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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.

Base Annotation Schemes

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Title: Base Annotation Schemes

Author: John Volan

Year: 11BB_z (2015_d)

Subject: History of base annotations in the Dozenal Societies as well as mainstream mathematics and computing. Proposes using single-character subscripts as base-independent base annotations that do not disrupt customary prose punctuation.

The Problem
An Early Expedient: Stylistic Marking
Humphrey’s Radical Radix-Point
Modularity in Design ... and Its Lack
Honourable (?) Mentions
The Mainstream Solution
Approaches from Programming Languages
Gene Zirkel’s “Unambiguous Notation”
A New/Old Solution
“Nominal” and “Digital” Annotations
To Subscript or Not to Subscript
International Neutrality
“Why Change?”

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., 2014_d = 11BA_z = 7DE_x.

(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., 0.5_d = 0.6_z = 0.8_x

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

EACH ONE, TEACH ONE.