Comments on Appendices

Appendix A

Page 620

In the first line of the proof of Theorem A.4, the wording “the identity \(0=a\cdot b\)” is confusing. The word “identity” can be an abbreviation for either “additive identity” or “multiplicative identity”, or it can mean “an equation that is true for all values of the variables”. The intent here, however, seems to be “the equation \(0=a\cdot b\)”.

In equation (1), the word “to” is in slanted type for no apparent reason.

Page 621

In line 5, the verb “satisfies” should be “satisfy”, since the subject “\(\mathbf{N}\) and \(\mathbf{Z}\)” is plural.

Page 622

In the proof of Theorem A.9, line 4, the reference to “Lemma A.8” should be to “Theorem A.8”.

In the next line, “belong to \(\mathbf{N}\)” should say “belong to \(\mathbf{N}\cup\{0\}\)”.

In lines 6 and 7 of the proof, the reference to “Lemma A.8” should be to “Lemma A.7”.

The second-to-last sentence in the paragraph should say “\(n+m \in\mathbf{Z}\)” rather than “\(n-m\in\mathbf{Z}\)” because, under the current hypothesis, the number \(m\) is a negative integer. The sentence is completing the proof that \(n+m\in\mathbf{Z}\) for all three cases of \(m\).

The final sentence in the paragraph should then say, “Similar arguments show that \(n-m\in\mathbf{Z}\) and \(nm\in\mathbf{Z}\).”

In the proof of Theorem A.10, Step 1, there should be a hypothesis that \(b>0\). [The statement of the theorem does say “\(b\in(0,\infty)\)”, but this property is a conclusion, not a hypothesis.]

Page 623

Step 2 should say a square root rather than the square root. Of course the positive square root is unique, but an additional (easy) argument is needed to verify the uniqueness.

Page 624

In the first line of the proof of Theorem A.11, the reference to “Theorem 1.11” should be to “Theorem 1.23”.

Appendix B

Page 628

Two lines from the bottom of the page, the reference to “Theorem B.2ii” should be to “Theorem B.2iii”.

Page 629, proof of Theorem B.4

Contrary to the claim in the first line of the proof, it is a loss of generality to assume that the angle \(\theta\) is acute. Since the argument uses the geometry of acute triangles, it is not evident a priori that the same method applies for obtuse triangles (although an analogous argument actually does work).

The argument for acute triangles is incorrect as written. The altitude \(h\) cuts out two right triangles, one having \(a\) and \(h\) as sides and one having \(c\) and \(h\) as sides. The length \(d\) should be the base of the second right triangle, not (as stated) the first triangle.

Appendix F

Page 644

Three lines from the bottom of the page, the quotation marks around “parallel to” have styles that do not match. A similar comment applies at line 4 on the next page.

Harold P. Boas