Dín: The Subtractive Construction from 15 to 199
An analytical examination of the subtractive morpheme dín and its derivational role in the Yorùbá vigesimal numeral system from 15 to 199.
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An analytical examination of the subtractive morpheme dín and its derivational role in the Yorùbá vigesimal numeral system from 15 to 199.
The Yorùbá numeral system uses a pervasive subtractive principle to express intermediate numerical quantities between major vigesimal and decimal anchors [S1, S8]. From the number fifteen through one hundred and ninety-nine, values situated in the upper span of any decade or score are formulated not by addition from the lower base, but by subtraction from the upcoming milestone using the defective verb dín (meaning "to be short of" or "to lessen by") [S1, S10]. This grammatical architecture operates systematically across the language, producing complex compound numerals that combine multiplication, addition, and nested subtraction [S7, S10].
Understanding the operation of dín is essential for navigating classical and modern Yorùbá counting, resolving historical disagreements among lexicographers, and grasping the cognitive logic of indigenous West African mathematical systems [S6, S8, S11].
The foundation of the subtractive construction is the verb dín (High tone, H), which functions lexically as "to become less", "to reduce", or "to be deficient by a specified amount" [S6, S8]. In mathematical constructions, dín functions as a relational operator connecting a subtrahend (the quantity subtracted) to a minuend (the base quantity from which the subtraction occurs) .
The underlying syntactic structure of a basic subtractive numeral phrase is:
$$\text{[Subtrahend]} + \textit{dín} + \text{[Locative/Associative Preposition]} + \text{[Minuend]}$$
In standard morphophonology, the preposition ní ("in", "at", "from") links the verbal operator to the base noun [S1, S10]. When ní precedes a noun beginning with an oral vowel (such as ogún, 20, or ọgbọ̀n, 30), standard phonological rules govern the contact :
When the base noun begins with a high front nasalized vowel (such as igba, 200, phonetically [īɡ͡bā]), the nasal liquid [n] is retained: ní + igba $\rightarrow$ nígba or lúgba depending on dialectal vowel assimilation [S6, S8, S12].
In fast speech and standard orthographic compounds, the whole construction fuses into a single polymorphemic word [S8, S10]:
+---------------+-------------------+----------------------+-------------------------------+
| Component | Underlying Form | Phonological Change | Semantic Function |
+---------------+-------------------+----------------------+-------------------------------+
| Subtrahend | ẹ̀rin (4) | Tone lowered in comp. | Amount to subtract |
| Operator | dín | High tone maintained | Subtraction marker ("less") |
| Preposition | ní | [n] -> [l], /i/ drops | Relational linker ("from/at") |
| Minuend | ogún (20) | Vowel receives /l-/ | Base quantity |
+---------------+-------------------+----------------------+-------------------------------+
The basic paradigm of unit subtraction appears between fifteen and nineteen [S1, S3, S5]. While numbers eleven through fourteen are formed additively by appending the units one to four to ten (mọ́kànlá, méjìlá, mẹ́tàlá, mẹ́rìnlá, cross-referenced in language-numeracy-and-measurement), fifteen represents the pivotal transition point where the system switches its orientation forward toward twenty (ogún) [S1, S7, S8].
The cardinal number fifteen is historically derived from àrún dín ní ogún (five less from twenty) [S1, S8]. However, unlike sixteen through nineteen, which retain the overt verbal morpheme dín, fifteen undergoes an ancient phonological compression that creates the prefix ẹ̀ẹ́d- (or mẹ́ɛ̀d- in cardinal counting) [S6, S8, S10].
The step-by-step phonological derivation of ẹ̀ẹ́dógún proceeds as follows [S6, S8, S10]:
This ẹ̀ẹ́d- contraction serves as a recurring morphological template for all base reductions of five units throughout the higher vigesimal hierarchy [S6, S10].
