Numerals

Ŋarâþ Crîþ has two sets of cardinal numerals: the long numerals and the short numerals.

Long numerals

The long numerals up to 6 are inflected for case and gender (Tables 2 to 7).

The rest of the long numerals (Tables 8 and 9) are inflected for case only.

The lemma form of a long numeral is its attributive form. Because long numerals inflect for case and possibly gender, they can be separated from their heads.

The following types of numerals can be derived from long numerals:

• To derive the pronominal form, prefix the particle ⟨a⟩ (as a separate word): ⟨prêno⟩ three; ⟨a prêno⟩ three of them.
• To refer to a number itself rather than the quantity it represents, prepend ⟦âħ-⟧ before a vowel or ⟦â-⟧ before a consonant to the celestial form, leniting the numeral: ⟨meþos⟩ sixteen; ⟨âm·eþos⟩ the number ‘16’
• To create an adverb denoting the number of times something occurs, append ⟦-el⟧ to the dative celestial form. Change any final ⟦-s⟧ to ⟦-r⟧, unless there is any ⟦r⟧ earlier in the numeral: ⟨nefas⟩ two (dative); ⟨nefarel⟩ twice. Note that ⟨plamiþa⟩ is changed to ⟨plamiþel⟩.

Ordinal long numerals

Ordinal numerals start from zero, such that number 0 refers to the first object, 1 refers to the second, and so on. Ordinal long numerals occur before the noun phrase they modify but need not be adjacent to them.

Short numerals

The short numerals up to 16 are listed in Table 14.

Short numerals up to $16^2=256$ of the form $16x+y$ are roughly formed as ⟦$x$-sraþ-$y$⟧. $x$ is omitted if it equals 1, and $y$ is omitted if it equals 0. ⟦-sraþ-⟧ fuses with certain values of $y$, yielding the forms in the last column of Table 14.

Short numerals up to $16^4=65536$ of the form $256x+y$ are formed as ⟦$x$-flen-$y$⟧, where $x$ is omitted if equal to 1 and $y$ is omitted if zero.

Numerals beyond 65536 are formed by splitting the digits into groups of four from the least significant digit and using the words for powers of 65536 in Table 15. A coefficient of one on the highest power of 65536 in a short numeral is omitted. Any power of 65536 with a coefficient of zero is omitted.

Like long numerals, short numerals are used primarily as determiners. Unlike long numerals, they require classifiers for most nouns.

The following nouns do not have classifiers, but they must use a long numeral if available:

• Units of measurement
• The nouns ⟨sar⟩ thing or ⟨ðên⟩ occurrence

The numeral ⟨ces⟩, as well as any numeral that is or ends in ⟨ħas⟩ or ⟨sreþas⟩, triggers a lenition in the classifier.

Adding the particle ⟨ceþe⟩ before a short numeral negates it, eclipsing it. Since there are no negative long numerals, a negative short numeral can always be used for unclassified nouns.

To refer to the quantity described by a short numeral itself, a short numeral can be compounded with ⟨mener⟩ as the head. The nominative form of a nominal short numeral may be abbreviated using figures followed by a ⟨/⟩, especially in data.

Short numerals other than the ones for 0, 1, 2, and 3 can be used as ordinal numerals by placing them immediately before the noun phrase being modified with no classifier. A negative short numeral refers to an object from the last one: −1 refers to the last object, −2 to the second-last object, and so on. For conventionalized ordinal numerals, the short numerals are used even for 0, 1, 2, and 3:

Non-integral numerals

Rational numerals

Some commonly-used fractions have long forms: ½ for each gender (Table 18), and ⅓, ⅔, and ⅕ without distinctions in gender (Table 19).

Rational numbers with a numerator of 1 are derived from the short numeral of the denominator plus a suffix that depends on case (Table 20). When the short numeral is monosyllabic, an extra syllable ⟦-râ-⟧ is infixed between the numeral and the suffix. Because of the presence of case agreement, these numerals are also considered long numerals.

Vulgar fractions (i.e. of the form $n⁄d$) are formed using the unit fraction for $1⁄d$ immediately followed by the numeral for $n$. Since $n$ does not receive a classifier, it must be long if there is a long form available for $n$, in which case its case and gender matches that of $1⁄d$.

A mixed number $m+n⁄d$ is expressed by using the numeral for $m$ (the long form if one is available, otherwise the short form), followed by the conjunction ⟨i⟩ plus the numeral for $n⁄d$.

If a mixed number is used pronominally, then only the numeral for $m$ assumes its pronominal form:

Likewise, a mixed number used nominally receives changes only to the numeral for $m$:

Inexact numerals

Inexact numerals are represented in scientific notation with base 16. The short-form digits of the significand are listed (with an implied decimal point after the first digit), followed by a suffix representing the exponent.

