Surprisingly Accurate Rational Approximations Surprisingly
... that Pn/qn will representa with at least 2K-digit accuracy.Many such examples exist,
some of which were noted above. It is also possible to construct examples that agree
in more than 2K digits. For example, the third convergent of the continued fraction
expansion of the irrationalnumber
a = 10(11 - ...
Location arithmetic
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.
