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
    Additive inverse
    Multiplicative inverse
        Common multiples
            Least common multiple
        Decimal fraction
        Proper fraction
        Improper fraction
        Vulgar fraction
        Common denominator
            Lowest common denominator
        Fundamental theorem of arithmetic
        Prime number
            Prime number theorem
            Distribution of primes
        Composite number
            Common factors
                Greatest common divisor
    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
            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.

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    $\begingroup$ 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. $\endgroup$ Aug 10, 2015 at 23:49
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    $\begingroup$ 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. $\endgroup$
    – Zach Stone
    Aug 10, 2015 at 23:50
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    $\begingroup$ 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... $\endgroup$ Aug 11, 2015 at 0:04
  • $\begingroup$ 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. $\endgroup$ Aug 11, 2015 at 0:25
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    $\begingroup$ If such a book doesn't exist, try writing your own. At the very least you'll gain an appreciation for the difficulties involved. $\endgroup$ Aug 13, 2015 at 3:01

1 Answer 1


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."

  • $\begingroup$ Yes, indeed. ... although in fact I'd claim there's even more disconnect, since much of the "logic/rigor" was created many years/decades/centuries after the events/understanding... so that it's not just that later points in any sort of logical development of mathematics explain things with hindsight, but also that the imperatives for explanations were from the environment as much as internal. $\endgroup$ Aug 11, 2015 at 0:06
  • $\begingroup$ @paulgarrett Yes, I think I see where you are coming from. A lot of the mechanics of calculus was developed well before the rigorous justifications of real analysis were provided; Newton and Leibniz did not, for instance, know of (or would have even cared for), say, the formalisms of Dedekind or Cauchy. To the inventors of the calculus, it was in many respects a means to an end; a tool. It took some time before mathematicians sought to place it on more solid foundations. And the same could be said of arithmetic. $\endgroup$
    – heropup
    Aug 11, 2015 at 0:26

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