# Senary

**Senary** is the positional numeral system with base six: it uses the six digits 0, 1, 2, 3, 4, and 5, and each place value is a power of 6 rather than a power of 10.<sup>[1](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)</sup> The base is also called base-6, heximal, seximal, sextal, or simply base six.<sup>[2](https://mathworld.wolfram.com/Base.html)</sup><sup> • </sup><sup>[3](https://sezimal.tauga.online/en)</sup> Senary is rare among the world's spoken number systems, but it is attested in several languages of southern [New Guinea](https://www.edgepedia.org/new-guinea), where it is tied to the ritual counting of yams, and it has a small following among base enthusiasts who value its divisibility properties.<sup>[4](https://research.chalmers.se/en/publication/253103)</sup><sup> • </sup><sup>[5](https://www.seximal.net/home)</sup>

| Key fact | Detail |
|---|---|
| Digits and places | Six digits (0–5); place values are powers of 6, so 3,024₆ = 3 × 6³ + 0 × 6² + 2 × 6¹ + 4 × 6⁰ = 664 in decimal<sup>[1](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)</sup> |
| Powers of six | 10₆ = 6, 100₆ = 36, 1000₆ = 216, 100000₆ = 7,776, 1000000₆ = 46,656<sup>[1](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)</sup><sup> • </sup><sup>[6](https://langsci.github.io/212/main.pdf)</sup> |
| Other names | Heximal, seximal, sextal, base six<sup>[3](https://sezimal.tauga.online/en)</sup> |
| Natural attestation | Base-6-36 numeral systems in the Kanum languages of New Guinea, arising in the Tonda group, probably for counting yams<sup>[4](https://research.chalmers.se/en/publication/253103)</sup> |
| Komnzo roots | näbi (1), eda (2), etha (3), asar (4), tabuthui (5), nibo (6); higher powers include fta (36) and taruba (216)<sup>[6](https://langsci.github.io/212/main.pdf)</sup><sup> • </sup><sup>[7](https://escholarship.org/content/qt3xf465n5/qt3xf465n5.pdf)</sup> |
| Prime test | Every prime greater than 3 is congruent to 1 or 5 modulo 6, so in senary every prime above 3 ends in the digit 1 or 5<sup>[8](https://en.wiki.beta.math.wmflabs.org/wiki/Senary)</sup> |
| Terminating fractions | A fraction terminates in base 6 exactly when its reduced denominator has no prime factors other than 2 and 3<sup>[9](https://carmamaths.org/florianbreuer/MATH3170/Chap-Some_Base_Results.html)</sup> |

## How the system works

Counting in senary follows the same pattern as counting in decimal, truncated to six symbols. After 0, 1, 2, 3, 4, 5 the next number is written 10, then 11 through 15, then 20, and so on; when both places are exhausted at 55, a third digit is added, giving 100.<sup>[1](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)</sup> A numeral is read by expanding it in powers of six: 3,024₆ means 3 × 216 + 0 × 36 + 2 × 6 + 4 × 1, which equals 664 in decimal.<sup>[1](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)</sup> Conversion from decimal to senary proceeds by repeated division by 6, with the remainders read in reverse order; 298 in decimal becomes 1214₆ by this method.<sup>[1](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)</sup>

Because six is a small base, numbers grow quickly in length.

## Arithmetic advantages

**Divisibility.** Six has the prime factors 2 and 3, so senary shares with decimal the property that its base is a product of small primes, and it improves on decimal by including 3. A fraction written in lowest terms has a terminating senary expansion if and only if every prime factor of its denominator divides 6, that is, the denominator is of the form 2ᵃ · 3ᵇ.<sup>[9](https://carmamaths.org/florianbreuer/MATH3170/Chap-Some_Base_Results.html)</sup> Thus one-third, which repeats forever in decimal, is simply 0.2 in senary.<sup>[10](https://wiki.rus.family/content/wiktionary_en_all_nopic_2025-07/senary)</sup> Fractions from 1/2 to 1/10 in base 6 can all be represented with three digits or fewer.<sup>[11](https://thoughtthrill.com/every-weird-number-system-explained-part-2/)</sup>

