ConseggioLigure/seed-instruct-eng-lij huggingface dataset:
Translate from English to Ligurian:
Arithmetic expressions must be evaluated according to the intended sequence of operations.
The Ligurian (Genoese) translation is:
E eprescioin aritmetiche dev’ësan valuæ segondo a sequensa d’operaçion che se veu effettuâ.
Translate this sentence to Ligurian:
For example, digital computers can reuse existing adding-circuitry and save additional circuits for implementing a subtraction, by employing the method of two's complement for representing the additive inverses, which is extremely easy to implement in hardware (negation).
The Ligurian (Genoese) translation is:
Pe fâ un exempio, i computer digitali peuan deuviâ torna i circuiti d’addiçion existenti e economizzâ quelli azzontivi pe implementâ unna sottraçion, deuviando o metodo do completemento à doî pe representâ i inversci additivi, che son ben ben façili da implementâ in sce hardware (negaçion).
What’s the Genoese translation of this sentence?
Multiplication also combines two numbers into a single number, the product.
The Ligurian (Genoese) translation is:
A moltiplicaçion ascì a combiña doî numeri inte un numero solo, o produto.
What’s the Ligurian translation of this sentence?
Any dividend divided by zero is undefined.
The Ligurian (Genoese) translation is:
Tutti i dividdendi, divixi pe-o zero, son indefinii.
Translate to Genoese:
The fundamental theorem of arithmetic was first proven by Carl Friedrich Gauss.
The Ligurian (Genoese) translation is:
O teorema fondamentale de l’aritmetica o l’é stæto provou pe-o primmo da Carl Friedrich Gauss.
Translate to Ligurian (Genoese):
Positional notation (also known as "place-value notation") refers to the representation or encoding of numbers using the same symbol for the different orders of magnitude (e.g., the "ones place", "tens place", "hundreds place") and, with a radix point, using those same symbols to represent fractions (e.g., the "tenths place", "hundredths place").
The Ligurian (Genoese) translation is:
A nottaçion poxiçionale (conosciua ascì comme «nottaçion di valoî de poxiçion») a se referisce a-a representaçion ò a-a codifica di numeri, pe mezo do mæximo scimbolo pe-i varri ordini de grandessa (comme «o pòsto de unitæ», «o pòsto de dëxeñe», «pòsto de çentenæa») e con un ponto reixe, pe mezo de quelli mæximi scimboli pe representâ e fraçioin (comme «o pòsto de dexime parte», «o pòsto di çenteximi»).
Translate to Genoese:
The result is calculated by the repeated addition of single digits from each number that occupies the same position, proceeding from right to left.
The Ligurian (Genoese) translation is:
O resultato o ve carcolou con l’azzonze repetuo de giffre sence da tutti i numeri ch’òccupan a mæxima poxiçion, da-a drita a-a manciña.
Translate from English to Genoese:
The rightmost digit is the value for the current position, and the result for the subsequent addition of the digits to the left increases by the value of the second (leftmost) digit, which is always one (if not zero).
The Ligurian (Genoese) translation is:
A giffra ciù a-a drita o l’é o valô da poxiçion do momento e o resultato pe l’addiçion succesciva de giffre a-a manciña a l’aumenta o valô da segonda giffra (ciù a-a manciña), ch’a l’é sempre un (se no zero).
Translate from English to Genoese:
A multiplication table with ten rows and ten columns lists the results for each pair of digits.
The Ligurian (Genoese) translation is:
Unna tabella de moltiplicaçion da dexe righe e dexe colònne a l’elenca i resultati pe tutte e cobbie de giffre.
Translate this sentence to Genoese:
Similar techniques exist for subtraction and division.
The Ligurian (Genoese) translation is:
De tecniche scimili existan pe-a sottraçion e a divixon.
What’s the Genoese translation of this sentence?
In mathematical terminology, this characteristic is defined as closure, and the previous list is described as .
The Ligurian (Genoese) translation is:
Inta terminologia matematica sta caratteristica a l’é definia «ciösua», e l’elenco preçedente o l’é definio comme .
What’s the Ligurian translation of this sentence?
The total in the pence column is 25.
The Ligurian (Genoese) translation is:
O totale, inta colònna di penny, o l’é 25.
Translate to Ligurian:
This operation is repeated using the values in the shillings column, with the additional step of adding the value that was carried forward from the pennies column.
The Ligurian (Genoese) translation is:
St’operaçion a ven repetua deuviando i valoî da colònna di scellin, pe azzonze dapeu o valô stæto reportou inta colònna di penni.
