green and blue areas along the damage factor of 3, then it is as if you were multiplying by 3 the ratio 5: 2. In fact, the top three on the scale of 7.5 falls right on the bottom, which is the exact result!

Obviously with the divisions and mixed with the calculations (multiplication and division together) could happen to leave "out of scale", but there is always a way to fix the stairs to get the result, as we saw for multiplication. ▲

the regular "modern"

With subsequent improvements, and the fact that from the mid-nineteenth century, the precision engineering industry has allowed the construction in series, the slide rule has become a tool for calculating substantially standardized and widely available. Slide rules consist of the following elements:

- a body on which the scales are fixed
- sliding rod with escalators
- a slider with one or more reference lines



scales are of various types, conventionally referred to by some letters. In simple steps, such as the Wingate there are always two, a sliding rod (scale C) and the other on the body (D ). Other scales are used to simplify the calculations when you are in the presence of squares, cubes, square and cube roots, trigonometric functions ... etc. .. The scales are usually sorted out between the front and back of the rod and the body of the ruler, in this rule are the steps:



K - Scale the cube (body )
A - Units of squares (body)
B - Scale of squares (auction)
ST - Breasts and Bribes for small angles (Auction) T
- Scale of tangents for angles> 6 ° (pole vault)
S - Scale of the breasts (and cosines) for angles> 6 ° (pole )
C - Scale of numbers (auction)
D - Scale of numbers (the body)
OF - Units of the inverse of the numbers (1 / x) ( body)

the reverse auction, at the top, there are three other scales:

CI - Inverse scale of numbers (1 / X)
CF - scale "bent back", starting from pi-greek instead of 1
CIF - inverse scale, which starts by pi-greek

For greater clarity of the photos that follow, setting out the calculations so that all numbers are in a short stretch of straight edge, and no escape from the stairs, also for the sake of brevity I do not explain how to determine the zeros or decimal point: my aim is only to show what were capable engineers when they were using a straight edge! ▲

Calcoli avanzati

Abbiamo visto che con due semplici scale si riesce a fare un'operazione mista di moltiplicazione-divisione in un colpo solo. E se invece volessi fare una divisione-divisione? Per fare questo bisogna ricondurre l'operazione che vogliamo fare a quella "standard" di moltiplicazione-divisione usando la scala dei reciproci; con questa scala si converte la divisione per B in una moltiplicazione per 1 / B :



In verde indico i numeri di partenza e il risultato reale (calcolato con la calcolatrice); in rosso il risultato ottenuto with the straight edge. In making the calculation of all operands must be traced to numbers between 1 and 10, then 2100 becomes 2.1, 1.7 and 17 becomes 86 becomes 8.6. Let us see how to use the stairs of the slide rule:



It starts by superimposing the value C (17) C value of the scale A (2100) of the D scale, as we have seen, the notch 1 left the auction falls on the quotient A / C. Move the cursor to the value B (86) of the scale we multiply this quotient by means of the inverse B, then divide this quotient still B . The reading of the cursor on the scale Q gives the desired result.

Now we see a variant of the previous calculation, a multiplication-multiplication. Again we must bring the desired operation to the standard case, then proceed two factors multiplying and dividing by the reciprocal of the third, as shown in formulas:





Overlap value B (65) of the stairs to the value A (19) of the D scale A means to divide by the reciprocal of B: To make this alignment using the cursor, because the CI and D scales are not adjacent. Now read the number on the scale corresponding to the value D C (12) means multiplying the C scale again for C here found the result you want!

Another type of calculation: a simple division, but the number is a square root.





Move the cursor to the value A (350) At the scale of the squares (this position is on the normal scale D, the value 18.7 which is precisely the root square 350). If we now match the value B (1.51) the C scale of the cursor on the line, I will have the value of the D scale at the left of the auction is a division of the square root A for B, which is the result.

Finally, a very complex calculation, which requires the use of four different scales. It is the relationship between a cube root and square root, all multiplied by a normal number. To make this calculation must fit the shaft of the slide rule to the contrary, in order to provide the scale of the square rod (scale B).





You place the cursor on the value A (7400) on the scale of the cubes K (this position is on the normal scale D, the value of 19.487 that it is the cube root of 7400). He then moved the rod to align the cursor value B (290) Scale B. At this point the value of the D scale at the left of the auction is a division of the cube root of A the square root of B , but if I try the value C (1.3) on the scale C, the corresponding value on the scale D will be the product of the quotient previous C, which is the value sought. ▲

few curious detail

There are endless variations of these types of computation that can involve trigonometric functions, logarithmic, exponential ... I now understand how to design engineers did what they were able to build over the last two centuries before the advent of electronics (at least until the early 70s of last century).

Among the nineteenth and twentieth century were built of massive proportions even slide rules: he had climbed two meters in length, and a microscope mounted on the slider. With these tools monumental could appreciate up to 6 significant digits, both in the operands that results!

last interesting note: when the astronauts took to the Moon (1969), electronic calculators had not yet been invented, in fact the first scientific calculator (the HP35) is only 1972. Then the astronauts went to the moon ... carrying a slide rule! The manufacturer of this rule "extraterrestrial" in fact written on the packaging of its rules: "5 moon flights, five flights to the Moon! ▲