# Unit 7. Numerals I

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

## 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:

Verb | Gloss | Digit | Keyboard | Transcription | Value |
---|---|---|---|---|---|

nĂ . | not | 0 | 0 | 0 | 0 |

rĂ . | one | 1 | 1 | 1 | 1 |

dwĂ . | two | 2 | 2 | 2 | 2 |

trĂ . | three | 3 | 3 | 3 | 3 |

gwĂ q. | four | 4 | 4 | 4 | 4 |

pnĂ . | five | 5 | 5 | 5 | 5 |

swĂ h. | six | 6 | 6 | 6 | 6 |

qĂ f. | seven | 7 | 7 | 7 | 7 |

xtĂ . | eight | 8 | 8 | 8 | 8 |

nĂ h. | nine | 9 | 9 | 9 | 9 |

dĂ h. | ten | A | A | A | 10 |

omĂ j. | eleven | B | B | B | 11 |

frĂ . | twelve | C | C | C | 12 |

Ă c. | thirteen | D | D | D | 13 |

Ă b. | fourteen | E | E | E | 14 |

gcĂ . | fifteen | F | F | F | 15 |

Verb | Gloss | Symbol | Keyboard | Transcription | Meaning |

kĂ . | opposition | _ | _ (underscore) | â | negation (minus) |

xĂ k. | point | , | , | , | hexadecimal separator |

liRnĂ . | â | # | # | Êł (for ârecurringâ) | beginning of the recurring part |

- Here is a simple example: 36 = 36
_{hexadecimal}= 3Ă16+6 = 54. - Large numbers are organised in blocks of four digits each: 19-E022 = 19,E022
_{hex}= 1,695,778. - Negative numbers are preceded by a negation sign: _1 = â1. The negation verb is none other than the opposition negator from the previous unit.
- The âhexadecimal separatorâ 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 = 0.3
_{hex}= 3/16 = 0.1875. - The beginning of the recurring part is marked with its own symbol: ,1#9 = 0.19Ì
_{hex}= 0.1999âŠ_{hex}= 0.1. This symbol can also be placed in the integer part: #3 = 3.3Ì_{hex}= 3.33âŠ_{hex}= 3.2. (The verb given in the table above is 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 âmake/become one individual, make/become two individualsâ, etc. With an inner dative, we have âsomething made into one/two individual(s)â, with an inner accusative â and this is again the most useful case â âindividual(s) with the property of being one/twoâ. In other words, the number of objects behaves just like a property, and numerals are adjectival verbs: rĂjd. is a red thing or red things, swĂh. are six things. swĂh. can also be written 6Ă. 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 âThe Abyssâ. 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 isnât 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 plantâs reproduction. A nightly hour can still be called ânightâ if we donât focus on the entirety of a night â say, sundown till morning â 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.

### Multidigit numbers

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

Verb | Gloss | Value |
---|---|---|

rĂ j. | 16 | 16 |

skmĂ . | 256 | 16^{2} = 256 |

jĂ s. | 4096 | 16^{3} = 4096 |

mrĂ j. | 65536 | 16^{4} = 65,536 |

Round numbers are multiples of an exponential number. We achieve multiplication by forming abstract nouns from numbers â e.g. seven*ĂŹl*. âthe consequence of making seven, seven-nessâ.

rĂj qĂŹlfy. â qilfrĂj. | sixteen consequences of making seven individuals; sixteen seven-nesses | 70_{hex} = 112 | |

16-acc^{1} seven-cons-acc^{2}. â seven-cons-16-acc^{1}. | |||

70Ă. | (written short form, pronounced the same as above) | ||

112-acc^{1}. |

Note how the epenthetic consecutive acts as a multiplication. Just to make the point clear: **rĂj seven*Ă*y. is nonsensical because it would mean âthe sixteen are sevenâ. 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 âandâ.

This example shows a number as a nominative object: | |||
---|---|---|---|

â skmynĂš gcĂne. | 10F_{hex} = 271 | ||

âŠ 256-partacc-nom^{2} fifteen-partacc-nom^{2}. | |||

â 10FĂe. | |||

âŠ 271-acc-nom^{2}. |

Sometimes it is useful to conflate such numbers into a single object via mĂ .-desporption.

mĂ skmynĂ gcĂny. | 271 | ||

make-acc 256-partacc-acc^{1}^{2} fifteen-partacc-acc^{2}. | |||

10FĂ. | |||

271-acc^{1}. |

The convention for round numbers is to place the exponential number as the modifier of the compound. This demotes it or â so to speak â 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 âfour*ty*â. This convention is also in line with the negative numbers below, negated and weighted adjectives, and similar constructions.

