|
Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations (such as Roman numerals) for its use of the same symbol for the different orders of magnitude (for example, the "ones place", "tens place", "hundreds place"). This greatly simplified arithmetic, leading to the rapid spread of the notation across the world. With the use of a radix point (decimal point in base-10), the notation can be extended to include fractions and the numeric expansions of real numbers. The Babylonian numeral system, base-60, was the first positional system developed, and is still used today to count time and angles. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations. ==History== Today, the base-10 (decimal) system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today. For example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a ''space'' between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish (dating from about 700 BC), the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges.〔Kaplan, Robert. (2000). ''The Nothing That Is: A Natural History of Zero''. Oxford: Oxford University Press.〕 The Babylonian placeholder was not a true zero because it was not used alone. Nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60), looked the same because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them. Before positional notation became standard, simple additive systems (sign-value notation) such as Roman Numerals were used, and accountants in ancient Rome and during the Middle Ages used the abacus or stone counters to do arithmetic.〔Ifrah, page 187〕 Counting rods and most abacuses have been used to represent numbers in a positional numeral system. With counting rods or abacus to perform arithmetic operations, the writing of the starting, intermediate and final values of a calculation could easily be done with a simple additive system in each position or column. This approach required no memorization of tables (as does positional notation) and could produce practical results quickly. For four centuries (from the 13th to the 16th) there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the latter continues to be used in Japan and other Asian countries. Georges Ifrah concludes in his ''Universal History of Numbers'': Aryabhata stated "''sthānam sthānam daśa guṇam''" meaning "From place to place, ten times in value". Indian mathematicians and astronomers also developed Sanskrit positional number words to describe astronomical facts or algorithms using poetic sutras. A key argument against the positional system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing (e.g.) 100 into 5100, or 100 into 1000. Modern cheques require a natural language spelling of an amount, as well as the decimal amount itself, to prevent such fraud. For the same reason the Chinese also use natural language numerals, for example 100 is written as 壹佰, which can never be forged into 壹仟(1000) or 伍仟壹佰(5100). After the French Revolution (1789-1799), the new French government promoted the extension of the decimal system.〔 L. F. Menabrea. Translated by Ada Augusta, Countess of Lovelace. ("Sketch of The Analytical Engine Invented by Charles Babbage" ). 1842. 〕 Some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. Many of the advantages claimed for the metric system could be realized by any consistent positional notation. Dozenal advocates say dozenal has several advantages over decimal, although the switching cost appears to be high. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Positional notation」の詳細全文を読む スポンサード リンク
|