# 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 . | eleven | B | B | B | 11 |

frĂ . | twelve | C | C | C | 12 |

Ă hs. | 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 . | (none) | # | # | 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 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 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 or individuals. 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.

### 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 isnât normally a problem (a queue, rice, water, walking). Regarding Lemizh grammar, an inâdividual of a discrete thing is just what is says: something that cannot be further divided while remaining what it is. Thus, âthree roomsâ or âtwo sneezesâ 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 âtwo queuesâ, but also âtwo rices, waters, walkingsâ in Lemizh. Such individuals are context dependent: âBuy two ricesâ will be understood as two packages, âCook two ricesâ as two servings; âtwo watersâ can be two servings or two bodies of water; âtwo walkingsâ can be two steps or going for a walk twice, depending on the situation.

In fact, there isnât 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 plantâs 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 âtwo packages of riceâ.

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.

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

Ă j. | 16 | 16 |

skmĂ . | 256 | 16^{2} = 256 |

jĂ s. | 4096 | 16^{3} = 4096 |

mrĂ j. | 65536 | 16^{4} = 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 ĂŹlj. âthe consequence of making sixteen, sixteenânessâ, building them into an accusative bracket, and compounding.

qĂf ĂŹljy. â iljqĂf. | seven consequences of making sixteen individuals; seven sixteen-nesses | 70_{hex} = 112 | |

seven-acc^{1} 16-cons-acc^{2}. â 16-cons-seven-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: **qĂf Ăjy. seven-acc^{1} 16-acc-acc^{2}. is nonsensical because it would mean âthe seven are sixteenâ. 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 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 .-desorption.

mĂ skmynĂ gcĂny. | 10F_{hex} = 271 | |

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

10FĂ. | ||

271-acc^{1}. |

### Larger numbers

Numbers from 10,0000_{hex} = 16^{5} upwards involve multiplication of mrĂj. â1,0000_{hex} = 65536â with a multidigit number, e.g. 13,0000_{hex} = 13_{hex} Ă 1,0000_{hex}. If the multidigit number is constructed with partitive âandâ, i.e. if it is not an exponential or a round number, compounding is impossible.

â mriljynjĂš skmynĂš gcĂne. | sixteen 65536-nesses (compounding works) | 10,010F_{hex} = 1,048,847 | |

âŠ 65536-cons-16-partacc-nom^{2} 256-partacc-nom^{2} fifteen-partacc-nom^{2}. | |||

â mrĂnje ynjĂŹl trynĂl skmynĂš gcĂne. | sixteen and three 65536-nesses (no compounding) | 13,010F_{hex} = 1,245,455 | |

âŠ 65536-partacc-nom^{2} 16-partacc-cons^{3} three-partacc-cons^{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: dyprĂ 2Ăi. cosine-fact^{1} 2-acc-dat^{2}. means âto calculate the cosine of twoâ. The result of the function is the accusative object, just as the lace is the result of the beaverâs action: dyprĂ 2Ăi. cosine-acc^{1} 2-acc-dat^{2}. is the cosine of two. Functions with two arguments have one of them in the dative and one in the nominative: lrĂ 4yĂš 8Ăi. exponentiate-acc^{1} 4-acc-nom^{2} 8-acc-dat^{2}. is âthe 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 concept, we form new exponential numbers that are powers of 65536: we first compound the verb meaning âto exponentiateâ with some power, e.g. two, in the nominative; and then compound the resulting word with 65536 in the dative.

lrĂ dwĂe. â dwĂ lrĂšy. â lredwĂ. | the second power of some number-dat | |

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

lredwĂ mrĂji. â mrĂj lredwĂŹy. â lredwimrĂj. | the second power of 1,0000_{hex} = 1,0000,0000_{hex} = 4,294,967,296 | |

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

â lredwimrĂnje yjĂl rĂne. | 10,0000,0001_{hex} = 68,719,476,737 | |

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

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 dyprĂŹ dwĂy. 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 (âunwiseâ). You can think of frilkĂ. as ââ1 Ă 12â with the consecutive as the multiplication sign and kĂ. as â1 (just as iljqĂf. is 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). Other examples 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 an inner consecutive case for multiplication: RĂ qĂŹlfyn. each-acc^{1} seven-cons-partacc^{2}. âeach of the seven-nessesâ. If we used an accusative as in RĂ qĂfyn. each-acc^{1} seven-acc-partacc^{2}., we would get âeach of seven individualsâ, which is not what we want. Instead, the inner accusative is reinstated by compounding, which gives us qilfRĂ. seven-cons-each-acc^{1}. âseven individuals eachâ. 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. 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.

qilfRĂ nĂhyn. | seven each of nine | 7â9 | |

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

nilhRĂ skmĂyn. | 0.09_{hex} = 9â256 â 3.5% | ||

nine-cons-each-acc^{1} 256-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 denominator with RĂ. 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 . âgroupâ 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 srĂ q. âqueueâ. Grouping numerals combine a number of individuals (in the dative) to form a âsuper-individualâ (the accusative: groups, sorts or ensembles). We can translate qmĂ . as âto make a group-acc from individuals-dat, to turn individuals-dat into a group-accâ; 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 âgroup of peopleâ can in principle 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}. |

The phrases âa group [made] of people / of threeâ are exactly parallel to âlace made from threadâ.

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

mĂ h wzuhkĂŹn 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 wzuhkĂŹn 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 wzuhkĂŹn 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. bvĂ. 1/2-acc^{1}. 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 RĂbv. 3/8-acc^{1}. and less than dmĂj. 5/8-acc^{1}.. Objects with weighting numerals are typically compounded.

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. dmĂ. 3/4-acc^{1}. 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. one-acc^{1}. and the indefinite numeral mlĂ. several-acc^{1}., 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 qmĂy. | they are few / it is a little, and they are groups | ambiguous: a few groups; a small group (a bit, which is a group) | |

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

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

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

crĂ qmĂŹy. â qmĂ 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 qmĂŹyn. | 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 **srĂ q nĂaR. would violate inversion ban. | |||
---|---|---|---|

nĂ srĂ Rqy. â sraRqnĂ . | He never queues. | ||

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

nĂ srĂŹqy. â sriqnĂ . | Nobody is queuing. | ||

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

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

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