Rational or not? This basic math question took decades to answer

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.