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

In 1844, Joseph Liouville demonstrated that the decimal representations of certain numbers were infinitely long and lacked pattern. This idea, which suggests that numbers do not necessarily have an exact and finite value, was first proposed by Greek philosopher Zeno in the 5th century BCE. Zeno’s paradoxes are based on the infinite divisibility of space. The resolution to these paradoxes was not found until the development of calculus by Leibniz. They demonstrated that an infinite geometric series can converge, balancing the infinite number of “half-steps” traveled with the increasingly short amount of time needed to cross the decreasing distances.

A rational number is a number that can be expressed as a fraction, p / q, where p and q are integers. Familiar examples of irrational numbers include pi and root 2. Transcendental numbers, a subset of irrational numbers, cannot be expressed using algebra; they are not roots of a polynomial with rational coefficients. While Liouville failed to prove that e is a transcendental number, he did construct an infinite class of transcendental numbers using continued fractions. In 1851, he produced an example of a transcendental number now known as Liouville’s constant, an infinite string of zeros and ones with a one positioned at every value of the exponential factorial, n!. In 1873, e was shown to be transcendental, and pi was proven to be so in 1882. Most numbers are transcendental; those with definable patterns are in the minority.

References

• Liouville, J. (1851). Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques. Comptes Rendus de l’Académie des Sciences, 32, 135.
• Hermite, C. (1873). Sur la fonction exponentielle. Comptes Rendus de l’Académie des Sciences, 77, 18-24, 74-79, 226-233, 285-293.
• Lindemann, F. (1882). Über die Zahl π. Mathematische Annalen, 20(2), 213-225.

The Fundamental Theorem of Algebra states that the field of complex numbers is algebraically closed, implying that every polynomial equation of degree n has n roots within the complex numbers, with at least one being a solution where the polynomial evaluates to zero. Historically, the theorem’s origin traces back to the conjectures by Albert Girard in 1629 and René Descartes in 1637, though neither provided proofs including complex numbers. A first significant proof attempt was made by Jean d’Alembert (my personal hero) in 1746. The most notable advancements were made by Carl Friedrich Gauss, who, in 1799, offered a proof that, despite its gaps, marked a significant step forward in understanding the theorem. Gauss’s efforts were refined by Jean Robert Argand in 1814, who introduced an existence proof that set the stage for more concrete proofs, including those by Gauss himself in subsequent years using Euler’s earlier work. The theorem’s proof evolution highlights the gradual understanding and acceptance of complex numbers in solving polynomial equations, an understanding put further by Hellmuth Knesser (1940).

References

• Girard, A. (1629). Invention Nouvelle en l’Algèbre.
• Descartes, R. (1637). La Géométrie.
• d’Alembert, J. (1746). Recherches sur le calcul intégral.
• Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse.
• Argand, J. R. (1814). Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques.

1796, Maskelyne, Astronomer Royal of Britain, dismissed his assistant for what he believed to be persistent inaccuracies in the timing of observations, specifically delays of approximately half a second. This decision unknowingly set the stage for the development of an important concept in measurement science: the personal equation. Maskelyne’s published their mixed observational results. After Maskelyne’s death, the issue was investigated further by F. Bessel, a German astronomer. Bessel’s research revealed a consistent measurable difference in the timing of observations made by different astronomers. He termed this difference the “personal equation,” highlighting its role in scientific measurements.

The concept of the personal equation stimulated significant research in the newly emerging field of psychology in the late 19th century. Researchers focused on understanding individual differences in reaction times, which could affect observational accuracy. The term personal equation entered general use, applied to any personal influence on objective situations. This illustrates how a specific scientific concept can expand beyond its original context to gain wider societal acceptance.

Important References

• Astronomische Beobachtungen auf der Koniglichen Universitiits-Sternwarte F. W. Bessel, 8. Abtheilung voe 1. Januar bis 31. December, 1822.

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