Translate from English to Genoese:
One typical booklet that ran to 150 pages tabulated multiples "from one to ten thousand at the various prices from one farthing to one pound".
The Ligurian (Genoese) translation is:
Un tipico libbretto ch’o l’arrivava a-e 150 pagine o tabulava di moltipli «da un à dexemia a-i prexi da un farthing à unna sterliña».
Translate this sentence to Ligurian:
This study is sometimes known as algorism.
The Ligurian (Genoese) translation is:
Sto studio o l’é ciammou de vòtte co-o nomme de «algorismo».
Can you translate this text to Ligurian?
Also, arithmetic was used by Islamic Scholars in order to teach application of the rulings related to Zakat and Irth.
The Ligurian (Genoese) translation is:
L’aritmetica a l’é stæta deuviâ da-i studioxi islamichi co-a finanlitæ de mostrâ e applicaçioin di decreti ligæ à Zakat e Irth.
Translate from English to Ligurian (Genoese dialect):
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.
The Ligurian (Genoese) translation is:
L’addiçion (pe-o sòlito representâ da-o scimbolo relativo) a l’é uña de operaçion a-a base de l’aritmetica; e atre træ son a sottraçioin, a moltiplicaçion e a divixon.
Can you translate this text to Ligurian?
In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces and subgroups.
The Ligurian (Genoese) translation is:
Inte l’algebra, unn’atra ramma da matematica, l’addiçion a se peu realizzâ ascì pe di oggetti astræti comme vettoî, matrixe, sottospaçi e sottogruppi.
Can you translate this text to Ligurian?
Using the gerundive suffix -nd results in "addend", "thing to be added".
The Ligurian (Genoese) translation is:
Pe mezo do suffisso gerundivo -nd s’ottëgne addend ‘cösa da azzonze’.
Translate from English to Genoese:
"Sum" and "summand" derive from the Latin noun summa "the highest, the top" and associated verb summare.
The Ligurian (Genoese) translation is:
Sum e summand, in ingleise, ne vëgnan da-o sostantivo latin summa ‘a çimma’ e da-o verbo conligou summare.
What’s the Ligurian translation of this sentence?
The later Middle English terms "adden" and "adding" were popularized by Chaucer.
The Ligurian (Genoese) translation is:
I termini ciù tardi de l’ingleise medio, adden e adding, en stæti portæ inta lengua de gente da Chaucer.
Translate to Genoese:
As an example, should the expression a + b + c be defined to mean (a + b) + c or a + (b + c)?
The Ligurian (Genoese) translation is:
Giusto pe fâ un exempio, l’esprescion a + b + c a doviæ ëse definia comme (a + b) + c ò a + (b + c)?
What’s the Genoese translation of this sentence?
Even some nonhuman animals show a limited ability to add, particularly primates.
The Ligurian (Genoese) translation is:
Sciña çerti animæ a-o de feua de l’òmmo dan à divedde unna capaçitæ limitâ pe fâ de addiçioin, comme i primati.
Translate to Ligurian (Genoese):
With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, five."
The Ligurian (Genoese) translation is:
Con ancon ciù tanta esperiensa, i figgeu imprendan à fâ de somme ciù a-a lesta pe mezo da propietæ commutativa de l’addiçion co-o contâ da-o numero ciù grande, inte sto caxo comensando da-o trei e contando «quattro, çinque».
Translate from English to Ligurian:
Zero: Since zero is the additive identity, adding zero is trivial.
The Ligurian (Genoese) translation is:
Zero: da-o momento che o zero o l’é l’identitæ additiva, azzonze un zero no cangia o resultato.
Translate from English to Genoese:
One aligns two decimal fractions above each other, with the decimal point in the same location.
The Ligurian (Genoese) translation is:
Un o l’allinia doe fraçioin deximæ uña in çimma à l’atra, co-o ponto deximâ inta mæxima poxiçion.
Translate this sentence to Ligurian:
If the addends are the rotation speeds of two shafts, they can be added with a differential.
The Ligurian (Genoese) translation is:
Se i addendi son a lestixe de rotaçion de doî erbi, se peuan sommâ con un differençiale.
Translate from English to Ligurian:
It made use of a gravity-assisted carry mechanism.
The Ligurian (Genoese) translation is:
O deuviava un meccanismo de traspòrto ascistio da-a gravitæ.
What’s the Genoese translation of this sentence?
To subtract, the operator had to use the Pascal's calculator's complement, which required as many steps as an addition.
The Ligurian (Genoese) translation is:
Pe sottrae, l’operatô o l’aiva da deuviâ o complemento de carcolatoa de Pascan, ch’a domandava tanti pasci comme unn’addiçion.
