S. A. E. Miller, Ph.D.

Graph theory emerged in the 18th century, connecting geometry with fields like topology and set theory. Leonhard Euler formulated graph theory during his time in Königsberg, now Kalingrad. His seminal work began with the 1736 paper, “The Seven Bridges of Königsberg.”

Residents of Königsberg enjoyed crossing the city’s seven bridges in one outing – considered impossible, but not proven. Euler approached this challenge not by traditional geometry or algebra, but through an abstraction where cities and bridges became vertices and edges in a graph. This approach showed that a path crossing each bridge once was impossible if more than two vertices had an odd number of edges.

Euler’s insights created the foundation for graph theory, now used in various applications from computer algorithms for facial recognition to network and process design. Or perhaps a bit of frustration by myself in graduate school studying math in the physics department at Penn State. Crossing all the bridges became possible later on in Kalingrad due to war, as they were partly destroyed.

See

• Euler, Leonhard, “Solutio problematis ad geometriam situs pertinentis,” 1741.
• https://maa.org/press/periodicals/convergence/leonard-eulers-solution-to-the-konigsberg-bridge-problem

\(e\), defined by the limit \(\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n\) and approximating 2.71828, is characterized by its irrational and transcendental nature, indicating it cannot be depicted as a quotient of two integers nor as a solution to any non-trivial polynomial equation with rational coefficients.

John Napier, a Scottish mathematician and theologian, significantly contributed to the field of mathematics through his invention of logarithms, detailed in his seminal work “Mirifici Logarithmorum Canonis Descriptio,” (1614). This development facilitated the transformation of cumbersome multiplicative calculations into simpler additive operations. Napier’s logarithms laid the groundwork for the natural logarithm concept, where \(e\) emerged as the fundamental base that equates the value of the logarithm to unity.

One hundred years later, Jacob Bernoulli, Swiss, explored the concept of compound interest, leading to the discovery of \(e\)’s significance in financial mathematics. His investigations, as detailed in his posthumously published research “Ars Conjectandi,” (1713), demonstrated that continuous compounding yields a growth factor of \(e\), showing \(e\)’s utility in not only financial models but also in natural processes like population dynamics and radioactive decay.

References

• Napier, John (1614), Mirifici Logarithmorum Canonis Descriptio.
• Bernoulli, Jakob (1713), Ars conjectandi, opus posthumum. Accedit Tractatus de seriebus infinitis, et epistola gallicé scripta de ludo pilae reticularis.

Binary numbers were originally used for encryption and communication, a fact recognized as early as the 17th century by Francis Bacon. Bacon used the binary system for encoding the alphabet using strings of binary characters. This laid the framework for subsequent developments in coded communication, such as technologies like the telegraph (Samuel Morse), which relied on a binary tones of ‘dots’ and ‘dashes.’ Binary’s mathematical formulation was created by Gottfried Wilhelm Leibniz, who recognized the system’s elegance. Leibniz’s research created a formal basis for binary arithmetic, outlining methods for converting between binary, decimal, and other number systems.

References:

• Bacon, F. (1605), “The Advancement of Learning.” Book VI. London: Henrie Tomes.
• Leibniz, G.W. (1703), “Explication de l’Arithmétique Binaire.” Mémoires de l’Académie Royale des Sciences, pp. 85-89. Paris.