ConseggioLigure/seed-instruct-eng-lij huggingface dataset:
Translate this sentence to Genoese:
However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers.
The Ligurian (Genoese) translation is:
À tutti i mòddi Pitagora o creddeiva inte l’assolutessa di numei e o no poeiva açettâ l’existensa de numei irraçionali.
Translate to Ligurian (Genoese):
By the 17th century, mathematicians generally used decimal fractions with modern notation.
The Ligurian (Genoese) translation is:
Do secolo dïsette i studioxi de matematica deuviavan, pe commun, de fraçioin deximæ con unna nottaçion moderna.
Can you translate this text to Ligurian?
In 1872, the publication of the theories of Karl Weierstrass (by his pupil E. Kossak), Eduard Heine, Georg Cantor, and Richard Dedekind was brought about.
The Ligurian (Genoese) translation is:
Do 1872 gh’é stæto a pubricaçion de teorie de Karl Weierstrass (da-o seu allevo E. Kossak), Eduard Heine, Georg Cantor e Richard Dedekind.
Translate from English to Ligurian:
Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of real numbers, separating all rational numbers into two groups having certain characteristic properties.
The Ligurian (Genoese) translation is:
Weierstrass, Cantor e Heine basan e seu teorie in sce de serie infinie, tanto che Dedekind o fonda a seu in sce l’idea de un taggio (Schnitt) into scistema di numei reali, co-o spartî tutti i numei raçionali inte doî gruppi con de propietæ caratteristiche determinæ.
Translate to Ligurian (Genoese):
Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations).
The Ligurian (Genoese) translation is:
Donca l’ea neçessäio consciderâ l’insemme ciù largo di numei algebrichi (tutte e soluçioin d’equaçioin polinomiali).
Translate from English to Ligurian:
Aristotle defined the traditional Western notion of mathematical infinity.
The Ligurian (Genoese) translation is:
Aristotele o l’à definio a noçion tradiçionale de infinio matematico deuviâ in Occidente.
Translate this sentence to Genoese:
But the next major advance in the theory was made by Georg Cantor; in 1895 he published a book about his new set theory, introducing, among other things, transfinite numbers and formulating the continuum hypothesis.
The Ligurian (Genoese) translation is:
Ma o grande progresso succescivo inta teoria o l’é stæto fæto da Georg Cantor; do 1895 o l’à pubricou un libbro in sciâ seu neuva teoria di insemmi e o l’à introduto, tra e atre cöse, i numei transfinii, formolando l’ipòtexi do continuum.
Translate to Genoese:
A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity", one for each spatial direction.
The Ligurian (Genoese) translation is:
Unna verscion geometrica moderna de l’infinio a l’é dæta da-a geometria proiettiva, ch’a l’introduxe di «ponti ideali à l’infinio», un pe ògni direçioin spaçiale.
Translate from English to Ligurian (Genoese dialect):
In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers.
The Ligurian (Genoese) translation is:
Do 240 a.C. Eratostene o l’à deuviou o Siasso d’Eratòstene pe isolâ a-a lesta i numei primmi.
Translate this sentence to Ligurian:
In this base 10 system, the rightmost digit of a natural number has a place value of 1, and every other digit has a place value ten times that of the place value of the digit to its right.
The Ligurian (Genoese) translation is:
Inte sto scistema à base 10, a giffra ciù a-a drita de un numeo naturale a l’à valô de leugo de 1 e tutte e atre giffre an un valô de leugo dexe vòtte quello da giffra a-a drita.
What’s the Ligurian translation of this sentence?
Negative numbers are usually written with a negative sign (a minus sign).
The Ligurian (Genoese) translation is:
Pe-o sòlito i numei negativi en scriti con un segno de negativitæ (o segno meno).
Translate from English to Ligurian:
Here the letter Z comes .
The Ligurian (Genoese) translation is:
Chì Z a vëgne .
What’s the Genoese translation of this sentence?
Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0.
The Ligurian (Genoese) translation is:
E fraçioin peuan ëse maggioî, menoî ò pæge à l’1; peuan ëse ascì poxitive, negative ò dâ 0.
Translate from English to Genoese:
The following paragraph will focus primarily on positive real numbers.
The Ligurian (Genoese) translation is:
O pròscimo paragrafo o l’approfondià primma de tutto i numei poxitivi reali.
What’s the Ligurian translation of this sentence?
Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.
The Ligurian (Genoese) translation is:
Coscì, pe fâ un exempio, unna meitæ a l’é 0,5, unna quinta parte a l’é 0,2, unna dexima parte a l’é 0,1 e unna çinquantexima parte a l’é 0,02.
