TL;DR
A roundup by Cliff Pickover lists fifteen well-known transcendental numbers, explains their constructions and historical proofs, and flags several constants whose transcendence remains unproven. The piece highlights classical results (Liouville, Hermite, Lindemann) and includes examples ranging from explicit constructions to constants tied to computability and chaos.
What happened
Cliff Pickover compiled a list of fifteen widely discussed transcendental numbers, combining historical notes, concrete examples, and commentary about open status for some constants. The essay recalls Joseph Liouville’s 19th-century breakthrough proving the existence of a specific transcendental number and highlights Hermite’s proof for e (1873) and Lindemann’s proof that pi is transcendental (1882). The list contains familiar constants (pi, e), constructed examples (Liouville’s and Champernowne’s numbers), algorithmic/semantic constants (Chaitin’s halting probability), special-function values (zeta(3), ln(2)), and more exotic entries (i^i, the Dottie number, Feigenbaum constants). Pickover flags several numbers as widely believed to be transcendental but not proven—Euler’s constant and Catalan’s constant among them—and points to theorems (for example, Gelfond–Schneider) that generate many transcendental examples. The article also includes a short thought experiment about an “ant” encoding of pi and a reader’s correction of a misconceived implication of that metaphor.
Why it matters
- Transcendental numbers lie outside algebraic equations with rational coefficients, showing limits of algebraic description.
- Proving transcendence is often difficult; historic proofs (e.g., for e and pi) mark major milestones in number theory.
- Theorems like Gelfond–Schneider turn many familiar expressions into transcendental numbers, linking elementary constants to deeper results.
- Some listed constants connect number theory with other areas—computability (Chaitin’s constant) and dynamical systems (Feigenbaum constants).
Key facts
- Joseph Liouville in the 1840s produced the first proof that a specific number is transcendental.
- Charles Hermite proved the transcendence of e in 1873; Ferdinand von Lindemann proved pi transcendental in 1882.
- Transcendental numbers are not roots of any polynomial equation with rational coefficients.
- Liouville’s number is an explicit decimal specially constructed to guarantee transcendence (ones placed at factorial-based positions).
- Champernowne’s number is formed by concatenating positive integers (0.1234567891011…), and is an example of a constructed transcendental.
- Chaitin’s constant (the halting probability) is described as both transcendental and incomputable by Noam Elkies in Pickover’s survey.
- Some constants on the list—Euler’s constant (gamma) and Catalan’s constant—are widely believed to be transcendental but lack proofs.
- The Feigenbaum constant appears in period-doubling bifurcations in dynamical systems; it is believed transcendental but not proven in the source.
- The value i^i equals e^(−pi/2) numerically (~0.207879576…) and is discussed as an elegant example arising from complex exponentiation.
- The Dottie number (the unique real solution of cos x = x, ≈ 0.739085…) is cited and noted to be transcendental via Lindemann–Weierstrass consequences.
What to watch next
- Attempts to prove whether Euler’s constant (gamma) is transcendental — currently not proven in the source.
- Work on the transcendence status of Catalan’s constant and the Feigenbaum constants — currently not proven in the source.
- Further investigations into algorithmic constants such as Chaitin’s constant and their formal properties (incomputability and transcendence).
Quick glossary
- Transcendental number: A real or complex number that is not a root of any nonzero polynomial with rational coefficients.
- Algebraic number: A number that is a root of some nonzero polynomial with rational (or equivalently integer) coefficients.
- Gelfond–Schneider theorem: A result saying that if a and b are algebraic (with a ≠ 0, a ≠ 1) and b is irrational algebraic, then a^b is transcendental.
- Incomputable: A property of a number or function for which no algorithm exists that can produce its digits or values to arbitrary precision in finite time.
- Continued fraction: An expression for a number as the sum of its integer part and the reciprocal of another number, repeated; often used to represent irrational numbers.
Reader FAQ
What makes a number transcendental?
It cannot satisfy any polynomial equation with rational coefficients; transcendence is a stronger property than irrationality.
Is pi transcendental?
Yes — Lindemann proved pi is transcendental in 1882.
Are Euler’s constant and Catalan’s constant known to be transcendental?
Not confirmed in the source; both are widely believed by mathematicians to be transcendental but lack proofs.
What is Chaitin’s constant?
In this piece it is described as the probability that a randomly chosen algorithm halts; the source reports it as incomputable and transcendental.
The 15 Most Famous Transcendental Numbers Cliff Pickover. Follow me on Twitter I am in love with the mysterious transcendental numbers. Did you know that there are "more" transcendental numbers…
Sources
- Fifteen Most Famous Transcendental Numbers
- 44 The 15 Most Famous Transcendental Numbers
- either pi is algebraic or some journals let in an incorrect …
- Transcendental number
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