By John E. Maxfield

ISBN-10: 0486671216

ISBN-13: 9780486671215

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Read them aloud in unison. ” Mathematics teachers need all the techniques they can borrow from language teachers to help students with new vocabulary. CHAPTER 2 OTHER ABSTRACT ALGEBRAS In Chapter 1 we introduced an important abstract algebra, the group, and showed that we have in the integers with binary operation + a familiar example. The group postulates can, indeed, be thought of as abstracted from various properties of the integers. However, the group properties under addition do not exhaust the possibilities offered by the integers, which have a structure with respect to a second operation, multiplication, and even a distributive property linking + and \ In this chapter we introduce several abstract algebras—rings, integral domains, and fields—and show that their various postulates can be abstracted from the properties of various number systems—the integers, the rational numbers, the real numbers, and the complex numbers.

3. The rational numbers Q . By introducing the integers we made it possible to solve equations of the form x + b = a, where a, b e N. O T H E R A BSTR A CT A LG EB R A S 31 Now we look at equations of the form bx = a, where a, b e I. For some number pairs (

Definition 2-8. The mathematical induction property of the counting numbers N can be stated: If a subset S of N contains 1 and contains the successor of each of its members, then it contains (and equals) all of N. 0 o o o o 2+ 3= 5 o o 3+ 2= 5 FIG U R E 2-2 H 22 O T H E R A B S T R A C T A LG E B R A S This property is used again and again in proofs and also in definitions. For instance, we said loosely that addition can be “ based” on counting. This can be done precisely by an inductive definition (n + 1 = n+, the successor of n e N, n + m+ = (n -f m)+.

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