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The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom - Using the Poem, 'The Enigmatic Number e,' in the Mathematics Classroom

Sarah Glaz (University of Connecticut)

The poem "The Enigmatic Number e" may be used in the mathematics classroom to enrich pedagogy or course content in any of the number of ways discussed in the previous section. In particular, the poem may be offered to students as a classroom handout, or as a link to an electronic posting, in mathematics courses that mention the number e. These courses range from high-school and college Intermediate Algebra, to Pre-Calculus and Calculus courses, to Business Mathematics courses, to more advanced college courses such as Probability, Differential Equations, Abstract Algebra, Complex Variables, and Number Theory, and, of course, History of Mathematics, or courses exploring the connections between mathematics and poetry. The appearances of e mentioned in the poem are sufficiently varied to engage students with the related material taught in the course, as well as to pique students' curiosity about the mathematics that lies ahead—in courses they have not yet taken.

"The Enigmatic Number e" may also be used to supplement classroom material and generate classroom activities on topics present in its text. The Links to Resources section of this article (see page 6) lists the hyperlinks appearing in the poem's text, as well as links to additional online resources, sorted by biographies and topics. Each link leads to a source of information about a particular mathematician or topic that can assist educators with the preparation of supplementary classroom material and activities. The Biographies links contain, in addition to biographical information about the mathematicians, historical information about the mathematical work of each mathematician. Contemporary approaches to the mathematics of a topic may be found in the Topics links. The links may also be used to drive course content, by combining several related topics into units of study. Below are two examples of such study units. Each unit consists of a number of parts that can also be used on their own or in different combinations with each other.

Study Unit: The Development of Computation

This unit may start with the definition of Napier's logarithm (links: Napier, references: [1], [30]), and his early computation devices called Napier's bones and counting board (links: Napier, reference: [7]). In particular, the second link of Napier contains information for the preparation of classroom activities and homework projects on the use of these calculation devices.

The unit may continue with the development of the slide rule (links: Oughtred, Slide rule). The top two links of Slide rule provide both historical information and ideas for classroom or homework activities on the historical uses of the various slide rules. The third link of Slide rule is an online slide rule powered by Mathematica. Asking students to do both manual and online slide rule calculations will provide a transition to the last part of this unit.

The unit may now transition to modern calculating devices (links: Logarithm and natural logarithm, e). In particular, this is a good place to introduce students to both calculator and computer software capabilities (for example, EXCEL or Mathematica), including the capability of computing logs with great ease and e to great accuracy. An online calculator is available at: Online computations of e and logs can be carried out at the third link of Logarithm and natural logarithm and the fourth link of e (click on Programs, then click on Online computation of some constants). One may conclude this unit with a project asking students to conduct an internet search to discover the latest accomplishment in accuracy for the computation of e and, if time permits, also of π. As of the writing of this article, e has been calculated to 200 billion digits after the decimal point (see the first link for e).

Study Unit: The Number System

This unit may start with a classical exposition of the real number system (see, for example, reference [6]) supplemented with the topic of continued fractions (links: Continued fractions, reference: [28]). Continued fractions are not a part of any standard college course's syllabus. Their introduction will add a new perspective on the representation of numbers and the difference between rational and irrational numbers. An engaging hands-on approach to this subject, complete with ready made activities and homework exercises, may be found at the top link of Continued fractions (scroll up all the way to the top of the site). The other links of Continued fractions provide additional information to enrich this material with applications and history of the topic.

The unit may continue with the extension of the real number system to complex numbers (links: Complex numbers, i, Roots of unity). The link Complex numbers provides an interesting self-contained short course on complex numbers—their history and their properties. It can be used to supplement or replace a course's textbook as a basis for lesson plans and assignments.

Keeping e as the main focus, the unit may now explore some properties and relations of the three special numbers mentioned in the poem, e, π, and i (links: Euler, Hermite, Euler Identity, i, Pi (π), e, Irrational numbers, Transcendental numbers). One can start by introducing the concept of a transcendental number as early as the study of roots of polynomials in an Intermediate Algebra course. The second link of Tanscendental numbers provides a gentle introduction and several examples of the concept. For more advanced courses, proofs of e's irrationality and transcendentality may be found in the second and third links of e. The first link of Trancendental numbers brings up, without proof, the transcendentality of π and the open questions regarding transcendentality of powers (and addition or multiplication) of π and e. The links for Euler's Identity can be used to introduce this identity and explore some of its implications for the calculations of powers of the three special numbers. One way to involve students with the material is to send them on an exploration project through these links for the purpose of finding out what is known about a list of powers of the three numbers, in terms of value, irrationality, or transcendentality. The project may request a justification for each answer, whenever possible (for example, when the conclusion comes out from a direct application of Euler's Identity) and a history of the answer. Such a list may include ei, eπ, ee, ii, iπ, ie, πi, ππ, πe, and other more complicated powers.

The Links to Resources section's 19 biographical links and 43 topics links, and the References section's 34 references, may be combined in many ways to make coherent units of study to suit instructor's tastes, time availability, and the content of the course in which they are intended to be used.

Sarah Glaz (University of Connecticut), "The Enigmatic Number [i]e:[/i] A History in Verse and Its Uses in the Mathematics Classroom - Using the Poem, 'The Enigmatic Number [i]e,[/i]' in the Mathematics Classroom," Loci (November 2010), DOI:10.4169/loci003482