**Prime recognition.** For every prime number p greater than 3, p is congruent to 1 or 5 modulo 6, so the final senary digit of any prime above 3 is a 1 or a 5.<sup>[8](https://en.wiki.beta.math.wmflabs.org/wiki/Senary)</sup> Divisibility tests carry over by analogy with decimal: a senary number is divisible by 7 if the difference of the sums of its alternate digits is divisible by 7, mirroring the decimal test for 11.<sup>[8](https://en.wiki.beta.math.wmflabs.org/wiki/Senary)</sup>

The trade-off is the short reach of each digit position.

## Senary in natural languages

The best-documented senary systems are found in southern New Guinea. Base-6-36 numeral systems, which the linguist Harald Hammarström of Chalmers University of Technology describes as a typological rarity, occur in the Kanum languages; his 2009 analysis in *Linguistic Typology* shows the system must have emerged specifically in the Tonda group, and, since the base-6 forms show no body-part etymologies, he suggests the systems arose for counting yams.<sup>[4](https://research.chalmers.se/en/publication/253103)</sup> Yams are a staple food in the region and are grouped in sextets for counting, an interpretation that both Williams (1936) and Lean (1986) adduced independently.<sup>[12](https://citeseerx.ist.psu.edu/document?doi=f50da75b532fc59419e767f312a60134a313987d&repid=rep1&type=pdf)</sup>

**Komnzo.** In Komnzo, a language of the Tonda subgroup of the Yam family spoken by roughly 150 to 250 people, a special base-six numeral system is used for the yam-counting ritual and is unique to the Yam languages.<sup>[6](https://langsci.github.io/212/main.pdf)</sup> The roots are näbi (1), eda (2), etha (3), asar (4), tabuthui (5), and nibo (6), with 7 expressed as nibo a näbi (six and one).<sup>[7](https://escholarship.org/content/qt3xf465n5/qt3xf465n5.pdf)</sup> The counting procedure involves two men who carry three yams each per unit of six while a designated counter calls out the number word three times; the system extends recursively through named powers of six, with one wärämäkä equal to 7,776 (6⁵) and six wärämäkä making one wi, equal to 46,656 (6⁶).<sup>[6](https://langsci.github.io/212/main.pdf)</sup> The grammar notes that nobody in the village of Rouku remembered the last time the largest unit was actually reached.<sup>[6](https://langsci.github.io/212/main.pdf)</sup>

**Ndom.** Ndom, spoken by approximately 1,200 people on Kolopom (Yos Sudarso) Island off the south coast of Papua province, Indonesia, has a senary, six-cycle counting system.<sup>[13](https://lingweb.eva.mpg.de/channumerals/Ndom.htm)</sup> The language featured as Problem 4 at the Fifth International Olympiad in Theoretical, Mathematical and Applied Linguistics in St. Petersburg in 2007, where contestants deduced from the number words that the base is 6 and that both 36 and 18 receive special names, with 18 used as an intermediate stepping stone instead of multiplying 6 by 4 or 5 directly.<sup>[14](https://ioling.org/booklets/iol-2007-indiv-sol.en.pdf)</sup>

Hammarström notes that whether the Kanum base-6 systems are historically connected with the base-6 systems of the Kolopom languages, near but not adjacent to the west, remains an open question; if a connection exists it would be areal rather than genetic.<sup>[4](https://research.chalmers.se/en/publication/253103)</sup>

