Unit 7. Numerals I
An Englishman, even if he is alone, forms an orderly queue of one.
Lemizh uses the hexadecimal system, the numeral place-value system with a base of 16. The 16 digits and three additional symbols, along with their names, are:
|kà.||opposition||_||_ (underscore)||−||negation (minus)|
|liRnà.||(none)||#||#||ʳ (for ‘recurring’)||beginning of the recurring part|
- Here is a simple example: 36 = 36hexadecimal = 3×16+6 = 54.
- Large numbers are organised in blocks of four digits each: 19-E022 = 19,E022hex = 1,695,778.
- Negative numbers are preceded by a negation sign: _1 = −1. The negation verb is none other than the opposition negator from the previous unit.
- The ‘hexadecimal separator’ corresponds to our decimal separator or period: it is placed between the fractional and integer parts of a number. Fractions without an integer part are written without a leading zero: ,3 = 0.3hex = 3/16 = 0.1875.
- The beginning of the recurring part is marked with its own symbol: ,1#9 = 0.19̅hex = 0.1999…hex = 0.1. This symbol can also be placed in the integer part: #3 = 3.3̅hex = 3.33…hex = 3.2. (The verb given in the table above is a kind of compound called a negated topic, discussed in unit 10.)
- The verb for zero is the nonexistence negator.
Now we can write numbers; but 36 is not a grammatical entity and so cannot be part of a sentence.
Numbers in grammar
The definite numeral verbs up to fifteen are shown in the table above. They denote the actions ‘make/become one individual, make/become two individuals’, etc. With an inner dative, we have ‘something made into one/two individual(s)’, with an inner accusative – and this is again the most useful case – ‘individual(s) with the property of being one/two’. In other words, the number of objects behaves just like a property, and numerals are adjectival verbs: rÌjd. is a red thing or red things, swÌh. are six things or individuals. swÌh. can also be written 6Ì. for short.
Numeral verbs imply making/becoming a number of individuals one after the other, as opposed to making them simultaneously. This has no effect on the inner accusative because the property of being a certain number of individuals is independent of how they came into existence; we will, however, need this subtlety in the chapter on ordinal numerals in the next unit.
In the physical world there are discrete things, which cannot be divided without losing their identity (a person, a room, an ant colony, a sneeze), and continuous ones for which this isn’t normally a problem (a queue, rice, water, walking). Regarding Lemizh grammar, an in‑dividual of a discrete thing is just what is says: something that cannot be further divided while remaining what it is. Thus, ‘three rooms’ or ‘two sneezes’ is as unambiguous in Lemizh as it is in English.
But nothing hinders us from using something continuous with definite numerals: we can of course say ‘two queues’, but also ‘two rices, waters, walkings’ in Lemizh. Such individuals are context dependent: ‘Buy two rices’ will be understood as two packages, ‘Cook two rices’ as two servings; ‘two waters’ can be two servings or two bodies of water; ‘two walkings’ can be two steps or going for a walk twice, depending on the situation.
In fact, there isn’t a clear-cut distinction between discrete and continuous things. Rice is continuous in terms of food, but grains of rice are discrete in terms of the plant’s reproduction. Night can be an individual time span from sundown till morning or the state of the sun being down without regard to duration. Whatever we are talking about, an individual is always something that functions as one in the given context (even if it is internally unconnected such as an ant colony). And in ambiguous cases we can always be more specific and say ‘two packages of rice’.
The English distinction between countable and uncountable (mass) nouns does not carry over to Lemizh: there is no meaningful difference between peas and rice beyond English grammar.
To form numbers larger than fifteen, we need these four verbs expressing exponential numbers.
|skmà.||256||162 = 256|
|jàs.||4096||163 = 4096|
|mràj.||65536||164 = 65536|
Round numbers are multiples of an exponential number. We construct them by forming abstract nouns from exponential numbers with an inner consecutive, such as ìlj. ‘the consequence of making sixteen, sixteen‑ness’, building them into an accusative bracket, and compounding.
|qÌf ìljy. ⇒ iljqÌf.||seven consequences of making sixteen individuals; seven sixteen-nesses||70hex = 112|
|seven-acc1 16-cons-acc2. ⇒ 16-cons-seven-acc1.|
|70Ì.||(written short form, pronounced the same as above)|
Note how the epenthetic consecutive acts as a multiplication. Just to make the point clear: **qÌf Ìjy. seven-acc1 16-acc-acc2. is nonsensical because it would mean ‘the seven are sixteen’. This difference between consecutive and accusative in the context of numerals is closely related to the abstract/concrete distinctions we met in the chapter on negators in the previous unit.
