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In real analysis, I am aware of James Propp's "Real Analysis in Reverse" which does "naive reverse mathematics", showing the equivalence of various theorems of analysis to the completeness property of the real line.

Recently I've been thinking about the various geometric results that are typically covered in a precalculus course, such as:

  • The Pythagorean theorem
  • The law of cosines
  • The equivalence between algebraic and geometric dot product ($u\cdot v = \|u\|\|v\|\cos \theta$)
  • The formula for projections: $\operatorname{proj}_u(v) = \frac{v\cdot u}{\|u\|^2}u$
  • The sine and cosine angle addition formulas
  • The geometric interpretation of complex multiplication ("add the angles and multiply the lengths")
  • Determinant of a $2\times 2$ matrix as the signed area of the parallelogram spanned by its columns

...and so on. It seems to me that these results are all saying "basically the same thing" (whatever that means); in particular it's often possible to prove one of them using some of the others.

I am wondering if there is a textbook or paper that does something like "naive reverse mathematics of precalculus". What would the "base axioms" be in this setting? What is the relationship between these theorems? Why are they so closely related? Are there other "equivalent" theorems that are missing from this list? Are there any keywords I should be using in my search?

IssaRice
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  • If I'm to make a conclusion, all those except the complex multiplication are direct results of "special geometric properties of $\mathbb R^2$," while the complex multiplication has more to do with series expansions. – Yuqiao Huang Dec 22 '20 at 22:15

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There are many algebraic statements like the ones you list that are equivalent to the Pythagorean theorem. That is essentially because the Pythagorean theorem is exactly what you need to show that the Euclid's geometric plane can be modeled as $\mathbb{R}^2$.

Even more interesting is the fact that there are many geometric equivalences. Among those are

  • The parallel postulate.
  • The angles of a triangle sum to $2\pi$.
  • Similar triangles that are not congruent exist.

See

Is Pythagoras' Theorem a theorem?

which axiom(s) are behind the Pythagorean Theorem

https://www.cut-the-knot.org/triangle/pythpar/PTimpliesPP.shtml

Ethan Bolker
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  • None of the pages you linked to seem to talk about the theorems I listed (other than the Pythagorean theorem). – IssaRice Dec 23 '20 at 00:41
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    @riceissa At least half those bullet points are equivalent to the Pythagorean theorem, which is equivalent to the distance formula in the coordinatized plae. – Ethan Bolker Dec 23 '20 at 12:07
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From what I have seen here (and I have contributed to this), just about about any inequality is a consequence of that most elementary inequality, Bernoulli's inequality.

marty cohen
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