By Arjeh M. Cohen

The 1st interactive direction masking first and moment yr algebra. ranging from such primary subject matters as integers and divisions, modular mathematics and polynomials the content material extends to earrings, fields and permutation teams. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the speculation whereas additionally contributing interactive calculators for computing with integers, polynomials and diversifications. the pc algebra approach hole is built-in all through, permitting the calculation and manipulation of mathematical gadgets. moreover, collections for Mathematica notebooks and Maple worksheets assessment the algorithms awarded. a number of selection routines offer clients with rapid suggestions, whereas amenities for tracking scholars and a bulletin board whole this electronic direction.

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**Sample text**

The relation congruence modulo n is reflexive, symmetric, and transitive; in particular, it is an equivalence relation. 1) The set of equivalence classes of Z modulo n is denoted by Z/nZ. Proof. We need to verify that the relation is reflexive, symmetric, and transitive. This implies congruence modulo n to be an equivalence relation. The other statements of the proposition follow easily. The relation is reflexive. Let a be an integer. Then a ≡ a (mod n) as n divides a − a = 0. The relation is symmetric.

For a = 0, a b-ary representation is [0]b . Now suppose that a > 0 and that the existence assertion is true for all non-negative integers less than a. Let r be the remainder of division of a by b. Then 0 ≤ r and r < b. Moreover, b|a − r. Since a−r b < a, we can apply the induction k i hypothesis. , a0 , r]b . Uniqueness of the representation. , c0 ]b are both b-ary representations of a. By the assumption on the most significant digit we have ak = 0 and cl = 0. According to the first representation, the remainder when a is divided by b is equal to a0 and, according to the second, it equals c0 .

We look at the time of day as a quantity determined up to a multiple of 24. Clock arithmetic 37. Algebra Interactive We will analyze arithmetic modulo an integer. 1. Let n be an integer. On the set Z of integers we define the relation congruence modulo n as follows: a and b are congruent modulo n if and only if n|a − b. We write a ≡ b (mod n) to denote that a and b are congruent modulo n. If a and b are congruent modulo n, we also say that a is congruent to b modulo n, or that a is equal to b modulo n.