"Rather we are going to introduce multiplicative inverses." Use the definition of an additive inverse to write down a possible definition for a multiplicative inverse. Explain why you believe this is an appropriate definition.
Page 52
"Consider b ⋅ a ⋅ c." Explain why is it OK to write b ⋅ a ⋅ c without parentheses.
Theorem 4.1.5: What are the analogs of the 3 statements making up this theorem for addition (and additive inverses)?
Equation (4.2): Explain why at this point, we do not know that the "1" in this equation is the same as the "1" of the natural numbers.
Theorem 4.1.6: Explain why this theorem allows us to identify the multiplicative identity of F to the first natural number.
Section 4.2
Page 55
Prove that the order defined in (4.7 - 4.8) is transitive and has trichotomy.
Page 56
Last line of Equation (4.16): Why can't they divide top and bottom by n to conclude that the distributive property holds?
Page 57
Equation (4.20): Say in words what the right hand side of this equation represents. Your answer should be of the form "This is the set of ... such that ...."
Section 4.3
Page 58
Equation (4.22): Explain why we no longer need a statement like (4.8) to define what f1 < f2 means.
Page 59
"The three formal definitions given earlier do give us well-defined notions." Where, in the definition of an ordered field (see Section 4.1.2), does it say that addition and multiplication are well defined?
Page 60
"All these are statements about integers." How do we know that n 1, m 1, q 1, etc are integers?
Line 3 from the bottom of the page: "Thus, n1 > 0; n2 > 0; q1 > 0 and q2 > 0." How do we know that these numbers are positive?
Page 63
Line 5 from the bottom of the page: "Then, m, q, and t are positive." Explain why this statement is true.
Page 64
Just above Equation (4.43): "Consider 0/1 ∈ ℚ." Explain why 0/1 is in ℚ.
Page 65
"We can write a = n/m, b = p/q, and c = s/t ." Explain why it is necessary, for all of the proofs in this section, to always rewrite elements of ℚ as ratios of the form n/m, p/q, etc.
Proof of Claim 13: Why do they say 1/1 and not just 1?
Page 66
"Then, m n / n 2 ∈ ℚ." Explain this step, i.e. why is m n / n 2 in ℚ?
Comment just below (4.47): Why is it important to check that m n 2 . 1 = m n 2 . 1 instead of simplifying the fraction? In other words, what is this comment about?