What’s the Ligurian translation of this sentence?
Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral.
The Ligurian (Genoese) translation is:
No solo che sti exempi prominenti, ma squæxi tutti i numei reali son irraçionali e donca n’an de schemi repetitivi, e pe conseguensa manco nisciun numeo deximâ correspondente.
Translate from English to Genoese:
Since not even the second digit after the decimal place is preserved, the following digits are not significant.
The Ligurian (Genoese) translation is:
Dæto che manco a segonda giffra dòppo o deximale a ven conservâ, quelle seguente no son scignificative.
Translate to Genoese:
For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1.
The Ligurian (Genoese) translation is:
Pe fâ un exempio, 0,999..., 1.0, 1,00, 1,000, ..., representan tutti o numeo naturale 1.
Translate from English to Ligurian:
Thus the real numbers are a subset of the complex numbers.
The Ligurian (Genoese) translation is:
Donca i numei reali son un sottoinsemme di numei complesci.
Translate to Ligurian:
The fundamental theorem of algebra asserts that the complex numbers form an algebraically closed field, meaning that every polynomial with complex coefficients has a root in the complex numbers.
The Ligurian (Genoese) translation is:
O teorema fondamentale de l’algebra o l’afferma che i numei complesci forman un campo algebricamente serrou; questo o scignifica che tutti i polinòmmi con di coeffiçenti complesci an unna reixe inti numei complesci.
What’s the Genoese translation of this sentence?
Real numbers that are not rational numbers are called irrational numbers.
The Ligurian (Genoese) translation is:
I numei reali che no son raçionali en diti numei irraçionali.
Translate this sentence to Ligurian:
The computable numbers are stable for all usual arithmetic operations, including the computation of the roots of a polynomial, and thus form a real closed field that contains the real algebraic numbers.
The Ligurian (Genoese) translation is:
I numei carcolabili en stabili pe tutte e operaçioin aritmetiche abituale, compreiso o carcolo de reixe de un polinòmio, e donca forman un campo reale serrou ch’o contëgne i numei algebrichi no reali.
Translate this sentence to Ligurian:
One reason is that there is no algorithm for testing the equality of two computable numbers.
The Ligurian (Genoese) translation is:
Unna raxon a l’é che no l’existe un algoritmo pe verificâ l’egualiansa de doî numei carcolabili.
What’s the Ligurian translation of this sentence?
The number system that results depends on what base is used for the digits: any base is possible, but a prime number base provides the best mathematical properties.
The Ligurian (Genoese) translation is:
O scistema numerico ch’o ne sciòrte o depende da-a base ch’a s’addeuvia pe-e giffre: tutte e base en boñe, ma un numeo primmo o fornisce e megio propietæ matematiche.
Translate to Ligurian:
The former gives the ordering of the set, while the latter gives its size.
The Ligurian (Genoese) translation is:
O primmo o dà l’ordine de l’insemme, mentre che o segondo o ne dà a dimenscion.
Translate to Ligurian:
This standard basis makes the complex numbers a Cartesian plane, called the complex plane.
The Ligurian (Genoese) translation is:
Sta base standard a fa di numei complesci un cian cartexian, ciammou cian complesso.
Translate from English to Ligurian (Genoese dialect):
The complex numbers of absolute value one form the unit circle.
The Ligurian (Genoese) translation is:
I numei complesci de valô assoluo un forman o çercio de l’unitæ.
What’s the Ligurian translation of this sentence?
In domain coloring the output dimensions are represented by color and brightness, respectively.
The Ligurian (Genoese) translation is:
Inta coloraçion de dominio e dimenscioin de sciortia en representæ da-o cô e da-a luminoxitæ respettivamente.
Translate to Ligurian:
Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.
The Ligurian (Genoese) translation is:
O travaggio in sciô problema di polinòmmi generali o l’à portou a-o teorema fondamentale de l’algebra, cösa ch’a demostra che con i numei complesci existe una soluçion pe tutte e equaçioin polinomiale de graddo un ò ciù grande.
Translate from English to Ligurian (Genoese dialect):
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.
The Ligurian (Genoese) translation is:
A memöia de Wessel a l’é stæta pubricâ inti Proceedings of the Copenhagen Academy, ma pe-a ciù parte a l’é passâ inosservâ.
Translate this sentence to Ligurian:
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
The Ligurian (Genoese) translation is:
Tra i autoî clascichi vegnui dòppo a teoria generale s’attreuva Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass, e ben ben di atri ancon.
