987654321 / 123456789

Original Article ↗ Hacker News Discussion ↗

Near-integer ratio and related curiosities

  • Thread centers on why 987654321 / 123456789 ≈ 8.0000000729 and similar “almost integer” coincidences.
  • People compare it to other near-integer expressions like e^π - π ≈ 20, noting that some have deep reasons while others seem more mysterious.
  • A simple approximate argument is: adding 123456789 twice to 987654321 gives ~10×123456789, so the ratio should be close to 8.

Series and exact rational explanations

  • Several comments analyze recurring decimals 0.123456… and 0.987654… as infinite series:
    • Show 0.123456… = Σ k·10⁻ᵏ = 10/81, via geometric series and derivatives or via squaring (1/9).
    • Then 0.987654… = 1 − 0.012345… = 1 − 1/81 = 80/81, giving an ~8 ratio.
  • 1/81 = 0.012345679… is discussed; the missing 8 is explained by carries in …789(10)(11)… causing 8→9 and eliminating an 8 and an extra 0.
  • Multiple equivalent derivations are compared; some find the “intuitive” ones less obvious than implied.

Base‑b generalization and formulas

  • The pattern is generalized to base b, with ascending digits 123…(b−1) and descending (b−1)…321.
  • Definitions of num(b), denom(b) are given and the exact identity
    num(b)/denom(b) = (b−2) + (b−1)³ / (bᵇ − b² + b − 1)
    is derived and also expanded via geometric series.
  • Approximation (b−2) + (b−1)³ / bᵇ is shown to have very small relative error ~(b²−b+1)/bᵇ; examples for bases 8, 9, 10, 16 illustrate how error shrinks with larger b.
  • Edge cases like b=2 and b=3 are explored; special behavior in base 2 is discussed.

Patterns in products and calculators

  • Noted patterns:
    • 12345679×8 = 98765432 and 123456789×8 ≈ 987654312 (swap of last two digits).
    • General base‑n formula: ascending-sequence×(n−2) + (n−1) = descending-sequence.
    • Classic calculator tricks: (1…1)² giving palindromes (111×111=12321), and 12345679×(9k)=k repeated 9 times.
  • “Center of mass” keypad patterns via averaging digit-wise (e.g., (147+369)/2 = 258) are discussed; some see them as trivial per-digit averages, others as delightful numerological structure.

Floating point, exact arithmetic, and scripts vs proofs

  • Comments link to floating-point references and show using arbitrary precision / rational arithmetic (e.g., Python’s Fraction) to get exact decompositions like 14 + 1/5465701947765793.
  • Discussion on using scripts to support proofs: scripts catch different errors than proofs and make details explicit, but aren’t substitutes for formal, explanatory proofs.
  • Brief digression on Curry–Howard and what exactly “code as proof” does and doesn’t mean.

Meta and aesthetics

  • OEIS entries are used to categorize “error terms”; curiosity about an OEIS-like resource for analytic observations.
  • Several remarks frame these coincidences as examples of the “fun”, “hidden magic”, and imperfect beauty of mathematics.