### Larger numbers

Numbers from 16^{5} upwards include a multiple of 65536Ă., e.g. 13,010F_{hex} = 10F_{hex} + 1,0000_{hex} Ă 13_{hex}. Compounding of the multiple is impossible if it is constructed with partitive âandâ.

mrĂj rĂŹljy. â riljmrĂj. | 10,0000_{hex} = 1,048,576 | ||

65536-acc^{1} 16-cons-acc^{2}. â 16-cons-65536-acc^{1}. | |||

â mrĂnje rilnjĂ trilnĂœ skmynĂš gcĂne. | 13,010F_{hex} = 1,245,455 | ||

âŠ 65536-partacc-nom^{2} 16-partcons-acc^{3} three-partcons-acc^{3} 256-partacc-nom^{2} fifteen-partacc-nom^{2}. |

### 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 beaverâs action: dypr*Ă* dwĂ*i*. (cosine*-acc*^{1} two-acc*-dat*^{2}.) is the cosine of two. Functions with two arguments have one of them in the dative and one in the nominative, as with lrĂ 8y*ĂŹ* 4Ă*e*. âthe content (=result) of 4 exponentiating 8; the 4^{th} power of 8; 8^{4}â. 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 lrĂ 65536Ăi. â 65536Ă lrĂŹy. â lri65536Ă . âraise 65,536 to some-nom powerâ is used to form new exponential numbers.

lrimrĂj dwĂe. | 1,0000,0000_{hex} = 4,294,967,296 | ||

exponentiate-dat-65536-acc^{1} two-acc-nom^{2}. | |||

â lrimrĂnje dwyĂš riljĂœ rĂne. | 10,0000,0001_{hex} = 68,719,476,737 | ||

âŠ exponentiate-dat-65536-partacc-nom^{2} two-acc-nom^{3} 16-cons-acc^{3} one-partacc-nom^{2}. |

Inverse functions have the accusative and dative switched: the arccosine of two is dypr*ĂŹ* dwĂ*y*. (cosine*-dat*^{1} two-acc*-acc*^{2}.).

### Negative numbers

Negative numbers are formed with the opposition negator kĂ..

kĂ frĂŹly. â frilkĂ. | the opposite of twelve-ness â1 consequences of making twelve individuals | â12 | |

opposition-acc^{1} twelve-cons-acc^{2}. â twelve-cons-opposition-acc^{1}. |

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 fr*il*kĂ. as â12 Ă â1â with the consecutive as the multiplication sign and kĂ. as â1 (just as seven*il*rĂj. was â7 Ă 16â). Generally speaking, the consecutive case is necessary whenever both verbs define certain quantities, as with numerals (including the negators nĂ. = 0 and kĂ. = â1). Another example will follow in the next unit.

Again, compounding is impossible if the number is constructed with a partitive âandâ.

## Indefinite numerals

Verb | Gloss | Value |
---|---|---|

mlĂ . | several | more than one individual |

RĂ . | each | each (separate, respective) individual |

The indefinite numeral verbs work like the definite numerals save they donât specify certain numbers. mlĂ . is the super-category verb for all definite numerals larger than one. We have already learned RĂ . when we were talking about reflexive and reciprocal pronouns in the previous unit.

### Fractions

We express fractions along the lines of âseven of nine eachâ.

âseven eachâ is translated with a multiplication consecutive â rhĂ seven*ĂŹl*yn. â seven*il*rhĂ. âeach of the seven-nessesâ â as rhĂ seven*Ă*yn. would mean âeach of the seven individualsâ. The inner accusative is reinstated by compounding to express seven *individuals* each (as opposed to the abstract concept of seven-ness). Finally, âof nineâ 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.

qilfRĂ nĂhyn. | 7â9 | ||

seven-cons-each-acc^{1} nine-acc-partacc^{2}. | |||

trilRĂ rĂjyn. | 0.3_{hex} = 3â16 | ||

three-cons-each-acc^{1} 16-acc-partacc^{2}. | |||

xtilRĂ pnĂyn. | 1.9Ì_{hex} = 1.6 = 8â5 | ||

eight-cons-each-acc^{1} five-acc-partacc^{2}. |

We do not need to compound the object ânineâ with rhĂ. â it already refers to each of the seven-nesses separately.

## Grouping numerals

Verb | Translation | ||
---|---|---|---|

Inner factive | Inner accusative | Inner dative | |

qmĂ . | to group | a group | something grouped |

mĂ h. | to sort, to group according to type | a sort, a type | something sorted |

krĂ j. | 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 |

qmĂ . is the topmost grouping numeral in terms of the semantic tree; sortĂ . and krĂ j. denote sub-categories. A further sub-category verb would be âqueueâ. Grouping numerals combine a number of individuals (in the dative) to form a âsuper-individualâ (the accusative: groups, sorts or ensembles). The plot resembles that of the verb âsplitâ. The groups, sorts or ensembles can be counted just like ordinary individuals.

