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numeration

For numeration is understood a whole of rules for enunciate and write the numbers. When in
times and different places the numbers did their appearance, bore the demand of a system that allows to point out them, to voice and in writing, using few words and few fundamental signs. Each people effected an own system of spoken and written numeration, in the course of the history was many the systems that asserted and they then disappeared. Today there are different systems, but most diffused it in the world is the system of the decimal numeration.
The base:
Each system is differentiated from the others for the base, the more used numeration is that to base 10. In most ancient system of numeration the base was 5, because 5 is the fingers of the hand. The numbers, for written, they were suitable with of the dots or of the lines engraved on tablets of clay or papyruses.The Roman notation was base 5 and 10. These two numbers are pointed out with the V signs and X and the single unity, suitable with the sign I, additions were considered or subtracted one had positioned to their right or left.
I II III IV V VI VII VIII IX X XI XII
If we define those suitable from the small numbers of the base like unity of 1° order,the same base is an unity of 2° order. The ancient considered unity of order also superior and they pointed out it with special signs. The Romans, for example they hollow out with the letters L, C, D, M, the superior groups to 50, 100, 500, 1000 unit.

additive numeration and principle of position

The ancient numerations of they define additives because for read the written numbers in those systems needed to add the values of all the signs with which they had represented. For example in the Roman notation the number 673 has represented from the symbols DCLXXIII and that is 500+100+50+10+10+1+1+1=673. This additive writing of the numbers made however complex the execution of the operations arithmetics, but with the years was been affirming the beginning of position. As for perform the calculations they were traced of the lines for earth and between a line and the other one they put of the small stones that constituted the unities you count. The spaces understood between the lines they corresponded the varieties orders and each small stone was worth a unity of the corresponding order to the space in which situated era, as his value depended on his position as regards the lines. In succession instead of tracing lines on the ground, a tool was used, that the Romans they called abacus: constituted from a tablet of wood or terracotta with of the grooves parallel in which they got ready the small stones, said calculi, from which the word calculates for point out any operational procedure.

positional notation

In the Middle Age the abacus had simplified in this way: they on the tablet had affected four parallel lines and they in the spaces between them placed of the diskettes marked from nine signs different, the "figures" for the unities from 1 to 9. In each space placed an only diskette and this represented the number from the figure suitable from the same diskette but of relative order to the space in which the diskette was situated. In succession when was begun to write on the paper, was thought about doing without of the abacus, leaving that each figure points out his only position with the the order of the unities. For point out the lack of any unity in anybody of the spaces of the abacus was used a sting soldier and then finally the zero. The invention and introduction of the zero is the end of the additives numerations and the beginning of the positional notation. The positional notation with him uses of the figures, is from India, where the first traces of such system go up again the VI sec. A.D., but his origin is unknown. After the year 1000 did his appearance in West from the Arabs, whose nearer civilization to the Indian one. In the 1202 the Italian Leonardo Pisano, nicknamed Fibonacci, after have discovered the used arithmetic from the Arabs during his trips in East, the diffused in Europe with his treatise Liber Abaci (book of the art of do the accounts). The figures use for the positional writing from the Indians they took different form depends on the places and the times. Their modern form goes up again at 1300, they besides come definite Arabic, but is uncertain the origin.

Roman notation

The Roman notation is based on 7 fundamental figures:

I	V	X	L	C	D	M
1	5	10	50	100	500	1000

and 6 groups base:

IV	IX	XL	XC	CD	CM
4	9	40	90	400	900

In these groups the figure to left has stolen from that to the right (IV=4=5-1). The other numbers are gotten with varied combinations of the seven fundamental symbols and
of the six groups base. The groups base is the only cases in which a number with a subtraction is pointed out, 49 is not written with the IL symbol (50-1), but must be written XLIX (40+9). The Roman numbers are written putting to the right of a fundamental figure or of a basic group the other necessary figures, so that the add of all the written figures are equal to the number that wants to be pointed out.
XCIX = 99
CI = 101
CIV = 104
DXXXVII = 537
DCXL = 640
Any particular rules:
The first 3 multiples of the symbols base The, X, C, M, is gotten repeating the symbols. For instance XX= 20, XXX= 30, these symbols could be repeated only three times.
The symbols V , L, D, ever is not repeated.
With these rules the taller number that could be written is 3999=MMMCMXCIX
(3000+900+90+9) and in fact the M symbol could not be repeated more than three times, and there is not no fundamental superior symbol to M. For go on the numeration, the Romans used a particular adroitness, setting a line above the symbol ¯ and they with this understood to multiply the value of the numbers for 1000. For multiply a number for 100000 then, beyond to the superior line two vertical lines were added him |¯| such from frame it.

binary notation

A system of more used numeration in the antiquity, it is the binary system, to base 2, constituted alone two figures, 0 and 1. In this system the numbers have gotten adding the multiplications of the figures 0 and 1 for the decreasing powers of 2. For instance the number
9 in the binary system is suitable from the sequence 1 0 0 1 and is equal to:
1 x 2³ + 0 x 2² + 0 x 2¹ + 1 x 2⁰ = 8 + 0 + 0 + 1 = 9.
From which by means of the first four powers of 2: 2⁰= 1; 2¹= 2; 2²= 4; 2³= 8, it is possible define the first 16 numbers.

2³	2²	2¹	2⁰	decimal num.
0	0	0	0	0
0	0	0	1	1
0	0	1	0	2
0	0	1	1	3
0	1	0	0	4
0	1	0	1	5
0	1	1	0	6
0	1	1	1	7
1	0	0	0	8
1	0	0	1	9
1	0	1	0	10
1	0	1	1	11
1	1	0	0	12
1	1	0	1	13
1	1	1	0	14
1	1	1	1	15

Such system is to the base of the calculuses performed with the electric circuits, in fact attributing to the value " 1 " an impulse of current, and to the value " 0 " his lack, it is possible transmit or record of the numbers by means of the binary system. Combining more electric based circuits on this system you has built the calculator.