Sommario:
+ Contare
+ Sistemi di numerazione
+ Addizione e Sottrazione
+ Moltiplicazione
+ Divisione
- Square Root
Why the square root?
The classic method of calculating
I "rods Napier"
irrational numbers
imaginary numbers
cubic equations
Complex numbers
; the complex plane
Equation cyclotomic
Equation cyclotomic
Niccolo Tartaglia
+ Exponentiation
+ + Logarithms
slide rule
When I was a kid I hated the square roots, not so much in itself, but because I did not understand the reason for that method so complicated to calculate. If you remained in this curiosity, read below: it was not something so far-fetched!
Why the square root?
The square root not used often in everyday use, while in science jumps out from all sides! But we see a (un) likely event that could happen to you as you need it!
An uncle of America has left me an inheritance of land perfectly square shape and surface area of \u200b\u200b56,250 square meters. I want to fence: what network should I buy? I want to know exactly, I do not want to waste even an inch!
The perimeter of this square is four times the side, and as the area of \u200b\u200ba square is ... the square of the side, then read the side I do the reverse, that I must take the square root of the area.
The calculation is performed for approssimazioni successive, come si fa con con la divisione: si ricava la cifra più significativa del risultato e si calcola il resto; poi si ricava una nuova cifra, un nuovo resto, e così via. ▲
Il metodo classico di calcolo
Il calcolo va impostato come mostro qui sotto, raggruppando a coppie le cifre del numero di cui si cerca la radice (radicando) partendo da destra: prima le cifre delle unità e decine, poi centinaia e migliaia, e così via. Il gruppo di sinistra alla fine sarà costituito da una o due cifre (in questo caso dalla sola cifra 5, indicata in black) is from here that will start the calculation. (This group the digits in pairs due to the fact that the square of 10 is 100. Each digit of the result is two digits of rooting, and the result of the root will have many whole numbers as there are groups of two figures in which it was divided by the roots).
The first digit of the result must be determined by looking up the number (from 1 to 9) whose square is less than or equal to the group on the left of the digits of rooting. In our case, 2 x 2 = 4 that is less than 5. Under the root sign is calculated in the rest, the difference between the 5 del radicando e il 4 che abbiamo appena trovato.
Questa prima cifra del risultato, che sappiamo essere la cifra delle centinaia (di metri), indica proprio il numero intero di ettometri che stanno nel lato del quadrato.
Nel disegno sopra le righe orizzontali e verticali rappresentano divisioni del terreno in quadrati di 100 metri di lato, a partire dall'angolo inferiore sinistro: dal disegno è subito evidente che il numero di ettometri che stanno nel lato è due, e che gli ettari interi che stanno nel quadrato è 4. Dal disegno appare ovvio che nel quadrato non ci potrebbero stare 3 x 3 = 9 ettari: ecco perché, come primo passo nel calcolo della radice quadrata, si cerca proprio il numero più alto il cui quadrato non sia maggiore del primo gruppo di cifre del radicando.
Per andare avanti con il calcolo bisogna considerare non più le centinaia ma le decine di metri (decametri). Aggiorno lo schema:
La griglia ora rappresenta decametri; l'area verde ha le stesse dimensioni del disegno precedente, solo che il lato è espresso in decametri invece che in ettometri; ogni quadratino quindi è un'ara (decametro quadrato), e l'area verde (4 ettari) ha adesso una superficie di 400 are.
Per trovare la cifra successiva del risultato bisogna prendere in esame even the next group of two digits of rooting (62): the total number 562 in our case represents the entire surface are (no decimals) of land.
To express the rest are in calculating the difference between less than 562 are 400 are (area corresponding to the first digit of the result already found it): the rest is then of 162 acres. Now we find the second digit (c) of the result.
c This figure determines the value of the two blue areas (each of which is a rectangle with a side of 20 decameter and the other is just c) and of the red (which is a square side c). These three areas are shown in their respective colors, nell'espressione sulla destra.
