Chapter 1
- We began the book with Zeno's paradoxes, and it turns out there is a similar paradox in quantum mechanics. Here's what I think I have learned: Physicists argue that a decaying particle (which usually decays continuously and exponentially) in a quantum system exhibits short-time deviations from this usual decay law; in fact, frequent or continuous observations seem to induce these periods in which the the particle is not decaying exponentially. But this seems to imply that simply by measuring the system frequently enough, one can inhibit decay. This give a Zeno-type paradox which is called the quantum Zeno effect. And a paper on this from 1976 has been cited 2,188 times (!!) as of this writing. So it seems important. That paper is Zeno's paradox in quantum theory and it is by Misra and Sudarshan. (And if you're a physicist and I did a laughable job at explaining this, please let me know.)
Chapter 2
- The (translated) 1874 paper entitled On a Property of the Collection of All Real Algebraic Numbers was Georg Cantor's first paper on set theory. In this paper he proves "Cantor's revolutionary discovery" that the real numbers are not countable, and hence that there are multiple infinities; in the book this is Theorem 2.9. It does not his his diagonalization argument, though, as that would come later. This paper also proves the existence of transcendental numbers. The paper is short but exceptional. It even his its own Wikipedia page that is longer than many biographical pages.
- The original proof that the reals are uncountable is different than the standard proof today, which is by Cantor's diagonalization argument, and is the book's proof of Theorem 2.9. Cantor's original (untranslated) paper of this argument is provided here. If you have a translated version (especially a PDF), please send it to me!
- For more on Question 1 of the Open Questions on page 63, see Combinatorial Cardinal Characteristics of the Continuum by Adreas Blass.
Chapter 3
- On Page 108 there is a discussion on now the axiom of completeness, with which we started the whole theory in Chapter 1, is equivalent to several of the other major theorems that we proved later on. For a survey of all these, check out Toward a More Complete List of Completeness Axioms by Holger Teismann (pdf file at bottom of this page). Also see Real Analysis in Reverse by James Propp.
Chapter 4
- On page 129 we discussed the Basel problem: 1 + 1/4 + 1/9 + ... = π²/6. For a 2010 paper by Johan Wästlund, see Summing Inverse Squares by Euclidean Geometry, or check out the amazing video on this paper by 3Blue1Brown here.
Chapter 5
- Section 5.4 is called The Greatest Definition in Mathematics, but great definitions don't always come out of nowhere. To see where the definition of compactness (Definition 5.16) came from, check out A Pedagogical History of Compactness by Raman-Sundström.
- A sequence of blog posts on the Heine-Borel Theorem (Theorem 5.19) is here.
Chapter 6
- The modern definition of continuity (Definition 6.16) is in terms of a δ and ε. These did not appear in the definition until the 1820s in the work of Cauchy. For more on the path to our modern day approach, check out Who Gave You The Epsilon? Cauchy and the Origins of Rigorous Calculus by Judith Grabiner.
Chapter 7
- On page 259 we mention that the circumference divided by the diameter is the same for every circle. Who proved this? See David Richeson's paper Circular reasoning: who first proved that C/d is a constant?
- Section 7.3 deals with the interaction between differentiability and continuity. In particular, Theorem 7.6 proves that if f is differentiable and a point, then it is also continuous at that point. For much more on this interaction, check out Differentiability Versus Continuity: Restriction and Extension Theorems and Monstrous Examples by Ciesielski and Seoane-Sepúlveda.
- In 1872 Weierstrass published the first example of a function which is continuous everywhere but differentiable nowhere (pdf of original paper is at the bottom of this page; let me know if you have an English translation). Theorem 7.6 implies that if a function is differentiable everywhere then it is also continuous everywhere, and this example shows that the converse is false.
- The article The Changing Concept of Change: The Derivative from Fermat to Weierstrass won the MAA's annual Carl B. Allendoerfer Award for best paper in Mathematics Magazine in 1984. It deals with the history of measuring change.
Chapter 8
- The article Integrals Don’t Have Anything to Do with Discrete Math, Do They? by Mark Kayll won the MAA's annual Carl B. Allendoerfer Award for best paper in Mathematics Magazine in 2012.
Chapter 9
- In the page 355 footnote I mentioned the work of mathematicians in Kerala, India, who developed the series of sine and cosine long before the Europeans. For much more, check out A Comparative Study of Kerala and European Schools of Mathematics by Rejikumar and Indukala.
Bonus Papers
- For an article on the origins of analysis in India, check out Development of Calculus in India by Ramasubramanian and Srinivas.
- What's a weird thing you can do with Cantor's diagonalization argument? Here's a blog post on Diagonalizing the Psamls.
- The article Descartes and Problem-Solving by Judith Grabiner (pictured above or to the left) won the MAA's annual Carl B. Allendoerfer Award for best paper in Mathematics Magazine in 1996.
- The article Higher Trigonometry, Hyperreal Numbers and Euler's Analysis of Infinities by McKinzie and Tuckey won the MAA's annual Carl B. Allendoerfer Award for best paper in Mathematics Magazine in 2002.
- The article The Lost Calculus (1637–1670): Tangency and Optimization without Limits by Jeff Suzuki won the MAA's annual Carl B. Allendoerfer Award for best paper in Mathematics Magazine in 2006.
- The article A Brief History of Impossibility by Suzuki won the MAA's annual Carl B. Allendoerfer Award for best paper in Mathematics Magazine in 2009.
- The article Blood Vessel Branching: Beyond the Standard Calculus Problem by John Adam is an interesting application of calculus to medicine.
- The article Various Proofs of the Cauchy-Schwarz Inequality contains 12 different proofs of this classic inequality.
- Finally, the MAA's Committee on the Undergraduate Program in Mathematics (CUPM) publishes a report on undergraduate real analysis courses, as well as a guideline for writing mathematical proofs from Book of Proof by Richard Hammack, which is pages 107-109 here.