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
Addition
    Sum
    Additive inverse
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:


*

*Show why things are the way they are (why are they true).

*Be succinct as possible.

*Contain no annoying images and distractions (which are everpresent in 99% of today's textbooks on arithmetic)

*Be lucid.

*Contain zero fluff.


That's it! I hope such a book even exists. 
 A: The problem is that your criteria are contradictory (and to a lesser extent, subjective):  you want "the very basics," and you say you don't want a treatise on elementary number theory or (abstract) algebra, yet your very first criterion is "Show why things are the way they are."  These are not mutually compatible requirements.
The reason why they are not mutually compatible is because the properties of arithmetic (natural numbers: identity, associativity, commutativity, the distribute property;  operators:  addition, multiplication, and their inverses;  equivalence relations;  and properties of rationals as an extension of integers) as they are taught at their most basic and fundamental axiomatic level precisely are those concepts you would learn in an algebraic and/or number-theoretic context.
What I suppose you have in mind is something based in calculation and computation.  A text for children is focused on stating the properties of arithmetic as postulates to be accepted, then using those to do calculation.  But at the same time, you want something that explains the "why."  Sorry, but if you really want to understand the "why" you will need to learn the theoretic foundations.
Your request is a little bit like asking to learn why calculus "works" but at the same time saying that you don't want to study real analysis.  You are essentially saying, "I want to know why this is true but I don't want to learn why it is true."
