Summary:
+ Count
- Numbering system
; Roman numerals
Numbers Babylonians
Numbers Decimal Numbers Binary
+ Addition and Subtraction
+ Multiplication + Division
+ + Square Root Exponentiation
+ + Logarithms
slide rule
In previous chapter we saw how the operation of the count is sufficient to solve many arithmetic problems. Of course, when the numbers become too large to be treated should find some faster system, but if the numbers get too big ... must also find a way to write! In fact you can not think of going on and on just doing the signs are all equal:
Numbers
Romans The Romans decided to group the notches in groups of five with the adoption of the V sign . Legend has it that it is a sign that represents an open hand:
as well as X, the number 10, would be two hands together
la cosa più probabile è comunque che tali segni siano stati scelti perché facili da intagliare su un palo di legno; in effetti sono stati trovati esempi di numeri scritti con simboli diversi da quelli "moderni", e l'adozione di lettere tutte facenti parte dell'alfabeto latino è stato un processo piuttosto lungo. Per esempio:
il 50 era nato come una V con una tacca in più (sempre un simbolo facile da intagliare); nel tempo è evoluto fino a diventare la L che conosciamo. Analogamente:
il 100 è nato come una X con una tacca in più, poi si has evolved to take the form of the letter C (possibly from the Latin "centium" ).
was finally codified a standard form to represent the "base numbers" in order to reach one million:
The problem with these numbers was that, although seemingly intuitive, it was very difficult us calculations. ▲
Numbers Babylonians
While the Romans, who were great scientists, they "made do" with these numbers, neighboring civilizations have implemented systems that are far better, such as greek, especially Egyptian and Babylonian
one shown in the photo is a Babylonian clay tablet found in the nineteenth century, known as Plimpton 322 tablet (from name of the collection of GA Plimpton at Columbia University).
Dating back to 1800 BC, in cuneiform writing contains numbers arranged in a table of four columns by 15 rows. According to one interpretation, the numbers are a list of Pythagorean triples whose numbers are the solutions of the Pythagorean Theorem, a ² + b ² = c ², (for example, 3 ² + 4 ² = 9 + 16 = 25 = 5 ²) but placed in such a way as to give rise to a series of trigonometric values!
The Babylonians built a system of numbering a genius: use for the first time a positional system in which a small number of symbols change meaning according to their relative positions. The thing for us today is clear: if I write the number 2222, means that I'm adding two thousand two hundred two scores with two units: total 2222, precisely where the figure 2 employs four different meanings.
The remarkable thing is that, unlike all other systems developed in history and are based on numbers 5 and 10 (from the fingers of our hands), and sometimes on the number 20 (even those of the feet ...), i babilonesi scrivono i loro numeri basandosi sul numero 60. Insomma inventano un sistema di scrittura dei numeri in base 60: usando 59 simboli, da 1 a 59 (più il simbolo su sfondo verde, di cui parlerò fra poco):
erano in grado di scrivere la cifra delle unità (da 1 a 59 appunto), delle "decine" (in realtà a ogni unità della seconda cifra corrispondeva il valore 60), delle "centinaia" (a ogni unità della terza cifra corrispondeva il valore 3600) e così via. Non solo: scrivevano anche i numeri decimali, usando gli stessi simboli per i sessantesimi, poi i tremilaseicentesimi... Faccio un esempio:
As you can see from the accounts, the number 2 13 44 15 12 8024.265 meant. It seems to us obscure ... but it's only a matter of habit, perhaps to the Babylonians would have seemed abstruse our decimal system!
way: why use just the number sixty? Perhaps because it has many divisors: 60 can be divided by 2 and 5 (like our 10), but also for 3, 6, 12, 15, 20 and 30, then the Babylonians were much more likely for us to write numbers exact decimal (for us even the simple division 1: 3 gives an infinite decimal).
Over the centuries, the Babylonians have found that the lack a symbol for zero was creating confusion usually tell from the speech if you had a two-digit number representing unity and "tens" (sixty) or unit and "hundreds" (tremilaseicentine). In fact at some point is also created the symbol for zero (green in the background of the figures above), but do not be fooled: This is the zero digit, not a number! In fact it is one thing to use a symbol that has only the function of "placeholder" (as in number 101, which is to say, a hundred and ten no units) and another account is to design a completely number zero : to get used to conceive the number zero (which in fact is not a natural number) must wait until the middle of the sixteenth century!