There is no rule limiting the number of combined individuals, so a âgroup of peopleâ can comprise a single person or even none at all.

cOĂc qmĂŹy. â qmĂ cOĂci. | grouped people â a group of people | ||

human-acc^{1} group-dat-acc^{2}. â group-acc^{1} human-acc-dat^{2}. | |||

trĂ qmĂŹy. â qmĂ trĂi. | three grouped â a group of three | ||

three-acc^{1} group-dat-acc^{2}. â group-acc^{1} three-acc-dat^{2}. | |||

qmĂ trĂy. | three groups | ||

group-acc^{1} three-acc-acc^{2}. | |||

Sentences with a grouping numeral as main predicate: | |||

mĂ h wzrukĂŹn RĂcjy. | The umbrellas (tool noun) are sorted, the sorts are coloured things. | He sorts the umbrellas by colour. | |

sort-fact^{1} umbrella-ins-partdat^{2} colour-acc-acc^{2}. | |||

mĂ h wzrukĂŹn wyncgĂ lĂnxwy. | He sorts the umbrellas into black and green ones. | ||

sort-fact^{1} umbrella-ins-partdat^{2} black-partacc-acc^{2} green-partacc-acc^{2}. | |||

qtrĂ wzrukĂŹn lĂty. | He sorts/arranges the umbrellas by weight. | ||

arrange-fact^{1} umbrella-ins-partdat^{2} heavy-acc-acc^{2}. |

The umbrellas in the last example are not âsortedâ (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 âmake/become much, quite a lot, a bitâ 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 shouldnât be difficult to substitute the missing items, as well as translations for the remaining cases. The topmost weighting numeral is rĂ w. âmake an amountâ.

Verb | Gloss | Relative weight | Approximate translation with inner accusative and various outer cases | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Plot cases | Temporal (aR) | Durative (yR) | Locative (ar) | Extensive (yr) | |||||||

rĂ w. | 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 | ||||

nĂ . | not | 0 | none, nothing | never | momentary | nowhere | punctiform | ||||

cĂ wb. | 1â8 | 1â8 approx. | hardly any(thing) | hardly ever | hardly anywhere | ||||||

crĂ . | 1â4 | 1â4 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 | ||||

RĂ bv. | 3â8 | 3â8 approx. | some, a fairly small number/amount | sometimes | in some places, in a fairly small area | ||||||

bvĂ . | 1â2 | 1â2 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 | ||||||

dmĂ j. | 5â8 | 5â8 approx. | quite a lot | quite often | in quite a number of places, in a fairly large area | ||||||

dmĂ . | 3â4 | 3â4 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 | ||||

xpĂ j. | 7â8 | 7â8 approx. | almost every, nearly all | nearly always | nearly everywhere | ||||||

jnĂ . | 1â1 | 1 | every, all, the whole | always | always | everywhere | everywhere |

The weights given in the table only describe relations to the other weighting numerals. 1/2Ă. 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 3/8Ă. and less than 5/8Ă..

srĂ q cĂwbaR. â cĂ wb srĂ Rqy. â sraRqcĂ wb. | He hardly ever queues. | ||

queue-fact^{1} 1/8-acc-temp^{2}. â 1/8-fact^{1} queue-temp-acc^{2}. â queue-temp-1/8-fact^{1}. |

Weighting numerals do not have a singular or plural connotation. 3/4Ă. 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 âmuchâ and âmanyâ, we can use compounds from brackets with the definite numeral rĂ. â1â and the indefinite numeral mlĂ. âseveralâ, respectively. The latter needs a partitive to express âmany of the (several) individualsâ, 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).

rĂ dmĂy. â dmyrĂ. | much, a large amount, much space or time, etc. | ||

one-acc^{1} 3/4-acc-acc^{2}. â 3/4-acc-one-acc^{1}. | |||

rĂ jnĂy. â jnyrĂ. | the whole (thing, amount, space, time, etc.) | ||

one-acc^{1} 1/1-acc-acc^{2}. â 1/1-acc-one-acc^{1}. | |||

dmĂ mlĂyn. â mlĂ dmĂny. â dmynmlĂ. | many (individual) things, places or times, etc. | ||

3/4-acc^{1} several-acc-partacc^{2}. â several-acc^{1} 3/4-partacc-acc^{2}. â 3/4-partacc-several-acc^{1}. |

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.

crĂ groupĂy. | they are few / it is a little, and they are groups | a few groups; a small group (a bit, which is a group) | |

1/4-acc^{1} group-acc-acc^{2}. | |||

crĂ groupĂyn. | a few (of the) groups | ||

1/4-acc^{1} group-acc-partacc^{2}. | |||

crĂ groupĂŹy. â groupĂ crĂi. | a few things grouped â a small group | ||

1/4-acc^{1} group-dat-acc^{2}. â group-acc^{1} 1/4-acc-dat^{2}. | |||

crĂ groupĂŹyn. | a few of the grouped things | a few / little of a group | |

1/4-acc^{1} group-dat-partacc^{2}. |

### Zero

nĂ ., 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 âuninvertedâ forms are omitted in the following examples since **queueĂ nĂaR. would violate inversion ban. | |||
---|---|---|---|

nĂ queueĂ Ry. â queueaRnĂ . | He never queues. | ||

not-fact^{1} queue-temp-acc^{2}. â queue-temp-not-fact^{1}. | |||

nĂ queueĂŹy. â queueinĂ . | Nobody is queuing. | ||

not-fact^{1} queue-dat-acc^{2}. â queue-dat-not-fact^{1}. | |||

nĂ queueanĂ Ry. | He never does not queue. | He does nothing but queue. | |

not-fact^{1} queue-fact-not-temp-acc^{2}. |