What's a good rigorous Arithmetic book? Specifically one on which the classic arithmetic algorithms are rigorously proven?

Question

I’ve been looking for some time for a book that rigorously builds up arithmetic from basic axioms, like the Peano axioms for example, and ends up showing why our classic algorithms for additions, subtraction, multiplication and division, the ones learned in primary school, work. So far the only arithmetic books I’ve been able to find limit themselves to stating the steps of the algorithms (for addition, first align the numbers such that their rightmost digits concur, then add up each digit individually and carry if necessary), or they don’t even mention the algorithms such as this book, which is really nice and shows properties of arithmetic operations such as commutative law, but doesn’t show the correctness of the algorithms typically used to perform such operations. How can we know that the algorithms produce the desired result? I’m looking for a book that answers this question in a rigorous way. Here is an example of what I mean. Thanks in advance!

Answer 1

Two different things going on here.

Treatment of the Peano axioms is very much in Fundamentals of Mathematics, which often starts with axiomatic set theory and works towards the establishment of the rules of commutativity and associativity. There are many such books on this, although all take the subject slightly differently from every other such book. Perhaps you would get on with Keith Devlin's "The Joy of Sets".

The "classical algorithms", on the other hand, are a different thing altogether. By this time, the construction of the standard number systems, along with commutativity and associativity, are already assumed to be proven.

What we are now about is the exercise of arithmetic on numbers expressed in basis representation form (i.e. a string of digits whose values depend on the position in the string). This is the stuff which you learn at age 6. Not so many books on this. As suggested in a comment, Knuth covers them in detail in his The Art Of Computer Programming Volume 2: Seminumerical Algorithms, Chapter 4 - Arithmetic: 4.3.1 The Classical Algorithms.

Answer 2

I would check out Set Theory: The Structure of Arithmetic by Hamilton and Landin. I love this book. They define addition formally and prove the addition tables and all the usual properties of addition. Unfortunately they do not go into algorithm, but I think you will find the content very enlightening and interesting. It’s very close to what you are looking for I believe.

Answer 3

I'd recommend Understanding Numbers in Elementary School Mathematics by Hung-Hsi Wu. Be sure to reference the errata listing on Wu's website.