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I would really never ask a question about resources, noting that it is a soft-question, unless I thought it was very difficult to find elsewhere, and I have looked. Furthermore, I believe that this is a useful question that may benefit other users as well.

Is there a resource which explains low-level maths using complex concepts?

What I am asking for is a resource which returns to old, elementary-level concepts such as arithmetic and describes it using all of the complex jargon, working its way up to the calculus level. This would allow a student to work through and say "oh, that's the tie between this basic idea and this abstract way to look at it". For example, some of the revelations I've had:

  1. Math is actually about patterns, not numbers (8th grade)
  2. Not all variables need to be one letter in length (9th grade)
  3. Oh, and units are actually just variables, too (11th grade, believe it or not)
  4. Lines are really visuals of a set of numbers which satisfy the function (12th grade)
  5. Slopes are the gradients of a function (12th grade)

I believe there shouldn't be any revelations in math, because it should (ideally) be obvious from the beginning. These are the reasons why it's good to have a resource without generalizations or over-simplification. That's why I'm asking for a resource like this.

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"I believe there shouldn't be any revelations in math, because it should (ideally) be obvious from the beginning". Revelations in maths are what makes it interesting... – fretty Oct 12 '13 at 7:19
Do you expect that this stuff was obvious to the mathematicians that invented/discovered it hundreds of years ago? How about todays unsolved problems? Should they be "obviously" solvable without a "revelation"? – fretty Oct 12 '13 at 7:21
A name that comes to mind is Bourbaki, but I'm not sure that this is what you are looking for. Could you explain what you mean by high-level language? Is it a language of abstraction or something else? Notably high-level seems to be contradicted by "without generalization"; typically high level discussions talk about something rather general. – Marc van Leeuwen Oct 12 '13 at 7:22
@fretty It should be obvious because it is a wholly understood concept, refined, optimized, and proven. Before you make the obvious claim that not all concepts are going to have these traits, duly note that I only asked for topics ranging up to entry-level calculus, which I would expect to be well understood (by top-level mathematicians) at this point. If the concept is understood in full detail, then it should be set up in such a way that when an identity is given, it does not shock the student, because all of the foundational concepts are already well-grounded and present. – Joshua Lamusga Oct 12 '13 at 7:52
Originally, I gave a wall of text for both low-level and high-level, but I deleted it due to illegibility (way too long). By high-level I meant that we revisit topics like $1 + 1$ and $2 / 4$ with words like "binary operations", and topics usually taught in a simplistic manner (such as $y=mx+b$) as an example of a function, and not as the introduction to functions, etc. In other words, all the new terminology usually learned in higher mathematics is applied to lower-level mathematics so that conclusions can be drawn easily. It "works its way up" to single-variable calculus. – Joshua Lamusga Oct 12 '13 at 7:59
up vote 1 down vote accepted

The purpose of Elementary Mathematics from an Advanced Standpoint (vol. 1 and vol. 2) by Felix Klein was to do what the OP asks. It was directed towards school teachers and mature students, who would know the technicalities of school mathematics, but might lack vision of the "big picture". Note that it was written about a century ago, which affects the language and some of the selections. It still gets good reviews on Amazon, though.

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I will see this and look it over, since I found an online version here: (this link is from and therefore not prone to link rot, so I figure this will be around for some good years to come). – Joshua Lamusga Oct 14 '13 at 23:04

The book Mathematics Made Difficult by Carl E. Linderholm is a wonderful book for this, and it has the extra advantage of being one of the funniest books I have ever read. For example, Linderholm constructs the natural numbers as a coequalizer in the category of categories, uses quadratic forms to prove that 3 is not divisible by 7, gives a non-circular proof that $2$ is prime using the fact that $\mathbb{Z}/2\mathbb{Z}$ is a field...

The one huge disadvantage of this book is that it is out of print. But it is worth tracking it down in whatever form you can find it. While clearly absurd, it's a very intelligent book, and the proofs are usually insightful and deep, even though they usually look like overkill at first glance (and I suppose that another disadvantage of this book is that you really do need to be quite advanced to understand or appreciate why the math isn't as ridiculous as it looks).

If $2a$ ends in $5$, then of course so does $10a$, since $5$ is idempotent modulo $10$.


To find the rational number associated to a numerator $N$ and a denominator $D$, one simply maps $(N,D)$ to $\mathbb{Q}$, considered as $\rm{End}(\mathbb{Q})$, by taking the product in the latter of the endomorphism associated with the integer $N$ and the inverse of the automorphism associated with the integer $D$. By previous remarks and exercises, the resulting map from fractions to rationals, $\mathbb{Z} \times (\mathbb{Z} \setminus \{0\}) \to \mathbb{Q}$, is surjective.


A bagel is a torus, and has been encountered already in the chapter on topology. It is eaten with lox, and is a topological group $\mathbb{R}^2 / \mathbb{Z}^2$.

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This is useful because it allows people to draw conclusions about how they can use their complex definitions to redefine simple operations and meanings, which leads to a much more intuitive understanding of the concepts. This is exactly what I am looking for, albeit a bit too high-level (I'm still in High-school, of course). – Joshua Lamusga Oct 12 '13 at 8:05
The funny thing about arithmetic is that most of it isn't defined at all until college level, and then it's usually glossed over with the assumption that everybody knows it. For example, how do we really know that every number factors uniquely into primes? It seems obvious, until you start learning that it fails to be true in all systems of arithmetic. – Slade Oct 12 '13 at 8:46
@user33433: A worse example even than that: how do we know arithmetic with rational numbers is well defined (and therefore coherent)? Rational numbers are equivalence classes of fractions, so one must prove for every arithmetic operation, and for such things as comparisons, that when applied to equivalent inputs it will give equivalent results. Now where is that taught in primary school? And almost all math (except that involving integers only) is built upon arithmetic of the rationals. – Marc van Leeuwen Oct 12 '13 at 9:36

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