Rational or not? This basic math question took decades to answer
Mathematicians’ efforts to prove whether specific constants are rational or irrational open onto deeper questions about how numbers are defined, used in models, and interpreted in physics and computation. Commenters explore the distinction between algebraic, transcendental, computable and physical numbers, why almost every real number is irrational in a probabilistic sense, and how proofs of irrationality often demand highly specialized techniques. The thread also touches on historical myths about the Pythagoreans, philosophical views on the “existence” of irrationals, and the role of clear popular science writing in making such abstract work accessible.
Why irrationality matters
- Several comments ask why mathematicians care if a constant is rational or irrational.
- Answers:
- Irrationality/transcendence often signals hidden structure; a rational result where irrational is expected can reveal unexpected symmetry or simplification.
- Some see results like irrationality proofs as filling gaps in our proof toolkit; the new methods are often more important than the specific constant.
- In applications (cryptography, simulation) people sometimes lean on properties of “random-looking” digit expansions of famous irrationals, though practical PRNGs use rationals on computers.
Algebra, constructibility, and terminology
- Confusion between “constructed from basic algebra” vs “constructible number” and between “algebra” and “an algebra.”
- One view: algebraic operations are just addition and multiplication; exponentials and roots belong to analysis or other fields.
- Others push back: this conflates technical term “algebra” with broader informal “algebra” and ignores areas like group theory.
Rational vs. irrational in practice and physics
- One side: in a discrete physical universe, all measurable quantities are effectively rational; irrationals are idealized limits, like complex numbers.
- Opposing side: current physical theories treat space/time as continuous; trajectories/angles are not quantized, and thinking only rationals are “real” is unjustified.
- Debate over whether a “1m square” genuinely has diagonal √2 m or only some rational approximation.
Random points and probability zero
- Clarification that if you choose a real number uniformly in an interval, the chance of hitting a rational is exactly zero, despite rationals being possible outcomes.
- Long subthread struggles with intuition: difference between finite “things in my pocket” vs. uncountable sets; need for measure-theoretic reasoning.
- Example constructions with infinite random digits illustrate that rationals (eventually periodic decimals) are “almost never” hit.
π, e, and transcendental curiosities
- Interest in whether π+e or π·e are irrational; known that at least one must be, but neither individually is proved so.
- People find a rational value for either especially “mind-blowing” because π and e are “not supposed” to be simply related, though others question that intuition.
- Discussion of “almost integers” like expressions close to integers (e^π−π, e^(√n π)), with clarification that some joking claims of exact integrality are false.
History and Pythagoreans
- Thread disputes the popular story that a Pythagorean was drowned for discovering √2 is irrational.
- Some call the story ahistorical/libel; others label it apocryphal but not definitively debunked, noting ancient sources mention a drowning over other mathematical “impieties.”
Mathematical intuition and communication
- Several comments describe advanced mathematical thinking as accessing a “garden” of ideas beyond step-by-step rigor, developed after learning enough concepts.
- Comparisons drawn between this intuition, famous notebooks of great mathematicians, and the behavior of modern AI systems that sometimes make “incredible leaps.”
- Quanta’s articles and related podcasts are widely praised for making deep topics accessible without being overly simplified.