The exponent suffixes are listed in Table 21. Odd powers over $16^6$ are derived from the power below, prepended with ⟦-lî-⟧. For instance, $16^{11}$ has the suffix ⟦-lîflerico⟧, from the suffix for $16^{10}$.

Powers below $16^{-4}$ are derived from their corresponding reciprocals by prefixing ⟦-maga-⟧. For instance, $16^{-12}$ has the suffix ⟦-magaacþeno⟧, from the suffix for $16^{12}$.

Note that adding or removing trailing zeroes in the significand changes the precision of the numeral. A leading zero in the significand is not allowed unless it is the only digit.

Examples:

• ⟨ensentarenłarmino⟩ $3.243F_{16}\times 16^0$
• ⟨senvilcescenroto⟩ $2.10_{16}\times 16^2$
• ⟨senvilhenroto⟩ $2.1_{16}\times 16^2$, which is a different number.
• ⟨ceþeħarpo⟩ $0\times 16^{-3}$; i.e. a quantity whose absolute value is less than $8\times 16^{-4}$.
• ⟨lenłarfemjalîdasvito⟩ $7.F86_{16}\times 16^{19}$

Inexact numerals are considered short numerals and thus take classifiers.

An inexact numeral with ⟨pen⟩ in the significand is an indefinite inexact numeral, which shows only the order of magnitude of something. Unlike an inexact numeral with ⟨ces⟩ as the magnitude, an indefinite inexact numeral establishes a lower bound on the quantity.

The inexact exponents are also used as prefixes for units of measure in systematic systems.

While the inexact numerals are the preferred set for expressing inexact quantities, exact numerals are sometimes used for this purpose in poetic language. Thus, we can prefix ⟨pen⟩ to an exact numeral such as ⟨dara⟩ to get an indefinite exact numeral.

Complex numerals

Complex numerals are composed of the real part, followed by ⟦-perin-⟧ or ⟦-ŋilis-⟧, then the imaginary part. ⟦-perin-⟧ is used when the imaginary part is positive and ⟦-ŋilis-⟧ is used when it is negative. Both the real and the imaginary part are expressed as short numerals, and the resulting numeral is short.

If the real part is omitted, then it is assumed to be zero. If the imaginary part is omitted, then it is assumed to be either 1 or −1, depending on its sign. ⟦-ŋilis-⟧ has the form ⟦-ŋils⟧ word-finally.

The particle ⟨ceþe⟩ affects only the real part of the numeral.

If one or both components are exact but not integral, then a rational numeral is used with a common denominator, in which the numerator is a complex numeral.

Alternatively, one or both components may be inexact, in which case the other must either be integral or inexact. If both components are inexact, then they must still be listed in full.

Interrogative quantities

Ŋarâþ Crîþ has the determiner ⟨met⟩ how many?, how much?. The nominal counterpart is ⟨metos, mjotos, medot⟩ (IIIc.m).

Number agreement

The morphological number of a noun modified by a numeral agrees with that numeral. All nouns take the generic number for the numeral 0 and may optionally take it for other numerals. In addition, numerals other than 0 are compatible with a different number depending on the clareþ of the noun:

• Singular nouns take the singular for −1 and 1, the dual for ±2, and the plural for all other numerals.
• Collective nouns take the singulative for 1 and the collective for all other numerals.
• Mass nouns always take the direct number when modified by a numeral.

Clitics for numerals

The distinctness clitic ⟨=’ot⟩ indicates that the items being counted are unique. This clitic can be used on all numerals as well as on the determiners ⟨mel⟩ and ⟨dân⟩ but makes sense only with integral quantities.

The bounding clitic ⟨=’ocþaf⟩ specifies the the numeral specifies an upper bound for the number of items (no more than). This clitic can be used on all numerals. It cannot be used on ⟨mel⟩, nor can it be used on ⟨dân⟩ (as it would be redundant).

Numeric prefixes

Ŋarâþ Crîþ has a set of prefixes used for derivation (e.g. in order to describe an entity of some number of parts). These are not based on the ordinary cardinal numeral system but rather on the prime factorization of a number.

An ordinary prefix phrase consists of one or more prefixes from Table 22, such that the prefixes are sorted first by ascending base, then by descending exponent. A power greater than 6 or less than −2 is expressed by compounding multiple prefixes of the same base until the desired power is reached. A prefix phrase is either an ordinary prefix phrase or a prefix from table Table 23.

If a prefix phrase modifies a noun that is monosyllabic in the nominative case, then an infix ⟦-i-⟧ is added between them.

