By Frederick M. Goodman
This creation to trendy or summary algebra addresses the traditional issues of teams, earrings, and fields with symmetry as a unifying subject matter, whereas it introduces readers to the lively perform of arithmetic. Its available presentation is designed to educate clients to imagine issues via for themselves and alter their view of arithmetic from a procedure of ideas and systems, to an area of inquiry. the quantity presents ample workouts that provide clients the chance to take part and examine algebraic and geometric rules that are fascinating, very important, and value puzzling over. the amount addresses algebraic topics, simple idea of teams and items of teams, symmetries of polyhedra, activities of teams, jewelry, box extensions, and solvability and isometry teams. For these attracted to a concrete presentation of summary algebra.
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Extra resources for Algebra. Abstract and Concrete
We can do computations in these algebraic systems in order to answer natural (or unnatural) questions, for example, to find out the order of a perfect shuffle of a deck of cards. In this section, we return to more familiar mathematical territory. We study the set of integers, probably most familiar algebraic system. The integers have two operations, addition and multiplication, but as you learned in elementary school, multiplication in the integers can be interpreted in terms of repeated addition: For integers a and n, with n > 0, we have na D a C C a (n times), and .
There exist exactly n distinct residue classes modulo n, namely Œ0; Œ1; : : : ; Œn 1. These classes are mutually disjoint. 7. 5. mod n/. mod n/. Proof. a C b/ a0 and b b 0 are divisible by n. b a0 b 0 / b0/ is divisible by n. ■ We denote by Zn the set of residue classes modulo n. The set Zn has a natural algebraic structure which we now describe. Let A and B be elements of Zn , and let a 2 A and b 2 B; we say that a is a representative of the residue class A, and b a representative of the residue class B.
In the former case, a D b D 0, and in the latter case, v D ˙1 by part (a), so a D ˙b. The proofs of parts (c) and (d) are left to the reader. ■ As you know, some natural numbers like 2 and 13 are special in that they cannot be expressed as a product of strictly smaller natural numbers; these numbers are called prime. Natural numbers that are not prime are called composite; they can be written as a product of prime numbers, for example, 42 D 2 3 7. 6. 3. A natural number is prime if it is greater than 1 and not divisible by any natural number other than 1 and itself.
Algebra. Abstract and Concrete by Frederick M. Goodman