Il nostro scopo è trovare il massimo valore di c tale che la somma delle aree dei due rettangoli e del quadrato non superi il valore del Resto (162). Semplifichiamo l'espressione delle aree: nella riga sottostante sostituisco la somma di due volte 20 per c con 40 per c; poi prendo c a fattor comune ricavando l'espressione (40+c) per c; ma siccome c può essere solo una cifra compresa fra 0 e 9, riduco il tutto all'espressione 4c per c, in cui 4c è un numero di due cifre composto dal 4 (decine) e da c (unità): in practice to replace the figure with the figure of 40 0 \u200b\u200bc.
The value "4c to c" is the area of \u200b\u200bthe two blue rectangles over the red square, and this total must not exceed the remaining area (the number shown in black), the second figure (c) the result must be sought as the highest value which, when placed in the expression 4c to c, gives a value less than or equal to Rest (162). The search value is three:
The result, which now has two digits, tells us that the green side of the square is 23 decameter, and that once removed from the Rest of the previous (162) in the rectangular areas we There is over (129), the new Rest is 33 ares (shown in purple).
Again, to go ahead with the calculations necessary to reduce the scale, going from decameter to meters, and are in square meters.
Here the grid is no more, because it would be too dense. The green area is the same as the previous design, only that the side is expressed in meters instead of decameter. Under the root sign should be updated calculation of the rest in order to consider square meters here earlier than the rest, which was of 162 are once dropped the last two digits of the rooting becomes 16,250 square meters, minus the 12,900 square meters previous calculation (129 acres), is a remnant of 3350 square meters (purple): Rest on the basis of this new look is the third figure (c) of the result.
The calculation is performed exactly as before: the units digit of the result (m) is 7.
This is the final calculation of the integer part. But at first I said I wanted the exact value centimeter : you can proceed as you do with the division, "dropping" of the zeros to be added to the rest, except that, unlike the division, in the calculation of the root are added zeros in pairs. We see the two steps successivi del calcolo:
Alla fine, il risultato della radice è di 237 metri e 17 centimetri. Moltiplicando questo risultato per quattro ottengo il perimetro del terreno: 948,68 metri, ed è proprio la quantità di rete che acquisterò per realizzare la mia recinzione!
Qui di seguito ricapitolo le regole per il calcolo della radice, così come ce le hanno insegnate a scuola:
1 — Si raggruppano le cifre del radicando a coppie, da destra a sinistra
2 — Si calcola il numero più grande il cui quadrato non sia superiore al gruppo di cifre di sinistra; questo numero è la prima cifra del risultato, and its square is subtracted from the group left two digits of the roots, in order to get a rest.
3 - This fall the next two digits from the root so as to complete the rest, if the digits of the roots are exhausted, you add two zeros.
4 - digit c next result should be calculated by looking up the value of c such that, by calculating the product of c for a number that is twice the result of the root already calculated and which is flanked on the right the figure c , yields a result that does not exceed remains determined in the preceding paragraph. Found that the new figure is the result.
5 - Subtract the product in the preceding paragraph I remain determined by the point [3].
6 - resumed from the point [3] until you finish the figures of rooting, in case with many runs in every other decimal places you want. ▲
I "rods Napier"
I have already explained the operation of "Napier's rods" with regard to multiplications and the divisions now is the turn of the square roots.
To perform this calculation, a stick in more than twice the width (the right in the photo below):
The right stick has two columns: the left square of the numbers 1 through 9, the right one twice the same numbers. Let's see how you perform with this equipment, the calculation of the same root we have seen above.
The first digit of the result is calculated as described in the previous episode: using only the stick of the roots, is what is the largest square that lies in the first group of digits of the root (five in red).
This square is 4 (highlighted in yellow), corresponding to line 2 (shown in green). The two then the first digit of the result, the square must be subtracted from the number 5 of rooting, resulting in a remainder of 1. Now go
used the right hand column of the cotton roots: the value is highlighted in blue written on the right (we'll see a little 'what is it):
stick to the left of the roots must be added pole corresponding to the number shown in blue (4), then complete the rest of dropping two more figures from the root (62, shown in red):
Attempts sticks on the "Total" as high as possible is in the rest. This total is 129 (Highlighted in yellow), which is on line 3 (highlighted in green): 3 is the new figure of the result, which is transcribed next to the two found earlier. Eventually the total must be subtracted from the rest.