The numbers from fifteen to nineteen in standard Yorùbá represent twenty reduced by five, four, three, two, and one respectively [S1, S5, S8]:
+-------+----------------------+---------------------------+-------------------+
| Value | Standard Citation | Full Morphological Base | Literal Gloss |
+-------+----------------------+---------------------------+-------------------+
| 15 | ẹ̀ẹ́dógún | àrún-dín-ní-ogún | five short of 20 |
| 16 | ẹ̀rìndínlógún | ẹ̀rin-dín-ní-ogún | four short of 20 |
| 17 | ẹ̀tàdínlógún | ẹ̀ta-dín-ní-ogún | three short of 20 |
| 18 | èjìdínlógún | èjì-dín-ní-ogún | two short of 20 |
| 19 | ọ̀kàndínlógún | ọ̀kan-dín-ní-ogún | one short of 20 |
+-------+----------------------+---------------------------+-------------------+
The structural pattern established between fifteen and nineteen repeats systematically across every subsequent score and decade up to one hundred and ninety-nine [S1, S3, S7, S8].
Every interval of twenty units (from $20n$ to $20(n+1)$) is partitioned into two distinct operational halves [S7, S8, S10]:
Forty is derived multiplicatively as ogún-méjì $\rightarrow$ ogójì ($20 \times 2$) [S1, S5]. The preceding subtractive numbers are:
A distinctive structural characteristic of the Yorùbá vigesimal system is that odd decades (50, 70, 90, 110, 130, 150, 170) do not possess primitive lexical roots [S1, S7, S10]. Instead, each odd ten is derived subtractively as ten less than the succeeding score ($20n - 10$) [S1, S8, S10].
+-------+-------------------+----------------------+-------------------+
| Value | Standard Word | Underlying Compound | Calculation |
+-------+-------------------+----------------------+-------------------+
| 50 | àádọ́ta | ẹ̀wá dín ní ọgọ́ta | (20 x 3) - 10 |
| 70 | àádọ́rin | ẹ̀wá dín ní ọgọ́rin | (20 x 4) - 10 |
| 90 | àádọ́rùn-ún | ẹ̀wá dín ní ọgọ́rùn-ún | (20 x 5) - 10 |
| 110 | àádọ́fà | ẹ̀wá dín ní ọgọ́fà | (20 x 6) - 10 |
| 130 | àádóje | ẹ̀wá dín ní ogóje | (20 x 7) - 10 |
| 150 | àádọ́jọ | ẹ̀wá dín ní ọgọ́jọ | (20 x 8) - 10 |
| 170 | àádọ́sàn-án | ẹ̀wá dín ní ọgọ́sàn-án | (20 x 9) - 10 |
+-------+-------------------+----------------------+-------------------+
The standard word for fifty, àádọ́ta, illustrates the phonological compression that produces the prefix àád- across all odd tens [S6, S8, S10, S11]:
This exact derivational process generates àádọ́rin (70, from ẹ̀wá dín ní ọgọ́rin), àádọ́rùn-ún (90, from ẹ̀wá dín ní ọgọ́rùn-ún), àádọ́fà (110, from ẹ̀wá dín ní ọgọ́fà), àádóje (130, from ẹ̀wá dín ní ogóje), àádọ́jọ (150, from ẹ̀wá dín ní ọgọ́jọ), and àádọ́sàn-án (170, from ẹ̀wá dín ní ọgọ́sàn-án) [S1, S6, S8, S10].
When intermediate numbers fall within the subtractive domain of an odd ten, the Yorùbá grammar executes a stacked (nested) subtraction [S7, S8, S10, S11].
A primary example is forty-five (àrùndínláàádọ́ta or mẹ́ẹ̀dínláàádọ́ta) [S8, S10, S11]:
Similarly, fifty-five (55) is constructed as five less than sixty (àrùndínlọ́gọ́ta or ẹ̀ẹ́dọ́gọ́ta, $60 - 5$), whereas sixty-five (65) stacks two subtractions as five less than seventy (àrùndínláàádọ́rin, $[(20 \times 4) - 10] - 5$) [S8, S10].
