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Numerals and Basic Operations

The basics of Thernese mathematical serve the basis of the modern Thernese writing system, and was in fact later adapted to write the Thernese language. As such, many of the symbols are present in the modern writing system and likewise fully pronounceable with Thernese readings. This generally leads to two ways to read out Thernese math notation: the mathematical reading where it is pronounced literally and the linguistic reading where it is read as a proper sentence or phrase using proper Thernese grammar. In the case of mathematical readings, some more archaic sounds are maintained that are no longer present in colloquial Thernese due to some sound mergers creating ambiguity in the mathematical reading if applied.

One peculiarity of Thernese math notation is that numbers and basic operations are often written as one unit. In other words, a given numeral or symbol representing a number usually has a basic operation built into it which is applied when combining the numeral with another numeral. Other unique features include the preference for working in hexadecimal and basic numeral symbols that can be broken down into binary constituents.

The table below outlines the basic neutral numerals of Thernese. These numerals imply neither positive nor negative values, and are denoted by the neutral vowel base ɛဝɜ a. Their linguistic counterparts are also given.

Modern	Classical	Value	Math Reading	Linguistic Reading
ɛဝɜ (ဝ)	𝌆 (〇)	0	xa	.ɛခုံƨ. (中) glurn
ɛဝs	𝌡	1	ha	.sစƨ. (周) zal
ɛဝƨ	𝌏	2	ga	.ɛပုံs. (閑) hlien
ɛဝɛ	𝌪	3	ka	.sဗ့s. (少) tharq
sဝɜ	𝌉	4	nga	.sပံs. (羨) thien
sဝs	𝌤	5	tha	.sဝုɛ. (差) sla
sဝƨ	𝌒	6	za	.sဓံƨ. (增) zrn
sဝɛ	𝌭	7	sa	.ƨပɜ. (銳) lie
ƨဝɜ	𝌇	8	la	.ɛဓုံƨ. (更) glrn
ƨဝs	𝌢	9	dha	.ƨဒံs. (斷) luon
ƨဝƨ	𝌐	10	ta	.sဒုံƨ. (裝) zluon
ƨဝɛ	𝌫	11	na	.sခံƨ. (眾) zurn
ɜဝɜ	𝌊	12	va	.sပုs. (睟) slie
ɜဝs	𝌥	13	ba	.sဓံɛ. (盛) srn
ɜဝƨ	𝌓	14	pa	.ɜဝ့ɜ. (法) vaq
ɜဝɛ	𝌮	15	ma	.ɛမံɜ. (應) yirn

Thernese numerals also have additive and multiplicative forms. The additive numbers and their inverses are given in the table below. Additive numbers are denoted by the vowel base ɛပɜ ye and imply addition when combined with another number. Their inverses are denoted by the vowel base ɛဒɜ wo.

Additive				Inverse Additive
Modern	Classical	Value	Pronunciation	Modern	Classical	Value	Pronunciation
ɛပɜ (ပ)	䷀ (乙)	+0	ye	ɛဒɜ (ၥ)	䷌ (兀)	-0	wo
ɛပs	䷪	+1	hie	ɛဒs	䷰	-1	huo
ɛပƨ	䷍	+2	gie	ɛဒƨ	䷝	-2	guo
ɛပɛ	䷡	+3	kie	ɛဒɛ	䷶	-3	kuo
sပɜ	䷈	+4	ngie	sဒɜ	䷤	-4	nguo
sပs	䷄	+5	thie	sဒs	䷾	-5	thuo
sပƨ	䷙	+6	zie	sဒƨ	䷕	-6	zuo
sပɛ	䷊	+7	sie	sဒɛ	䷣	-7	suo
ƨပɜ	䷉	+8	lie	ƨဒɜ	䷘	-8	luo
ƨပs	䷹	+9	dhie	ƨဒs	䷐	-9	dhuo
ƨပƨ	䷥	+10	tie	ƨဒƨ	䷔	-10	tuo
ƨပɛ	䷵	+11	nie	ƨဒɛ	䷲	-11	nuo
ɜပɜ	䷼	+12	vie	ɜဒɜ	䷩	-12	vuo
ɜပs	䷻	+13	bie	ɜဒs	䷂	-13	buo
ɜပƨ	䷨	+14	pie	ɜဒƨ	䷚	-14	puo
ɜပɛ	䷒	+15	mie	ɜဒɛ	䷗	-15	muo

When denoting higher place values using zero, the base vowel of zero harmonizes with the number representing the highest place value, and the number of non-ones place values is glottalized. For example, .ɛပ့ƨ.ɛပɜ. gieq ye would be 20₁₆ where zero .ɛဒƨ. xa changes to positive .ɛပɜ. ye and .ɛပƨ. gie is glottalized as .ɛပ့ƨ. gieq. On the other hand, -20₁₆ would be represented as .ɛဒ့ƨ.ɛဒɜ. guoq wo where zero .ɛဒƨ. xa changes to negative .ɛဒɜ. wo and .ɛဒƨ. guo is glottazlied as .ɛဒ့ƨ. guoq. When zero is replaced by a different number filling in that place value, then the number also harmonizes vocalically. For example, 23₁₆ would be .ɛပ့ƨ.ɛပɛ. gieq kie and -23₁₆ would be .ɛဒ့ƨ.ɛဒɛ. guoq kuo.

