By John E. Maxfield

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Definition 2-17. The complex numbers C are number pairs (j, /), where s, t e R (alternatively, forms s + ti, where i2 = —1). Two complex numbers (s, t) and (u, v) are equal jf and only if s equals u (their real parts are equal) and t equals v (their pure imaginary parts ti and vi are equal). Addition is defined by (s, t) + (w, v) = (s + w, t + v), and multiplication by (s, t) • (u, v) = (su — tv, sv + tu). We let i stand for the pair (0, 1), noting that (0, l)2 = ( - 1 , 0). Theorem 2-7. The complex numbers with + and •, as defined in Definition 2-17, constitute a field.

Definition 2-4. An integral domain is a commutative ring with unit element with no proper divisors of zero; that is, a product a • b cannot equal zero unless one of the factors ¿7 or is zero. 20 O T H E R A B S T R A C T A LG E B R A S Definition 2-5. ” These definitions are more restrictive than Definition 1-2 of groups; that is, they impose more than the group requirements. One abstract algebra with fewer restrictions than a group is the semigroup. It is important to mathe maticians and receives study even though it has no identity nor inverse prop erties.

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.