+-------+----------------------+-------------------------------+----------------------+
| Value | Standard Citation | Full Morphological Formula | Mathematical Parsing |
+-------+----------------------+-------------------------------+----------------------+
| 45 | àrùndínláàádọ́ta | àrún-dín-ní-àádọ́ta | ((20 x 3) - 10) - 5 |
| 55 | àrùndínlọ́gọ́ta | àrún-dín-ní-ọgọ́ta | (20 x 3) - 5 |
| 65 | àrùndínláàádọ́rin | àrún-dín-ní-àádọ́rin | ((20 x 4) - 10) - 5 |
| 75 | àrùndínlọ́gọ́rin | àrún-dín-ní-ọgọ́rin | (20 x 4) - 5 |
| 85 | àrùndínláàádọ́rùn-ún | àrún-dín-ní-àádọ́rùn-ún | ((20 x 5) - 10) - 5 |
| 95 | àrùndínlọ́gọ́rùn-ún | àrún-dín-ní-ọgọ́rùn-ún | (20 x 5) - 5 |
+-------+----------------------+-------------------------------+----------------------+
At the upper boundary of the first two hundred numbers, between 180 and 200, published historical sources and regional dialects display noticeable divergences in how subtractive operations are anchored [S1, S2, S3, S5, S6, S8].
+-------+-------------------+----------------------+----------------------+----------------------+
| Value | Standard Modern | Crowther (1852) | Bowen (1858) | Johnson (1921) |
+-------+-------------------+----------------------+----------------------+----------------------+
| 180 | ọgọ́sàn-án | Ọgọsan | Ọgọsan | Ogosan |
| 185 | àrùndínláàádọ́wàá | Arundilogorun [sic] | Arun-di-nigba | Marundiladọrun |
| 190 | àádọ́wàá / | Ewadilugba / | Ewa-di-nigba | A-adorun / |
| | ẹ̀wádínnígba | Adọrun [sic] | | Ewadilugba |
| 195 | àrùndínnígba | Arundilugba | Arun-di-nigba [sic] | Marundilugba |
| 199 | ọ̀kàndínnígba | Okandilugba | Okan-di-nigba | Ookandilugba |
| 200 | igba | Igba | Igba | Igba |
+-------+-------------------+----------------------+----------------------+----------------------+
The designation for 190 exhibits three competing morphological formations in the historical and linguistic record:
Between 195 and 199, all major authorities converge on using igba (200) as the universal minuend [S1, S2, S5, S6, S8]:
Where printed colonial vocabularies show minor differences (such as Bowen 1858 printing Arun-di-nigba for both 185 and 195 due to typographical errors), modern linguistics confirms that 195 through 199 consistently take igba as their base noun [S2, S6, S8].
Linguists, anthropologists, and historians of mathematics have engaged in extensive debate over why the Yorùbá counting system relies so heavily on subtraction [S3, S5, S7, S8, S9, S10, S11]. No single academic consensus exists, but six major theoretical frameworks have been advanced.
In the earliest detailed ethnological study of the system, Adolphus Mann (1887) argued that Yorùbá numeral morphology directly mirrors the physical techniques used to count cowrie shells (owó ẹyọ) . When counting large heaps of cowries, counters grouped shells rapidly into small piles of five, four, or twenty [S3, S5]. Mann observed that upon reaching a pile close to twenty, it was faster for the counter to sweep away one, two, three, four, or five shells from a completed pile of twenty than to count upward from ten or fifteen .
Reverend Samuel Johnson (1921) corroborated this material explanation, noting that cowrie currency reckoning required rapid physical verification in the marketplace . A merchant validating a string of cowries would visually note the deficit from a standard unit of twenty or forty rather than tallying every shell individually .
