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For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. | Ye tarəqan iyad, sund was 5e, imusənən dǎɣ tigzaten anmərdən fal magrad iyan, izar adabara ən magrad atwərdən atwəkfa, mashan ilant taraqan waren ikna as tiɣda tartayat. |
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. | Ilant taraqan assin waren ɣar atwəkna, mashan adobanen iji ən eqal awaren adobat amukən dǎɣ ijitan ashraynen. |
The other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance. | Sanatət təmərwen ad iyan ən taraqan iyad ijrawən kul aniyat atiwənan, ad ɣor samdo ən awatay was sanatət təmərwen, ashəɣəlan fal taraqan aqalan harwa atiwəjan sund eqal dǎɣ tahuskət təknat təmɣare. |
Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work. | Hilbert idar 12 iwətyan darat as Kurt Gödel azizjar tamusne iness, mashan war oleh das iktab asudmar iknan ɣor ashəɣəl wan Gödel. |
In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible. | Dǎɣ tamadasht ən tənna iness tas ataraq kul ən maɗin ad ajrəw adabara, Hilbert isay tadabit as adabara adobat oɣəl ən tamətert as ataraqa iknan war tila. |
The first of these was proved by Bernard Dwork; a completely different proof of the first two, via ℓ-adic cohomology, was given by Alexander Grothendieck. | Tazart dǎɣ təmtar atiwəsəkna as Bernard Dwork; tamətert təknat azalay ad tin sanatət azarnen, as tartit ən milhaw -adique, atiwəkfa as Alexander Grothendieck. |
However, the Weil conjectures were, in their scope, more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. | Aguden, tənaten tin Weil aqalnat, dǎɣ taləsse nasnat, ogdahən ad ataraqa iyan ən Hilbert, izar Weil kala war ermess sund takən ye imaɗinan kul. |
Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem. | Erdős ihak alwaq iyan alhəkan ən azrif; təzəjrət ən alhak təlad ɣor assuhu atiwənhayən ən ataraqa. |
At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute. | Andəran dǎɣ salan ən təməte maqarat,emelah ən faəto ən taraqan win Hilbert ɣor 21e awatay eqal lay wan əssa taraqan ən hebu ən efdan anifrənan ɣor 2000 as Clay Mathematics Institute. |
The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise. | Tənna tan Riemann tatiwəjrah as azajor iness dǎɣ lay wan taraqan win Hilbert, lay wan Smale, lay ən taraqan ən hebu wan efdan, ad deɣ tənnaten tin Weil, dǎɣ ini wan səgdah. |
1931, 1936 3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second? | 1931, 1936 3e falas sanatət təssəjwen amosnen ən iyat tazayt,eqal harkuk tadabit ən akrəwi ən wazaran as adin imdan ən kərumatən ajətnen adobatnen tamsədawt ye ihuk ən was assin? |
— 12th Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field. | 12e asjət ən tamusne tan Kroneəker-Weber fal asjət ən abélientan ən maɗinan oɣadnen dǎɣ edag kul ən maɗinan maqornen. |
1959 15th Rigorous foundation of Schubert's enumerative calculus. | 1959 15e sənto assohen ən aɗin atiwəɣrən wan Schubert. |
1927 18th (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?(b) What is the densest sphere packing? | 1927 18e (a) ilet aylan təssəjwen war izey ariyən ashrut waren ogdeh as karad segdahan?(b) mamos edag ən tabəlaɣen tiknanen tajuten? |
A number is a mathematical object used to count, measure, and label. | Ajət eqal harat aɗinan atiwətkalən ye assedən, eket ad akatab. |
"More universally, individual numbers can be represented by symbols, called numerals; for example, ""5"" is a numeral that represents the number five." | "As alməɣnat iyat, imaɗinan iyad adoben atwəsəkən as eshwələn, sitawəna edan; sund, ""5"" eqal amaɗin assiknən edan səmos". |
Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. | Issidinən ad maɗinan aqalan ayjan ad iketən ən maɗinan, wi əknanen milhaw aqalnen assiwəd, afənaz, assənətfəs, tazunt ad təmɣare. |
Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas, John Wiley & Sons, Feb 24, 2012.Restivo, S. Mathematics in Society and History, Springer Science & Business Media, Nov 30, 1992. | Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Studies in the Otomies and Incas, John Wiley & Sons, Feb 24, 2012.Restivo, S. Mathematics in Society and History, Springer Science & Business Media, Nov 30, 1992. |
During the 19th century, mathematicians began to develop many different abstractions which share certain properties of numbers, and may be seen as extending the concept. | Ɣor sənto ən 19e awatay, iməssedənan assəntan asjət ibəɗitan ajəten azlaynen tuzanen təlaten tiyad ən maɗinan, izar adobətnat adəkəlnat sund təzəjrət ən muzyat. |
A tallying system has no concept of place value (as in modern decimal notation), which limits its representation of large numbers. | Tartit ən asseɣəd war təla muzyat ən tembe ən edaj (sund dǎɣ akatab ən tamara təshrayat), awa ifənazən assəkən ən maɗinan maqornen. |
Brahmagupta's Brāhmasphuṭasiddhānta is the first book that mentions zero as a number, hence Brahmagupta is usually considered the first to formulate the concept of zero. | Brāhmasɗhuțasiddhānta ən Brahmaguɗta eqal alkad azaran waytəjən wəla sund maɗin, dǎɣ alkum Brahmaguɗta eqal sund wazaran ye iji ən muzyat tan wəla. |
In a similar vein, Pāṇini (5th century BC) used the null (zero) operator in the Ashtadhyayi, an early example of an algebraic grammar for the Sanskrit language (also see Pingala). | Dǎɣ azar iyan, Pānini (5e awatay Aɣ.J.-C.) Itkəl enəmetkəl ibrəran (wəla) dǎɣ Ashtadhyayi, assəkən waren imda ən məgridən əmaɗinən ye magrad wan sanskrite (ahanay deɣ Pingala). |
By 130 AD, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for 0 (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. | Ɣor 130 darat J.-C., Ptolémée, iswadən as Hiɗɗarque ad Babylontan, tajən eshwal ye 0 (taɣləlwayt əndarat ad təjətewt zəjret safəla) dǎɣ tartit ən adin sexagésimal itajən dǎɣ ashrut iyan maɗinan ən shəkelan fin grecs. |
Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. | Tihza hintokayat ən Dioɗhante təmədash tənna təjet as awmadinan ən Inde Brahmaguɗta, dǎɣ Brāhmasɗhuțasiddhānta ɣor 628, itkalən imaɗinan ədarətnen ye iji ən iri quadratique ən ini niyan aqimən ijan ashali. |
At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit of the corresponding positive number's numeral. | Dǎɣ azaman iyan, chinoitan saknen imaɗinan əndarətnen ad assərad ən tassəret ən tədnaɣ as maɗin waren eɣshed ibret hulen ən aɣil ən adin wan maɗin maqaran anhəjan. |
Classical Greek and Indian mathematicians made studies of the theory of rational numbers, as part of the general study of number theory. | Iməssidan win Grèce təknat ad win Inde aɣran tamusne maɗinan adutətnen, dǎɣ taberat ən taɣare kul ye tamusne ən maɗinan. |
The concept of decimal fractions is closely linked with decimal place-value notation; the two seem to have developed in tandem. | Muzyat ən ikəruma déəimales oɣad arətay as akətab déəimale ən təmɣar ən edaj ; win assən olahən das ajətan dǎɣ tandem. |
However, Pythagoras believed in the absoluteness of numbers, and could not accept the existence of irrational numbers. | Aguden, Pythagore iflas as emədi ən maɗinan izar tərədawt ən emel ən maɗinan waren adutət. |
By the 17th century, mathematicians generally used decimal fractions with modern notation. | Ɣor 17e awatay,iməssidan tajən ikəruma déəimales ad akətab ən išray. |
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. | Ɣor 1872,nəjiha ye azajor ən təmusnawen tin Karl Weierstrass, (as analmid iness E. Kossak),Eduard Heine, Georg Cantor ad Riəhard Dedekind. |
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. | Weierstrass, Cantor ad Heine əkrassən timusnawen nassən fal inəlkaman waren emədi,aguden Dedekind ikrass taness fal anyat ən akrəwi (Schniƭ) dǎɣ tartit ən maɗinan adutətnen,azunen imaɗinan kul dǎɣ assin jirəbətan lanen tikraš olahnen. |
Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations). | Eqal awen tahašilt ye iji ən tadiwt təknat taharut ən maɗinan ən təlaten ( adabatan kul ən équations ən tartay ajətnen). |
