... 2’s Complement Number “line”: N = 5
Residue Number Systems
... • Magnitude Comparison
• Overflow Detection
• Generalized Division
Suffices to discuss first three in context of being able to
do magnitude comparison since they are essentially same
if M is such that M=N+P+1 where the values represented
are in interval [-N,P].
... How do we know that two fractions are
We cannot tell whether two fractions are the same until
we simplify them to their lowest terms.
A fraction is in its lowest terms (simplified) if we cannot find a
whole number (other than 1) that can divide into both its
numerator and denominator (A c ...
... powers) will take place within the groups Z(p)* and G(q), which we will define and
explain in this section.
The two groups Z(p)* and G(q) are very important for public key cryptography
and digital cash. They play roles in Diffie-Hellman key exchange, in the Schnorr
signature scheme, in the Digital S ...
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.