Translate this sentence to Ligurian:
Both XOR and AND gates are straightforward to realize in digital logic allowing the realization of full adder circuits which in turn may be combined into more complex logical operations.
The Ligurian (Genoese) translation is:
Tanto e pòrte XOR comme quelle AND en façile da realizzâ in lògica digitale, ch’a permette a realizzaçioin de çircuiti d’addiçion completi che, à seu vòtta, peuan vegnî combinæ inte de operaçioin lògiche ciù complesse.
Translate to Genoese:
Unanticipated arithmetic overflow is a fairly common cause of program errors.
The Ligurian (Genoese) translation is:
L’overflow aritmetico imprevisto a l’é unna raxon pe di erroî de programmaçion ch’a s’attreuva assæ de lengê.
Translate to Ligurian (Genoese):
Taken literally, the above definition is an application of the recursion theorem on the partially ordered set N2.
The Ligurian (Genoese) translation is:
Piggiâ a-a lettia, a definiçion stæta mensunâ de d’ato a l’é unn’applicaçion do teorema de recorscion in sce l’insemme parçialmente ordinou N2.
Translate from English to Ligurian (Genoese dialect):
If either a or b is zero, treat it as an identity.
The Ligurian (Genoese) translation is:
Se a ò b o l’é pægio a-o zero, beseugna trattâli comme de identitæ.
Translate from English to Ligurian (Genoese dialect):
Here, the semigroup is formed by the natural numbers and the group is the additive group of integers.
The Ligurian (Genoese) translation is:
Chì o semigruppo o l’é formou da-i numeri naturali e o gruppo o l’é o gruppo additivo di intreghi.
Translate to Ligurian (Genoese):
The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity.
The Ligurian (Genoese) translation is:
A propietæ commutativa e assoçiativa de l’addiçion reale en immediæ; se se definisce o numero reale 0 comme l’insemme di raçionali negativi se vedde de lengê l’identitæ additiva.
Translate from English to Genoese:
One must prove that this operation is well-defined, dealing with co-Cauchy sequences.
The Ligurian (Genoese) translation is:
Beseugna demostrâ che st’operaçion a l’é ben definia, trattandose de sequense co-Cauchy.
Translate to Ligurian:
The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the "exclusive or" function.
The Ligurian (Genoese) translation is:
L’insemme di numeri intreghi mòdolo 2 o l’à solo che doî elementi: l’operaçion d’addiçion ch’o l’eredita a l’é conosciua, inta lògica booleaña, comme fonçion «exclusive or».
Translate this sentence to Ligurian:
These give two different generalizations of addition of natural numbers to the transfinite.
The Ligurian (Genoese) translation is:
Questi dan doe generalizzaçion despæge de l’addiçion de numeri naturali a-o transinfinio.
What’s the Genoese translation of this sentence?
There are even more generalizations of multiplication than addition.
The Ligurian (Genoese) translation is:
Gh’é ancon ciù tante generalizzaçioin inta moltiplicaçion che inte l’addiçion.
Translate from English to Ligurian:
In fact, if two nonnegative numbers a and b are of different orders of magnitude, then their sum is approximately equal to their maximum.
The Ligurian (Genoese) translation is:
Defæti, se doî numeri no negativi a e b en despægi pe grandessa, aloa a seu somma a l’é pöcassæ pægia a-o seu mascimo.
Translate to Genoese:
It includes the idea of the sum of a single number, which is itself, and the empty sum, which is zero.
The Ligurian (Genoese) translation is:
O comprende l’idea da somma de un numero sencio, ch’o l’é lê mæximo, e a somma veua, ch’a l’é pægia do zero.
Can you translate this text to Ligurian?
Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold.
The Ligurian (Genoese) translation is:
L’integraçion a l’é unna speçie de «sommatöia» in sce un continuo ò, con ciù precixon e in generale, in sce un collettô differençiabile.
Translate from English to Ligurian:
Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.
The Ligurian (Genoese) translation is:
E combinaçioin lineare an unna particolare utilitæ inte quelli contesti donde l’azzonta diretta a l’anieiva contra à quarche regola de normalizzaçion, comme a mescciua de strategie inta teoria di zeughi ò a sorvepoxiçion di stati inta mecanica quantistica.
Translate to Ligurian (Genoese):
Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers.
The Ligurian (Genoese) translation is:
A divixon a l’é uña de quattro operaçioin de base de l’aritmetica, saieiva à dî e mainee che permettan de combinâ di numeri pe cavâne di atri.
Translate to Genoese:
Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate (which define multiplication and addition over single-variabled formulas).
