Summary:
Count + + Numbering system
- Addition and Subtraction
Roman numerals
Decimal numbers
Stealing
commutative property and Negative Numbers
Negative numbers today
+ Multiplication + Division
+ + Square Root Exponentiation
+ + Logarithms
slide rule
In the first chapter we saw how the operation of counting is sufficient to solve many arithmetic problems. Of course, when the numbers become large, this system is unthinkable!
Roman numerals Roman numerals The system started out as simply "additive" means accumulated symbols from more largest to smallest until you reach the desired value (IV = IIII equivalence is an addition to the Middle Ages, see below). So for example, numbers 297 and 728 are represented by:
CCLXXXXVII
DCCXXVIII
Be the addition of these numbers was relatively easy (the total is 1025): Just concatenate all the symbols in the two numbers, and then replaced by right figures redundant (I write in bold figures to be replaced: The five become a V a V two X, etc.):
DCCCCLXXXXXXVV IIIII
DCCCCLXXXXXX VV V
DCCCCL XXX XX XXV
DCCCC LL XXV
D CCCCC DD
XXV XXV
MXXV
Things are further complicated when they were introduced "shortcuts" that I mentioned before, such as IV = IIII, VIIII = IX, XL = XXXX and so on. In short, we wanted a good practice ... ▲
Decimal numbers
In previous chapter we saw how we got to use decimal numbers that have been born in Babylon and then come to us, passing through India: Do not think I have to explain how these numbers with an addition! ▲
Stealing
the sum operation, which derives directly from the count, it is not sufficient to solve any transaction problem arithmetic, and it was not even for men of antiquity.
Many human activities have developed in prehistoric times in the Fertile Crescent, ranging from Egypt to Mesopotamia, the cradle of civilization Mediterranee. Agriculture in these areas is characterized by grains (which give dry seeds) and vegetables (which give dried fruits), all easily storable commodities.
The need to store these commodities required different institutions: a system of laws which force farmers to surrender part of their harvest, the construction of special storage areas and, with regard to our history, scribes can keep track of stocks . It is possible that the scribes to "encourage" farmers to Privas part of the crop, have invented some "sacred rite" in order to strike the proper fear of divine punishment, hence a rise priests di chissà quale culto il passo è breve... (insomma, eccoci belli e sistemati con una casta in grado di dominare conoscenza, coscienze, economia... e, anche se forse non sempre per vie dirette, di gestire anche il potere politico!)
In questo collage, fra farro e lenticchie rosse ho inserito la statua in diorite dedicata al dio Ningishzida, 2120 a.C. (periodo neo-sumerico), ritrovata tra le rovine di Girsu, Tellō (Iraq meridionale).
L'aritmetica inizia a svilupparsi in epoche antichissime: forse una delle prime applicazioni del contare può essere stata quella dei cacciatori che incidevano tacche sul manico della loro ascia per tenere il conto degli animali abbattuti, ma per fare questo avevano bisogno solo di aggiungere (sommare) una tacca ogni tanto. Per gestire un magazzino invece occorre non solo sommare... ma anche sottrarre, a seconda delle entrate e delle uscite! ▲
Proprietà commutativa e Numeri negativi
La sottrazione complica la vita molto più dell'addizione, per almeno due motivi: intanto la sottrazione non gode della proprietà commutativa dell'addizione. Se nell'addizione
calcolare 4 + 3 è lo stesso che 3 + 4,
nella sottrazione
calcolare 4 – 3 is not the same as 3 to 4
addition, subtraction 3-4 result which gives?
Indeed, the result of the subtraction even 3-3?
Let's start with the latter question. When the scribe had to turn the empty warehouse, who had gone to ask "how many lentils you" would not have heard some say "zero" rather than "no" or "does not exist anymore." The zero is not a natural number, and to accept the row of numbers it took not centuries, but millennia. The Babylonians, the inventors of the existing system positional numbering only to the third century. BC had invented a symbol to represent zero as a digit (as in our numbers, in which 101 means one hundred, zero tens and units), but not as a number to itself.
