Sommario:
+ Contare
+ Sistemi di numerazione
+ Addizione e Sottrazione
+ Moltiplicazione
- Division
Primitive man
Ancient Egypt
; The Babylonians "
Roman numerals: the Abaco
; I "rods Napier"
rational numbers
recurring decimal
The Hamlet Generating
historical curiosity
+ + Square Root Exponentiation
power
+ + Logarithms
slide rule
Primitive man
Among our ancient ancestors have certainly had some primitive man dedicated to hunting and gathering. Which, being beneath an apple tree and having collected seven beautiful mature apples, he goes home all happy.
at home are his wife and daughter, so you have to make the parts, although it is likely that the man has eaten all of it, maybe just giving one to each of its women. But suppose that man would be fair: some will not have said
Here, women hold each of your two point three periodic apples!will have just given them two apples and one third. In this sense he is certainly right in with the natural numbers: two apples (natural number) and a (always a natural number) one third of apple: apple parts (third) would be considered as a unit pieces of apple . ▲
Ancient Egypt
We move in ancient Egypt. For Sunday lunch with the family, the chef has prepared a cake to be divided between father, mother and three children. The cake is presented on the table already divided into five slices, but the eldest daughter on a diet, and the younger son immediately took possession of the slice more. Question: how much cake he ate the child?
The obvious answer would be "two fifths," but to the ancient Egyptian was not. The value found in 2 / 5 would be needed for an ancient Egyptian, a further "simplification" for a fraction of them had meaning only if the numerator was the number one. Then:
In this papyrus, dated around 1700 BC, that copy back another 100 years earlier than others, there is a conversion table of all fractions of the type 2 / N in sums of unit fractions, and this for all odd numbers N between 5 and 101. I do a couple of examples:
Happy
them. Sure, I'd gone to school in those days I think I should earn my 2 less less because I simply said that:
▲ The Babylonians
Contemporary Egyptians, Babylonians solved some problems in a way so brilliant that we take advantage today. Any fraction calculation was fatto non con l’unità al numeratore, ma con il numero 60 al denominatore (per esempio la torta egiziana sarebbe stata divisa in 60 parti: 12 parti ciascuno; ma siccome il figlio minore ne mangia due parti, a lui toccano 24/60 di torta).
In una tavoletta babilonese contemporanea al papiro egizio di cui sopra, c’è una tabella di conversione per il calcolo dei reciproci (ovvero, dato un numero N, calcolare il valore di 1/N). Alcuni esempi:
in cui alla destra dell'uguale è indicato il numero di “sessantesimi”. Occhio all’ultima frazione a destra: cosa significa 7,5? I decimali non potevano esistere… quindi i babilonesi came up with a second "number" decimal. The result was written:
where 7 stands for seven sixtieths, 30 for thirty sixtieth of a sixtieth, namely:
As I explained here , the Babylonian numbering system included a group of symbols for the numbers from 1 to 60, more was also invented zero. The numbers were written with more "digits", always taking the same set of symbols. In other words, they were able to write numbers on how many numbers they wanted, always using the same symbols. For example, the number 253.125 to the Babylonians would have written 4 13 7 30 :
The only thing that they had introduced in their numbering system was a clear method to determine where he was going put a decimal point. In fact, the numbers above could be interpreted as 4, or 4 13 7 30 13 7 30 13 7 or 4, 30 or even as a whole number, without "figures" decimal (it must be said that the convention of using the decimal point is a conquest of the late sixteenth century ...). short point position was understood from the text which was written the number.
Too complicated? Really that is what we do every day, but using 10 instead of 60. Example:
using 60 instead of our 10 I think it was just a matter of habit. It is certain that the Babylonians were far fewer journals of us: for we have just a 1 / 3 0.3 is periodic, for their worth 0.20 (in the sense of 20 / 60).
Some additional curiosity: the sexagesimal system (so you call the number system in base 60) is still used in the measurement of time (an hour is 60 minutes, worth 60 seconds) and the extent of the corners (a grade that is 60 / 1, worth 60 seconds). ▲
Roman numerals: the Abaco
I have already explained here as you did the multiplication with the abacus: Now the turn of the divisions. Let's say we want to divide
256432: 723. In relatively few steps and without the aid of either paper or pen, you come quickly to the result. This method is also used when the only numbering system common in Europe was that of the Roman numerals ... certainly there would be how to calculate with pencil and paper division, expressed it this way: ______
CCLVIMCCCCXXXII: DCCXXIII
So we begin to one side. In the following photos for clarity I show, step by step, the procedure numerically, indices in the blue key figures for each step, and the arrows next to the bullets indicate the lines on which it is operating.
Meanwhile, we report the dividend on the abacus, as shown on the left. As the operation is performed for successive differences, we must first subtract the divisor from the most significant digits of the dividend should then consider the first four digits, since otherwise it would be more than 723 256. On the right side I am subtracting 3 (units digit of the divisor) from 4 (least significant digit of the 4-digit):
continue subtracting 2 from 5 (left) and 7 .. . from 25: I update the abacus while putting eight shots, then ...