Other numbers are expressed as sums of round numbers. They are added up with a partitive ‘and’.
|This example shows a number as a nominative object:|
|— skmynè gcÌne.||10Fhex = 271|
|… 256-partacc-nom2 fifteen-partacc-nom2.|
Sometimes it is useful to conflate such numbers into a single object via mà.-desorption.
|mÌ skmynÌ gcÌny.||10Fhex = 271|
|make-acc1 256-partacc-acc2 fifteen-partacc-acc2.|
Numbers from 10,0000hex = 165 upwards involve multiplication of mrÌj. ‘1,0000hex = 65536’ with a multidigit number, e.g. 13,0000hex = 13hex × 1,0000hex. If the multidigit number is constructed with partitive ‘and’, i.e. if it is not an exponential or a round number, compounding is impossible.
|— mriljynjè skmynè gcÌne.||sixteen 65536-nesses (compounding works)||10,010Fhex = 1,048,847|
|… 65536-cons-16-partacc-nom2 256-partacc-nom2 fifteen-partacc-nom2.|
|— mrÌnje ynjìl tryníl skmynè gcÌne.||sixteen and three 65536-nesses (no compounding)||13,010Fhex = 1,245,455|
|… 65536-partacc-nom2 16-partacc-cons3 three-partacc-cons3 256-partacc-nom2 fifteen-partacc-nom2.|
Mathematical functions; very large numbers
A mathematical function acts on a number, which makes this number, in terms of the Lemizh plot, the dative object: dyprà 2Ìi. cosine-fact1 2-acc-dat2. means ‘to calculate the cosine of two’. The result of the function is the accusative object, just as the lace is the result of the beaver’s action: dyprÌ 2Ìi. cosine-acc1 2-acc-dat2. is the cosine of two. Functions with two arguments have one of them in the dative and one in the nominative: lrÌ 4yè 8Ìi. exponentiate-acc1 4-acc-nom2 8-acc-dat2. is ‘the result of 4 exponentiating 8; the 4th power of 8; 84’. This distribution of cases might be a bit arbitrary, but then it was introduced by mathematicians and is not part of the natural grammar.
Based on this concept, we form new exponential numbers that are powers of 65536: we first compound the verb meaning ‘to exponentiate’ with some power, e.g. two, in the nominative; and then compound the resulting word with 65536 in the dative.
|lrÌ dwÌe. ⇔ dwÌ lrèy. ⇒ lredwÌ.||the second power of some number-dat|
|exponentiate-acc1 two-acc-nom2. ⇔ two-acc1 exponentiate-nom-acc2. ⇒ exponentiate-nom-two-acc1.|
|lredwÌ mrÌji. ⇔ mrÌj lredwìy. ⇒ lredwimrÌj.||the second power of 1,0000hex = 1,0000,0000hex = 4,294,967,296|
|exponentiate-nom-two-acc1 65536-acc-dat2. ⇔ 65536-acc1 exponentiate-nom-two-dat-acc2. ⇒ exponentiate-nom-two-dat-65536-acc1.|
|— lredwimrÌnje yjíl rÌne.||10,0000,0001hex = 68,719,476,737|
|… exponentiate-nom-two-dat-65536-partacc-nom2 16-acc-cons3 one-partacc-nom2.|
We will touch on an informal way of expressing large numbers on the page about units of measurement in the appendix.
Inverse functions have the accusative and dative switched: the arccosine of two is dyprì dwÌy. cosine-dat1 two-acc-acc2..
Unit fractions are formed with the function làgz. ‘calculate the reciprocal’ in the way described above. Outside mathematical contexts, this verb means ‘to belittle, to make dear (in a patrinising way)’. Other fractions are simply whole numbers multiplied with unit fractions by means of a consecutive case.
A number with a hexadecimal separator can be expressed as a fraction with an exponential number in the denominator. This is commonly done with 256 in situations where we would use percent. Numbers with a recurring part are often simpler expressed as fractions, but with a non-exponential denominator.
|lÌgz nÌhi. ⇔ nÌh lìgzy. ⇒ ligznÌh.||1⁄9|
|little-acc1 nine-acc-dat2. ⇔ nine-acc1 little-dat-acc2. ⇒ little-dat-nine-acc1.|
|nÌh ligzskmìly.||0.09hex = 9⁄256 ≈ 3.5%|
|xtÌ ligzpnìly.||1.9̄hex = 1.6 = 8⁄5|
Compounding is impossible if the denominator is constructed with a partitive ‘and’.
