Decimal Position System
The decimal position system, also called the Hindu-Arabic or Arabic system, is a numeral system in which all derived units are based on the number 10 and the powers of 10. The decimal system in nearly universal use today requires ten different symbols, or digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and a decimal point (dot) to represent numbers. In this scheme, the numerals used in denoting a number take different values depending upon position. A number is written as a row of digits, with each position in the row corresponding to a certain power of 10. A decimal point in the row divides the row into those powers of 10 equal to or greater than zero and those less than zero. Positions farther to the left of the decimal point correspond to increasing positive powers of 10 and those farther to the right to increasing negative powers.
As previously stated, the decimal number system is based on the powers of the base 10, that is (..., 1000, 100, 10, 1, 1/10, 1/100,...) = (..., 103, 102, 101, 100, 10-1, 10-2,...). Any real number can be represented in a variety of positional number systems, each of a unique base "b". Since the decimal number system has 10 as its base (i.e., b = 10), any real number x can be represented as: x = an10n + an-110n-1 + ... + a1101 + a0100 + a-110-1 + ... + a-(n-1)10-(n-1) + a-n10-n. The coefficients of this equation (i.e., the values ai, where i = {0, 1, 2, ... , n}) may take on the numeric values of 0 through 9; or stated another way, 0 ai b. As a particular example of the general equation just given, the decimal notation "647.25" represents the sum (6 x 102) + (4 x 101) + (7 x 100) + (2 x 10-1) + (5 x 10-2) = (600 + 40 + 7 + 2/10 + 5/100) = (600 + 40 +7) + (0.20 + 0.05).
The decimal system is derived from the Arabic system and can be traced back to ancient Egyptian, Babylonian (Sumerian), and Chinese roots. This Near East (Arabic digits) system was first developed by the Hindus and was in use in India in the third century b.c.. At that time the numerals 1, 4, and 6 were written in substantially the same form that is used today. The Hindu numeral system was probably introduced into the Arab world about the seventh or eighth century. The early Egyptians used a base-10 system that had different symbols for each power of 10 up to 106, and lacked a place-value notation and an explicit number zero. The Chinese, Cretans, Greeks, Hebrews, and Romans also used similar decimal systems that lacked the features of our modern decimal system, but helped to formulate that system. The important innovation in the Arabic system was the use of positional notation, in which individual number symbols assume different values according to their position in the written number system. Positional notation was made possible beginning around AD 800 by the use of a symbol for zero. The creation of the symbol "0" made it possible to differentiate between 11, 101, and 1001, for instance, without the use of additional symbols. Therefore, all numbers could be expressed in terms of only ten symbols, the numerals from 1 to 9 plus 0. Positional notation also greatly simplified all forms of written numerical calculation. The majority of the credit for the origin of the modern base 10 system goes to the Hindu-Arabic mathematicians between the eighth and eleventh centuries. The first recorded use of the system in Europe was in AD 976. The notation of the modern-day decimal system is credited to the work of Leonardo of Pisa (Fibonacci) in AD 1202. Thomas Harriot (1560-1621) and Gottfried Wilhelm von Leibniz (1646-1716) both independently developed the generalized treatment of positional number systems that included the decimal number system.
Decimal is derived from Latin as "decem" (10) and Greek "deka" (10). It is probable, but not certain, that the special position occupied by 10 is biologically connected to the number of human fingers. In fact, the word "digit" is from the Latin digitus and means "finger". The base 10 system is still evident in modern usage not only in the logical structure of the decimal system, but also in the English names for numbers. Thus, eleven comes from Old English endleofan that literally means "[ten and] one left [over]", and twelve from twelf that is defined as "two left". In addition, the endings "-teen" and "-ty" both refer to 10, and the word "hundred" comes originally from a pre-Greek term meaning "ten times [ten]". The powers of ten commonly possess their own names such as (within the United States) 1 million for 106 = 1,000,000; 1 billion for 109 = 1,000,000,000; and 1 trillion for 1012 = 1,000,000,000,000.
In the base-10 number system, the number 983.75, for instance, is a compact way of writing (900 + 80 + 3 + 7/10 + 5/100) = (9 x 102) + (8 x 101) + (3 x 100) + (7 x 10-1) + (5 x 10-2). For comparison, the number 983.75 is expressed in base 2 as 1111010111.11 (that is, (1 x 29) + (1 x 28) + (1 x 27) + (1 x 26) + (0 x 25) + (1 x 24) + (0 x 23) + (1 x 22) + (1 x 21) + (1 x 100) + (1 x 2-1) + (1 x 2-2) and in base 8 as 1727.6 (that is, (1 x 83) + (7 x 82) + (2 x 81) + (7 x 80) + (6 x 8-1).
The decimal system is widely used in various systems employing numbers. The metric system of weights and measures, used in most of the world, is based on the decimal system, as are most systems of national currency. In the course of history, the decimal system finally overshadowed all other positional number systems, and it is now found in all technologically advanced nations.
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