Aristotle defined the traditional Western notion of mathematical infinity. | Aristote aləɣat tamusne ən kal ejədəl b tafuk tigozt ən maɗin waren imədi. |
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. | Mašan ikuy ad imalan maqaran dǎɣ tamusne tatiwəja as Georg Cantor ; en 1895, azizjar alkaɗ fal tamusne iness taynayat ən tidawen, tajənen, jar iyad, imaɗinan anilkamnen waren imda ad simətuyan tənna tan tadiwt. |
"A modern geometrical version of infinity is given by projective geometry, which introduces ""ideal points at infinity"", one for each spatial direction." | "Tənna ən maɗin təšrayat waren imədi tatiwahəka as maɗin asdat,itajən ““idagan aknanen ɣor iban samədo””,ye taberat kul ən afəla”. |
The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De algebra tractatus. | Anyat ən assəkən assarad ən maɗinan imdnen eqal aguden azajor ɣor 1685,dǎɣ De algebra traətatus ən Wallis. |
In 240 BC, Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers. | Ɣor 240 darat J.-C., Eratosthène itaj tamay tan Eratosthène ye azalay šik ən maɗinan azarnen. |
Other results concerning the distribution of the primes include Euler's proof that the sum of the reciprocals of the primes diverges, and the Goldbach conjecture, which claims that any sufficiently large even number is the sum of two primes. | Igarawən iyan oraknen as təzunt ən maɗinan azarnen sigušən tamətert tan Euler as amərtiy milhaw ən maɗinan azarnen nakəš, izar tənna tan Goldbaəh, ijənen as maɗin assin igdahən maqar eqal tartit ən assin maɗinan azarnen. |
Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) | Dǎɣ tigozst,anəlkəm ən maɗinan aknanen issənta as 1 (0 war atwəja eqalan sund maɗin ɣor Greətan arunen). |
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. | Dǎɣ tartit ən ider 10,adin ihan aɣil ən maɗin iknan ikraš təmɣare ən edaj ən 1, aɗin kul ikraš təmɣare ən edaj maraw dəjan ən tijra ye tan aɗin dǎɣ aɣil iness. |
Negative numbers are usually written with a negative sign (a minus sign). | Imaɗinan madroynen eqalən ayiktabən as ašəkel ifanazən (ašəkel ifanəzan). |
Here the letter Z comes . | Diha, z malid. |
Fractions can be greater than, less than, or equal to 1 and can also be positive, negative, or 0. | Ikəruma adoben adəqəlan tijra,təmədrit meɣ migdəhaw as 1 izar adoben deɣ adəqəlan ajit,afənaz meɣ migdəhaw. |
The following paragraph will focus primarily on positive real numbers. | Tazune adəmalat tatiwətkal ɣas fal imaɗinan adutətnen maqornen. |
Thus, for example, one half is 0.5, one fifth is 0.2, one-tenth is 0.1, and one fiftieth is 0.02. | Dǎɣ awen,sund,tazune togdah ad 0,5,was səmos ogdah ad 0,2,was assin ogdah ad 0,1 ad was simossat təmərwen ogdah 0,02. |
Not only these prominent examples but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral. | Iden ɣas issəkna janen,mašan išwar imaɗinan adutətnen kul aqalan aduten izar warlen tənaten amisaɣəlnen falas war tila aɗin ən təfrəst anihəjan. |
Since not even the second digit after the decimal place is preserved, the following digits are not significant. | Falas imaness aɗin was assin darat tafrist war atwagəz,aɗinan win asdat war ənfen. |
For example, 0.999..., 1.0, 1.00, 1.000, ..., all represent the natural number 1. | Sund, 0,999…,1,0,1,00,1,000,…, saknen imaɗinan kul aknanen 1. |
Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place. | Dǎɣ samdo afəl aɗinan kul ən aɗin aqalan 0, maɗin eqal 0, afəl deɣ aɗinan kul ən aɗin aqalan anəlkəm waren imədu ən 9, tədobem ad təyəm tedut ən 9 as aɣil ən təfrəst, izar ad assiwəd ən iyan ye anəlkəm ən 9 as tašalje ən təfrəst. |
Thus the real numbers are a subset of the complex numbers. | Darat awen, imaɗinan iknanen eqal daw tartit ən maɗinan imdanen. |
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. | Tamusne tazarat ən assedən təlaɣat as imaɗinan əmdanen tajən edag ən adin eharən,awa stanitwəna as tartitayen kul ən təməhakaten ən təmɣar timdat ye raəine dǎɣ maɗinan əmdanen. |