Claudius Ptolemy (about 100-175), the author of the famous Almagest (the book describing the geocentric system) uses all his admirable work in their numerical notation, the more versatile than previously known. For more glaring coincidence occurred in relation to pi-greek:
By definition π is the ratio of circumference to diameter of a circle, and since man has started studying the geometry sought to determine its value with the accurately as possible. On π entire books have been written, and to determine the nature of numbers (the famous problem of squaring the circle ") it was not until the late nineteenth century! At the time of Ptolemy it was found a good approximation in the fraction 377 / 120: the quotient of division 377: 120 gives 3.1416 with 6 magazine, a close approximation of π (which actually is 3.14159265. ..), with an error of only 0.000074 (precision even 0,002%). Ptolemy must have been obvious to choose the value 60 for the radius of the unit circle by which to do all its calculations, not only was the basis of the numbers he used, but with its dual obtained 120, ie the diameter of a circle whose circumference is beautiful (almost) exactly a whole number 377! (The exact calculation is 376.991)
(For completeness, it must be said that 355: 113 gives π to an accuracy better than 250 times compared to 377: 120, but 113 of this fraction with the denominator I never found any trace anywhere)
A numbering based on 60 may sound strange, but the numbers Babylonians have left a legacy that is still alive: on our watch, with the hours that are divided into sixty minutes sixty secondi; e negli angoli, in cui ogni grado si divide in 60 primi di 60 secondi. ▲
Numeri Decimali
In Europa per molti secoli rimasero in uso solo i numeri Romani; e le cose peggiorarono ulteriormente nei secoli bui del medioevo, quando diventò mentalità corrente che lo studio si dovesse applicare solo alla teologia e alla filosofia: le scienze "esatte" furono accantonate quasi del tutto. Nel frattempo il sistema posizionale babilonese veniva adottato in India, cambiando da sessagesimale a decimale e dando luogo a ciò che oggi chiamiamo numeri Indian or Indo-Arabic, or Arabic (in other words, the numbers that we know today).
So the Roman numerals in Europe continued to be used to deal, as in everyday life, without scientific problems to solve, is not frequently have to make things more complicated than addition and subtraction. Unless ...
... one does not operate a currency exchange desk! This job required her to master concepts such as fractions, decimals, multiplication and division (all arguments which I will soon), and here the Roman numerals are a disaster: it was essential to make the calculations using the abacus (which I'll from next chapter).
The turning point came in early 1200 to the work of Leonardo da Pisa, called Fibonacci ( filius Bonaca , 1170-1250). He lived several years with his father William the Bonacci, who was representative of the merchants of the Republic of Pisa in the region of Bejaia (Algeria), where he studied arithmetic processes that Muslim scholars were spreading in the different regions of the Islamic world. Here he also had contacts with the world of traders and mathematical techniques unknown in the West. Some of these methods were introduced for the first time in India, just to refine these skills, Fibonacci traveled widely, reaching Constantinople, alternating trade with the mathematical studies.
In 1202
Fibonacci writes a fundamental text for Western culture: the "Liber Abaci", a weighty manual of arithmetic and algebra, with which it introduces in Europe the Hindu-Arabic decimal number system and the main methods of calculating. The crucial innovation is the following:
There are nine Indian figures: 9 8 7 6 5 4 3 2 1. With these nine figures, and the symbol 0, which the Arabs call Zephiro [which later the name "zero"] , any number can be written, as will be shown. FibonacciIn this book explains the fundamentals of the new numbers, teaching the algorithms to calculate the four operations without having to use the abacus, it is from here that we develop the arithmetic, algebra and mathematics in the West.
Another story to tell would be the evolution of symbols used to write different numbers, because even today the figures used in the Arab world are very different from ours:
In Actually the numerical method already in the West Indian had arrived the previous century, by the translator, British mathematician and philosopher Adelard of Bath (1080-1152). Adelard translated into Latin the book "Algorithms number of Indorum" the Persian mathematician Al-Khwarizmi (Full name Abu Ja'far Muhammad ibn Musa Khwarizmi, about 780-850). The first word of the title is a transliteration of the name, but then got lost in the origin of its name and the word algorithm has become a common noun, taking on the meaning (in information) of the standard method to accomplish a complex on the basis of elementary processes (sum the digits, write the result, reports the carryover amount ...).
Adelard's translation, although it preceded the work of Fibonacci, was not very successful. Fibonacci was luckier, perhaps because the Liber Abaci was written in a Tuscany dedicated to trafficking of all kinds ... where this nuovo sistema ha trovato subito un certo numero di estimatori. Attenzione però: le abitudini sono dure a morire, e coloro che non avevano bisogno di compiere calcoli complicati continuarono a utilizzare i numeri romani ancora per secoli! ▲
Numeri Binari
Tutti i sistemi di numerazione che abbiamo visto più sopra sono di fatto equivalenti: in qualche modo consentono di scrivere gli stessi numeri (almeno, i numeri interi). Quindi la scelta del sistema di numerazione viene fatta in base alla sua praticità: ed ecco che, per certi usi, neanche il sistema decimale è the most suitable.
Since the electricity was used for calculations (through relays, solenoids, transistor or microprocessor) each circuit can be found in two different states: on and off. These two states are associated with the symbols 0 and 1 here is the birth of the binary system! The
which works exactly like the decimal, except that instead of running for powers of 10 (tens, hundreds, thousands) works for powers of 2. Then:
1011 = 1 x 2 ³ + 0 x 2 ² + 1 x 2 ¹ + 1 x 2 = 8 + 0 + 2 + 1 = 11
computer programmers are abituatissimi to make calculations of various kinds on numbers like these but when the numbers become very large it is not very practical. In fact, to express the number (decimal) 1234567 need the following number (binary):
1234567 (d) = 100101101011010000111 (b)
To simplify the writing of these big numbers were "invented" the other two systems for numbering
- Octal , in which the binary digits are grouped in threes, then encode the digits 0 to 7.
1234567 (d) = 4553207 (o)
- Hex, in which the binary digits are grouped in groups of four. In this case the values \u200b\u200bthat can take each digit from 0 to 15: for each digit then assigns a symbol that goes from 0 to 9 or A to F.
1234567 (d) = 12D687 (h) ▲
Next Chapter: Addition and subtraction
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