A beautiful book on arithmetic doesn't treat you like a little baby

The state of arithmetic today is disgusting. The textbooks on it are absolutely repelling, the authors treat it like a subject that will be of concern to only babies. They don't show any love, they treat the subject like a dirty rug. It's been two years since I majored in mathematics, since then, I have been programming very wildly and would like to relearn arithmetic in a way that Leonhard Euler and Euclid would personally enjoy.

Arithmetic is actually very rigorous, there exist theorems on even the most basic of the components and it's a very beautiful topic, if you're being taught by the right author.

I seek a complete book on arithmetic, how old it may be, that deals with it in an elegant manner and covers the following topics;

Order of operations
Sum
Subtraction
Multiplication
Multiplicative inverse
Multiples
Common multiples
Least common multiple
Division
Quotient
Fraction
Decimal fraction
Proper fraction
Improper fraction
Vulgar fraction
Ratio
Common denominator
Lowest common denominator
Factoring
Fundamental theorem of arithmetic
Prime number
Prime number theorem
Distribution of primes
Composite number
Factor
Common factors
Greatest common divisor
Fractions
Equivalent Fractions and Elementary Continued Fractions
Square root
Cube root
Properties of operations
Associative property
Commutative property
Distributive property


And if possible...

Real number
Rational number
Integer
Natural number
Irrational number
Odd number
Even number
Positive number
Negative number
Prime number
Whole number
Natural number


What I am describing is a treatise on arithmetic and I do not want a book on Calculus because it covers some of the topics above in it's first few chapters. I want a book that deals with arithmetic only. And no, I don't want a number theory book. I have been suggested this many times before and the books are not at all elementary, they discuss many advanced topics and all I am asking for is the very basics, the very very basics.

The book also must:

1. Show why things are the way they are (why are they true).
2. Be succinct as possible.
3. Contain no annoying images and distractions (which are everpresent in 99% of today's textbooks on arithmetic)
4. Be lucid.
5. Contain zero fluff.

That's it! I hope such a book even exists.

• Algebra by Israel M. Gelfand covers a lot of this and its approachable for someone in junior high but also doesn't treat you like a child. I liked it a lot as a high school student. It doesn't cover everything you want though. Aug 10, 2015 at 23:49
• What you've described is not a book, but a pamphlet. You want a deep understanding of why arithmetic principles make sense? Study algebra, category theory, or something like that. How can you want the basics and show why things are the way they are? Each item in your syllabus is a paragraphs at most of you exclude any non-basic theory. Aug 10, 2015 at 23:50
• Because this is not an answer in the form requested, it is a comment: the real "problem" (which is not really a problem) is that the true explanations are not elementary. They are practical-historical + clear with hindsight (from considerably more sophisticated mathematics). It is true that quite a few authors have found enthusiasm for (false... too bad...) pseudo-elementary accounts for "why things are they way they are", but, regardless of the satisfaction they provide, are factually inaccurate. A complicated "human" situation... Aug 11, 2015 at 0:04
• I think Euclid would enjoy to see how the arithmetic most basic results are derived from an axiomatic point of view. I would take a look to the meta math theorem list (us.metamath.org/mpegif/mmtheorems.html), where the most classic results (fund. theorem of arithmetic, square root of 2 irrational, etc) are derived. The problem of course is that less rigorous treatments of the basics appeal to intuition. Aug 11, 2015 at 0:25
• If such a book doesn't exist, try writing your own. At the very least you'll gain an appreciation for the difficulties involved. Aug 13, 2015 at 3:01