## Number Names

George Lakoff has pointed out that we do not normally distinguish numbers from what might be more properly be called number names. (Lakoff 1989) The most common number naming systems adopt base-10 and use ten single-digit number names, for instance (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). They then form other multi-digit number names (21, 1342 etc) from these basic number names, or numerals, using a positional representation. With Arabic numerals this starts with the rightmost numeral being the quantity of units and the next leftward numeral being the quantity of tens and so on.

But many other bases are possible, the next most common probably being base-2 or binary, with just two basic number names (0 and 1). The difference between numbers and number names can therefore perhaps best be understood by realising that the number name ‘3’ in base-10 (3 × 10^{0}) represents the same quantity or numerosity as the number name ’11’ in base-2 ((1 × 2^{1 }) + (1 × 2^{0})). That is the base used, as well as the digits or glyphs adopted, can change the way any particular quantity, numerosity or number is represented.

Cardinal numbers measure the **size** of collections or sets and therefore include the number zero needed to represent the size of an empty collection or set. In English cardinal numbers are nouns.

Ordinal numbers represent **position** or rank in a sequential, spatial or temporal lists or order and therefore do not include zero, there is no zeroth element in a sequential list. In English ordinal numbers are adjectives.

## Number Words

Number names in this sense are different and distinct from number words, the verbal version of numbers, the way numbers are spoken or transliterated, (one, two, three etc.). see Five Finger Exercises

In English, verbal numbers are organised as a hybrid series of additions and multiplications summarised, for the cardinal Arabic number 350172, by the graph below where the plus signs indicate addition and the X signs multiplication.

After Dehaene (1992) *Varieties of Numerical Abilities *Cognition, 44 1-42

So that ((((three is multiplied by a hundred) and added to fifty) which is then multiplied by a thousand) and added to ((one multiplied by a hundred) added to (seventy added to two)))

This system involves a combination of simple number words; one, two, three etc., some special multiplier words like hundred, thousand etc and the particularly English **-ty** words like six**ty**, seven**ty**, eigh**ty **and nine**ty** plus the slightly modified twen**ty**, thir**ty**, for**ty** and fif**ty**. And **-teen** words like thir**teen**, four**teen** etc. plus the unique eleven and twelve.

With Arabic numerals the same cardinal number (350172) is represented positionally; starting with the rightmost numeral being the quantity of units (2) and the next leftward numeral being the quantity of tens (7) etc. Note that Arabic numbers are read, or more accurately generated, from right to left, perhaps betraying their origin.

Chinese number words follow a similar but somewhat simpler, more regular pattern.

## Comparison of English and Chinese Number Words

Some of the extra complexity of English number words derive from spelling conventions rather than word sound, for instance **eigh**[]**teen**, **fo**[]**rty** and **eigh**[]**ty**. There is also some evidence of pronunciation slippage. Thus **twelve** and **twenty** to avoid the awkwardness of **twoteen** and **twoty**, **thirteen** and **thirty** to avoid **threeteen** and **threety** and **fifteen** and **fifty** to avoid **fiveteen** and **fivety**.

## Ordinal Number Words

In English the initial verbal ordinal words are the unique **first**, **second**, and **third**, but typically ordinals have a **th** suffix added to the cardinal name for the number, so four**th**, six**th**, seven**th**, nine**th**, and ten**th** plus the slightly modified in spelling terms fif[]**th**, and eigh[]**th**. The multiple powers of ten have an **ieth** suffix replacing the **y** ending of the cardinal name, so twent**ieth **from twent**y**, thirt**ieth** from thirt**y**, fort**ieth** from fort**y** etc. Again these are organised as a hybrid series of additions and multiplications.

In English the Arabic version of ordinals borrow their suffices from the end of their verbal equivalents, so we have 1**st **(from fir**st**),** **2**nd** (from seco**nd**) and 3**rd** (from thi**rd**), followed by 4**th** .. 20**th** then 21**st**, 22**nd** and 23**rd **etc.

The first three English ordinals have interestingly varied etymologies. *First* derives from the Old English **fyr(e)st **and Old Norse** ****fyrst**, having the sense of *furthest* forward, and the German **Fürst**, a prince, that is *furthest* forward in rank. *Second* derives from the Latin **sequi*** follow*, **secundus** *following* and **second** via Old French into Middle English. *Third* derives from Old English **thridda** via English **thrid** which was the most common spelling until the 16th century.

A Latinate ordinal system is also used to represent importance and precedence, *primary, secondary, tertiary, quaternary* etc. which are rarely used beyond the first four. So **primary**, **secondary** and **tertiary** education. This system is also used to indicate a sequence of dependent effects, thus **secondary** picketing.

In technical and academic practice Greek ordinals are also used as prefixes *proto-, deutero-, trite- and tetarto-, *thus **proto**-renaissance, **prot**agonist and **deuter**ium.

## Fractions

When speaking of fractions a **half** is used for 1/2, a **quarter** for 1/4 and **three quarters **for 3/4 but a **fourth **is also used in music. Otherwise ordinals are used as in a **third **for 1/3, a **fifth**, a **sixth** etc. In the more general case a cardinal number is used for the numerator and an ordinal for the denominator, so 2/3 is **two thirds** and 19/32 is **nineteen thirty seconds** etc.

## Transcoding

Literate English speakers have no problem reading, writing, comprehending or producing all these systems and transcoding between them even though there is evidence, through the study of patients with deficits in one or more of these capacities, of a neurological dissociation between the verbal and written systems (McConey 1992)

## Bibliography

Dehaene, S. (1992) *Varieties of Numerical Abilities *Cognition, 44 1-42

Lakoff, G. (1987) *Women, Fire, and Dangerous Things* University of Chicago, Chicago Page 150

Lakoff, G. and Núñez, R. (2000) *Where Mathematics Comes From* Basic Books, New York

McCloskey, M. (1992) *Cognitive mechanisms in numerical processing: Evidence from acquired dyscalculia,* Cognition 44 107-157