## Historical traces and modern uses

The nineteenth-century literature on number origins records scattered possible traces of counting by sixes, such as the Breton word for 18, *triouec'h* (3-6), which the philologist Levi Leon Conant's survey treats as an accidental formation rather than evidence of a true senary scale, while noting that the Mosquito tribe of [Central America](https://www.edgepedia.org/central-america) used a quinary-vigesimal system containing three senary compounds.<sup>[15](https://ajhw.co.uk/books/book415.html)</sup> One hypothesis about the origin of Sumerian sexagesimal holds that it arose from an encounter between base-6 and base-10 cultures, with higher orders of magnitude formed by alternating multiplication by 10 and by 6 (10, 60, 600, 3600); the written Sumerian numerals from 1 to 59 are visibly decimal clusters of tens and ones, which supports this mixed-radix reading.<sup>[16](https://www.cambridge.org/core/journals/journal-of-institutional-economics/article/countless-ways-to-count-the-functional-heterogeneity-of-number-systems/ADCDF578D7A9919E0E582DA773B231B7)</sup>

Senary also appears in laboratory-scale technology. A 2016 study realized ternary and senary digit representations using DNA double-crossover tiles, controlling the length and position of DNA hairpins on the tiles and verifying the base-3 and base-6 digits by atomic force microscopy.<sup>[17](https://arxiv.org/abs/1601.01194)</sup> Outside linguistics and such demonstrations, senary survives mainly as a subject of recreational and advocacy literature: the seximal.net project by jan Misali promotes base six with a full nomenclature, and related projects offer converters, calculators, clocks, and calendars.<sup>[5](https://www.seximal.net/home)</sup><sup> • </sup><sup>[18](https://www.seximal.net/resources)</sup>

## References

1. [4.3 Converting with Base Systems, Contemporary Mathematics, OpenStax](https://openstax.org/books/contemporary-mathematics/pages/4-3-converting-with-base-systems)
2. [Base, Wolfram MathWorld](https://mathworld.wolfram.com/Base.html)
3. [Sezimal Number System – Base Six Number System](https://sezimal.tauga.online/en)
4. [Harald Hammarström (2009). Whence the Kanum base-6 numeral system? Linguistic Typology 13(2), 305–319](https://research.chalmers.se/en/publication/253103)
5. [seximal, jan Misali](https://www.seximal.net/home)
6. [Christian Döhler (2018). A grammar of Komnzo, Language Science Press](https://langsci.github.io/212/main.pdf)
7. [Learning a Novel Number System: The Role of Compositional Rules and Counting Procedures, Cognitive Science (2025)](https://escholarship.org/content/qt3xf465n5/qt3xf465n5.pdf)
8. [Senary, testwiki](https://en.wiki.beta.math.wmflabs.org/wiki/Senary)
9. [Some Base Results, MATH3170, Florian Breuer](https://carmamaths.org/florianbreuer/MATH3170/Chap-Some_Base_Results.html)
10. [senary, Wiktionary mirror](https://wiki.rus.family/content/wiktionary_en_all_nopic_2025-07/senary)
11. [Every Weird Number System Explained Part 2, ThoughtThrill](https://thoughtthrill.com/every-weird-number-system-explained-part-2/)
12. [Whence the Kanum base-6 numeral system? (PDF), CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=f50da75b532fc59419e767f312a60134a313987d&repid=rep1&type=pdf)
13. [Ndom numeral system, MPI EvA Linguistic Database](https://lingweb.eva.mpg.de/channumerals/Ndom.htm)
14. [IOL 2007 Individual Contest Solutions, Ndom problem](https://ioling.org/booklets/iol-2007-indiv-sol.en.pdf)
15. [Levi Leon Conant. The Number Concept: Its Origin and Development](https://ajhw.co.uk/books/book415.html)
16. [Countless ways to count: the functional heterogeneity of number systems, Journal of Institutional Economics](https://www.cambridge.org/core/journals/journal-of-institutional-economics/article/countless-ways-to-count-the-functional-heterogeneity-of-number-systems/ADCDF578D7A9919E0E582DA773B231B7)
17. [Kim et al. (2016). Ternary and senary representations using DNA double-crossover tiles, arXiv](https://arxiv.org/abs/1601.01194)
18. [seximal resources](https://www.seximal.net/resources)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Integers and rational numbers*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026; Oct 25, 2026 · Last review: Oct 15, 2026*

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