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By Arjeh M. Cohen

The 1st interactive path masking first and moment yr algebra. ranging from such basic issues as integers and divisions, modular mathematics and polynomials the content material extends to jewelry, fields and permutation teams. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the idea whereas additionally contributing interactive calculators for computing with integers, polynomials and diversifications. the pc algebra method hole is built-in all through, permitting the calculation and manipulation of mathematical gadgets. furthermore, collections for Mathematica notebooks and Maple worksheets overview the algorithms provided. a number of selection workouts supply clients with rapid suggestions, whereas amenities for tracking scholars and a bulletin board entire this electronic path.

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Example text

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.

Congruence modulo 3 splits the integers in three disjoint subsets. These subsets are represented by columns. Integers in the same subset differ by a multiple of 3. Let n be an integer. Consider Z/nZ, the set of equivalence classes of Z modulo n. Addition and multiplication with these classes can be defined in the following way. 39. Algebra Interactive Addition of congruence classes is defined in terms of representatives. For instance, to add the two congruence classes modulo 5 above take any representatives in each of these classes, say 6 in the first and 3 in the second.

So, by the Characterization of the gcd, we find gcd (a, n) = 1. 17. The invertible elements in Z/2n Z are the classes x(mod 2n ) for which x is an odd integer. Indeed, the gcd of x and 2n equals 1 if and only if x is odd. An arithmetical system such as Z/pZ with p prime, in which every element not equal to 0 has a multiplicative inverse, is called a field, just like Q, R, and C. Suppose that n and a are integers with n > 1 and gcd (a, n) = 1. The Characterization of Modular Invertibility not only gives the existence of the inverse of a(mod n) in Z/nZ, but also a way to compute this inverse.

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Algebra Interactive by Arjeh M. Cohen


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