|
Location arithmetic (Latin ''arithmeticæ localis'') is the additive (non-positional) binary numeral systems, which John Napier explored as a computation technique in his treatise ''Rabdology'' (1617), both symbolically and on a chessboard-like grid. Napier's terminology, derived from using the positions of counters on the board to represent numbers, is potentially misleading in current vocabulary because the numbering system is non-positional. During Napier's time, most of the computations were made on boards with tally-marks or jetons. So, unlike it may be seen by modern reader, his goal was not to use moves of counters on a board to multiply, divide and find square roots, but rather to find a way to compute symbolically. However, when reproduced on the board, this new technique did not require mental trial-and-error computations nor complex carry memorization (unlike base 10 computations). He was so pleased by his discovery that he said in his preface : ... ''it might be well described as more of a lark than a labor, for it carries out addition, subtraction, multiplication, division and the extraction of square roots purely by moving counters from place to place.'' == Location Numerals == Binary notation had not yet been standardized, so Napier used what he called location numerals to represent binary numbers. Napier's system uses sign-value notation to represent numbers; it uses successive letters from the English alphabet to represent successive powers of two: a = 20 = 1, b = 21 = 2, c = 22 = 4, d = 23 = 8, e = 24 = 16 and so on. To represent a given number as a location numeral, that number is expressed as a sum of powers of two and then each power of two is replaced by its corresponding digit (letter). For example, when converting from a decimal numeral: : 87 = 1 + 2 + 4 + 16 + 64 = 20 + 21 + 22 + 24 + 26 = abceg Using the reverse process, a location numeral can be converted to another numeral system. For example, when converting to a decimal numeral: : abdgkl = 20 + 21 + 23 + 26 + 210 + 211 = 1 + 2 + 8 + 64 + 1024 + 2048 = 3147 Napier showed multiple methods of converting numbers in and out of his numeral system. These methods are similar to modern methods of converting numbers in and out of the binary numeral system, so they are not shown here. Napier also showed how to add, subtract, multiply, divide, and extract square roots. 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「Location arithmetic」の詳細全文を読む スポンサード リンク
|