ConseggioLigure/seed-instruct-lij-eng huggingface dataset:
Ti peu tradue sto testo in ingleise?
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi sto testo in ingleise:
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.
A traduçion in ingleise do testo a l’é:
A modern geometrical version of infinity is given by projective geometry, which introduces "ideal points at infinity", one for each spatial direction.
Traduxi in lengua ingleise:
Do 240 a.C. Eratostene o l’à deuviou o Siasso d’Eratòstene pe isolâ a-a lesta i numei primmi.
A traduçion in ingleise do testo a l’é:
In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers.
Traduxi da-o ligure à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi sto testo da-o ligure à l’ingleise:
Pe-o sòlito i numei negativi en scriti con un segno de negativitæ (o segno meno).
A traduçion in ingleise do testo a l’é:
Negative numbers are usually written with a negative sign (a minus sign).
Ti peu tradue sto testo in ingleise?
Chì Z a vëgne .
A traduçion in ingleise do testo a l’é:
Here the letter Z comes .
Traduxi da-o zeneise à l’ingleise:
E fraçioin peuan ëse maggioî, menoî ò pæge à l’1; peuan ëse ascì poxitive, negative ò dâ 0.
A traduçion in ingleise do testo a l’é:
Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0.
Traduxi sto testo da-o zeneise à l’ingleise:
O pròscimo paragrafo o l’approfondià primma de tutto i numei poxitivi reali.
A traduçion in ingleise do testo a l’é:
The following paragraph will focus primarily on positive real numbers.
Traduxi da-o zeneise à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02.
Traduxi sto testo da-o ligure à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral.
Traduxi in lengua ingleise:
Dæto che manco a segonda giffra dòppo o deximale a ven conservâ, quelle seguente no son scignificative.
A traduçion in ingleise do testo a l’é:
Since not even the second digit after the decimal place is preserved, the following digits are not significant.
Traduxi sto testo da-o zeneise à l’ingleise:
Pe fâ un exempio, 0,999..., 1.0, 1,00, 1,000, ..., representan tutti o numeo naturale 1.
A traduçion in ingleise do testo a l’é:
For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1.
Traduxi sto testo in ingleise:
Donca i numei reali son un sottoinsemme di numei complesci.
A traduçion in ingleise do testo a l’é:
Thus the real numbers are a subset of the complex numbers.
Traduxi sto testo da-o zeneise à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Ti peu tradue sto testo in ingleise?
I numei reali che no son raçionali en diti numei irraçionali.
A traduçion in ingleise do testo a l’é:
Real numbers that are not rational numbers are called irrational numbers.
Traduxi sto testo in ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi sto testo da-o ligure à l’ingleise:
Unna raxon a l’é che no l’existe un algoritmo pe verificâ l’egualiansa de doî numei carcolabili.
A traduçion in ingleise do testo a l’é:
One reason is that there is no algorithm for testing the equality of two computable numbers.
Comm’à l’é a traduçion ingleise de sto testo?
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.
A traduçion in ingleise do testo a l’é:
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.
Comm’à l’é a traduçion ingleise de sto testo?
O primmo o dà l’ordine de l’insemme, mentre che o segondo o ne dà a dimenscion.
A traduçion in ingleise do testo a l’é:
The former gives the ordering of the set, while the latter gives its size.
Traduxi da-o ligure à l’ingleise:
Sta base standard a fa di numei complesci un cian cartexian, ciammou cian complesso.
A traduçion in ingleise do testo a l’é:
This standard basis makes the complex numbers a Cartesian plane, called the complex plane.
Traduxi sto testo da-o ligure à l’ingleise:
I numei complesci de valô assoluo un forman o çercio de l’unitæ.
A traduçion in ingleise do testo a l’é:
The complex numbers of absolute value one form the unit circle.
Ti peu tradue sto testo in ingleise?
Inta coloraçion de dominio e dimenscioin de sciortia en representæ da-o cô e da-a luminoxitæ respettivamente.
A traduçion in ingleise do testo a l’é:
In domain coloring the output dimensions are represented by color and brightness, respectively.
Traduxi da-o ligure à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi in lengua ingleise:
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.
A traduçion in ingleise do testo a l’é:
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.
Traduxi da-o zeneise à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
Traduxi da-o ligure à l’ingleise:
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).
A traduçion in ingleise do testo a l’é:
The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).
Traduxi sto testo da-o zeneise à l’ingleise:
I numei intreghi forman o gruppo ciù piccin e l’anello ciù piccin ch’o contëgne i numei naturali.
A traduçion in ingleise do testo a l’é:
The integers form the smallest group and the smallest ring containing the natural numbers.
Ti peu tradue sto testo in ingleise?
