Modular Arithmetic 2

# Modular Arithmetic 2 - Modular Arithmetic Differences and...

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Modular Arithmetic Differences and Similarities Modular arithmetic is very similar to the Arithmetic we are so much used to. In line with the Sylvester's pronouncement , let's write down for the record the differences and similarities found between Euclidean (regular) and Gaussian (modulo) arithmetic. Differences 1. Numbers Euclidean arithmetic operates on infinite set of all integers; Gaussian only works with finite sets {0,1,...,N-1} (or with sets of residue classes .) 2. ab = 0 In usual arithmetic, ab = 0 is only possible when either a or b is zero. In Gaussian case we have, e.g., 2×3 = 0 (mod 6), whereas neither 2 = 0 (mod 6) nor 3 = 0 (mod 6). 2 and 3 are zero divisors in the arithmetic modulo 6. Euclidean arithmetic has no zero divisors. Arithmetic modulo a prime, too, has no zero divisors. 3. Polynomial roots Fundamental Theorem of Algebra implies that every polynomial of order n over the field C of complex numbers has exactly n roots. In the Gaussian arithmetic all depends on a polynomial. For example, in modulo arithmetic even linear equations may behave

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