0.9999 ≊ 1

Original Article ↗ Hacker News Discussion ↗

Equality of 0.999… and 1 in the real numbers

  • Many comments reiterate standard results: in ordinary real-number math, 0.999… = 1.
  • Common proofs mentioned:
    • Algebraic: let x = 0.999…, then 10x = 9.999…, subtract to get 9x = 9 ⇒ x = 1.
    • No-gap argument: there is no real number strictly between 0.999… and 1; if it’s not < 1 and not > 1, it must equal 1.
    • Limit/series view: 0.999… is the limit of 0.9, 0.99, 0.999, …; that geometric series converges to 1.
  • Several point out that the key is understanding what the notation “0.999…” means (a limit of a sequence), not viewing it as “a process that never finishes in time.”

Numbers vs their decimal representations

  • Distinction emphasized: a real number is an abstract object; decimal strings are one way to represent it, often non-uniquely (e.g., 1.0 and 0.999…).
  • Some stress that repeating decimals should be treated as alternate notations for fractions (e.g., 0.333… = 1/3), which makes 0.999… = 3/3 obvious.
  • Others argue that elevating fractions as “more real” than decimals is just pedagogical convention, not mathematics. Both are just representations.

Hyperreals and infinitesimals

  • Several criticize bringing hyperreals into a basic question: in standard hyperreal constructions that respect the transfer principle, 0.999… still corresponds to 1 if the sum ranges over all (hyper)naturals.
  • Some note a subtle distinction: if you only sum over standard naturals, you can get 1 − ε in the hyperreals, but then you must be precise about what “…” indexes.
  • A recurring objection: using an undefined “eps” object (1 = 0.999… + ε) without adjusting all related definitions breaks standard calculus and is mathematically confused.

Pedagogy, intuition, and bases

  • Multiple comments describe student intuition: they picture digits being “added one by one” and insist there is always a tiny gap.
  • Effective teaching strategies mentioned:
    • Forcing consistency: if 0.333… = 1/3, then 3×0.333… must equal 1.
    • Emphasizing that numbers lack a time dimension; the infinite expansion is taken as a whole via limits.
  • Some discuss base-10’s awkwardness for thirds and note that different bases (e.g., 12) change which fractions get finite expansions.