Download Algebra Interactive by Arjeh M. Cohen PDF

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 variations. the pc algebra procedure hole is built-in all through, permitting the calculation and manipulation of mathematical items. additionally, collections for Mathematica notebooks and Maple worksheets overview the algorithms provided. a number of selection routines supply clients with quick suggestions, whereas amenities for tracking scholars and a bulletin board whole this electronic path.

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Example text

N} |gcd (m, n) = d} where d runs through the set of positive divisors of n (in which case also dn runs through the set of positive divisors of n). For multiples m, n of d, we have gcd (m, n) = d if and only if gcd therefore also equals d·V dn , 1 . m n d,d = 1. The set V (n, d) But |V (m, 1)| = Φ(m), so V (n, d) contains precisely Φ( dn ) elements. Consequently, n = ∑d∈{d∈N|d|n} Φ( dn ) = ∑d∈{d∈N|d|n} (Φ(d)). Taking apart the summand Φ(n) (occurring for d = n), and bringing the remaining summation to the other side, we find the required formula.

The integers 1222, 124211, 2113 and 4121 are given in their decimal representation. Give the representation in base 2, 4, and 8, respectively. 36. Algebra Interactive Chapter 2 Modular arithmetic It frequently happens that we prefer to ignore multiples of a given number when we do calculations. Just think of the days in the week or the hours in a day; in the first case we ignore multiples of seven, in the second case multiples of 12 or 24. In this chapter we will describe this ‘arithmetic modulo n’.

The decimal and binary system are two examples in a series. 1 (b-ary representation). Let b > 1 be an integer. , a0 ]b . We speak of the b-ary number system. 2. Besides the binary system, the octal (base 8) and hexadecimal (base 16) systems are often used in computer science. In base 8 we use the digits 0 to 7, but in base 16 we need more digits. Apart from the digits 0 to 9, it is customary to use the symbols A, B, C, D, E, F to represent the decimal numbers 10, 11, 12, 13, 14, and 15, respectively.

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