Chapter 54The Vigesimal Numeral System

54.1 Overview

Hokkaido Ainu numerals fall into three grammatical categories — numeral prenominals (数連体詞), numeral nouns (数名詞), and a counting-off form (数え上げ) — and the three are not interchangeable in syntax Nakagawa (2024: 461). Underlying all three is a shared set of numeral roots from 1 to 10, with the integer 20 (hotne) serving as the base of a vigesimal counting system: even decades are formed multiplicatively from hotne, while the odd tens (30, 50, 70, 90) are built subtractively, expressing how many units fall short of the next even decade Nakagawa (2024: 462–463); Refsing (1986: 110–111).

The vigesimal base sets Saru–Chitose Hokkaido Ainu apart from the surrounding Japanese decimal system and from several of its own dialects — Asahikawa and Sōya use a decimal base with the same root hot at the value of ten rather than twenty Nakagawa (2024: 467). Sakhalin Ainu is also decimal. A shift in modern usage away from the subtractive architecture and toward simpler additive or decimal expressions is documented by Ochiai (2021).

This chapter covers the numeral roots and their three form classes, the multiplicative and subtractive decade structure, the syntax of numeral phrases, the debate over classifier status, and the dialect geography of the base. Ordinals, the interaction of numerals with quantifier syntax, and the day- and time-counting constructions are treated in the sibling chapter Chapter 55 (Numeral Classifiers, Ordinals, and Quantifier Syntax); numeral phrases and the noun suffix -utar are set out in Chapter 41 (Nominal Number: Transnumerality, -utar, and the Locus of Plurality).

54.2 The numeral roots 1–10 and the three form classes

Each numeral value from 1 to 10, and 20, is represented in three paradigmatic shapes. The prenominal precedes the noun directly; the numeral noun stands after the noun or alone; and the counting-off form is used for reckoning aloud in sequence Nakagawa (2024: 461, 464–466); Refsing (1986: 110).

The suffix -p (after vowel-final stems 1–5) and -pe (after consonant-final stems 6–10) are instances of the formal noun p ‘thing’ treated in Chapter 44 (Concept Form vs. Affiliative (Possessed) Form). The human counterparts -n and its suppletive partner -iw (6–10) are the same -n that appears in the interrogative nen 'who' and in hempak-iw 'how many people' — see Chapter 51 (Interrogative, Indefinite, and Negative-Polarity Pro-forms) Nakagawa (2024: 464). The choice between -p and -n for human referents is weak: -p is described as "more brusque" but neither is ungrammatical with persons Nakagawa (2024: 464). The community observation that the numeral-noun form rep cannot be placed before its noun — that 'three worlds' must be re mosir, not *rep mosir — is consistent with Nakagawa's syntactic description aynu-corpora Discord (2023–2026) (nukopoli, aynu-corpora Discord, 2024-11-09); Nakagawa (2024: 461).

A secondary '1' root ar- appears before certain nouns: ar-suy (also assuy) 'once', ar-itak 'one word' Nakagawa (2024: 461). The same element appears in arwan (7) and in the iteratives arsuy, tusuy, resuy 'once, twice, three times' Refsing (1986: 108).

The etymology of roots 6–9 is historical, not synchronic. Working from Chiri's finger-counting data and Patrie's comparative materials, Refsing analyses these as old "ten-minus" compounds: iwan (6) < i(ne)+wan "4+ten"; arwan (7) < ar+wan (where ar is glossed as an old '3' or old '1' — contested); tupesan (8) and sinepesan (9) as tu/sine+pesan, or in Patrie's reconstruction tu-upis/sine-upis, with upis an old '10' attested in Kuril Ainu Refsing (1986: 111). These reconstructions are ‹speculative›; synchronically the forms are opaque roots. The root hot (20) is analysed as 'a set' — a full complement of fingers and toes, the somatic basis commonly attributed to vigesimal systems Refsing (1986: 111).

