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.