The Ligurian (Genoese) translation is:
Quelli pe-i quæ se definisce unna divixon euclidea (con do resto) piggian o nomme de dominni euclidëi e comprendan di anelli polinomiali inte un indeterminou (che definiscian moltiplicaçion e addiçion in sce de formole con unna variabile solo).
Translate from English to Ligurian:
This division sign is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator.
The Ligurian (Genoese) translation is:
Sto segno de divixon o se deuvia da solo ascì pe representâ a mæxima operaçion de divixon, comme etichetta in sce un pommello de unna carcolatoa.
Translate from English to Genoese:
Distributing the objects several at a time in each round of sharing to each portion leads to the idea of 'chunking' a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself.
The Ligurian (Genoese) translation is:
A distribuçion d’oggetti despægi inte tutti i gii de condivixon pe tutte e porçioin a pòrta à l’idea de chunking, unna forma de divixon donde se sottrae de forma repetua di moltipi do divisô da-o dividdendo mæximo.
Translate from English to Genoese:
A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
The Ligurian (Genoese) translation is:
Unna persoña a peu deuviâ e tabelle di logaritmi pe dividde doî numeri, co-o sottrae i logaritmi di doî numeri, e dapeu ammiando l’antilogaritmo do resultato.
Translate to Genoese:
Some programming languages, such as C, treat integer division as in case 5 above, so the answer is an integer.
The Ligurian (Genoese) translation is:
Çerti lenguaggi de programmaçion, comme o C, trattan a divixon di numeri intreghi comme into caxo 5 mensunou chì in çimma; donca a respòsta a l’é un numero intrego.
Translate from English to Ligurian (Genoese dialect):
Similarly, right division of b by a (written ) is the solution y to the equation .
The Ligurian (Genoese) translation is:
Do mæximo, a divixon drita de b pe a (scrita ) a l’é a soluçion y de l’equaçion.
Translate from English to Genoese:
Examples include matrix algebras and quaternion algebras.
The Ligurian (Genoese) translation is:
Di exempi comprendan l’algebra de matrixe e quella di quaternioin.
What’s the Genoese translation of this sentence?
Entry of such an expression into most calculators produces an error message.
The Ligurian (Genoese) translation is:
Se unn’esprescion comme questa a ven digitâ inte unna carcolatoa s’ottëgne un messaggio d’errô.
Translate to Ligurian:
Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal.
The Ligurian (Genoese) translation is:
Da-o momento che sta sostituçion a redue o ciù grande di doî numeri, co-a repetiçion de sto processo s’ottëgne de cobbie de numeri succescivamente ciù piccin fin à quande i doî numeri vëgnan pægi.
Translate this sentence to Genoese:
The fact that the GCD can always be expressed in this way is known as Bézout's identity.
The Ligurian (Genoese) translation is:
O fæto che o GCD o pòsse vegnî delongo espresso à sta mainea o se conosce comme identitæ de Bézout.
Translate from English to Ligurian:
With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer.
The Ligurian (Genoese) translation is:
Con sta megoria, l’algoritmo o domanda nint’atro che çinque vòtte a quantitæ de giffre (à base 10) de l’intrego ciù piccin.
Translate from English to Genoese:
The Euclidean algorithm has many theoretical and practical applications.
The Ligurian (Genoese) translation is:
L’algoritmo d’Euclide o l’à ben ben d’applicaçion, tanto teòriche comme prattiche.
What’s the Ligurian translation of this sentence?
The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers.
The Ligurian (Genoese) translation is:
L’algoritmo d’Euclide o peu ëse deuviou pe resòlve de equaçioin diofantiñe, pe exempio pe attrovâ di numeri che satisfan de cogruense multiple segondo o teoremo do resto cineise, pe construe de fraçioin continuæ e pe attrovâ d’approscimaçioin raçionale accuræ pe-i numeri reali.
Translate from English to Ligurian:
The greatest common divisor is often written as gcd(a, b) or, more simply, as (a, b), although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers, which is closely related to GCD.
The Ligurian (Genoese) translation is:
O mascimo divisô commun o ven scrito de spesso comme gcd(a, b) ò, con ciù sempliçitæ, comme (a, b), sciben che quest’urtima a l’é ciù ambigua e deuviâ ascì pe di atri conçetti, comme un ideale inte l’anelli di numeri intreghi, ligou à curto fî co-o MDC.
What’s the Genoese translation of this sentence?
For example, neither 6 nor 35 is a prime number, since they both have two prime factors: 6 = 2 × 3 and 35 = 5 × 7.