The first traces of zero as a number indicating the found nothing in the mathematics of Ptolemy (second century), which sometimes uses a separate symbol to indicate zero, but the first systematic study of the zero number is due to the Indian mathematician Brahmagupta (598-668), who in a text of 628 not only sets out the rules for the use of zero as a number, but also negative numbers:
- The sum of zero with a negative number is negative
- The sum of zero with a positive number is positive
- The sum of zero by zero is zero
- The sum of a positive and a negative is their difference [1]
- If a positive number is divided by zero as a fraction with zero in the denominator [2]
- Zero divided by a number pos. or neg. is zero or a fraction with zero in the numerator
- Zero divided by zero gives zero [3]
[1] is the algebraic sum, which I'll discuss later, [2] the author does not express what this means fraction with zero in the denominator, [3] diciamo che questa è... una "semplificazione eccessiva"!
Veniamo a tempi più recenti. Leonardo Fibonacci (di cui ho già parlato qui ), nel suo "Liber Abaci" introduce lo zero come cifra, ma anche come numero; inoltre inizia ad usare i numeri negativi, ma solo per distinguere i debiti dai crediti. L'uso dei numeri negativi in questo senso ha comunque tardato molto a diventare una pratica comune, anche perché i mercanti continuavano ad usare i numeri romani; la partita doppia come sistema di contabilità prevedeva il calcolo in due colonne distinte per il dare e l' avere , precisely to avoid having to resort to negative numbers.
Unlike merchants, among scholars of the mathematical sciences Arabic numerals spread faster, but the doubts about zero and negative numbers continued for centuries.
A (small) step forward makes the French mathematician Nicolas Chuquet (about 1445-1500), which incorporates the Fibonacci numbers
saying that "the tenth figure has no means no value, and therefore nothing is known figure or shape of any value. "Chuquet still uses a very verbose when expressing the four operations with plus, moins, multiplier par, par partyr (the expressions with operators + - x: we are used to have required centuries of adjustments). Chuquet's work is especially important because it lays the foundations of algebra, as somehow sets the modern method of studying the values \u200b\u200bof the unknowns in their various powers (classic example: the quadratic equations where the unknowns X appears simple in its value and its square). Here, not only Chuquet invent symbols to express the exponents, but also uses negative exponents, and this is the first time ever in the West are used intentionally negative numbers in algebra! Beware, though: negative exponents are equivalent to the calculation of the mutual, for example x ˉ ² vuol dire 1/x², xˉ³ vuol dire 1/x³ e così via. Se x è positivo, queste operazioni danno sempre come risultato numeri positivi: i numeri negativi in quanto tali, per Chuquet, continuavano ad essere assurdi.
Per vedere il primo studio che non rifiuta del tutto i numeri negativi bisogna aspettare Girolamo Cardano (1501-1576) che accettò tali numeri come soluzione alle equazioni di terzo grado... ma sempre chiamandoli "numeri ficti" ! ▲
I numeri negativi oggi
Al giorno d'oggi i numeri negative are no longer a big problem. At least, until you come across in the teaching of arithmetic in school, where the introduction of negative numbers, associated with the concept of "algebraic numbers", it creates anxiety and, often, rejection. But I believe that any child will understand the difference between a temperature of -8 ° and +8 ° ...
fact is that as soon as you hear the word "algebra", usually the curtain falls and your brain refuse to cooperate! Yet the adoption of the negative numbers involved, in the context of abduction we are dealing with a simplification : if instead of considering a subtraction expression
4-3
see it as a sum,
4 + (-3)
then return to have the commutative property except that I had at the beginning, because this expression above is exactly equivalent to:
-3 + 4
short, all the progress being made in mathematics are aimed at simplifying , that regular use increasingly synthetic as general as possible. The use of numbers that include negative values \u200b\u200band zero, not only agrees with all the possible results of subtractions, but allows to generalize any expression of mixed addition and subtraction in a single addition of terms, each to be considered with its positive or negative. ▲
Next Chapter: Multiplication
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