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

[Ankel]

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:

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.

 

[micromass]

So on the high school level, you should look at Gelfand's Algebra book and Euler's algebra book.
If you are looking for a more advanced treatise, then you will have to start from the Peano axioms and work up from there. A good book that does this is Bloch's real numbers and real analysis. It doesn't quite cover everything you mentioned though.

 

[Ankel]

Doe

So on the high school level, you should look at Gelfand's Algebra book and Euler's algebra book.
If you are looking for a more advanced treatise, then you will have to start from the Peano axioms and work up from there. A good book that does this is Bloch's real numbers and real analysis. It doesn't quite cover everything you mentioned though.

Do Gelfand/Euler cover up all the topics I mentioned, including equivalent fractions?

 

[micromass]

I have no idea what you mean with equivalent fractions since it lead to exponentiation. Do you just mean $a/b = c/d$ iff $ad = bc$. Then yes, they cover this.

 

[micromass]

Euler is available for free online too, so check it out

 

[berkeman]

You might also enjoy taking a step back to the more basic foundations of mathematics, for example Peano's Axioms

https://en.wikipedia.org/wiki/Peano_axioms

 

[micromass]

You might also enjoy taking a step back to the more basic foundations of mathematics, for example Peano's Axioms

https://en.wikipedia.org/wiki/Peano_axioms

In my opinion, if you're interested in doing arithmetic in a very rigorous setting, then you have no other choice but to start from the Peano axioms. Gelfand and Euler both do not start from the Peano axioms.