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)|
|recurringà.||recurring||#||#||r (for ‘recurring’)||beginning of the recurring part|
- The digits in a number are reversed compared to our system: the unit position is placed first, followed by the sixteens position, and so on: 63 = 36hexadecimal = 3×16+6 = 54. This order corresponds to the direction of counting or the arrangement of Lemizh levels – lower numbers come before higher ones.
- Large numbers are organised in blocks of four digits each: 220E-91 = 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 trailing zero: 3, = 0.3hex = 3/16 = 0.1875.
- The end of the recurring part is marked with its own symbol: 9#1, = 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 for zero is the nonexistence negator.
Now we can write numbers; but 63 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. 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.
There is no generally applicable definition of an individual, especially as Lemizh grammar does not distinguish between countable and uncountable (mass) words: two individuals of rice are just as acceptable as two individuals of peas, or two individuals of sneezing. We can even speak of an individual of water if context allows. Think of a body of water, or the film ‘The Abyss’. An individual need not even be coherent: a counterexample would be an ant colony.
Of course there still are discrete things, which cannot be divided without losing their identity (a person, an ant colony, a sneeze), and continuous ones for which this isn’t generally a problem (a queue, water, walking). There are also intermediates; half a grain of rice is still rice in terms of food (although not much of it), but not in terms of the plant’s reproduction. A nightly hour can still be called ‘night’ if we don’t focus on the entirety of a night – say, sundown till morning – but on the state of the sun being down. (A good question is whether something functions as a unit in the given context.) The point of all this is that the concept of an individual can also be applied to continuous things.
To form numbers larger than fifteen, we need four verbs expressing exponential numbers.
|skmà.||256||162 = 256|
|jàs.||4096||163 = 4096|
|mràj.||65536||164 = 65,536|
Round numbers are multiples of an exponential number. We achieve multiplication by forming abstract nouns from numbers – e.g. sevenìl. ‘the consequence of making seven, seven-ness’.
|rÌj qìlfy. ⇒ qilfrÌj.||sixteen consequences of making seven individuals; sixteen seven-nesses||70hex = 112|
|16-acc1 seven-cons-acc2. ⇒ seven-cons-16-acc1.|
|07Ì.||(written short form, pronounced the same as above)|
Note how the epenthetic consecutive acts as a multiplication. Just to make the point clear: **rÌj sevenÌy. is nonsensical because it would mean ‘the sixteen are seven’. This difference between consecutive and accusative in the context of numerals is actually the same as the abstract/concrete distinction 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:|
|— gcynè skmÌne.||10Fhex = 271|
|… fifteen-partacc-nom2 256-partacc-nom2.|
Sometimes it is useful to conflate such numbers into a single object via mà.-desporption.
|mÌ gcynÌ skmÌny.||271|
|make-acc1 fifteen-partacc-acc2 256-partacc-acc2.|
The convention for round numbers is to place the exponential number as the modifier of the compound. This demotes it or – so to speak – reduces its visibility, just as the exponent is invisible in digit notation. The digit, being the more salient part (visible in digit notation), is placed as the head. English and other languages do something similar in words such as ‘fourty’. This convention is also in line with the negative numbers below, negated and weighted adjectives, and similar constructions.
Numbers from 165 upwards include a multiple of 65536Ì., e.g. 13,010Fhex = 10Fhex + 1,0000hex × 13hex. Compounding of the multiple is impossible if it is constructed with partitive ‘and’.
|mrÌj rìljy. ⇒ riljmrÌj.||10,0000hex = 1,048,576|
|65536-acc1 16-cons-acc2. ⇒ 16-cons-65536-acc1.|
|— gcynè skmynè mrÌnje trilnÌ rìlnjy.||13,010Fhex = 1,245,455|
|… fifteen-partacc-nom2 256-partacc-nom2 65536-partacc-nom2 three-partcons-acc3 16-partcons-acc3.|
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. The result of the function is the accusative object, just as the lace is the result of the beaver’s action: squarerootÌ twoÌi. is the square root of two. Functions with two arguments have one of them in the dative and one in the nominative, as with powerÌ 8yì 4Ìe. ‘the content (=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 consideration, the verb powerà 65536Ìi. ⇔ 65536Ì powerìy. ⇒ poweri65536à. ‘raise 65,536 to some-nom power’ is used to form new exponential numbers.