Now the number of roots of the right of the stick (highlighted in yellow) is added to 4 according to the calculation shown in light blue that you see on the right. What you get is the value of double digits already found the result (23 x 2 = 46):
This 46 is composed of the roots to the left of the stick, then complete the rest dropping two more figures from the root (3350, shown in rosso):
Di nuovo si cerca sui bastoncini il Totale più alto possibile che stia nel Resto. Questo totale è 3269 (evidenziato in giallo), che si trova sulla riga 7 (evidenziata in verde): il 7 è la nuova cifra del risultato, che va trascritta accanto al 23 trovato in precedenza. Alla fine il Totale va sottratto dal Resto.
Il numero di destra del bastoncino delle radici (evidenziato in giallo) va aggiunto al 46 evidenziato in celeste secondo il calcolo che si vede sulla destra. Quello che si ottiene è il valore doppio delle cifre già trovate del risultato (237 x 2 = 474).
The 474 is composed of the roots to the left of the stick, then complete the rest by adding a pair of zeros (8100):
The new figure of the result is 1. Again,
The whole process is a bit 'tricky, however, makes sure the fly is what each digit to be added to the result, because the sticks allow you to easily calculate all the necessary products. Surely a better method compared to only paper and pen! ▲
I Numeri Irrazionali
L'aritmetica nasce come arte di fare calcoli sui numeri naturali . Nei capitoli precedenti abbiamo visto comparire nuove classi di numeri, rese necessarie proprio dai calcoli che si fanno in aritmetica. Infatti:
— Addizione e Moltiplicazione di numeri naturali sono le uniche operazioni che danno sempre come risultato altri numeri naturali .
— La Sottrazione può dare come risultato un numero naturale, oppure zero oppure numeri negativi: ecco nascere la classe dei numeri interi (con segno), che comprende ma espande i numeri naturali. Qualunque operazione di somma, sottrazione o moltiplicazione integer results in another integer.
- The division gives rise to rational numbers, ie those numbers, being the result of a division between integers, decimal digits after the decimal point have a finite number or infinite (in this case the decimal point is always periodicals). The class of rational numbers including those of the integers (and even more so the natural numbers), but any of the four arithmetic operations to be made on rational numbers results in a more rational number.
those circumstances, one would think that the class of rational numbers includes all numbers imaginable ... but it is not like this: now begins an adventure that opens up new, unexpected horizons!
distributed by the definitions given any number, its square root is that number multiplied by itself gives the number given (root).
the same way you define the cube root:
In general we can define the nth root of a number:
Now comes a question. These operations are producing results that always fall into the class of rational numbers? The answer is no, and the proof is relatively simple:
Let us assume that the nth root of a natural number to can be expressed as a fraction between natural numbers (integers) p and q , and that this fraction is reduced to a minimum (p and q no factors in common). Raised to the power n (green expression), to the left of the equal sign of the root disappears, while on the right we p and q nth power: seeing p and q had no factors in common After this operation are still not having any. Stirring the equation is obtained (right) p to n must essere multiplo di q alla n ; ma dato che p e q sono primi fra loro, l'espressione può essere vera solo se q vale 1 , nel qual caso la frazione p / q è un numero intero. Conclusione: la radice ennesima di un numero qualsiasi può dare come risultato solo un numero intero, oppure un numero irrazionale.
L'esistenza dei numeri irrazionali è nota dalla più remota antichità: già i pitagorici avevano scoperto che la radice quadrata di due (che da un punto di vista geometrico rappresenta il rapporto fra diagonale e lato di un quadrato) non poteva essere espressa sotto forma di frazione. Quindi le radici ampliano ulteriormente il concetto di numero, aggiungendo ai naturali, interi (con segno) e razionali, anche i numeri irrazionali .
Nota: i numeri irrazionali non possono essere mai periodici; se lo fossero, si potrebbe ricavarne una frazione generatrice ( vedi qui ), e allora... non sarebbero più irrazionali! ▲
I Numeri Immaginari
Passiamo ora a vedere una conseguenza davvero "rivoluzionaria" nel calcolo delle radici quadrate. Quanto fa la radice quadrata di -1? Se ricordate qualcosa dell'algebra, meno per meno fa più; quindi sia 1 che -1, elevati al quadrato, danno 1 positivo:
Sembra quindi che in nessun caso si potrà trovare un numero che, elevato al quadrato, dia il risultato -1. Invece la radice cubica di -1 si può fare, infatti meno per meno per meno (tre meni) dà meno:
Armeggiando con ragionamenti di questo tipo si vede come l'estrazione delle radici di ordine dispari (terza, quinta, ecc.) di un numero negativo sia sempre possibile, mentre le radice di ordine pari (seconda, quarta, ecc.) non lo è. Ma ne siamo proprio sicuri?