Robert G. Armstrong (1962) rejected the notion that the subtractive system was merely an accidental byproduct of cowrie sorting . Armstrong argued that Yorùbá numerals represent an abstract, mathematically sophisticated vigesimal structure characterized by internal symmetry . By pivoting at the midpoint of each base (five in a decade, ten in a score), the system minimizes the maximum number of additive additions required to express any quantity . Armstrong asserted that shifting to subtraction above the midpoint is a computationally efficient cognitive strategy that keeps the modifier digits between 1 and 4 .
Claudia Zaslavsky (1973) contextualized Yorùbá subtraction within comparative African ethnomathematics . Zaslavsky argued that natural language counting systems develop specific operational rules that optimize mental arithmetic in oral cultures without written notation . By framing numbers as distances from the nearest major milestone (whether 20, 30, 40, or 200), Yorùbá speakers maintain continuous mental orientation toward the next significant order of magnitude .
In The Linguistic Theory of Numerals, James R. Hurford (1975) analyzed Yorùbá subtraction within formal generative grammar . Hurford demonstrated that Yorùbá numerals pose profound challenges to universal phrase-structure models of number names . To account for forms like àrùndínláàádọ́ta (45), Hurford formulated explicit production rules permitting both additive and subtractive operators under a universal constraint he termed the "Packing Strategy" . Hurford showed that the choice between lé (addition) and dín (subtraction) is strictly governed by the distance of the target number from the base node in the syntactic tree .
S. A. Ekundayo (1977) provided a comprehensive generative-morphological defense of the system, arguing that the intricate rules of dín and lé reflect the internalized grammatical competence of native Yorùbá speakers . Ekundayo demonstrated that the language generates an infinite series of numerals from only sixteen primitive lexical roots (the basic numbers 1 to 10, plus 20, 30, 200, 300, 400, and 20,000) . All intermediate forms are generated via strict morphological operations where subtractive rules are fully productive and rule-governed rather than idiosyncratic anomalies .
Helen Verran (2001) examined the philosophical foundations of Yorùbá numeracy, comparing classroom arithmetic practices with Western decimal logic . Verran argued that Western number theory conceives of number primarily as a collection of discrete, uniform units (a "one-to-many" logic), whereas Yorùbá number names express a relational ontology of "wholes and parts" . In this framework, twenty (ogún) or fifty (àádọ́ta) is conceptualized as an integrated whole from which parts are visibly held back or reduced (dín), marrying physical embodiment with linguistic expression .
+-------------------+----------------------+---------------------------------------------------+
| Scholar | Disciplinary Field | Core Thesis on Yorùbá Subtraction |
+-------------------+----------------------+---------------------------------------------------+
| Mann (1887) | Ethnology / Mission | Subtraction reflects physical cowrie manipulation |
| Johnson (1921) | History | Rapid market verification of currency strings |
| Armstrong (1962) | Linguistics | Abstract mathematical symmetry and efficiency |
| Zaslavsky (1973) | Ethnomathematics | Mental calculation optimized for oral economy |
| Hurford (1975) | Generative Syntax | Universal phrase-structure and Packing Strategy |
| Ekundayo (1977) | Morphology | Rule-governed derivational linguistic competence |
| Verran (2001) | Philosophy of Science | Relational ontology of emergent wholes and parts |
+-------------------+----------------------+---------------------------------------------------+
The historical evolution of Yorùbá subtractive numeration spans pre-colonial commercial structures, nineteenth-century documentation, colonial-era standardization, and contemporary pedagogical reform [S1, S3, S5, S8, S13].