Negative numbers are formed with the opposition negator kÌ..
|kÌ frìly. ⇒ frilkÌ.||the opposite of twelve-ness|
−1 consequences of making twelve individuals
|opposition-acc1 twelve-cons-acc2. ⇒ twelve-cons-opposition-acc1.|
The use of the epenthetic consecutive should remind you of the consecutive we have been using for multiplication, and also of the use of negators with adjectives (‘unwise’). You can think of frilkÌ. as ‘−1 × 12’ with the consecutive as the multiplication sign and kÌ. as −1 (just as iljqÌf. is ‘7 × 16’). Generally speaking, the consecutive case is necessary whenever both verbs define certain quantities, as with numerals (including the negators nÌ. = 0 and kÌ. = −1). Other examples will follow in the next unit.
Again, compounding is impossible if the number is constructed with a partitive ‘and’.
|mlà.||several||more than one individual|
|Rà.||each||each (separate, respective) individual|
The indefinite numeral verbs work like the definite numerals save they don’t specify certain numbers. mlà. is the super-category verb for all definite numerals larger than one. We have already learned Rà. when we were talking about reflexive and reciprocal pronouns in the previous unit.
|Inner factive||Inner accusative||Inner dative|
|qmà.||to group||a group||something grouped|
|màh.||to sort, to group according to type||a sort, a type||something sorted|
|kràj.||to form an ensemble (a group that is meaningful or useful as a whole)||an ensemble||something forming an ensemble, a component of a whole|
qmà. ‘group’ is the topmost grouping numeral in terms of the semantic tree; the other two denote sub-categories. There are many other sub-category verbs, for example sràq. ‘queue’. Grouping numerals combine a number of individuals (in the dative) to form a ‘super-individual’ (the accusative: groups, sorts, ensembles). We can translate qmà. as ‘to make a group-acc from individuals-dat, to turn individuals-dat into a group-acc’; thus, grouping numerals are nominal verbs, or at least very similar to them. The groups, sorts or ensembles can be counted just like ordinary individuals.
There is no rule limiting the number of combined individuals, so a ‘group of people’ can in principle comprise a single person or even none at all.
|cOÌc qmìy. ⇔ qmÌ cOÌci.||grouped people ⇔ a group of people|
|human-acc1 group-dat-acc2. ⇔ group-acc1 human-acc-dat2.|
|trÌ qmìy. ⇔ qmÌ trÌi.||three grouped ⇔ a group of three|
|three-acc1 group-dat-acc2. ⇔ group-acc1 three-acc-dat2.|
|qmÌ trÌy.||three groups|
The phrases ‘a group [made] of people / of three’ are exactly parallel to ‘lace made from thread’.
|Sentences with a grouping numeral as main predicate:|
|màh wzuhkì RÌcjy.||The umbrellas (tool noun) are sorted, and the sorts are coloured things.||He sorts the umbrellas by colour.|
|sort-fact1 umbrella-ins-dat2 colour-acc-acc2.|
|màh wzuhkì wyncgÌ lÌnxwy.||He sorts the umbrellas into black and green ones.|
|sort-fact1 umbrella-ins-dat2 black-partacc-acc2 green-partacc-acc2.|
|qtrà wzuhkì lÌty.||He sorts/arranges the umbrellas by weight.|
|arrange-fact1 umbrella-ins-dat2 heavy-acc-acc2.|
The umbrellas in the last example are not ‘sorted’ (grouped into distinct types) but arranged into a continuous row, hence the different main predicate.
These verbs are like definite numerals except they describe more vague quantities such as ‘make/become much, quite a lot, a bit’ etc. In the following table, some translations in the durative and extensive columns have been omitted as the table is far too complicated anyway. It shouldn’t be difficult to substitute the missing items, as well as translations for the remaining cases. The topmost weighting numeral is ràw. ‘make an amount’. Two more weighting numerals will be treated in unit 11.
