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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=51). 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 (501),
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 2^{3} + 0 x 2^{2} + 0 x 2^{1}
+ 1 x 2^{0} = 8 + 0 + 0 + 1 = 9.
From which by means of the first four powers of 2: 2^{0}=
1; 2^{1}= 2; 2^{2}= 4; 2^{3}=
8, it is possible define the first 16 numbers.
2^{3}

2^{2}

2^{1}

2^{0}

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.


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