Resource for low level maths explained in high level perspectives

Question

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.

Answer 1

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.

Answer 2

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 $10$.

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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.

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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 $\mathbb{R}^2 / \mathbb{Z}^2$.