lemÌc. Lemizh grammar and dictionary

Unit 7. Numerals I

An Englishman, even if he is alone, forms an orderly queue of one.

(George Mikes. How to be a Brit)

Numbers

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:

VerbGlossDigitKeyboardTranscriptionValue
nà.not0000
rà.one1111
dwà.two2222
trà.three3333
gwàq.four4444
pnà.five5555
swàh.six6666
qàf.seven7777
xtà.eight8888
nàh.nine9999
dàh.tenAAA10
omà.elevenBBB11
frà.twelveCCC12
àhs.thirteenDDD13
àb.fourteenEEE14
gcà.fifteenFFF15
VerbGlossSymbolKeyboardTranscriptionMeaning
kà.opposition__ (underscore)negation (minus)
xàk.point,,,hexadecimal separator
liRnà.(none)##ʳ (for ‘recurring’)beginning of the recurring part

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.

Individuals

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.

Multidigit numbers

To form numbers larger than fifteen, we need these four verbs expressing exponential numbers.

VerbGlossValue
àj.1616
skmà.256162 = 256
jàs.4096163 = 4096
mràj.65536164 = 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-nesses70hex = 112
seven-acc1 16-cons-acc2. 16-cons-seven-acc1.
70Ì.(written short form, pronounced the same as above)
112-acc1.

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.
— 10FÌe.
… 271-acc-nom2.

Sometimes it is useful to conflate such numbers into a single object via mà.-desorption.

skmynÌ gcÌny.10Fhex = 271
make-acc1 256-partacc-acc2 fifteen-partacc-acc2.
10FÌ.
271-acc1.

Larger numbers

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 4yèi. 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..

Negative numbers

Negative numbers are formed with the opposition negator kÌ..

kÌ frìly. frilkÌ.the opposite of twelve-ness
−1 consequences of making twelve individuals
−12
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’.

Indefinite numerals

VerbGlossValue
mlà.severalmore than one individual
Rà.eacheach (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.

Fractions

We express fractions along the lines of ‘seven of nine each’.

‘seven each’ is translated with an inner consecutive case for multiplication: RÌ qìlfyn. each-acc1 seven-cons-partacc2. ‘each of the seven-nesses’. If we used an accusative as in RÌ qÌfyn. each-acc1 seven-acc-partacc2., we would get ‘each of seven individuals’, which is not what we want. Instead, the inner accusative is reinstated by compounding, which gives us qilfRÌ. seven-cons-each-acc1. ‘seven individuals each’. 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. This is commonly done with 256 in situations where we would use percent. Numbers with a recurring part also are often simpler expressed as fractions, but with a non-exponential denominator.

qilfRÌ nÌhyn.seven each of nine7⁄9
seven-cons-each-acc1 nine-acc-partacc2.
nilhRÌ skmÌyn.0.09hex = 9⁄256 ≈ 3.5%
nine-cons-each-acc1 256-acc-partacc2.
xtilRÌ pnÌyn.1.9̄hex = 1.6 = 8⁄5
eight-cons-each-acc1 five-acc-partacc2.

We do not need to compound the denominator with RÌ. – it already refers to each of the seven-nesses separately.

Grouping numerals

VerbTranslation
Inner factiveInner accusativeInner dative
qmà.to groupa groupsomething grouped
màh.to sort, to group according to typea sort, a typesomething sorted
kràj.to form an ensemble (a group that is meaningful or useful as a whole)an ensemblesomething 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 or 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
group-acc1 three-acc-acc2.

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 wzuhkìn RÌcjy.The umbrellas (tool noun) are sorted, the sorts are coloured things.He sorts the umbrellas by colour.
sort-fact1 umbrella-ins-partdat2 colour-acc-acc2.
màh wzuhkìn wyncgÌ lÌnxwy.He sorts the umbrellas into black and green ones.
sort-fact1 umbrella-ins-partdat2 black-partacc-acc2 green-partacc-acc2.
qtrà wzuhkìn lÌty.He sorts/arranges the umbrellas by weight.
arrange-fact1 umbrella-ins-partdat2 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.

Weighting numerals

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’.

VerbGlossRelative weightApproximate translation with inner accusative and various outer cases
Plot casesTemporal (aR)Durative (yR)Locative (ar)Extensive (yr)
ràw.amountundefinedsome amount, some quantity (including zero)some amount of timefor some time/durationin some number of places, in some areawith some extent
nà.not0none, nothingnevermomentarynowherepunctiform
càwb.1⁄81⁄8 approx.hardly any(thing)hardly everhardly anywhere
crà.1⁄41⁄4 approx.few, little, a bitseldomfor a short time/durationin a few places, in a small areawith a small extent, over a small area
Ràbv.3⁄83⁄8 approx.some, a fairly small number/amountsometimesin some places, in a fairly small area
bvà.1⁄21⁄2 approx.a medium number/amounta medium number of times, a medium amount of timein a medium number of places, in a medium-sized area
dmàj.5⁄85⁄8 approx.quite a lotquite oftenin quite a number of places, in a fairly large area
dmà.3⁄43⁄4 approx.many, muchoften, much of the timefor a long time/durationin many places, in a large areawith a large extent, over a large area
xpàj.7⁄87⁄8 approx.almost every, nearly allnearly alwaysnearly everywhere
jnà.1⁄11every, all, the wholealwaysalwayseverywhereeverywhere

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 groupsambiguous: a few groups; a small group (a bit, which is a group)
1/4-acc1 group-acc-acc2.
crÌ qmÌyn.a few (of the) groups
1/4-acc1 group-acc-partacc2.
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 thingsa few / little of a group
1/4-acc1 group-dat-partacc2.

Zero

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.
not-fact1 queue-fact-not-temp-acc2.

Exercises

Translate, optionally using a pocket calculator. Spell out the numbers:
48Solve
−60Solve
1,000,000Solve
1⁄17Solve
Translate:
The group of tortoises isn’t seen anywhere.Solve
Achilles is queuing quite often.Solve
dnirdmà.Solve
walk-ill-3/4-fact1.
eOlcàwb.Solve
laugh-psu-1/8-fact1.
spiRznà.Solve
happy-egr-not-fact1.