54.3 The vigesimal counting architecture

The counting system above 10 operates in three zones: an additive zone for 11–29, a multiplicative zone for even decades, and a subtractive zone for odd decades Nakagawa (2024: 462–463); Refsing (1986: 111).

11–19 are formed by adding a unit prenominal to the verb ikasma ‘exceed, surpass’ followed by wan '10': 11 = sine ikasma wan, 13 = re ikasma wan, 19 = sinepesan ikasma wan. The construction frames each value as the amount by which the count surpasses the last decade — 'three exceeds ten' Nakagawa (2024: 462).

20 = hotne, the vigesimal base. 21–29 replace wan with hotne: sine ikasma hotne (21) through sinepesan ikasma hotne (29).

Even decades from 40 to 100 are multiplicative: the unit prenominal compounds directly with hotne, yielding tuhotne (2×20=40), rehotne (3×20=60), inehotne (4×20=80), and asiknehotne (5×20=100) Nakagawa (2024: 463). The 100-value asiknehotnep is independently confirmed ‹corpus-confirmed› aynu-corpora Discord (2023–2026) (palestine_2026, aynu-corpora Discord, 2026-06-08).

Odd decades are subtractive. The pattern is wan + the prefix e- + the next even decade: wan etuhotne = "ten toward forty" = 30; wan erehotne = 50; wan einehotne = 70; wan easiknehotne = 90. The prefix e- marks the following even decade as the reference point from which ten units are counted off; this is the defining property that the literature labels vigesimal subtractive counting (二十進法; 下方算法) Nakagawa (2024: 463); Ochiai (2021).

The community discussion of the subtractive pattern (aomidori_cs_, aynu-corpora Discord, 2024-05-28) independently corroborates Nakagawa's 30 and 50 formulas; the same contributor extends the pattern analogically to propose expressions for negative quantities (*sine e-isam for −1, combining e- with isam 'be absent'), but that extension is a modern coinage ‹speculative› and lies outside the attested system aynu-corpora Discord (2023–2026).

In the counting-off form, the bare root hot replaces hotne: 20 = hot, 30 = wanpe etuhot, 40 = tuhot Nakagawa (2024: 466).

For composite numeral-noun phrases above 29, each component stands as an independent noun form: "31 things" is sinep ikasma wanpe etuhotnep; "11 loads" is sinep ikasma wan sike Nakagawa (2024: 466). Nakagawa observes that composites above 10 "almost never appear in recorded classical texts"; the attested vigesimal compounds in the broader corpus are concentrated in modern アイヌタイムズ prose from the 1990s–2000s Nakagawa (2024: 468).

54.4 Numeral syntax

Prenominals precede the head noun directly, without an intervening classifier or particle; numeral nouns follow the head noun in apposition or stand alone; and the counting-off forms are used only for enumerating in sequence, without an NP head Nakagawa (2024: 461–465); Refsing (1986: 115–117).

The error of placing the numeral-noun form before the noun — *sinep tuki for 'one cup' instead of sine tuki or tuki sinep — is explicitly flagged as a common learner mistake Nakagawa (2024: 465). The [noun + postposed numeral-noun + verb] construction, as in pasuy sinep uk 'take one chopstick', is common and has been noted as productive — arguably more so than its Japanese near-counterpart aynu-corpora Discord (2023–2026) (nukopoli, aynu-corpora Discord, 2024-12-12); Refsing (1986: 116) ‹corpus-suggested›. Whether a modifier can intervene between a head noun and its following numeral-noun is undocumented: Satō notes that no such example appears in his data and flags the question as open Satō (2008: 256) ‹contested›.

54.5 Special number uses and lexicalized expressions

Several numeral roots carry extended or bleached meanings in fixed constructions beyond their cardinal count function.

iwan (6) functions as an intensifier glossable as 'many' in oral literary registers and fixed expressions. The expression iwan suy means 'many times', and the form noiwan is an intensified variant Nakagawa (2024: 469).