Can you translate this text to Ligurian?
The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).
The Ligurian (Genoese) translation is:
L’utilizzaçion di numei imaginäi o no l’é stæto açettou de forma esteisa fin a-o travaggio de Leonhard Euler (1707-1783) e Carl Friedrich Gauss (1777-1855).
Translate from English to Ligurian:
The integers form the smallest group and the smallest ring containing the natural numbers.
The Ligurian (Genoese) translation is:
I numei intreghi forman o gruppo ciù piccin e l’anello ciù piccin ch’o contëgne i numei naturali.
Translate from English to Ligurian (Genoese dialect):
It is the prototype of all objects of such algebraic structure.
The Ligurian (Genoese) translation is:
O l’é o protòtipo de tutti i oggetti con unna struttua algebrica pægia.
Translate to Ligurian (Genoese):
Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).
The Ligurian (Genoese) translation is:
Inte tanti lenguaggi de programmaçion (comme Algol68, C, Java, Delphi, etc.) i tipi de dæti (ò sottinsemmi) d’approscimaçion di intreghi con longhixe fissa son indicæ comme «int» ò «Integer».
Can you translate this text to Ligurian?
These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.
The Ligurian (Genoese) translation is:
Ste chì en de propietæ di numei raçionali che se peuan demostrâ e no vëgnan deuviæ comme definiçioin inta matematica.
Translate from English to Ligurian (Genoese dialect):
Since the triangle is isosceles, a = b).
The Ligurian (Genoese) translation is:
Da-o momento che o triangolo o l’é isòscele, a = b).
Translate this sentence to Ligurian:
Since c is even, dividing c by 2 yields an integer.
The Ligurian (Genoese) translation is:
Dæto che c o l’é pari, se se dividde c pe doî s’ottëgne un intrego.
Translate from English to Genoese:
Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.
The Ligurian (Genoese) translation is:
A sostituçion de 4y2 con c2 inta primma equaçion (c2 = 2b2) a ne dà 4y2 = 2b2.
Translate from English to Genoese:
Since b2 is even, b must be even.
The Ligurian (Genoese) translation is:
Dæto che b2 o l’é pari, b o dev’ëse pari ascì.
Translate from English to Ligurian:
However this contradicts the assumption that they have no common factors.
The Ligurian (Genoese) translation is:
À tutti i mòddi questo o contradixe l’ipòtexi che n’aggian de fattoî commun.
Translate from English to Ligurian:
Hippasus, however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “...for having produced an element in the universe which denied the...doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.”
The Ligurian (Genoese) translation is:
À tutti i mòddi, Ippaso o no l’é stæto lödou pe-i seu sfòrsci: segondo unna lezzendia o l’à fæto a seu scoverta tanto ch’o l’ea pe mâ e dapeu o l’é stæto cacciou inte l’ægua da-i seu compagni pitagòrichi «pe avei produto un elemento de l’universo ch’o denegava a... dottriña che tutti i fenòmeni de l’universo peuan vegnî reduti à di numei intreghi e a-i seu rappòrti».
Translate this sentence to Genoese:
For example, consider a line segment: this segment can be split in half, that half split in half, the half of the half in half, and so on.
The Ligurian (Genoese) translation is:
Comme exempio, se conscidere un segmento de linea: sto segmento o peu vegnî diviso inte doe parte pæge; a meitæ a peu vegnî divisa ancon inte doe atre parte pæge, e coscì avanti.
Translate from English to Ligurian (Genoese dialect):
This is just what Zeno sought to prove.
The Ligurian (Genoese) translation is:
Questo o l’é quello che Zeno o l’à çercou de provâ.
Can you translate this text to Ligurian?
In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.
The Ligurian (Genoese) translation is:
Inta mente di Greghi, a confutaçion da validitæ de unna çerta vixon a no provava neçessäiamente a validitæ de unn’atra, e donca s’aiva da portâ avanti de atre reçerche.
Translate to Ligurian:
A magnitude “...was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously.
The Ligurian (Genoese) translation is:
Unna grandixe «a no l’ea un numeo, ma a l’indicava unn’entitæ comme segmenti de linee, angoli, aree, volummi e tempo che poeivan cangiâ, comme ghe diëscimo niatri, delongo».
What’s the Genoese translation of this sentence?
Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios.
The Ligurian (Genoese) translation is:
Dæto che no s’assegnava di valoî quantitativi pe-e grandixe, Exodus o poeiva rende conto tanto di rappòrti commensurabili che de quelli incommensurabili co-a definiçion de un rappòrto in termini da seu grandixe e a proporçion comme unn’egualiansa tra doî rappòrti.