When denoting an equation, the symbol “≺” (or “⋏” when written vertically) is used and is generally read as gr (個) when using mathematical pronunciation. Some examples of equations using addition and subtraction are given below.

Multiplicative numerals are denoted by rhoticizing the additive numbers as shown in the table below.

Multiplicative				Inverse Multiplicative
Modern	Classical	Value	Math Reading	Modern	Classical	Value	Math Reading
ɛမɜ (မ)	䷫ (乜)	x0	yir	ɛခɜ (ခ)	䷠ (夭)	1/0	wur
ɛမs	䷛	x1	hir	ɛခs	䷞	1/1	hur
ɛမƨ	䷱	x2	gir	ɛခƨ	䷷	1/2	gur
ɛမɛ	䷟	x3	kir	ɛခɛ	䷽	1/3	kur
sမɜ	䷸	x4	ngir	sခɜ	䷴	1/4	ngur
sမs	䷯	x5	thir	sခs	䷦	1/5	thur
sမƨ	䷑	x6	zir	sခƨ	䷳	1/6	zur
sမɛ	䷭	x7	sir	sခɛ	䷎	1/7	sur
ƨမɜ	䷅	x8	lir	ƨခɜ	䷋	1/8	lur
ƨမs	䷮	x9	dhir	ƨခs	䷬	1/9	dhur
ƨမƨ	䷿	x10	tir	ƨခƨ	䷢	1/10	tur
ƨမɛ	䷧	x11	nir	ƨခɛ	䷏	1/11	nur
ɜမɜ	䷺	x12	vir	ɜခɜ	䷓	1/12	vur
ɜမs	䷜	x13	bir	ɜခs	䷇	1/13	bur
ɜမƨ	䷃	x14	pir	ɜခƨ	䷖	1/14	pur
ɜမɛ	䷆	x15	mir	ɜခɛ	䷁	1/15	mur

Multiplicative numbers may combine with additive numbers to denote multiplication, while their inverse forms can be used to denote division. Some examples of equations using multiplication and division are given below.

It should be noted that in the examples so far, order does not matter. In fact, it may be better to think of both addition and subtraction as addition and multiplication and division as multiplication. Instead of subtract, the inverse additive of some number is added; and similarly, instead of division, the inverse multiplicative of some number is multiplied. Order does, however, matter when combining addition and multiplication, where multiplication is always solved first.

Combining two multiplicative numbers is also used to express exponents, while combining two inverse multiplicative numbers is used to express roots. This is exemplified in the equations below.

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Variables and Non-Fusional Operations

Variables standing in for some unknown number can be derived by adding a liquid medial to a number. Typically, each additional variable within an expression goes up in its numerical base. For example, where we might use the variables x, y, and z, the Thernese would use ɛဝုs hla, ɛဝုƨ gla, and ɛဝုɛ kla, with ɛဝုs hla denoting the first variable, ɛဝုƨ gla denoting the second variable, and so on. Variables can be given a non-neutral vowel base as well for denoting a subset of numbers. For example, hlie would denote only positive or additive numbers, hluo would denote only negative or inverse additive numbers, hlir would denote only positive multiplicative numbers, and hlur would denote only inverse multiplicative numbers. In classical notation, neutral variables are typically indicated by characters 元 and 次, or the characters of the decimal or duodecimal systems when more than two variables are involved.

In addition to fusional operations (operations expressed through the vowel bases of numerals), the non-neutral zeros (ye, wo, yir, and wur) can also be used to express basic operations. This is particularly useful for algebraic expressions with multiple variables. When non-neutral zeros are used as operators, they are typically replaced with a certain symbol. The additive zero is represented by ပ (乙) ye, the inverse additive zero by ၥ (兀) wo, the multiplicative zero by မ (乜) yir, and the inverse multiplicative zero by ခ (夭) wur. These symbols act as postpositions that are placed after a variable. Examples expressions are given below.

The non-neutral zero operators can also be placed in front of a series of numbers and apply to all that precede it. This also takes precedence in the order of operations and functions similarly to how we use parantheses. In the case of its application to more than two numbers, then the full set of numbers under the scope of the operator are followed by န (之) zlr to denote the end of the set.

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