Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 323 An immediate consequence of Definition 3 and the formula (x + yi) - ' = (X - yi)/ ...

324 PROPERTIES OF NUMBER SYSTEMS Chapter 9 Partial proof (e) If z2 = x + yi # 0, then z;' = (x - yi)/(x2 + y2), so that (z, ')* ...

9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 325 The primary use of the polar form is in connection with multiplication of complex ...

326 PROPERTIES OF NUMBER SYSTEMS Chapter 9 (b) The proof proceeds by induction on n. The case n = 1 is evident. If the conclusio ...

9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 327 (Imaginary axis) I Figure 9.5 Graphic view of the four complex 4th roots of z = 1 ...

PROPERTIES OF NUMBER SYSTEMS Chapter 9 (a) Prove, in detail, that (C, +, .) is a field. (b) Explain, on the basis of (a) and ma ...

Construction of the Number Svstems of CHAPTER 10 In Chapter 9 we studied the number systems N, 2, Q, R, and C from a descriptive ...

330 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 Philosophically, knowledge of this construction provides insight into the famous r ...

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 331 and yields ultimately to the existence of a complete ordered fiel ...

332 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 Axiom 1 the operations of addition and multiplication. The theorem by which we wil ...

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 333 and called "addition to m." Intuitively, our goal is that s,(n) s ...

334 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 We channel our efforts now toward verifying (a) and (b), relying heavily on the in ...

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 335 (i)' and (ii)'. Thus the inductive step (11) is proved and we con ...

336 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 T H E 0 RE M 4 (Principle of Inductive Definition) Let f: N -, N be an arbitrary m ...

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 337 Partial proof (a) Given n, p E N, denote by S,, the set (q E Nl(n ...

338 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 Proof For n, p E N, let Snp = {q E N (n(~ + q) = np + nq}. First, we have 1 E Snp, ...

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 339 must show that p(q + 1) # 1. Now p(q + 1) = pq + p = p + pq = p + ...

340 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 Partial proof We prove (b), with the others for you to verify in Exercise 5(b). To ...

10.1 AN AXlOMATlZATlON FOR THE SYSTEM OF POSITIVE INTEGERS 341 prove this fact and, for good measure, introduce another form of ...

342 .CONSTRUCTION OF NUMBER SYSTEMS Chapter 10 The second induction principle IP2 [(c) of Theorem 111 is occasionally useful in ...