# Fourier’s Infinite Series Proof of the Irrationality of e: A Mini-Primary Source Project for Second-Semester Calculus Students

Author(s):
Kenneth M Monks (Front Range Community College – Boulder County Campus)
 Three techniques central to the second-semester calculus curriculum are  arguing via comparison tests; applying the infinite geometric series formula $a+ar+ar^2+ar^3+\cdots =\frac{a}{1-r}, \, \,\,\,\,\, (|r|<1);$ and defining transcendental functions and constants via infinite series expansions, such as $e=1+1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots .$ Jean-Baptiste Joseph Fourier's short and beautiful proof that $e$ is irrational combines exactly these three techniques! Jean-Baptiste Joseph Fourier (1768–1830). Wellcome Library no. 3110i,  public domain.

The first written account of Fourier's proof was given by Janot de Stainville (1783–1828), in his Mélanges d’analyse algébrique et de géométrie (Miscellany of algebraic analysis and geometry) [de Stainville 1815, pp. 339–343]. The only idea required to understand this argument that is not typically in the first-year calculus student’s toolbox is that of proof by contradiction. The mini-Primary Source Project (mini-PSP) Fourier’s Infinite Series Proof of the Irrationality of $e$ introduces the student to this powerful proof technique via a passage from Aristotle and some gentler warm-up contradiction arguments before walking the student through de Stainville’s presentation of Fourier’s lovely argument.

More specifically, the sections of this mini-Primary Source Project entail the following:

• Section 1 (Proof by Contradiction). In this section, the student learns the general form of a proof by contradiction from a passage in Aristotle’s Prior Analytics [McKeon 1941, pp. 65–107]. A student task then analyzes a simple contradiction argument proving there are infinitely many natural numbers using the well-ordering principle.

• Section 2 (Some Fundamental Sets of Numbers). Here the project makes sure the student understands exactly what is meant by the words rational and irrational before attempting to prove statements involving these words! The template for an irrationality proof via contradiction is also given.

• Section 3 (A Warm-up Irrationality Proof). The project returns to the passage from Aristotle, in which he claimed that the side length and diagonal of a square are not commensurate since otherwise “odd numbers are equal to evens.'' The Greek geometers' notions of commensurability/incommensurabilty are briefly related to the rational and irrational numbers.1 The student then works through a proof of the irrationality of $\sqrt{2}$ corresponding to Aristotle's claim—a much easier warm-up before the main event in the next section!

Raphael, The School of Athens, 1509–1511, fresco at the Raphael Rooms, Apostolic Palace, Vatican City.
Aristotle is depicted in light blue in the center of the fresco standing next to a depiction of Plato in red.
Public domain, Wikimedia Commons.

• Section 4 (Fourier’s Proof of the Irrationality of $e$). Here the student works through de Stainville’s argument, which compares the series representation for $e$ against a geometric series to show that $e$ is a number between $2$ and $3$. Afterwards, the student works through Fourier’s proof by contradiction that proves $e$ is irrational (as communicated by de Stainville), which again uses a comparison to a geometric series.

• Section 5 (Transcendence of $e$). As a brief epilogue, the student explores the idea of transcendental numbers as an extension of irrationality, comparing the behavior of $\sqrt{2}$ with that of $e$.

The project also provides a short biography of Fourier, detailing his inspiring rise from young orphan to prominent mathematician and government official.

The complete project Fourier’s Infinite Series Proof of the Irrationality of $e$ (pdf) is ready for student use and the LaTeX source is available from the author by request. Instructor notes are provided to explain the purpose of the project and guide the instructor through its implementation. These notes also provide information about an extended version of this mini-project that instructors seeking a more in-depth experience for their Calculus 2 students may wish to consider [Monks 2022]. The longer project, in which Fourier’s proof of $e$’s irrationality is followed up with Joseph Liouville’s (1809–1882) more challenging proof of the irrationality of $e^2$, is also appropriate for use in an introduction to proofs course or as a part of a capstone experience for prospective secondary mathematics teachers.

This project is the twenty-fifth in A Series of Mini-projects from TRIUMPHS: TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources to appear in Convergence, for use in courses ranging from first-year calculus to analysis, number theory to topology, and more. Links to other mini-PSPs in the series appear below. The full TRIUMPHS collection also offers dozens of other mini-PSPs and a similar number of more extensive full-length PSPs. These include an additional twelve mini-PSPs for use in first-year calculus courses as well as four PSPs for use in a multivariable calculus course.

Notes

[1] In Section 2 of the extended version of this project (described in the penultimate paragraph of this article), the Greek geometers' notion of commensurate figures is explored in more depth and the student works through a proof of Aristotle's actual claim about the diagonal and the side of a square (albeit using a numerical notion of length and modern symbolism). The definitions of rational and irrational numbers are only given later, in Section 3, where the student is prompted to relate that incommensurabilty proof to the irrationality of $\sqrt{2}$.

Acknowledgments

The development of this project has been partially supported by the TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) Project with funding from the National Science Foundation’s Improving Undergraduate STEM Education Program under Grants No. 1523494, 1523561, 1523747, 1523753, 1523898, 1524065, and 1524098. Any opinions, findings, and conclusions or recommendations expressed in this project are those of the author and do not necessarily reflect the views of the National Science Foundation.

References

Richard McKeon. 1941. The Basic Works of Aristotle. New York: Random House.

Kenneth M Monks. 2022. Why $\sqrt{2}$ is Friendlier than $e$: Irrational Adventures with Aristotle, Fourier, and Liouville. TRIUMPHS Digital Commons Collection. Calculus. 22.

Janot de Stainville. 1815. Mélanges d’analyse algébrique et de géométrie (Miscellany of algebraic analysis and geometry). Courcier, Paris.

Kenneth M Monks (Front Range Community College – Boulder County Campus), "Fourier’s Infinite Series Proof of the Irrationality of e: A Mini-Primary Source Project for Second-Semester Calculus Students," Convergence (September 2022)