Translate this sentence to Genoese:
This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9.
The Ligurian (Genoese) translation is:
St’incommensurabilitæ a l’é trattâ inti Elementi d’Euclide, Libbro X, Propoxiçion 9.
Can you translate this text to Ligurian?
In fact, in many cases algebraic conceptions were reformulated into geometric terms.
The Ligurian (Genoese) translation is:
De fæto, inte tanti caxi e conceçioin algebriche vegnivan reformulæ con di termini geometrichi.
Can you translate this text to Ligurian?
The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that underlie that theory.
The Ligurian (Genoese) translation is:
O fæto de dâse conto che quarche conçeçion de base drento da teoria existente a l’ea in constrasto co-a realtæ a l’à fæto de mòddo che fïse neçessäio unna reçerca completa e approfondia di asciömi e di presuppòsti che s’attreuvan a-a base de quella teoria.
Translate from English to Ligurian:
However, historian Carl Benjamin Boyer writes that "such claims are not well substantiated and unlikely to be true".
The Ligurian (Genoese) translation is:
À tutti i mòddi, o stòrico Carl Benjamin Boyer o scrive che «de affermaçioin comme queste no son ben comprovæ e l’é improabile che seggian vee».
Translate to Ligurian:
Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed.
The Ligurian (Genoese) translation is:
Di matematichi comme Brahmagupta (do 628 d.C.) e Bhāskara I (do 629 d.C.) an dæto de contribuçioin inte sto campo, coscì comme di atri studioxi de matematica ch’en vegnui dòppo.
Translate to Genoese:
The year 1872 saw the publication of the theories of Karl Weierstrass (by his pupil Ernst Kossak), Eduard Heine (Crelle's Journal, 74), Georg Cantor (Annalen, 5), and Richard Dedekind.
The Ligurian (Genoese) translation is:
Do 1872 l’é stæto pubricou e teorie de Karl Weierstrass (da-o seu allevo Ernst Kossak), Eduard Heine (Crelle’s Journal, 74), Georg Cantor (Annalen, 5) e Richard Dedekind.
Translate from English to Ligurian:
Weierstrass, Cantor, and Heine base their theories on infinite series, while Dedekind founds his on the idea of a cut (Schnitt) in the system of all rational numbers, separating them into two groups having certain characteristic properties.
The Ligurian (Genoese) translation is:
Weierstrass, Cantor e Heine basan e seu teorie in sce de serie infinie, tanto che Dedekind o fonda a seu in sce l’idea de un taggio (Schnitt) into scistema de tutti i numei raçionali, co-o spartîli inte doî gruppi con de propietæ caratteristiche determinæ.
Translate from English to Genoese:
Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
The Ligurian (Genoese) translation is:
Dirichlet ascì o l’à contribuio a-a teoria generale, coscì comme tanti coblaboratoî a-e applicaçioin in sce st’argomento.
Translate from English to Genoese:
This asserts that every integer has a unique factorization into primes.
The Ligurian (Genoese) translation is:
Questo o l’afferma che tutti i intreghi an unn’unica fattorizzaçion inte di numei primmi.
Translate from English to Ligurian:
To show this, suppose we divide integers n by m (where m is nonzero).
The Ligurian (Genoese) translation is:
Pe demostrâlo, supponemmo de dividde i intreghi n pe m (donde m o no l’é pægio do zero).
Translate to Ligurian:
If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once.
The Ligurian (Genoese) translation is:
Se o 0 o no se verifica mai, aloa l’algoritmo o peu exeguî a-o mascimo m - 1 pasci sensa deuviâ nisciun resto ciù de unna vòtta.
Translate to Ligurian:
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
The Ligurian (Genoese) translation is:
Inta matematica, i numei naturali en quelli deuviæ pe contâ (comme «gh’é sëi monæe in sciâ töa») e mette in ordine (comme «sta chì a l’é tersa çittæ ciù grande do paise»).
Translate to Ligurian:
These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
The Ligurian (Genoese) translation is:
Ste cadeñe d’estenscion fan de mòddo che i numei naturali seggian incorporæ (identificæ) canonicamente inti atri scistemi de numei.
Translate to Ligurian (Genoese):
The first major advance in abstraction was the use of numerals to represent numbers.
The Ligurian (Genoese) translation is:
O primmo grande progresso inte l’astraçion a l’é stæta l’utilizzaçion de giffre pe representâ di numei.