The pair tu / re in oral-literary formulaic contexts carries the sense 'many' rather than exactly 'two' or 'three': tu cup ka re cup 'many months', tu ray onkami / re ray onkami 'bow many times' Nakagawa (2024: 468). By contrast, tutko rerko '2–3 days' genuinely denotes a short span Nakagawa (2024: 468).

Day-counting uses the combining form -ko: tutko '2 days', rerko '3 days', then ine rerko '4 days', asikne rerko '5 days'. In the last two expressions rerko has bleached from 'three-day-span' to a general 'days' noun Nakagawa (2024: 465).

The animal count noun atuyta is lexically restricted to Saru/Chitose and denotes 10 head of animals in Saru but 20 head in Chitose — a lexical split internal to the same dialect cluster Nakagawa (2024: 467).

54.6 The counter and classifier question

Whether -p/-pe and -n/-iw constitute numeral classifiers depends on the criterion applied. Nakagawa's answer is unambiguous: Ainu has no numeral classifiers (助数詞) of the Sinitic type; quantification uses the prenominal plus noun directly, analogous to English counting, and the thing/person suffix pair is instead the native derivational contrast parallel to Japanese ひとつ/ひとり, a trait shared with the neighbouring Nivkh Nakagawa (2024: 461–462, 464). Refsing explicitly calls -p(e) and -n/-iw "a rudimentary system of 'classifiers' or 'counters'" and reserves 'numeral' for the bare stem Refsing (1986: 110–111). The Handbook goes further, applying the label "word-classifiers" to to 'X days' and suy 'X times', and speculates about remnants of an earlier, more elaborate classifier system — though the passage is labelled as speculation in the source Bugaeva (2022).

The disagreement reduces to a terminological question: Nakagawa's criterion requires Sinitic-style morphological obligatoriness and categorial specificity (absent in Ainu); Refsing's and the Handbook's criterion requires only semantic categorization in regular numeral co-occurrence (present in Ainu) Refsing (1986: 110); Nakagawa (2024: 461) ‹contested›. The fuller treatment of the classifier debate, the status of to and suy, and the day-counting constructions are reserved for Chapter 55 (Numeral Classifiers, Ordinals, and Quantifier Syntax).

54.7 Dialect variation and the vigesimal-to-decimal shift

The vigesimal base is the defining property of the Saru–Chitose standard but is not pan-Hokkaido. Two dialect areas use a decimal base, and in those the root hot takes the value ten rather than twenty.

Asahikawa is decimal above ten: sinehot = 10, tuhot = 20, rehot = 30, and so on. The Sōya (宗谷) dialect similarly has wanhot = 100 (decimal 'ten-ten') Nakagawa (2024: 467–468). Community records confirm that the collision between vigesimal and decimal speakers produced misunderstandings in commercial exchange ‹corpus-confirmed› aynu-corpora Discord (2023–2026) (nukopoli, aynu-corpora Discord, 2024-12-25; Deleted User, 2024-05-13).

The form for 100 varies markedly across Hokkaido. Saru, Chitose, Horobetsu, and Nayoro use asiknehotne(p) (5×20); Yakumo and Obihiro use sineik; Bihoro uses ik; Sōya uses wanhot Nakagawa (2024: 468).

Sakhalin Ainu is decimal throughout Sakaguchi (2022). The root for 10 is kunkutu, plausibly an Itelmen loan ‹speculative› aynu-corpora Discord (2023–2026) (antitwilight, aynu-corpora Discord, 2024-11-22), and 100 is sinetanku, from Tungusic tanku Nakagawa (2024: 468). The Tungusic and Itelmen contact reflected in these forms has no counterpart in the Hokkaido system.

A directional shift in modern usage, from the subtractive architecture toward additive or decimal expressions, is documented by Ochiai Ochiai (2021). The subtractive structure is complex to produce and rare in classical oral texts Nakagawa (2024: 468), and the pedagogical grammar of Ijäs (2023) teaches both the vigesimal and a simplified decimal alternative, reflecting classroom practice. The direction of change belongs to the sociolinguistics of the revitalization context; the attested classical system is as described above.