The primes have been widely studied for more than 2000 years and have led to many questions, only some of which have been answered. | Imadinan azarnen atwəɣran səjət aru ajaran 2000 ans elan izar ifan edag ye assestanən ajətnen,iss iyad ɣas ajrəwnen asudmar. |
Real numbers that are not rational numbers are called irrational numbers. | Imaɗinan adutətnen waren eqel imaɗinan waren ija. |
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. | Imaɗinan adobatnen iɗan asuken ye sidinan kul ən maɗinan arunen,as iha aɗin wan raəines ən tartit,tajan edag iharan aduten etafan imaɗinan naɗin adutətnen. |
One reason is that there is no algorithm for testing the equality of two computable numbers. | Iyat dǎɣ tiditen taqal as war tila assedən ye tirma ən migdəhaw ən assin maɗinan adobatnen edan. |
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. | Tartit ən maɗinan təsjarat tazlay ašrut atwətkalən ye maɗinan: ašrut kul adobat,mašan ašrut ən maɗinan azarnen ihak tikraš ən maɗinan tihudkatnen. |
The former gives the ordering of the set, while the latter gives its size. | Ta tazarat haku təzart ən tadiwt, aguden ta sanatət haku tebəde iness. |
This standard basis makes the complex numbers a Cartesian plane, called the complex plane. | Ašrut wen darussən itaj imaɗinan iknanen ahanay raqissən, sitawəna ahanay raqissən. |
The complex numbers of absolute value one form the unit circle. | Imaɗinan raqisnen ən təmɣar təmdat tajan aɣlilway iyan. |
In domain coloring the output dimensions are represented by color and brightness, respectively. | Dǎɣ assəhəsku nədaj,iketan fin azagar atiwəsəknən dǎɣ alkum as iri ad amlulu. |
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. | Ašəɣəlan mertayən ajətnen amdən agad ən tamusne maqarat tan aɗin,səknet ad maɗinan raqisnen,ilet adabara ye təmətuyt tartayat tamɣar iyan meɣ ajen. |
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. | Takətawt tan Wessel tazjarid dǎɣ ijitan win Aəadémie wan Coɗenhague mašan tokay hulen as iban ahanay. |
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. | Məssawəs iknanen alkamnen ən tamusne tajen amossən Richard Dedekind, Oƭo Hölder, Felix Klein, Henri Poinəaré, Hermann Səhwarz, Karl Weierstrass ad iyad. |
The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). | Atkul ən maɗinan anizjamatnen war ikna atwərd dat ašəɣəlan win Leonhard Euler ( 1707-1783) ad Carl Friedriəh Gauss ( 1777-1855). |
The integers form the smallest group and the smallest ring containing the natural numbers. | Inəmda tajan jəruɗ wa iknan təmadrit ad aɣalay wa iknan təmadrit etafən imaɗinan iknanen. |
It is the prototype of all objects of such algebraic structure. | Eqal iri ən haratan kul ən təməkrust ən maɗin. |
Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, Delphi, etc.). | Iritan ən ihukan ohaznen ən maɗinan əmdanen ən təzəjrət təbdadət (meɣ madin ətəkan)atiwəktabən int meɣ Integer dǎɣ magradan ajətnen ən təqan (sund Alol68, C, Jaɣa, Delɗhi, etc.). |
These are provable properties of rational numbers and positional number systems, and are not used as definitions in mathematics. | Adar tidila akarašat tatiwəsəknat ən maɗinan gadəhnen, izar war atwəjənat sund tənaten dǎɣ maɗinan. |
Since the triangle is isosceles, a = b). | Falas triangle taqal aylan assin ašrutan, a=b. |
Since c is even, dividing c by 2 yields an integer. | Falas ə eqal assin, tazunt ən ə as assin haku amaɗin imdan. |
Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2. | Dǎɣ asəmutiy 4y2 as ə2 dǎɣ akatab wazaran (c2=2b2), nəjrəw 4y2=2b2. |
Since b2 is even, b must be even. | Falas b2 eqal assin, b adəqəl assin. |
However this contradicts the assumption that they have no common factors. | Aguden, awen isabəhaw tənna tas inta warlen maɗinan iyadǎɣ. |