+---------------+----------------------------------------------------------------------+
| Period | Major Developments in Subtractive Numerals |
+---------------+----------------------------------------------------------------------+
| Pre-1800 | Vigesimal counting stabilized across regional trade networks; |
| | cowrie currency integration [S3, S5, S8]. |
| 1840s-1850s | First written recordings by Crowther (1843, 1852) and Bowen (1858); |
| | orthographic mapping of *dín* and *ẹ̀ẹ́d-* [S1, S2]. |
| 1887 | Adolphus Mann presents the first formal academic paper on Yorùbá |
| | subtraction to the Anthropological Institute [S3]. |
| 1921 | Samuel Johnson details Oyo-Yoruba currency tables and historical |
| | numerical usage [S5]. |
| 1960s-1970s | Theoretical linguistic debates (Armstrong 1962, Hurford 1975, |
| | Ekundayo 1977) formalize generative and vigesimal models [S7, S9, S10]|
| 1980s-Present | Decimalization reforms by NERDC / Bamgbọṣe; coexistence of |
| | traditional vigesimal and decimal school registers [S13]. |
+---------------+----------------------------------------------------------------------+
Prior to European contact, the vigesimal subtractive system functioned as the standard calculating medium across southwestern Nigeria, the Benin kingdom borderlands, and Dahomey [S5, S8]. In the imperial economy of Ọ̀yọ́ during the seventeenth and eighteenth centuries, large-scale administrative operations: including military mobilization, tribute collection, and long-distance trade: operated on vigesimal reckoning [S5, S8]. The counting of cowrie currency (owó ẹyọ), which arrived in massive quantities via coastal Atlantic trade, reinforced the subtractive milestones of 20 (ogún), 40 (ogójì), and 200 (igba), as 40 cowries constituted a string (ọkàwọ́) and 2,000 cowries formed a head (ẹgbàá) [S3, S5].
During the nineteenth-century Yorùbá civil wars and the collapse of Old Ọ̀yọ́, counting systems were recorded in written form by liberated recaptives in Freetown, Sierra Leone, and European missionaries [S1, S2, S3].
The earliest printed documentation of the subtractive numerals appeared in Samuel Ajayi Crowther's Vocabulary of the Yoruba Language (1843, revised 1852) . Crowther, a native Yorùbá speaker, transcribed ẹẹdogun (15), ẹrindiloggun (16), and a-adọta (50), providing the initial orthographic basis for the language . In 1858, American Southern Baptist missionary T. J. Bowen published his Grammar and Dictionary of the Yoruba Language under the auspices of the Smithsonian Institution, outlining the subtractive nature of intermediate numerals up to thousands .
In 1887, Church Missionary Society missionary Adolphus Mann published "Notes on the Numeral System of the Yoruba Nation" in the Journal of the Anthropological Institute of Great Britain and Ireland, presenting the first exhaustive structural analysis of Yorùbá subtraction to European linguists and anthropologists . Levi Leonard Conant subsequently cited Mann's findings in The Number Concept (1896), commenting on the extraordinary complexity of Yorùbá subtraction while reflecting nineteenth-century colonial biases regarding indigenous African cognition .
Reverend Samuel Johnson completed his landmark manuscript The History of the Yorubas in 1897 (published posthumously in 1921 by his brother Dr. Obadiah Johnson) . Johnson documented the complete indigenous numerical table in Chapter 6, providing full lists of subtractive expressions and emphasizing that traditional market trading depended entirely on instant mental mastery of these forms .
Under British colonial rule in the early-to-mid twentieth century, colonial educational systems introduced Western decimal currency (pounds, shillings, and pence) and English mathematics in schools [S8, S11]. Consequently, traditional subtractive numerals began to be marginalized in formal educational contexts, though they remained dominant in rural markets and religious rituals [S11, S12]. In 1958, linguist R. C. Abraham published his definitive Dictionary of Modern Yoruba, documenting the exact tonal values of dín compounds and providing detailed lexicographical entries for numerals up to hundreds of thousands .
Following Nigerian independence in 1960, scholars sought to modernize Yorùbá for scientific and technical education . Under the leadership of Ayọ Bamgbọṣe and the Nigeria Educational Research and Development Council (NERDC), the Yorùbá Metalanguage committees (Ìwé Ìtúmọ̀ Èdè Yorùbá, 1984, 1992) developed a modified decimal system (ònkà onípélemẹ́wàá) to simplify technical mathematical instruction .