Nel corso dei secoli mathematicians have had to learn to not be scared by concepts uncomfortable, or anti-intuitive. So it was for the zero and negative numbers for the irrational numbers, as there was some difficulty in accepting the rational numbers with infinite decimals. As far as the square root of negative numbers, it is not understood the meaning, and it seemed obvious that he would never have needed them. But the history of mathematics ... is full of surprises! ▲
cubic equations
early sixteenth century, much of the method discussed for solving cubic equations. There were big arguments about it and "duels" between the mathematicians of the time (I speak below), until a general formula for the solution of these equations was found. But it was also found a big stumbling block with this case:
Solving this equation means finding the values \u200b\u200bof X for which the expression becomes the left of the equal zero. The quadratic formula was known to be this:
which appears twice in just the square root of -1 (highlighted in red). That's right it was known that this equation of third degree had a viable solution for X = 4, because:
in green indicates the replacement of X 4, the intermediate gray and result in red. So the question was: how can a valid result to jump off a formula seemingly absurd?
Over time, mathematicians, from Raphael Bombelli (1526-1572) have resigned themselves to admit the existence of a strange new entity, calling it the significant name "imaginary unit". In essence, then:
the imaginary unit i is that number which, when multiplied by itself gives -1. You need not "understand" their meaning, is a "subject" on which the mathematician can work even without knowing exactly what it is. For example, if you take the quadratic formula equation of third grade that I have shown above, and replacing the square root of -1 with the letter , and then develop appropriate bearing in mind that the formula itself the x the is -1, we find that the result is indeed X = 4.
short roots have added to the math the numbers not only irrational but also Imaginary ... but still not enough, the roots have another very interesting feature! ▲
Complex numbers
Earlier we tried testing the square root of -1 with the values \u200b\u200b1 and -1, but we saw that in no way was the desired result (so much so that it had to admit the existence of imaginary unit). Looking better in these attempts, it turns out, however, that the square roots of 1 are in fact two, not one, not just 1 is the square root of 1, but so is -1:
Now it is obvious that the fourth root of 1 has these same roots, in fact
But the fourth root of 1 has also two others: in fact even i and-i are roots of a fourth.
And the cube root? I mean, if the square root has two roots and the fourth root has as many as four, may the cube root has a root-only? Well, the cube root of 1, in addition to the unit, also has two odd roots:
(if you felt like it, I write to show that the cube of the first of these expressions, the red, gives really 1):
The expression is developed according to the cube of the binomial, and bearing in mind that the x i is -1, in the bottom line, the words gray cancel out, the green combine to form the unity of the result.
We see about the structure of these cube roots of 1 are made up of a real part (- ½) and imaginary (½ √ 3i). Here's another novelty: the sum of a real number, and gives rise to an imaginary object called a "complex number". Finally, we can say that the complex numbers are the last class that we is needed in mathematics was never found another type of operation that requires a further extension of the concept of numbers! ▲
the complex plane
The real numbers can be represented on a line, so that larger numbers are always to the right of smaller numbers :
(red integers, rational in green, blue irrational).
As we have seen, complex numbers are expressed by two quantities, one real and one imaginary. It does not suffice in a straight line to represent geometrically, we need a plan: it defines as "the complex plane" a Cartesian plane where the horizontal axis indicates the real part and the vertical axis the imaginary part of any complex number. Let me give an example of it with a cube roots of 1, we mentioned above:
Each point is identified by two coordinates: one that "falls" on the horizontal (real number, given by red) and that falls on the vertical axis (the imaginary part, shown in green, the letter I put the interests of clarity, though not andrebbe scritta in quanto implicita nell'asse verticale, che definisce appunto la parte immaginaria del numero complesso).
