We review the empirical research on numerical bases in the context of spoken and written numerical expressions, visual number symbols and the mappings between these expressions and symbols. Most of this research has been carried out with speakers of Indo-European languages, which have decimal systems. Children typically acquire spoken numerical expressions before they learn associated visual number symbols, but numerical expressions are language-specific. The structure of the numerical expressions influences number word learning, with evidence that an explicit base-10 structure in the spoken numerical expressions (i.e. <i>base transparency</i>) is advantageous. Later, when visual base-10 number symbols are acquired, inconsistencies in the mapping between those and the numerical expressions affect numerical and mathematical learning. We also review research on base-20 numeration systems and on inversion in number words: when the order of constituents in the spoken number word is inversed in relation to their order in the visual number symbols, e.g. 'fourteen' versus '14'. In conclusion, we discuss evidence about how using number symbols with a non-decimal base affects the processing of numbers and make suggestions for future research.This article is part of the theme issue 'A solid base for scaling up: the structure of numeration systems'.

This dissertation investigates whether and how the morphological transparency of number words in a child's language shapes the acquisition of number knowledge. Number words vary considerably across the world’s languages in how transparently they reflect the underlying Base-10 structure of the number system. In some languages, such as Mandarin and Tamil, number words are highly transparent - i.e., the word for ‘eleven’ literally translates as ‘ten-one,’ making the compositional logic of the counting system explicit. In others, such as English, Turkish, and Hindi, number words are partially or entirely opaque, requiring children to rely more heavily on rote memorization. This dissertation presents three empirical studies that examine how this cross-linguistic variation affects three key aspects of early numerical development: counting fluency, knowledge of the cardinal principle, and understanding of the Successor Function and how the count list implements it. Study 1 compared 363 children aged 3 to 6 acquiring Mandarin, English, and Turkish - three languages that differ systematically in the transparency of their number words. Study 2 compared 230 children aged 4 to 7 acquiring Hindi and Tamil - two languages spoken in India that differ dramatically in number word transparency while sharing comparable cultural, socioeconomic, and educational contexts, providing a uniquely controlled test of the causal role of number word transparency on number learning. In both studies, childr

'gate'). The traditional counting system used in the Welsh language is vigesimal, i.e. it is based on twenties. Welsh numbers from 11 to 14 are "x on ten" (e Welsh (Cymraeg [kəmˈraːiɡ] or y Gymraeg [ə ɡəmˈraːiɡ]) is a Celtic language of the Brittonic subgroup that is native to the Welsh people. Welsh is spoken natively in Wales by about 18% of the population, by some in England, and in Y Wladfa (the Welsh colony in Chubut Province, Argentina). Historically, it has also been known in English as British, Cambrian, Cambric and Cymric. The Welsh Language The traditional counting system used in the Welsh language is vigesimal, i.e. it is based on twenties. Welsh numbers from 11 to 14 are "x on ten" (e.g. un ar ddeg: 11), 16 to 19 are "x on fifteen" (e.g. un ar bymtheg: 16), though 18 is deunaw, "two nines"; numbers from 21 to 39 are "1–19 on twenty"(e.g. deg ar hugain: 30), 40 is deugain "two twenties", 60 is trigain "three twenties", etc. This form continues to be used, especially by older people, and it is obligatory in certain circumstances (such as telling the time, and in ordinal numbers). There is also a decimal counting system, which has become relatively widely used, though less so in giving the time, ages, and dates (it features no ordinal numbers). This system originated in Patagonian Welsh and was subsequently introduced to Wales in the 1940s. Whereas 39 in the vigesimal system is pedwar ar bymtheg ar hugain ('four on fifteen on twenty') or even deugain namyn un ('two twenty minus one'), in the decimal system it is tri deg naw ('three tens nine'). Although there is only one word for "one" (un), it triggers the soft mutation (treiglad meddal) of feminine nouns, where possible, other than those beginning with "ll" or "rh". There are separate masculine and feminine forms of the numbers 'two' (dau and dwy), 'three' (tri and tair) and 'four' (pedwar and pedair), which must agree with the grammatical gender of the objects being counted. The objects being counted appear in the singular, not plural form.