|Verb||Gloss||Relative weight||Approximate translation with inner accusative and various outer cases|
|Plot cases||Temporal (aR)||Durative (yR)||Locative (ar)||Extensive (yr)|
|ràw.||amount||undefined||some amount, some quantity (including zero)||some amount of time||for some time/duration||in some number of places, in some area||with some extent|
|càwb.||1⁄8||1⁄8 approx.||hardly any(thing)||hardly ever||hardly anywhere|
|crà.||1⁄4||1⁄4 approx.||few, little, a bit||seldom||for a short time/duration||in a few places, in a small area||with a small extent, over a small area|
|Ràbv.||3⁄8||3⁄8 approx.||some, a fairly small number/amount||sometimes||in some places, in a fairly small area|
|bvà.||1⁄2||1⁄2 approx.||a medium number/amount||a medium number of times, a medium amount of time||in a medium number of places, in a medium-sized area|
|dmàj.||5⁄8||5⁄8 approx.||quite a lot||quite often||in quite a number of places, in a fairly large area|
|dmà.||3⁄4||3⁄4 approx.||many, much||often, much of the time||for a long time/duration||in many places, in a large area||with a large extent, over a large area|
|xpàj.||7⁄8||7⁄8 approx.||almost every, nearly all||nearly always||nearly everywhere|
|jnà.||1⁄1||1||every, all, the whole||always||always||everywhere||everywhere|
The weights given in the table only describe relations to the other weighting numerals. bvÌ. 1/2-acc1. is not necessarily half the amount (or, as for that, it is not necessarily any portion of a given quantity, as this is the domain of the partitive), but at any rate more than RÌbv. 3/8-acc1. and less than dmÌj. 5/8-acc1.. Objects with weighting numerals are typically compounded.
|sràq cÌwbaR. ⇔ càwb sràRqy. ⇒ sraRqcàwb.||He hardly ever queues.|
|queue-fact1 1/8-acc-temp2. ⇔ 1/8-fact1 queue-temp-acc2. ⇒ queue-temp-1/8-fact1.|
Weighting numerals do not have a singular or plural connotation. dmÌ. 3/4-acc1. can not only mean a large amount of water, a long time, a large space and so on, but also a large number of things, times, places, etc. To distinguish ‘much’ and ‘many’, we can use compounds from brackets with the definite numeral rÌ. one-acc1. and the indefinite numeral mlÌ. several-acc1., respectively. The latter needs a partitive to express ‘many of the (several) individuals’, as opposed to a cumulative bracket, which simply denotes a large quantity that consists of several individuals (which could also be satisfied by, say, a small number of large areas).
|rÌ dmÌy. ⇒ dmyrÌ.||much, a large amount, much space or time, etc.|
|one-acc1 3/4-acc-acc2. ⇒ 3/4-acc-one-acc1.|
|rÌ jnÌy. ⇒ jnyrÌ.||the whole (thing, amount, space, time, etc.)|
|one-acc1 1/1-acc-acc2. ⇒ 1/1-acc-one-acc1.|
|dmÌ mlÌyn. ⇔ mlÌ dmÌny. ⇒ dmynmlÌ.||many (individual) things, places or times, etc.|
|3/4-acc1 several-acc-partacc2. ⇔ several-acc1 3/4-partacc-acc2. ⇒ 3/4-partacc-several-acc1.|
Brackets of weighting numerals with grouping ones are also useful, with inner accusative and dative of the latter expressing different relationships between the two. Cumulative brackets of weighting numerals with any other kind of word can be ambiguous, so partitive brackets are needed again. The next unit will bring more examples of such brackets.
|crÌ qmÌy.||they are few / it is a little, and they are groups||ambiguous: a few groups; a small group (a bit, which is a group)|
|crÌ qmÌyn.||a few (of the) groups|
|crÌ qmìy. ⇔ qmÌ crÌi.||a few things grouped ⇔ a small group|
|1/4-acc1 group-dat-acc2. ⇔ group-acc1 1/4-acc-dat2.|
|crÌ qmìyn.||a few of the grouped things||a few / little of a group|
nà., the nonexistence negator and definite numeral zero, is also a weighting numeral. For a reason discussed in unit 9, the so-called inversion ban for negators, it is mostly used as a predicate.
|The ‘uninverted’ forms are omitted in the following examples since **sràq nÌaR. would violate inversion ban.|
|nà sràRqy. ⇒ sraRqnà.||He never queues.|
|not-fact1 queue-temp-acc2. ⇒ queue-temp-not-fact1.|
|nà srìqy. ⇒ sriqnà.||Nobody is queuing.|
|not-fact1 queue-dat-acc2. ⇒ queue-dat-not-fact1.|
|nà sraqnàRy.||He never does not queue.||He does nothing but queue.|