Decimal System

The system is a decimal number system using base ten. In this system, powers of ten and their multiples enjoy a privileged representation.

decimal

The decimal system is the most widely spread. And consist, for example, counts:

Rating Systems

People with a radix decimal employed, over time, various techniques to represent numbers. Here are some examples.

• With numbers for one, ten, hundred, thousand, etc..

Number systems whose figures represent powers of ten are additive type. This is the case of the Egyptian numeration. Example: 1506 reads

hieroglyphic writing (1000 +100 +100 +100 +100 +100 +1 +1 +1 +1 +1 +1).

• With numbers for one, five, ten, fifty, a hundred, five hundred, etc..

Such systems are also count-up type, but involve a quinary system auxiliary. This is the case counts penthouse, Etruscan, Roman and Chuvash. Example: 2604 reads MMDCIIII. Roman numerals (1000 +1000 +500 +100 +1 +1 +1 +1). Roman numerals also knows an alternative additive and subtractive: 2604, in this way, written MMDCIV. (1000 +1000 +500 +100-1 +5).

• Avec des chiffres pour un, deux, ..., neuf, dix, vingt, ..., cent, deux cents, ..., neuf cents, etc.

Number systems employing nine digits for the units, and for tens, hundreds, etc.. are still type additive. This is the case counts Armenian , Arabic alphabet, Gothic , Greek and Hebrew. Example: 704 figures written in Greek Ionic (700 +4).

• With numbers from one to nine, ten, hundred, thousand, etc..

Number systems whose figures represent the units and the powers of ten hybrid type. It counts if Chinese and Japanese. Example: 41007 written in the Japanese system (4 10 000 1000 7). The Chinese system uses zero to indicate more empty positions before the units: 41007, written in Chinese figures (4 10 000 1000 0 +7).

• With numbers from zero to nine

Number systems whose figures represent the units are type positional. This is the case counts Arab non-alphabetic, European, most Indian counts and counts Mongolian and Thai. Example: 8002 8002 is written in Thai numerals (8002).

History

The base ten is very old. It follows a natural choice, dictated by the number of fingers of both hands. The Proto-Indo-European probably had in base ten. A decimal system was developed by:

• ^Third millennium BC. BC , the Egyptians . Note however that the Egyptian system was a decimal system without positioning ^, .
• -1350 before, the Chinese
• v. -650, The Etruscans
• v. -500, The Indusiens in Sanskrit

• Note that the ancient civilizations of Mesopotamia (Sumer, Babylon ,...) used a sexagesimal positional base (60) ^{, , ,} .
• Also note the use of proto-Elamite mixed systems, "said bisexagsimaux (binary, decimal or sexagesimal after qualifying objects or living things count) ^, .
• Civilization Maya used a base 20 system by introducing some variants .

Bases combined

Decimal Numeration combined with an auxiliary base

Counts may use decimal subsidiary bases:

• Quinary system auxiliary is used in some scoring systems (see above) and stating the numbers in some languages such as Wolof.
• Vigesimal auxiliary system is used for stating the numbers in some languages, like Basque, or "eighty" in French.

Decimal Numeration used as a backup system

• In French and in most world languages, the decimal system applies until 9999, every power of ten being, exposing one to the exponent three, designated by an appropriate term (10 ¹ = " ten "; 10 ² =" percent ", 10 ³ =" thousand "). Strictly speaking, the utterance numbers greater than 9999, however, is more decimal places when, among the Powers of ten exponent greater than three, do not benefit from a denomination that correspond to those powers of a mile.
• Based miles involved in writing the numbers in the integer part of large numbers , for easier reading (eg written 12,345,678 12,345,678 12,345,678 or by country), but we should not separate the digit decimal part of each other.
• The Babylonian numbering systems and measuring time and angles in minutes and seconds, sexagesimal , using a decimal system auxiliary.
• The Mayan count , although vigesimal , revealing a decimal system in the auxiliary statement numbers.
• The Chinese and Japanese uitilisent base ten thousand with ten as auxiliary base.

Systems of units

China's measures of capacity and fractional weights are around 170 BC. BC United States, the monetary system is decimal 1786. In Europe, the decimal units is initiated in France from 22 August 1790, when Louis XVI asked the Academy of Sciences to appoint a commission to set the weights and measures. The latter calls for decimal division.

Advantages and disadvantages

Most modern languages down the numbers in base 10 because of some advantages of this:

• on behalf of the ten fingers is very intuitive as has been mentioned above;
• its magnitude is satisfactory, because it significantly reduces the length of a large number compared to the base 2, while maintaining tables of additions and multiplications memorable.

However, it was not until the widespread use of positional notation, and the existence of a division algorithm adapted to this notation for units of measurement are gradually losing their non-decimal sub-multiples. When the book included 20 cents in France to 12 pence (or Great Britain 20 shillings 12 pence) economic agents appreciated that this unit could be divided exactly by 20 different divisors (including 1 and 240). In 1971, despite the computer who can now easily manage the heterogeneity of relationships between non-decimal sub-multiples, Great Britain did not hesitate to decimalize its currency.

Mathematics

Conversion to the base N of a number written in base 10

To place a number in base 10 to a number in base N, we can apply the following method:

Let K be the base-10 number to be converted into N-base.

1. Perform the integer division of K N. D is the result of this division and the remainder R
2. If D> = N, repeat in 1
3. Otherwise, writing in base N of K is equal to the concatenation of the last result and all the remains, starting with the last.

Example: Convert the number 3257 in base 16

• 3257/16 = 203, or 5625
• 3257 = 203 16 + 9
• 12 203 = 16 + 11

Knowing that 11 is a B and is denoted as 12 C, the writing of 3257 in base 16 is CB9.

Conversion to base 10 of a number written in base N

To place a number in base N is a number in base 10, we can apply the following method:

Let K be the number in base N to convert. For any number c of rank r in K, we calculate c N ^r. The representation of K in base 10 is the sum of all products.

R counting starts at zero from right to left.

Example
The number "10110" is written in base 2 to base 10:

1 2 ⁴ + 2 ³ + 0 1 1 ² 2 + 2 ¹ + 0 2 ⁰ = 22 (base 10)

Example
The number "3FA" base-16 written in base 10:

3 16 ^{2 +15 1 +10} 16 16 ⁰ = 1018 (base 10)

Reminder: F base sixteen is 15 in base ten, sixteen A base is 10 in base ten.

Library Resources

• Evolution of human culture (note, the information on this site are no longer updated.)
• Measure

See also

• Figure
• Number system
• Positional decimal notation
• Decimal expansion
• Decimal
• Quinary system
• System vigesimal
• Sexagesimal system
• Numbers in French
• Positional notation
• Decimal system without zero
• International Bureau of Weights and Measures

References

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