O l’é o protòtipo de tutti i oggetti con unna struttua algebrica pægia.
A traduçion in ingleise do testo a l’é:
It is the prototype of all objects of such algebraic structure.
Traduxi sto testo in ingleise:
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».
A traduçion in ingleise do testo a l’é:
Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.).
Traduxi da-o ligure à l’ingleise:
Ste chì en de propietæ di numei raçionali che se peuan demostrâ e no vëgnan deuviæ comme definiçioin inta matematica.
A traduçion in ingleise do testo a l’é:
These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics.
Comm’à l’é a traduçion ingleise de sto testo?
Da-o momento che o triangolo o l’é isòscele, a = b).
A traduçion in ingleise do testo a l’é:
Since the triangle is isosceles, a = b).
Traduxi in ingleise:
Dæto che c o l’é pari, se se dividde c pe doî s’ottëgne un intrego.
A traduçion in ingleise do testo a l’é:
Since c is even, dividing c by 2 yields an integer.
Traduxi sto testo in ingleise:
A sostituçion de 4y2 con c2 inta primma equaçion (c2 = 2b2) a ne dà 4y2 = 2b2.
A traduçion in ingleise do testo a l’é:
Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.
Traduxi da-o ligure à l’ingleise:
Dæto che b2 o l’é pari, b o dev’ëse pari ascì.
A traduçion in ingleise do testo a l’é:
Since b2 is even, b must be even.
Ti peu tradue sto testo in ingleise?
À tutti i mòddi questo o contradixe l’ipòtexi che n’aggian de fattoî commun.
A traduçion in ingleise do testo a l’é:
However this contradicts the assumption that they have no common factors.
Traduxi sto testo da-o ligure à l’ingleise:
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».
A traduçion in ingleise do testo a l’é:
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.”
Traduxi sto testo in ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Ti peu tradue sto testo in ingleise?
Questo o l’é quello che Zeno o l’à çercou de provâ.
A traduçion in ingleise do testo a l’é:
This is just what Zeno sought to prove.
Traduxi in lengua ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi da-o zeneise à l’ingleise:
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».
A traduçion in ingleise do testo a l’é:
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.
Traduxi da-o ligure à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi da-o zeneise à l’ingleise:
St’incommensurabilitæ a l’é trattâ inti Elementi d’Euclide, Libbro X, Propoxiçion 9.
A traduçion in ingleise do testo a l’é:
This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9.
Ti peu tradue sto testo in ingleise?
De fæto, inte tanti caxi e conceçioin algebriche vegnivan reformulæ con di termini geometrichi.
A traduçion in ingleise do testo a l’é:
In fact, in many cases algebraic conceptions were reformulated into geometric terms.
Traduxi in lengua ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi da-o ligure à l’ingleise:
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».
A traduçion in ingleise do testo a l’é:
However, historian Carl Benjamin Boyer writes that "such claims are not well substantiated and unlikely to be true".
Traduxi da-o zeneise à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed.
Traduxi in lengua ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Traduxi da-o zeneise à l’ingleise:
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.
A traduçion in ingleise do testo a l’é:
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.
Ti peu tradue sto testo in ingleise?
Dirichlet ascì o l’à contribuio a-a teoria generale, coscì comme tanti coblaboratoî a-e applicaçioin in sce st’argomento.
A traduçion in ingleise do testo a l’é:
Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject.
Traduxi in lengua ingleise:
Questo o l’afferma che tutti i intreghi an unn’unica fattorizzaçion inte di numei primmi.
A traduçion in ingleise do testo a l’é:
This asserts that every integer has a unique factorization into primes.
Quæ a l’é a traduçion ingleise de sto testo?
Pe demostrâlo, supponemmo de dividde i intreghi n pe m (donde m o no l’é pægio do zero).
A traduçion in ingleise do testo a l’é:
To show this, suppose we divide integers n by m (where m is nonzero).
Comm’à l’é a traduçion ingleise de sto testo?
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.
A traduçion in ingleise do testo a l’é:
If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once.
Traduxi in lengua ingleise:
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»).
A traduçion in ingleise do testo a l’é:
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").
Traduxi da-o ligure à l’ingleise:
Ste cadeñe d’estenscion fan de mòddo che i numei naturali seggian incorporæ (identificæ) canonicamente inti atri scistemi de numei.
A traduçion in ingleise do testo a l’é:
These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
Traduxi sto testo da-o ligure à l’ingleise:
O primmo grande progresso inte l’astraçion a l’é stæta l’utilizzaçion de giffre pe representâ di numei.
A traduçion in ingleise do testo a l’é:
The first major advance in abstraction was the use of numerals to represent numbers.