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

  1. Aug 10, 2015 #1
    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;

    And if possible...

    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.
  2. jcsd
  3. Aug 10, 2015 #2
    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.
  4. Aug 10, 2015 #3
    Do Gelfand/Euler cover up all the topics I mentioned, including equivalent fractions?
  5. Aug 10, 2015 #4
    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.
  6. Aug 10, 2015 #5
    Euler is available for free online too, so check it out
  7. Aug 10, 2015 #6


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    Staff: Mentor

  8. Aug 10, 2015 #7
    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.
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