Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.2 ORDERED FIELDS 303 (b) (i) Construct addition and multiplication tables for the field Z, of integers modulo 7. (ii) Find all ...

304 PROPERTIES OF NUMBER SYSTEMS Chapter 9 In the previous article, there was a lack of any reference to ordering of elements in ...

9.2 ORDERED FIELDS 305 (c) Let x be a nonzero element of the ordered field F. Then either x E 9 or -x E 8. If x E 9, then x2 = ( ...

306 PROPERTIES OF NUMBER SYSTEMS Chapter 9 Definition 1, (z - y) + (y - x) E 9. But (z - y) + (y - x) = z+(-y+y)-x=z-x,soz-x~9,a ...

9.2 ORDERED FIELDS 307 THEOREM 4 Let a, b, x, and y be elements of an ordered field F: (a) If x < y, then a + x < a + y (b ...

308 PROPERTIES OF NUMBER SYSTEMS Chapter 9 the first two sentences of this proof. Hence 1x1 = 0 implies x = 0, so that the concl ...

9.2 ORDERED FIELDS 309 THEOREM 6 Let F be an ordered field with x, y, ZE F. Then: We conclude this article with the reminder tha ...

310 PROPERTIES OF NUMBER SYSTEMS Chapter 9 (6) *(i) Prove that if a, b E F with a^2 0 and b^2 0, then a < b if and only if a2 ...

9.3 COMPLETENESS IN AN ORDERED FIELD 311 less than n, no matter how close to n, is less than some number in the sequence. Stated ...

312 PROPERTIES OF NUMBER SYSTEMS Chapter 9 DEFINITION 3 Let F be an ordered field and S c F. An element u of F is said to be a l ...

9.3 COMPLETENESS IN AN ORDERED FIELD 313 Solution Suppose there is a rational number u such that u = lubQ S. If u exists, then e ...

314 PROPERTIES OF NUMBER SYSTEMS Chapter 9 DEFINITION 5 An ordered field F is said to be Archimedean ordered if and only if for ...

9.3 COMPLETENESS IN AN ORDERED FIELD 315 we are willing to assume that any real number falls between two consecu- tive integers. ...

316 PROPERTIES OF NUMBER SYSTEMS Chapter 9 parts of Article 4.3 at this stage.) Furthermore, this theorem is the basis of import ...

9.3 COMPLETENESS IN AN ORDERED FIELD 317 The abstract definition of "interval" given earlier in the text (Definition 3, Article ...

318 PROPERTIES OF NUMBER SYSTEMS Chapter 9 Also, since I is not bounded below, there exists d E I such that d < x. Hence d &l ...

9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 319 *6. Prove that, in a complete ordered field F, any nonempty subset S of F that is ...

320 PROPERTIES OF NUMBER SYSTEMS Chapter 9 We interject at this point only one discordant note into the rather "tidy" view, just ...

9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 321 EXAMPLE 2 Use the quadratic formula to find the solution(s) to the equation 5x2 + ...

322 PROPERTIES OF NUMBER SYSTEMS Chapter 9 Because C satisfies the multiplicative inverse axiom, it is possible to de- fine divi ...