Height of harmonic numbers
A follow-up post examines the number of bits in the numerator and denominator of harmonic numbers written as reduced fractions, building on previous asymptotic estimates with base-2 plots.
Background
- Harmonic numbers (Hₙ) are the sum 1 + 1/2 + 1/3 + … + 1/n. They appear in analysis, number theory, and computer science, especially in algorithm analysis (e.g., the expected number of comparisons in quicksort).
- This post studies Hₙ expressed as a reduced fraction (numerator/denominator with no common factors). The "height" of a rational number is roughly its "size" — here, the total number of bits (binary digits) needed to write both the numerator and denominator.
- John D. Cook is a mathematician and blogger who writes accessible posts about applied math, scientific computing, and statistics. This is a follow-up to an earlier post on the same topic.
- The practical interest: when computing Hₙ exactly as fractions, numerators and denominators grow extremely fast. Understanding their bit-length helps gauge computational cost and storage needs.