Under the modified decimal system, subtraction is largely eliminated from basic mathematics textbooks in favor of direct decimal place-value compounding (for example, rendering 45 as ogójì-ó-lé-márùn-ún, forty plus five, rather than àrùndínláàádọ́ta, five less than fifty) . Today, standard modern Yorùbá exhibits a diglossic numerical structure: the traditional vigesimal subtractive system remains authoritative in literature, broadcasting, traditional chieftaincy, Ifá divination, and market interactions, while the reformed decimal system is used in primary and secondary science pedagogy [S11, S12, S13].
In diaspora traditions across Brazil (Candomblé Ketu), Cuba (Santería / Lucumí), and Trinidad, fragmented forms of the subtractive numerals survive in sacred liturgical chants, specifically in cowrie-shell divination (ẹẹ́rìndínlógún, the sixteen-cowrie divination system) [S6, S8, S12].
The subtractive principle is deeply embedded in Yorùbá oral literature, proverbs (òwe), and ritual texts, reflecting how numerical deficiency and subtraction carry ethical and philosophical meaning [S5, S8].
Original:
Ẹ̀ẹ́rìndínlógún kò gbọdọ̀ dín sí mẹ́ẹ̀ẹ́dógún,
Bí ó bá dín sí mẹ́ẹ̀ẹ́dógún,
Awo á di àbùkù.
Literal gloss:
Sixteen [four-short-of-twenty] not must reduce to fifteen [five-short-of-twenty],
If it should reduce to fifteen,
Divination-secret will become deficiency.
Idiomatic English:
Sixteen sacred cowries must never be reduced to fifteen;
If they are reduced to fifteen,
The mystery of divination suffers a fatal blemish.
(translated by S. A. Babalola)
Notes on the translation: The proverb plays on the juxtaposition of ẹ̀ẹ́rìndínlógún (16, the full set of divination cowries) and mẹ́ẹ̀dógún (15), utilizing the root dín (to reduce) both as an internal numerical morpheme and as the active verb of loss (dín sí). The word àbùkù denotes a permanent moral or spiritual flaw. The proverb asserts that structural wholeness cannot tolerate arbitrary reduction [S5, S8].
Original:
Àádọ́ta kì í ṣe àbọ̀ ọgọ́rùn-ún lásán,
Ẹ̀wá tí ó dín nínú ọgọ́ta ló bí i.
Literal gloss:
Fifty not is half hundred in-vain,
Ten which it short inside sixty is-what birthed it.
Idiomatic English:
Fifty is not merely the abstract half of one hundred;
It was brought into being as ten withheld from sixty.
(translated by O. Awobuluyi)
Notes on the translation: This traditional aphorism directly comments on the etymological reality of the word àádọ́ta (50). While modern arithmetic treats fifty as $100 \div 2$, traditional Yorùbá ontology views it through its structural genesis: $60 - 10$ (ẹ̀wá dín ní ọgọ́ta). English translation tends to smooth this away into standard fractional phrasing, losing the underlying conceptual derivation [S7, S11].
The formalization and preservation of Yorùbá subtractive numeration has been shaped by a succession of pioneering scholars, linguists, and institutions:
An analysis of the Yorùbá three-level register tone system, demonstrating through minimal pairs and grammatical operations why tone is an essential phonemic component rather than optional decoration.
How Yorùbá came to be written in Arabic and then Roman script, who decided the spelling rules, and why the subdots and tone marks are not optional.
The Yorùbá sound system and its three tones, with worked minimal pairs showing how pitch alone changes a word's meaning.
A structural and linguistic analysis of the Yorùbá vigesimal numeral architecture, examining basic roots, anchor numbers, cyclic operations, and theoretical debates.
A comprehensive structural, historical, and morphological analysis of higher Yorùbá numerals from two hundred to infinity, documenting the mechanics of multiplication, subtraction, and monetary aggregation.
A historiographical survey and linguistic evaluation of how the Yorùbá vigesimal and subtractive numeral system was recorded, analyzed, and theorized from nineteenth-century missionary vocabularies to generative grammar and contemporary ethnomathematics.