Decimal system

The decimal system is a Numbering system using the bases ten. In this system, the powers of ten and their multiples profit from a privileged representation.

Decimal notations

The decimal system is largely most widespread. Thus are made up, for example, numerations:

Marking systems

The people having a bases decimal numeration employed, during time, of the techniques varied to represent the numbers. Here are some examples.

With figures for one, ten, hundred, thousand, etc

The numbering systems whose figures represent the powers of ten are of additive type. It is the case of the Egyptian Numération. Example: 1506 is written in hieroglyphic writing (1000+100+100+100+100+100+1+1+1+1+1+1).

With figures for one, five, ten, fifty, hundred, five hundreds, etc

Such numbering systems are also of additive type, but utilize an auxiliary quinary system. It is the case of numerations attic, Roman Etruscan, and Chuvash. Example: 2604 is written MMDCIIII. in Roman numerals (1000+1000+500+100+1+1+1+1). Roman numeration also knows an additive and subtractive alternative: 2604, in this manner, is written MMDCIV. (1000+1000+500+100-1+5).

With figures for one, two,…, nine, ten, twenty,…, hundred, two hundreds,…, nine hundreds, etc

The numbering systems employing nine digits for the units, like tens, the hundreds, etc are still of additive type. It is the case of numerations Armenian, Arab alphabetical, gotic, Greek and Hebraic. Example: 704 is written ψδ in ionic Greek figures (700+4).

With figures from one to nine, and for ten, hundred, thousand, etc

The numbering systems whose figures represent the units and the powers of ten are of hybrid type. It is the case of numerations Chinese and Japanese. Example: 41007 is written 四万千七 in the Japanese system (4×10000+1000+7). The Chinese system uses in more the zero to indicate empty positions before the units: 41007, is written 四萬千〇七 in Chinese figures (4×10000+1000+0+7).

With figures from zero to nine

The numbering systems whose figures represent the units are of positional type. It is the Arab case of numerations not-alphabetical, European, of the majority of the Indian numerations and numerations Mongolian and Thai. Example: 8002 is written in Thai figures (8002).

History

The base ten is very old. It rises from a natural choice, dictated by the number of the fingers of the two hands. The Proto-indo-Europeans probably counted bases ten of them. A decimal marking system was developed by:

v. -4000, Élamites of the plate of Sumer,
v. -3200, Sumériens,
v. -3000, Akkadiens,
v. -3900, Proto-élamites in Mésopotamie (new numbering system),
v. -2700, the Egyptians
v. -2500, Semites of Mésopotamie
v. -2350, the whole of Mésopotamiens (base 10 supplants base 60)
v. -2000, Hittites and Indusiens
v. -1900, the Babylonians
v. -1500, the Assyrians
v. -1350, them Chinese
v. -650, the Etruscans
v. -500, Indusiens in Sanskrit

See also: positional decimal Writing .

Combined bases

The decimal notations use sometimes auxiliary bases:

an auxiliary quinary system is used in certain marking systems (see higher) and for the stating of the numbers in certain languages, like the wolof;
an auxiliary vigesimal system is used for the stating of the numbers in certain languages, as out of Basque, or “eighty” in French;
of the bases thousand and one auxiliaries million are often used for the stating of the numbers in the European languages, and a base thousand in the writing of the great numbers, in order to facilitate of it the reading, like, for example, 12 345 678, 12.345.678 or 12,345,678, according to the countries;
in Chinese and Japanese, a base ten thousand auxiliary is used.

Some other systems use an auxiliary decimal system:

the Babylonian Numeration and the systems of measurement of time and the angles in minutes and seconds, sexagesimal, use an auxiliary decimal system;
the Maya Numeration, although vigesimal, lets appear a decimal system auxilaire in the stating of the numbers.

Systems of units

In China measurements of capacity and weight are decimalized towards 170 av. J. - C. In the United States, the monetary system is decimal in 1786. In Europe, the decimalization of the units is initiated in France starting from August 22nd, 1790, date on which Louis XVI asks for the Academy of Science of name a commission to define the weights and measures. The latter recommends decimal division.

Advantages and disadvantages

The base ten comprises some assets:

the account on the ten fingers is very intuitive;
it is built on an even number, and division by two is most current;
its order of magnitude is satisfactory, because it makes it possible to reduce considerably the length of a great number compared to a base two, while preserving tables of additions and multiplications memorable;
it is most current;
the international standards are built on this basis.

However, the base ten is not that which offers the best benefit, because it is not based on a number having advantageous properties:

a number comprising much dividers (e.g.: 12 or 60) would have had a practical aspect, but ten has only two of them (2 and 5), and division by five is not most current;
a prime number would be interesting for mathematics, because, in such a base, the numbers with comma would be written easily in the form of irreducible fraction, but ten is not first;
a power of two would be adapted to data processing, but ten is not power of two.

Mathematics

Conversion into base NR of a number bases 10 of them

To pass from a number bases of them 10 with a number in base NR , one can apply the following method:

Either K the number bases 10 of them to convert into base NR .

To carry out the whole division of K by NR . Either D the result of this division and R the remainder
If D >= NR , to start again into 1
If not, the writing bases NR of it K is equal to the Concaténation last result and of all the remainders while starting with the last.

Example: conversion of number 3257 bases 16

of them 3257 = 203 × 16 + 9
203 = 12 × 16 + 11

Knowing that 11 notes B and that 12 notes C, the writing of 3257 bases 16 of them is CB9.

Conversion into base 10 of a number in base NR

To pass from a number in base NR to a number bases 10 of them, one can apply the following method::

Either K the number bases NR of it to be converted. For any figure C of row R in K , one calculates C × NR ^R. The representation of K bases 10 of them is the sum of all the products.

Example
The number “10110” bases 2 of them is written in base 10:

1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 0×2⁰ = 22 (bases 10)

Example
The number “3FA” bases 16 of them is written in base 10:

3×16² + 15×16¹ + 10×16⁰ = 1 018 (bases 10)

Recall: F bases sixteen of them is worth 15 bases ten of them, has in base sixteen is worth 10 bases ten of them.

Bibliographical resources

Evolution of the cultures of humanity
To measure

See too

Be-X-old: Дзесятковаясыстэмазьлічэньня Simple: Decimal

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