TL;DR
Cliff Pickover assembled a list of 15 well-known transcendental (and conjectured-transcendental) numbers and explained historical proofs and curiosities around them. The piece highlights that while transcendental numbers outnumber algebraic ones, proving any particular constant transcendental is often hard and many familiar constants remain open questions.
What happened
Writer Cliff Pickover collected and commented on fifteen of the best-known transcendental numbers, weaving historical context, constructions, and open problems. He notes Joseph Liouville as the first to exhibit a specific transcendental number (1844), followed by Hermite’s proof that e is transcendental (1873) and Lindemann’s proof that pi is transcendental (1882). The list includes classical constants (pi, e), purpose-built examples (Liouville’s number, Champernowne’s number), algorithmic and dynamical constants (Chaitin’s constant, Feigenbaum number), and several familiar constants whose transcendence has not been settled (Euler’s constant gamma, Catalan’s constant). The article explains short demonstrations and properties — for example, i^i equals e^{-pi/2} — and mentions general results such as the Gelfond–Schneider theorem that produce many transcendental values from algebraic inputs. Pickover also relays a few illustrative anecdotes, including an ants thought experiment and reader comments correcting its implications.
Why it matters
- Transcendental numbers expose limits of algebra: they are not roots of any polynomial with rational coefficients.
- Proving transcendence of specific constants is often difficult, so each proof marks a notable advance in number theory.
- Some transcendental or conjectured-transcendental constants arise naturally in analysis, dynamical systems, and computer science, connecting disparate fields.
- General theorems (e.g., Gelfond–Schneider, Lindemann–Weierstrass) can produce many transcendental numbers from algebraic inputs, making the class widespread in mathematics.
Key facts
- Joseph Liouville was the first to prove a specific number transcendental (circa 1844); Liouville also constructed an explicit transcendental number now called Liouville’s number.
- Charles Hermite proved that e is transcendental in 1873; Ferdinand von Lindemann proved that pi is transcendental in 1882.
- A transcendental number cannot be the root of any algebraic equation with rational coefficients (polynomial equations with integer coefficients).
- Liouville’s number is given by a decimal with ones at rapidly increasing positions (e.g., positions 1, 2, 6, 24, …), guaranteeing transcendence by construction.
- Champernowne’s number is formed by concatenating the positive integers (0.123456789101112…), a simple constructed example discussed in the list.
- Chaitin’s constant, the halting probability of a random program, is stated as transcendental and also incomputable in the article (attributed to Noam Elkies).
- Some constants on the list are not proven transcendental but are widely believed to be so, including Euler’s constant (gamma), Catalan’s constant, and certain Feigenbaum numbers.
- The Gelfond–Schneider theorem: if a and b are algebraic (a ≠ 0, a ≠ 1) and b is irrational algebraic, then a^b is transcendental — a source of many transcendental values.
- The expression i^i equals e^{-pi/2} (numerically about 0.207879576…), a compact example linking complex exponentiation and transcendence results.
- The remarkable rational 355/113 gives pi accurate to six decimal places, illustrating how algebraic approximations can closely match transcendental values.
What to watch next
- Proofs or counterexamples resolving whether Euler’s constant (gamma) is transcendental — currently not proven.
- Progress on the transcendence status of Catalan’s constant and various Feigenbaum numbers, which remain open or conjectural.
- Further computational refinements and theoretical work around Chaitin’s constant and its incomputability/transcendence properties.
Quick glossary
- Transcendental number: A real or complex number that is not the root of any nonzero polynomial equation with rational (equivalently integer) coefficients.
- Algebraic number: A number that is a root of some nonzero polynomial equation with rational (or integer) coefficients.
- Liouville’s number: A constructed decimal number with ones placed at rapidly increasing positions; historically the first explicit example used to prove the existence of transcendental numbers.
- Gelfond–Schneider theorem: A result that guarantees numbers of the form a^b are transcendental when a and b are certain algebraic numbers (a ≠ 0,1 and b irrational algebraic).
- Incomputable number: A real number for which there is no algorithm that can produce its digits to arbitrary precision in finite time; Chaitin’s constant is an example cited as incomputable.
Reader FAQ
What is a transcendental number?
A number that does not satisfy any polynomial equation with rational coefficients; examples include pi and e.
Are pi and e proven to be transcendental?
Yes — e was proved transcendental by Hermite (1873) and pi by Lindemann (1882), as noted in the source.
Are Euler’s constant (gamma) or Catalan’s constant known to be transcendental?
Not proven in the source; both are widely believed to be transcendental but remain open problems.
Is Chaitin’s constant computable?
The source reports Chaitin’s constant is incomputable and remarks that it is transcendental (attributed commentary by Noam Elkies).
Why are there so many transcendental numbers?
The article notes there are 'more' transcendental numbers than algebraic ones, and general theorems can generate many transcendental values, though it does not quantify cardinalities.
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
- The most famous transcendental numbers
- 44 The 15 Most Famous Transcendental Numbers
- Fifteen Most Famous Transcendental Numbers
- The Art of Mathematics
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