... I carry the most significant digit, which becomes 2 1 (left). At the end (right) move a ball in the top line, as a reminder that I made the first subtraction:
With the same procedure going forward with the subtractions, we see that the dots in the top line moved diventano prima 2 e poi 3:
A questo punto abbiamo finito con questa serie di sottrazioni, perché l'ultimo resto (395) è minore del divisore (723): allora la cifra più significativa del quoziente sarà un 3.
Ora "caliamo" il 6 dal dividendo (immagine a sinistra) e cominciamo con le sottrazioni da questo nuovo numero di 4 cifre; a destra si vede già il primo risultato, con il pallino spostato a destra nella seconda riga dall'alto:
Proseguiamo allo stesso modo...
fino to get a rest less than the divisor, in this case 341. The second digit of the quotient is then a 5.
It's time to "drop" the last digit of the dividend, 2, and continue with the subtractions:
and finally run out of the calculations to be done in the image of right you can see clearly how my division results in 534, while the rest 520:
For centuries this was the only way to calculate divisions. It does not matter if you would use Roman numerals or decimals, once transcribed the dividend on the abacus, all calculations were made only on the shot, the final result could then be entered either in decimal or Roman numerals. ▲
I "rods Napier"
let's see how you could do the same division with sticks Napier (hence I have already spoken here, regarding multiplication).
It starts by placing the rods in order to make the divisor with the numbers up (in this case, 723). As I explained above, initially to manage the figures of the dividend are the first four:
With sticks Napier is easy to see what is the largest multiple of 723 to less than 2564: this multiple is 3 x 723, 4 x 723 since 2892 would, too great. Then transcribe the 3 and the product of 3 x 723, then calculate the difference (bottom right), then "drop" the next digit of the dividend:
Now repeat the procedure : the largest multiple of 723 to less than 5 x 3956 is 723, as 6 x 723 4338 would, too large. Then transcribe the 5 and the product of 5 x 723, then calculate the difference (bottom right), then "drop" the last digit of the dividend:
repeated for 'last case: the last digit of the quotient is 4, the right result: the quotient 534, remainder 520.
ones I described above were the most common means of "mechanical" to do arithmetic calculations before the invention of mechanical machines. The development of mechanical calculators Pascal began and ended with only a few decades ago with the introduction of the first electronic calculators. ▲
rational numbers
Arithmetic is historically born to perform calculations on the natural numbers (positive integers without zero). Any operation of addition and multiplication on these numbers always gives a natural number: it is said that the system of natural numbers is closed with respect to the operations of addition and multiplication.
The subtraction instead può dare come risultato sia numeri negativi che lo zero: ecco comparire una nuova classe di numeri, quella dei "numeri interi relativi", o semplicemente dei "numeri interi", costituiti da tutti i numeri interi positivi, negativi e lo zero. Una volta introdotto questo nuovo sistema di numeri, si può dimostrare che esso è "chiuso" rispetto a alle operazioni di somma, sottrazione e moltiplicazione; queste operazioni, se fatte su numeri interi, possono dare come risultato solo altri numeri interi.
Negli esempi mostrati più sopra abbiamo calcolato divisioni fra numeri naturali che hanno dato un quoziente, ma anche un resto. Cosa significa questo resto that "advances"? That the division is not over, but could go looking for the digit "places" after the comma. The division then gives rise to a new class of numbers: the "rational numbers".
Rational numbers are those obtained by the division or ratio of two natural numbers (the term "rational" comes from the Latin "ratio", in the meaning of the report). Every rational number is the result of a division in / b where to is the numerator and the denominator b; b must obviously be different from zero, while if we b = 1 the result is an integer: integers (natural) are a subset of rational numbers. Same manner as for integers, it can be shown that the system of rational numbers is closed under all four arithmetic operations (I'll stop here ... because of the number ranges there would be several more!)
The result of a division between integers can give a whole number (if the dividend is a multiple of the divisor) or two other types of result:
- a limited number of decimal places (with a finite number of digits after the decimal point)
- a decimal number limit (infinite number of digits after the decimal point)
The type of result depends on the divisor. If a fraction reduced to lowest terms in the denominator has a number composed of only the prime factors 2 and 5, the quotient will be limited to a decimal number (this is because the number 10, behind the decimal number is divisible by both 2 and for 5), but if the denominator includes several other prime factors of 2 and 5, the result will be a recurring decimal.
In this regard, I want to prove two things that we were taught at school, and which perhaps not everyone knows the explanation:
1 - Division between integers: if the decimal result is unlimited, it is always periodic.
