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Dozenal Society of America

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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.

Dozenal Divisibility Tests Quick Guide

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Title: Dozenal Divisibility Tests Quick Guide

Author: Treisaran

Year: 11B9_z (2013_d)

Subject: Dozenal divisibility tests

An invaluable resource for anyone who wants to become more adept with dozenal arithmetic, particularly mental arithmetic. This fact sheet puts together all the major divisibility tests of the dozenal system into a single, easy-to-learn and easy-to-understand package.

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.