The Ligurian (Genoese) translation is:
Pe fâ un exempio, ni o 6 e ni o 35 en di numeri primmi, da-o momento che tutti doî an doî fattoî primmi: 6 = 2 × 3 e 35 = 5 × 7.
Translate from English to Genoese:
Factorization of large integers is believed to be a computationally very difficult problem, and the security of many widely used cryptographic protocols is based upon its infeasibility.
The Ligurian (Genoese) translation is:
A fattorizzaçion de numeri intreghi erti a l’é tegnua pe un problema be ben complesso da-o ponto de vista computaçionâ, e e a seguessa de tanti protocòlli criptografichi deuviæ in sce larga scâ a l’é basâ in sciâ seu inattuabilitæ.
Translate from English to Ligurian:
The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer).
The Ligurian (Genoese) translation is:
L’insemme de tutte e combinaçioin lineare integrale de a e b o l’é, in realtæ, o mæximo che l’insemme de tutti i moltipli de g (mg, donde m o l’é un intrego).
What’s the Genoese translation of this sentence?
In other words, multiples of the smaller number rk−1 are subtracted from the larger number rk−2 until the remainder rk is smaller than rk−1.
The Ligurian (Genoese) translation is:
Dita à un atro mòddo, i moltipli do numero ciù piccin rk-1 vëgnan sottræti da quello ciù grande rk-2 fintanto che o resto rk o l’é ciù piccin de rk-1.
Translate from English to Genoese:
Therefore, c divides the initial remainder r0, since r0 = a − q0b = mc − q0nc = (m − q0n)c.
The Ligurian (Genoese) translation is:
Coscì, c o dividde l’arregòrdo r0, dæto a-o comenso, da-o momento che r0 = a − q0b = mc − q0nc = (m − q0n)c.
Translate to Ligurian (Genoese):
We first attempt to tile the rectangle using b-by-b square tiles; however, this leaves an r0-by-b residual rectangle untiled, where r0 < b. We then attempt to tile the residual rectangle with r0-by-r0 square tiles.
The Ligurian (Genoese) translation is:
Pe primma cösa çerchemmo de impî o rettangolo con di quaddri b-by-b; à tutti i mòddi, à sta mainea o rettangolo r0-by-b (donde r0 < b) o l’arresta descoverto. Dapeu çerchemmo de impî o rettangolo ch’o l’arresta con di quaddri r0-by-r0.
Translate from English to Ligurian:
The theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique.
The Ligurian (Genoese) translation is:
O teorema a-a base da definiçion da divixon euclidea o l’assegua che quello quoçiente e o resto existan e son delongo unichi.
Translate from English to Genoese:
At the end of the loop iteration, the variable b holds the remainder rk, whereas the variable a holds its predecessor, rk−1.
The Ligurian (Genoese) translation is:
A-a fin de l’iteraçion do çiclo, a variabile b a tëgne o resto rk, mentre a variabile a a tëgne o seu predeçessô, rk-1.
What’s the Ligurian translation of this sentence?
The mathematician and historian B. L. van der Waerden suggests that Book VII derives from a textbook on number theory written by mathematicians in the school of Pythagoras.
The Ligurian (Genoese) translation is:
O matematico e stòrico B. L. van der Waerden o suggerisce che o Libbro VII o derive da un libbro de testo in sciâ teoria di numeri scrito da di studioxi de matematica da scheua de Pitagora.
What’s the Genoese translation of this sentence?
Centuries later, Euclid's algorithm was discovered independently both in India and in China, primarily to solve Diophantine equations that arose in astronomy and making accurate calendars.
The Ligurian (Genoese) translation is:
De çentanæa d’anni apreuvo, l’algoritmo d’Euclide o l’é stæto descoverto independentemente tanto inte l’India comme inta Ciña, ciù che tutto pe risòlve e equaçioin diofantiñe che nasceivan inte l’astronomia e inta realizzaçion de lunäi esatti.
What’s the Ligurian translation of this sentence?
In the 19th century, the Euclidean algorithm led to the development of new number systems, such as Gaussian integers and Eisenstein integers.
The Ligurian (Genoese) translation is:
Do secolo XIX l’algoritmo euclideo o l’à portou a-o sviluppo de neuvi scistemi de numeraçion, comme i intreghi gaussien e intreghi de Eisenstein.
Translate to Genoese:
Peter Gustav Lejeune Dirichlet seems to have been the first to describe the Euclidean algorithm as the basis for much of number theory.
The Ligurian (Genoese) translation is:
Pâ che Peter Gustav Lejeune Dirichlet o segge stæto o primmo à descrive l’algoritmo euclideo comme a base de gran parte da teoria di numeri.