2 - At each recurring decimal fraction may be associated with a generator, or that the relationship between integer division which gives the number given period. ▲
recurring decimal
We see the first point. Once you have made a division if the remainder is zero, the quotient is an integer and division end here. Otherwise you can continue the calculation "calando" gli zeri, che ci consentono di trovare le cifre dopo la virgola. Anche qui, appena ottengo un resto uguale a zero il calcolo finisce. Facciamo un esempio con la divisione 13 : 40
Vediamo il calcolo di questa divisione riga per riga: nella riga A imposto la divisione; dato che il divisore è maggiore del dividendo, nella riga B scrivo lo "zero virgola" e nella C calo uno zero. Il 40 nel 130 ci sta tre volte, infatti nella riga D aggiungo la cifra 3 al quoziente e scrivo il 3 x 40 = 120. Nella riga E calcolo la differenza e calo un altro zero; proseguo I like this until the line: the rest zero terminates the Division, whose ratio is 0.325. We now
Division 1 / 6
Again in line to set the division, the B write "zero point" and a decrease in C zero. The 6 in the 10's there once: D Add the figure in row 1 to the quotient and write the 6 x 1 = 6. In the line and calculate the difference and drop another zero, there is the 6 in 40 6 times and then add 6 to the quotient, transcribe the 6 x 6 = 36 and calculating the difference. Line G shows the same remainder there was the previous step (line E): this means that from now on, the procedure will be repeated ad infinitum, ad infinitum as they add even 6 in the quotient. In fact the quotient is 0.16, with 6 regular.
So far we have seen two cases: a division in which at some point the rest to zero, and another in which the rest is repeated unchanged indefinitely. But there are different remains that I get when I am calculating the digits after the decimal point? Well, the remains are all possible numbers between one and the value of the divisor minus one. Example: If the divisor is 7, the remains are possible only 1, 2, 3, 4, 5 or 6. Then it is clear that sooner or later necessarily happen to come across in fact already obtained earlier. Let us now own Division 1: 7
concentrate on the ruins: the first one that I find (line C) is a 1, then (line E) a 3, then 2, 6, 4, 5 and finally ( O line) to a new one. Clearly, this new one starts a sequence identical to the one I just described, and that will be repeated ad infinitum: the quotient then has a period of 6 digits.
recap: the ratios can be integers, decimal, limited or unlimited, in which case the period can not be made up of a number of digits greater than or equal to the number indicated by the divisor. Here's how it is that no division can never produce a quotient decimal not unlimited but periodic . ▲
The village generating
Let's see the second question left open because every recurring decimal fraction may have a generator?
If you remember something from the average, each number could be written as a decimal fraction. If the decimal number is limited as 0125 to create just a fraction where the numerator is the number given without the comma, the denominator is a 1 followed by many zeros as the number of decimal places: then we would have 125 / 1000 che, ridotto ai minimi termini, fa 1 / 8.
La regola per un numero decimale periodico semplice (come 0,57 con 57 periodico) è: scrivere una frazione in cui al numeratore si riporta il periodo (57) e al denominatore un numero composto da tanti 9 quante le cifre del periodo (99). Insomma la frazione generatrice di 0,57 periodico è 57 / 99: vediamo perché.
È semplice vedere che le divisioni 1 / 9, 1 / 99, 1 / 999 eccetera danno luogo a quozienti di questo tipo:
Calcolare 57 / 99 significa moltiplicare 57 per il reciproco di 99, quindi 57 per 0,01 con 01 periodico. Ecco qui il calcolo:
Da quanto sopra si vece che 57 x 0,01 periodico, che è come dire 57 / 99, dà proprio 0,57 periodico!
Per adesso mi basta "svelare" da dove saltano fuori quei 9 al denominatore della frazione generatrice; ma le cose si complicano un po' se ci troviamo davanti numeri con parte intera diversa da zero e/o antiperiodo (es. 12,425555... dove solo il 5 è periodico). Se poi la cosa vi interessa, diciamo che... non è impossibile ricavarle da quanto ho spiegato qui sopra!
One last question: what kind of number is the regular 0.9? Presto said:
0.9 is indistinguishable from the regular, then the same thing! ▲
Curiosity stiruche
a historical curiosity. When did you start writing decimal numbers, not in the sense of whole numbers in decimal notation, but numbers with decimal digits after the decimal point? The first to do this was the Belgian or Simone Simon Stevin of Bruges (1548-1620), also known as Simon Stevin. In one of his works from 1585 describes the arithmetic operations on numbers written in decimal notation (although, curiously, did not accept the existence of unlimited decimal numbers ...). Numbers appear in this publication are written in this way:
The figures in circles indicate the powers of 10 to put in the denominator. Here's what they represent in other words:
that simplify and
and in modern notation means:
point (or comma) was used for the decimal first by the mathematician, astronomer German theologian and Pitiscus Bartholomaeus (1561-1613, the one who coined the term "trigonometry", starting in 1595, then followed by other mathematicians as Napier that with which we already had (and we still have) to do. ; ▲
Next Chapter: Square Root
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