Arithmetics: Shane's comments to be discussed

[sylvia-blaho]

Summarized feedback:
The intro promises to be accessible to secondary school level. For me personally, I don’t feel this is the case beyond page 8. If this is indeed the aim, more terms need to be described. Suggest adding a much expanded reference table at the start of the chapter.
I think Chapter 1 could benefit from more references to how the particular maths you’re explaining relates to snarks and related concepts. There is some explanation of this in the introduction, but I would encourage you to sprinkle more reminders throughout the chapter. I believe it would help readers remain engaged and committed to fully learning the maths concepts discussed in the chapter.
The chapter could benefit from a conclusion that summarizes the chapter and introduces the following chapter.
I wonder if you could offer clearer signposting between: Bulk text, examples, example questions, jargon etc. Maybe not, sometimes adding different content types in boxes or different fonts/colours can be hard on the eye, but it could be worth playing around with.
General comment: Personally I haven’t seen the ‘dot’ as a multiplication symbol, only x and *. Maybe clarify in the table at the start of the chapter.

P.12.

Consider deleting “As a result, the reader….immature code” – I think the previous sentence is clear enough.
I think section 3.1.1. could be shortened. There are several ways this could be done, one simple suggestion is to avoid repetition. For instance, there is repetition of what is going to be covered in chapter 1 in the middle of the section “This is why we dedicated this chapter to…” and at the end “However…some mathematical groundwork is needed. This chapter therefore will help you focus…”

 Moved this to the intro chapter

PAGE 13
1.2. Expand the table. It’s clear and useful, but lacks enough terms. Based on the intro, this should be accessible to anyone with secondary-school level maths. I did pretty well in Maths at school, but it’s been maaaaany years since I’ve heard these terms. So I would err on the side of more terms, if someone like me is your target audience. It will also serve as a useful cheat sheet/reference table for anyone going through other sections of the MMM.

PAGE 5
Similarly, the definition of prime numbers is basic. But then it jumps to A000040 in ??OEIS – what is this? It’s not defined?
‘Natural Numbers’ vs ‘Counting Numbers’ – if they’re the same, make sure you pick one and stick with it. CN is in blod on the previous page (p4), but on this page it’s referred as NN.
“To see that, let n ε Ν >_ 2 be any natural number” (I understand this as every positive integer/natural number above and including 2. If so, how does this link to the previous and subsequent sentence about prime numbers).
“except for the order of the factors” – Don’t understand this.
Prime Factorization example 1: Why is it divided by multiple 2s, multiple 3s, 5s and other numbers.
Sage example: What does n = NN mean?
Explain why computation speed is such an important consideration or at least give an example the relative speed difference between factorization and computing the product of prime numbers.
“inverse process” sounds right to me (but I’m not fluent in maths terms)
Delete “classical” (or amend to ‘classic’ but in my opinion it’s not necessary to include at all) before “Turing machine.”
Consider defining “polynomial time”, “sub-exponentially,” and the formula presented at the end of that sentence.
Define the mutual direction arrow ⇔
PAGE 6
Typo “a basic cryptographic assumption” [delete the ‘s’]
Typo “that” not “hat”
Consider shifting the paragraph about Peter Williston Shor and usable quantum computers to a footnote. If these aren’t a near-term reality (5-10 years), it’s probably not a major consideration for most readers.
What does ‘absolute value’ mean in Example 1? Is this basic addition/subtraction?
Consider giving answers to the examples.
“A mod b” – what does “mod” mean? Presumably ‘modulus’ (as I learned on p10, but what does this actually mean?)
PAGE 7
The term “Truth Value” hasn’t been explained (although it’s not difficult to understand).
Long Division – I assume most people are familiar with this and it doesn’t need much more explanation.
PAGE 8
Grammar issue in the last line of Exercise 5: “and. In which”
Define ‘Pseudocode’.
Define ‘while-loop’.
I’m having trouble understanding the end of page 8 and Algorithm 1 (as well as examples 4-10) on p.9.
PAGE 10
Don’t understand ‘exponentiation function’.
Typo “it’s actually not that hard” not “its”
After defining “incongruent” you write “w.r.t”. I presume this stands for ‘with respect to’. Regardless, it should be written out in full.
I’ve never seen the triple horizontal line symbol.
Remove comma here “one particularly useful thing about congruencies is, that we can…”.

PAGE 11
The maths is beyond my level of comprehension from here on.

PAGE 13
	
“are congruent modulo the product” doesn’t sound right. 

PAGE 15
I actually understand the logic of the maths here.
PAGE 16
How does ‘Jargon 1’ relate to the Exercises?
PAGE 19
∑ - define this?
Univariate – define this = explain the equation in words

I like that you reference the application of polynomials in erasure codes and snarks – it helps the reader understand the practical importance of this concept. Suggest adding references to snarks, erasure coding etc throughout relevant passages of the chapter.