Se ci fate caso, queste tre radici 1 si dispongono sul piano complesso come i vertici di un triangolo equilatero inscritto in una circonferenza unitaria:
Qui le coordinate di ciascun punto sono indicate come coppie di numeri fra parentesi, separati da virgole. Il primo numero (rosso) definisce la parte reale, il secondo (verde) la parte immaginaria del numero complesso che definisce ciascun vertice.
Lo stesso accade anche con le radici quarte: ▲
L'Equazione Ciclotomica"
L'equazione ciclotomica consiste nel trovare tutte le radici dell'unità nel piano complesso. Tale equazione può essere espressa in due modi assolutamente equivalenti:
zⁿ = – 1 = 0 zⁿ = 1
in cui z è un numero complesso che rappresenta la radice dell'unità, n il grado dell'equazione.
It proves (mathematical methods with far too complicated to explain here), that the roots of 1, calculated to any degree, are always in the same issue of the degree of the root, and, if you draw on the complex plane, these roots are arranged according to the vertices of a regular polygon of many sides as is the degree of the root, hence the name of the equation "cyclotomic" or equation that "cuts the circle." Here are represented here the roots of the fifth degree:
Well, for those who want to check it, I give below details of all the vertices of the triangle, square, pentagon, hexagon and octagon inscribed in the regular circle of unit radius: good job! ▲
Niccolo Tartaglia
Commemorative plaque of Niccolo Tartaglia in Piazza Paolo VI (formerly Cathedral Square) Brescia, on the jamb of the main door of the Old Cathedral (or "Round").
During the taking of Brescia by the French in 1512, Niccolò Fontana (Brescia, about 1499 - Venice, 13 December 1557) was wounded in the jaw and palate. Barely survived these wounds, remaining segnato per tutta la vita da una evidente difficoltà ad articolare le parole. Per questo ebbe il soprannome di "Tartaglia", soprannome che usò lui stesso tutta la vita per firmare le proprie opere.
Fu matematico sommo, e il suo nome è legato ad alcune cose davvero notevoli:
— fu il primo traduttore in lingua italiana degli "elementi" di Euclide
— descrisse per primo in occidente il celebre triangolo che porta il suo nome (era già noto in Cina da almeno due secoli).
— scoprì la formula risolutiva delle equazioni cubiche. Ecco la storia di questa scoperta:
La formula termination for cubic equations was first discovered by Scipione dal Ferro, which is not easy in times when they were not yet addressed the negative numbers nor imaginary numbers. At that time, mathematicians jealously guarded their discoveries, in fact the results were not published by the Iron, but went to "legacy" to Antonio Maria del Fiore, its not a brilliant pupil. The latter began to boast of their ability to solve cubic equations, the Tartaglia heard of it, he began to study the problem and reached the same results then accepted a "sign of math challenge" by the Flower . In these
public duels each of the other contestants submitted a number of problems, then a jury chosen by common agreement the winner. On this occasion, the Tartaglia solved all the problems brought by the Flower in a couple of hours, while the Flower could not solve any of the problems proposed by Tartaglia.
The outcome of this duel was aware Gerolamo Cardano (the universal joint), which invited Tartaglia to Milan and he could be confident the formula behind the promise that was not disclosed. Years went by, and Tartaglia decided not to publish the formula, while the Cardano not only the best, but was also to know the results of Scipione dal Ferro: At this point, a little 'because in the formula there, he also had his own, a little' just because he found that the results of Tartaglia had already been found by someone else, Cardano felt released from the promise made, and included the famous formula in his "Ars Magna" in 1545. The
Tartaglia did not take it, and a dispute arose at times very harsh, which resulted in a series of "challenge" leveled by Ludovico Ferrari, a student of Cardano. Tartaglia, given his disability, he wanted to play the duels in writing, but was instead ordered to support the fighting in a hearing. In the last of these challenges, the Tartaglia was even impedito di esporre le proprie ragioni... sicché il poveretto ne uscì con il prestigio infangato, perse il lavoro ed ebbe grossi problemi finanziari per tutto il resto dei suoi giorni!
La formula risolutiva dell'equazione cubica porta oggi il nome di formula Cardano-Tartaglia. ▲
Prossimo capitolo: Elevamento a potenza
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