|powerimrÌj dwÌe.||1,0000,0000hex = 4,294,967,296|
|— rÌne powerimrÌnje dwyè rìljy.||10,0000,0001hex = 68,719,476,737|
|… one-partacc-nom2 power-dat-65536-partacc-nom3 two-acc-nom4 16-cons-acc4.|
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. You can think of frilkÌ. as ‘12 × −1’ with the consecutive as the multiplication sign and kÌ. as −1 (just as sevenilrÌj. was ‘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). Another example 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.
We express fractions along the lines of ‘seven of nine each’.
‘seven each’ is translated with a multiplication consecutive – eachÌ sevenìlyn. ⇒ sevenileachÌ. ‘each of the seven-nesses’ – as eachÌ sevenÌyn. would mean ‘each of the seven individuals’. The inner accusative is reinstated by compounding to express seven individuals each (as opposed to the abstract concept of seven-ness). Finally, ‘of nine’ needs another partitive bracket.
A number with a hexadecimal separator can be expressed as a fraction with an exponential number in the denominator. Numbers with a recurring part also are often simpler expressed as fractions, but with a non-exponential denominator.
|qilfRÌ nÌhyn.||7⁄9; 7 divided by 9|
|trilRÌ rÌjyn.||0.3hex = 3⁄16|
|xtilRÌ pnÌyn.||1.9̄hex = 1.6 = 8⁄5|
We do not need to compound the object ‘nine’ with eachÌ. – it already refers to each of the seven-nesses separately.
|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. is the topmost grouping numeral in terms of the semantic tree; sortà. and kràj. denote sub-categories. A further sub-category verb would be ‘queue’. Grouping numerals combine a number of individuals (in the dative) to form a ‘super-individual’ (the accusative: groups, sorts or ensembles). The plot resembles that of the verb ‘divide’. 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 comprise a single person or even none at all.
|humanÌ qmìy. ⇔ qmÌ humanÌi.||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|
|Sentences with a grouping numeral as main predicate:|
|sortà umbrellayì RÌcjy.||(the umbrellas are sorted, the sorts are coloured things)||He sorts the umbrellas by colour.|
|sort-fact1 umbrella-acc-dat2 colour-acc-acc2.|
|sortà umbrellayì wyncgÌ lÌnxwy.||He sorts the umbrellas into black and green ones.|
|sort-fact1 umbrella-acc-dat2 black-partacc-acc2 green-partacc-acc2.|
|arrangeà umbrellayì weighÌy.||He sorts/arranges the umbrellas by weight.|
|arrange-fact1 umbrella-acc-dat2 weigh-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 amountà..
|Verb||Gloss||Relative weight||Approximate translation with inner accusative and various outer cases|
|Plot cases||Temporal (aR)||Durative (yR)||Locative (ar)||Extensive (yr)|
|amountà.||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. 1/2Ì. 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 3/8Ì. and less than 5/8Ì..
|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. 3/4Ì. 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. ‘1’ and the indefinite numeral mlÌ. ‘several’, 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 groupÌy.||they are few / it is a little, and they are groups||a few groups; a small group (a bit, which is a group)|
|crÌ groupÌyn.||a few (of the) groups|
|crÌ groupìy. ⇔ groupÌ crÌi.||a few things grouped ⇔ a small group|
|1/4-acc1 group-dat-acc2. ⇔ group-acc1 1/4-acc-dat2.|
|crÌ groupìyn.||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 **queueà nÌaR. would violate inversion ban.|
|nà queueàRy. ⇒ queueaRnà.||He never queues.|
|not-fact1 queue-temp-acc2. ⇒ queue-temp-not-fact1.|
|nà queueìy. ⇒ queueinà.||Nobody is queuing.|
|not-fact1 queue-dat-acc2. ⇒ queue-dat-not-fact1.|
|nà queueanàRy.||He never does not queue.||He does nothing but queue.|
|Translate, optionally using a pocket calculator. Spell out the numbers:|
|The group of tortoises isn’t seen anywhere.|
|Achilles is queuing quite often.|