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.” | Hippasus aguden, war atwahəka ye tirmaten: dǎɣ tafust iyat,ija ajaraw iness dǎɣ ebanɣ ad dǎɣ alkum atwajarin falas ijim as midawan iness Pythagortan” … ye ajaraw ən iji ən harat dǎɣ təməte tasibəhawət as…tamusne tas inta ijitan kul dǎɣ təməte adoben afənaz as maɗinan imdanen ad agədah nassən.” |
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. | Sund, ajatanəɣ ašrut ən aɣil: ašrut wen adobat tazunt as assin, tašrut ten tazunat as assin, tazune ən tazune as assin, as dǎɣ ikuy. |
This is just what Zeno sought to prove. | Awen dǎɣ as imaɣ Zénon ye atissəkən. |
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. | Dǎɣ anyat ən greətan, adgul ən tarədawt ən tənna war assəkna əs tahašilt tarədawt ən iyat,tahašilt alkum ən umɣawən. |
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. | Tamɣare…war təkel maɗin mašan sakna haratan sund išrutən ən tasaret, tidənaɣ, idagan, tijutawen ad taqanen adobatnen asəmutiy, sund nəjane,as alməɣnat okayan. |
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. | Falas tamɣar tajen wala iyat war tətwəhaka ye tamɣaren, Eudoxe eqal dǎɣ tadabit tənna təjat ən ijitan atiwəsəntanen ad waren atwəsəta dǎɣ alaɣi iji ən təlat ən təmɣar, ad anəhəj sund migdahaw jar assin ijitan. |
This incommensurability is dealt with in Euclid's Elements, Book X, Proposition 9. | Iban tanətit ten tatiwəɣra dǎɣ haratan win Euəlide, alkad X, tenahəjit 9 |
In fact, in many cases algebraic conceptions were reformulated into geometric terms. | As tidit, dǎɣ dajan ajətnen, timusnawen ən maɗinan ašəzatnat as iritan. |
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. | Adkul nanyat adakal iyan nider dǎɣ amass ən tamusne titəlat war tərda ad tidit tašihəšal umax imdan ad zəjren fal tənaten ad issəkna azalnen tamusne. |
"However, historian Carl Benjamin Boyer writes that ""such claims are not well substantiated and unlikely to be true""." | "Mašan awtəfas Carl Benjamin Boyer iktab as”” tənaten tin war iknenat atwilal izar lanat tabaɣort əndarat ye oɣəl ən tidit””. |
Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed. | Iməssidan sund Brahmaguɗta (ɣor 628 AD) ad Bhāskara I (ɣor 629 AD) iwayanid tilalen dǎɣ adag wen, kul sund iməssidan iyad wi alkamnen. |
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. | Awatay wan 1872 ihanəy tazəjort ən timusnawen tin Karl Weierstrass, (as eməssaɣar iness Ernst Kossak), Eduard Heine (jornal wan Ərelle,74),Georg Cantor (Annalen,5)ad Richard Dedekind. |
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. | Weierstrass, Cantor ad Heine assiɣaymən timusnawen nassən fal anəlkəm waren imədi,aguden Dedekind assiɣayma taness fal anyat ən aɣatas (Schniƭ)dǎɣ tartit ən maɗinan kul atwəjanen,tan azunen as assin gruɗtan lanen tikraš olahnen. |
Dirichlet also added to the general theory, as have numerous contributors to the applications of the subject. | Dirichlet deɣ assewad ye tamusne maqarat, kul sund imilalən ajitnen ye ijitan ən awadəm. |
This asserts that every integer has a unique factorization into primes. | Awen ijraw alavi as mapin imdan kul ila anətfəs iyan as imapinan azarnen. |
To show this, suppose we divide integers n by m (where m is nonzero). | Awen aleɣat as maɗin kul imdan ila assənətfəs iyan dǎɣ maɗinan azarnen. |
If 0 never occurs, then the algorithm can run at most m − 1 steps without using any remainder more than once. | Ye assəkən iness, nətkal as nətuzan inəsumda n as m (dihad m war eqel wala). |
"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"")." | Dǎɣ maɗinan, imaɗinan aknanen eqalan wi atwəjanen ye assedən (sund dǎɣ”” ilanət saɗissət ɗieəeten ən amanu fal tabalt””)ad amzizər (sund dǎɣ ““awen tas karadat təknat təmɣare taɣrəmt ən akal””)”. |
These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems. | Imzizəran ajətnen təjan as imaɗinan iknanen aqalan tərəɣse as ugiš (atiwəzay)dǎɣ tirtay tiyaɗnen ən maɗinan. |
The first major advance in abstraction was the use of numerals to represent numbers. | Təzart maqarat ən ikuy dǎɣ Iban adatu eqal iji ən aɗinan ye assəkən ən maɗinan. |