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Dozenal Society of America
Promoting base twelve and alternative base mathematics
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The Dozenal Society of America
The DSA is a voluntary, nonprofit education corporation, organized for the conduct of research and education of the public in the use of dozenal (also called duodecimal or base-twelve) in calculations, mathematics, weights and measures, and other branches of pure and applied science.

The Duodecimal Bulletin

◀️ Issue: Whole Number 9Az (118d) | Vol. 4Bz (59d) No. 1 | Year 11B6z (2010d) ▶️



Click to view PDF

Editor: Michael DeVlieger

Cover Title: Digital Fractional Expansions

Contents:

  • President’s Message | Michael DeVlieger
  • Electronic Publication of the Bulletin
  • Robert Roy McPherson, DSA No. 4Bz
  • The Duodecimal Bulletin Online Archve
  • A Multipliction Oddity | Jean Kelly
  • Dozenal Counting on Your Fingers | Timothy F. Travis, DSA No. 342z
  • Problem Corner | Prof. Gene zirkel & Eugene M. Scifres
  • On Maximal Repeating Sequence of Decimal Expansions in Base Twelve: Feature | T. J. Gaffney, DSA No. 399z
  • Featured Figures → Dozenal Fractional Expansions
  • A Numeral Toolbox | Michael DeVlieger
    • Simple Graphical Tools
      • Rotation and Reflection
      • Elision and Appendage
      • Smoothing
      • Cultural Resonance
      • Justification
    • Strategy
      • The Analog Strategies
      • Additive Analog Strategy
      • Positional/Rotational Additive Analogs
      • Other Analog Strategies
      • Indexed Value Stategy
      • Modular Symmetry Strategy
    • Application to Subdecimal Bases
    • Exceedingly Transdecimal Bases
      • Least Change
      • Separate Identity
  • DozensOnline Forum: The Symbols Debate
    • Introduction
    • Selected Forum Statements Regarding Dozenal Numberalsa
    • Criteria for Numerals as Implied on the Forum
      • Distinction
      • Clarity
      • Ease of Writing
      • Visual Unity
      • Minimization of Impact
    • Further Seven Segment Evaluations
  • Mathematics and Computer Education (Advertisement)
  • Book Review: Here’s Looking at Euclid | Prof. Gene Zirkel
  • The Mailbag
    • John Earnest
    • Erich Kothe
    • Jeff Wells
    • Shaun Ferguson
  • Join the DSA Today!
Copyright © 11B9–120Az (2013–2026d). The Dozenal Society of America. Contact Us.
Creative Commons License
This site and the documents on it are licensed under a Creative Commons Attribution 4.0 International License.

This site expresses numbers in alternative bases, particularly dozenal (also called duodecimal, or base twelve).
The DSA does not endorse any particular notation or pronunciation scheme for alternative bases.
However this site defaults to the following to ensure online readability across all platforms:
“Computerese” uppercase letters represent transdecimal digits: A = ten, B = eleven, C = twelve, etc.
Single-character subscripts annotate bases: d = decimal, z = dozenal, x = hexadecimal, etc., e.g., 2014d = 11BAz = 7DEx.

(0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 ...)d
(0 1 2 3 4 5 6 7 8 9  A  B 10 11 12 13 14 15 16 17 18 19 1A 1B 20 21 22 23 24 25 26 27 28 29 2A 2B 30 ...)z
(0 1 2 3 4 5 6 7 8 9  A  B  C  D  E  F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20 21 22 23 24 ...)x

PDF documents on this site reflect changing practices among dozenalists over the years:
Various symbols have been used to represent digits ten and eleven:
Dwiggins 𝒳 and Ɛ (often rendered X and E); Bell Telephone * and #; Pitman turned 2 and 3 (↊ and ↋); etc.
Italics were once used to distinguish dozenal numbers from decimal: e.g. 0.5 = 0.6.
Later, semicolon became a separate‑identity “dozenal point”, versus period as “decimal point”: e.g., 0.5 = 0;6.
However, subscript base annotations provide a more versatile solution that is least‑change: e.g., 0.5d = 0.6z = 0.8x

Dozenal offers superior divisibility over decimal, which can make everyday arithmetic easier to learn:
⅙ = 0.2z | ¼ = 0.3z | ⅓ = 0.4z | ½ = 0.6z | ⅔ = 0.8z | ¾ = 0.9z | ⅚ = 0.Az
⅙ = 0.1666…d | ¼ = 0.25d | ⅓ = 0.333…d | ½ = 0.5d | ⅔ = 0.666…d | ¾ = 0.75d | ⅚ = 0.8333…d

EACH ONE, TEACH ONE.