Numerals in writing

The general rule for writing numerals is that anything from the short numeral sytem may either be written with digits or spelled out, and that anything else is always spelled out. For instance, ⟨navo glêmac⟩ may not be written as **⟨navo 5⟩ or even as **⟨navo 5âc⟩, but ⟨navo sensraþpinlaþ⟩ may be written as ⟨navo 2Blaþ⟩. Similarly, ⟨nefasaþ forþ⟩ cannot be written using figures at all, while ⟨len forþ⟩ may be abbreviated as ⟨7 forþ⟩.

The short-numeral parts of rational numerals, which are as a whole considered long numerals, may also be abbreviated: ⟨sraþfilm·ener i mjarâten glêma⟩ to ⟨11m·ener i 6râten glêma⟩ (but not **⟨11m·ener i 6râten 5⟩).

Since the digits in inexact numerals are read one by one instead of respecting place value, a łil is inserted after the first digit when an inexact numeral is abbreviated: ⟨senvilcescenroto⟩ to ⟨2·10henroto⟩.

Units of measure

For dimensions other than time, there are two systems of measure: the traditional system uses all units. The systematic system uses only one of the units, prefixed with inexact exponents; sometimes, the systematic unit is used without any prefixes, using inexact numerals instead.

Date and time

The year is approximated as having 403 5⁄24 days; that is, the year length varies between 403 and 404 days, with five leap years per 24-year cycle. (In particular, the leap years fall on years 0, 5, 10, 15, and 20 mod 24.) This approximation drifts from the true year by about 1 day per 1234 years.

The year is also divided into 13 half-months (senło; sg. senłas) of 31 or 32 days (approximating half of the lunar orbital period of 62.02 days). In non-leap years, each half-month is conveniently 31 days long. In leap years, the last half-month is extended by one day.

The calendar has a nine-day week. As a result, the day of week advances by two days every 24-year cycle, and the day-of-week pattern cycles every 216 years. The first six days of the week are considered work days, while the last two are rest days. The seventh day, in modern times, is a ‘half-work’ day.

Length

Currency

The traditional currency system, which was used in the Federation of Crîþja, is almost isomorphic to the British pre-decimal currency system. One erłol is equivalent to 20 mjari, and one mjare is equivalent to 12 edva. Unlike with the £sd system, one edvel is divided into seven seedva, although the seedvel is used primarily as a unit of accounting.

In the full abbreviation, each unit has its own symbol, which is followed by the number of that unit involved. If there is more than one unit involved, then each such string is separated by spaces. Units of which there are zero are omissible only from the left and the right; that is, **⟨9{rł2 e3}⟩ is not allowed and should be written as ⟨9{rł2 m0 e3}⟩ instead. Likewise, the unabbreviated reading lists the units with their quantities, except that all units of which there are zero are omitted (i.e. ⟨erłoc nefa edva prêno’ce⟩).

In the condensed abbreviation, the mjare and edva amounts are separated with a jedva, with no ⟨m⟩ or ⟨e⟩. If there are zero mjari, then the part before the slash is ⟨0⟩; if there are zero edva, then the part after the slash is the ŋos, ⟨’⟩. Note that the condensed abbreviation is not applicable if there are any seedva.

In the abbreviated reading, which also does not admit any seedva, amounts less than one mjare are expressed using a numeric prefix plus ⟨edva⟩. Amounts of one mjare or more are expressed as follows:

• If there are any erłer, then they are declared as they would in the unabbreviated reading. If there are no lower units, then the process ends here.
• The nominal form of the long numeral corresponding to the number of mjari is inserted. If there are no mjari, then ⟨inora⟩ emptiness is used instead. If there are 17, 18, or 19 mjari, then there is no appropriate long numeral and so the corresponding short numeral with the classifier ⟦-čis⟧ is used instead. Note that ⟨sraþfilčis⟩, ⟨sraþþenčis⟩, and ⟨sraþienčis⟩ are not declined.
• The conjunction ⟨i⟩ is used to separate the mjare and edva amounts.
• The short numeral corresponding to the number of edva is inserted.

Another format for expressing currency, used in accounting, is ⟨9{E mmes}⟩, in which the units below the erłol are expressed as fixed-width fields. For instance, £127. 19s. 11 6⁄7d. (⟨9{rł7F m13 eB s6}⟩ in the common format) can be expressed as ⟨9{7F 13B6}⟩. The right part of an amount expressed in the accounting format is read digit by digit, with the space read as ⟨inora⟩; thus the prior example would be read as ⟨lensratłar inora vilenpinmja⟩.

Rates of currency against currency (such as tax rates) are customarily given as an amount per erłol, giving a resolution of 1/1680 or roughly 0.06%.