I want to strike a better balance with this blog going forward — not just a raw log of my own reading reflections, but also weaving in summary overviews or proofs that I find particularly elegant.
To be honest, I originally had no intention of giving Chapter 4 its own post. I figured I’d either fold it into Chapter 5 or just skip it entirely — much like I did with Chapter 1. The content in this chapter really isn’t much; it only took me about 300 minutes to get through. That said, there are a few genuinely interesting bits worth writing about. And, to be honest, I’d also like to keep the site looking tidy — so I figured I might as well write something for every chapter.
Section 4.1 The integers
Definition 4.1.1 (Integers). An integer is an expression of the form a--b, where a and b are natural numbers. Two integers are considered to be equal, a-b = c-d, if and only if a + d = c + b. We let \mathbb{Z} denote the set of all integers.
This definition alone honestly threw me a bit. For example, I’d always thought of a non-zero natural number as just being a positive integer — but it turns out that’s a later definition. \mathbb{Z} should be something more like the set of expressions a--b, or strictly speaking, it should be written as:
\mathbb{Z} = (\mathbb{N} \times \mathbb{N}) / \sim = \{[(a,b)] \mid a,b \in \mathbb{N}\}, \quad (a,b) \sim (c,d) \iff a + d = b + c
I briefly looked up the definitions of equivalence relations and equivalence classes, but honestly — I don’t think bringing them in here is really necessary. The more intuitive notation the book uses is just easier to follow.
Also, starting from this chapter, I really began to feel the significance of verifying that every newly defined operation is well-defined. Sure, most of the time you can just push the definition back onto some previously proven well-defined operation and lean on the properties of equality. But if I ever tried to define an operation from scratch myself — satisfying well-definedness could actually be pretty hard, and “it absolutely has to be verified”.
Going back to embedding \mathbb{N} into \mathbb{Z} — I originally thought it would work just like the previous chapter: as long as there exists a bijection f: \mathbb{N} \to \mathbb{Z} that sends each n \in \mathbb{N} to something like f(n) = n--0, that would be enough. But actually, it also requires an algebraic isomorphism. That is, f(n + m) = f(n) + f(m) and f(n \times m) = f(n) \times f(m) must hold as well.
Another thing that surprised me a bit: the reason the book doesn’t just define \mathbb{Z} as \mathbb{N} \cup (-\mathbb{N}) is really just a matter of workload — that definition is honestly a lot more intuitive. I think what’s more important, though, is that the definition used in the book generalizes nicely to \mathbb{Q} and \mathbb{R} in a similar way.
The exercises in Chapter 4 as a whole are pretty straightforward — apart from the last few problems, there’s really not much worth mentioning.
Section 4.2 The rationals
The definition of \mathbb{Q} is actually pretty similar to \mathbb{Z} — the only thing worth keeping an eye on is that age-old rule: “0 cannot be a divisor.” Also, you really have to watch out for 0. For example, in Exercise 4.2.1, when proving transitivity, I completely forgot about the conditions required for the cancellation law.
This chapter briefly touches on the concepts of rings and fields — for instance, \mathbb{Z} is a commutative ring, while \mathbb{Q} is a field (since every non-zero element has a multiplicative inverse). And \mathbb{Q} as a field is also ordered — I’d actually heard that one before. \mathbb{C}, on the other hand, is not. My algebra background is pretty weak — I only really knew a bit about groups before. I don’t think I’ll go out of my way to fill in these gaps either; if I really need it later, I’ll deal with it then.
Section 4.3 Absolute value and exponentiation
Just seeing the section title already raised a question for me — why are these two things defined on \mathbb{Q} instead of \mathbb{Z} ? I figure exponentiation probably has a good reason, but for absolute value, it’s most likely just to save a bit of space.
Also — is the definition of \varepsilon-close really broken down that finely? I’m not sure how other textbooks handle this, but it definitely gives off that feeling.
Section 4.4 Gaps in the rational numbers
Every natural number is either even or odd, but not both (why?). If p is odd, then p^2 is also odd (why?).
This is part of Exercise 4.4.3. The latter proof is dead simple — the problem is, I actually spent quite a while thinking about the former, gave it a try, and still couldn’t solve it. But if you’re allowed to just use the division algorithm directly, it’s actually pretty easy to get.
As for the proof that \sqrt{2} is irrational — I learned that one back in high school. In fact, the proof given here isn’t really all that different. But this infinite descent method is genuinely quite interesting.
Then, I briefly looked into Dedekind cuts — for example, the definition of \sqrt{2}: L = \{q \in \mathbb{Q} \mid q^2 < 2\}, \quad R = \{q \in \mathbb{Q} \mid q^2 \geq 2\}
Of course, L also needs to satisfy some additional properties. Uh, honestly — I don’t really see the advantage of this definition.