The most famous transcendental numbers
Status of Euler’s and Catalan’s constants
- Multiple comments note that Euler–Mascheroni γ and Catalan’s constant are not known to be transcendental, or even irrational in γ’s case.
- Some argue they should not appear on a list titled “transcendental numbers,” even with a parenthetical caveat, because math standards require proof, not consensus.
Rigor vs. popularity in labeling numbers
- One side: titles like “most famous transcendental numbers” should only include numbers proven transcendental, just as we would not state “P ≠ NP” as fact.
- Other side: the article explicitly flags the uncertainty; the title is about numbers that “are” transcendental in reality, not “known to be,” and famous unproven candidates are part of that landscape.
- Several see the wording as misleading or “clickbait” for a mathematical topic.
Definitions and algebraic background
- Transcendental = not a root of any nonzero polynomial with rational coefficients.
- Clarifications:
- Irrational but algebraic (e.g., √2) vs. transcendental (e.g., π, e).
- Standard operations with radicals cannot express all algebraic numbers; Abel–Ruffini and Galois theory are briefly discussed and sometimes misunderstood.
Bases, fields, and transcendence
- Changing numeral base (even to a transcendental base like π or e) does not affect whether a number is transcendental.
- In abstract algebra, transcendence is relative to a base field: π is transcendental over ℚ but not over ℚ(π); whether e is transcendental over ℚ(π) is mentioned as open.
“Almost all numbers are transcendental,” randomness, and representation
- Comments stress: almost all reals are transcendental and even undefinable by finite expressions, though one user notes this “undefinability” depends on set-theoretic subtleties.
- Debate over whether one can “pick a real at random”:
- With finite digital representations you only get rationals.
- Some suggest bit-generating schemes or analog sampling; others counter you still only ever observe finite precision, so outcomes are indistinguishable from rationals.
- Distinction drawn between definable vs. computable vs. uncomputable numbers.
Physical reality vs mathematical numbers
- Several argue you never have an actually provable irrational from measurement; physical quantities are modeled by reals, but always measured to finite precision.
- Counterpoints cite pervasive use of trig, exponentials, and π in both classical and quantum physics; reply is that these are successful models, not evidence that specific transcendentals “exist” as physical magnitudes.
Importance and utility of e, π, 2π, and ln 2
- One participant claims e is practically unnecessary, and that ln 2 (and 2π rather than π) are the truly important constants, especially for numerical computation with binary exponentials and logarithms.
- Others strongly disagree, emphasizing:
- e as the natural base where derivatives of exponentials and logs simplify.
- Its central role in differential equations, Fourier transforms, probability, and finance.
- A technical subthread argues that numerical libraries implement e-based functions using ln 2 internally and that binary exponentials and cycle-based trig can be more efficient and accurate; critics respond that this doesn’t diminish e’s conceptual centrality.
Constructed constants and “utility”
- Some see numbers like Champernowne’s and other concatenation-based constants as “manufactured” with little use beyond existence proofs (e.g., normality).
- Others reply that fame can come from simplicity or conceptual role, not practical utility, and that essentially all explicit irrational/transcendental constants are “lab-made” in this sense.
Miscellaneous points
- Mention of Lévy’s constant as another likely transcendental candidate tied to continued fractions.
- Brief nods to iⁱ and its non-uniqueness; interest in “least famous” transcendental numbers; and connections to automata and continued-fraction-style representations as alternative ways to think about “simple” vs “complex” numbers.