ilyasaqit/en-tzm · Datasets at Hugging Face

en stringlengths 1 347	ber stringlengths 0 314	null sequencelengths 1 43 ⌀
Stone age cultures	including ancient indigenous American groups	[ " used tallies for gambling", " personal services", " and trade-goods.", "ssmrsnt tdlsiwin n uzmz-aggun g amunt trubba timirikanin tiqbuṛin", " imḍan i tqbbaḍt d twuriwin tinmɣurin d tɣawsiwin n mnziwt." ]
Beginning about 3500 BC	clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked.	[ "sg wattayn n 3500 dat tlalit n lmasiḥ ttusnflt tmatarin n waluḍ imik s imik ɣr tmatarin n wuṭṭun imssiggzn s uɣanib awrerray ilan tiɣmrin imzarayn g tfilitin n waluḍ ( ikkan gant tin ttuknat) nna yad inwan." ]
These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs.	aɣnt tmatarin n wuṭṭun akmam ad	[ " timitar n wuṭṭun awrerray nna illan g udɣar nns tḥḍu atig n uzmmem n tmitar n turnut i tmitar n wuṭṭun aqlallay." ]
Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons.	kkant wuṭṭun amṣḍiṣ-mraw gan angraw n ljidr iccarn iḥṭṭun talgamt 10 tawalit d talgamt 6 g tgffurt d tgusin tikmamin ibddan d cifrun.	null
Unique numbers of troops and measures of rice appear as unique combinations of these tallies.	dad tffɣn imḍan iẓlin n taggalin d imsɣaln n idgl am trubba istin g imḍan ad.	null
Conventional tallies are quite difficult to multiply and divide.	icqqa kigan usfukti d bṭṭu n imḍan izaykutn.	null
Jews began using a similar system (Hebrew numerals)	with the oldest examples known being coins from around 100 BC.	[ "ssntin wudayn assmrs n ungraw amm wa (uṭṭun n lɛibriya)", " acku kkant imdyatn iqburn ityassn adrim azaɣur attayn n 100 dat tlalit n lmasiḥ." ]
The Maya of Central America used a mixed base 18 and base 20 system	possibly inherited from the Olmec	[ " including advanced features such as positional notation and a zero.", "tssmrs lmaya g amrika n wammas; angraw iccarn gr tlgamt 18 d tlgamt 20", " iɣy is asntid udjan ulmk", " g tlla tbɣurt yattuyn zund azmmem asursan d umya." ]
Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century	and has not survived although simple quipu-like recording devices are still used in the Andean region.	[ "ttwamsay tussan n isntln n ugllay d ikʷlan g ɣur wazzaɣn n sbanya g uzmz wiss 16", " ur tnjim waxxa llan ingmam unziln n uzmmem yaɣn g “kibbu”", " tsul da ttussmras g wansa n “landiz”." ]
Zero was first used in India in the 7th century CE by Brahmagupta.	ittusmrs umya tiklt izwarn g lhind sg ɣur “brahmaguta”	[ " g uzmz wiss 7 ḍaṛt tlalit n lamsiḥ." ]
Arabic mathematicians extended the system to include decimal fractions	and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century.	[ "sbirwn imusnawn aɛrabn n tusnakt angraw mar ad yamẓ imtwaln imrawn", " d yaru muḥmmad bn musa lxawarizmi yat twuri istawhmman ɣf uynnaɣ", " g uzmz wiss 9." ]
The binary system (base 2)	was propagated in the 17th century by Gottfried Leibniz.	[ "ityafsar ungraw amsin (talgamt 2)", " g uzmz wiss 17 sg ɣur gutfrid libniz." ]
The variables for which the equation has to be solved are also called unknowns	and the values of the unknowns that satisfy the equality are called solutions of the equation.	[ "imskiln s nn iqn ad as tyafsay tgdazalt", " ttuyassn s warism", " d tinditin tilimslitin nna iskarn asiksl g ifssayn n tgdazalt." ]
A conditional equation is only true for particular values of the variables.	tagdazaltn tfada tga day tamddadt mar ad tg tazlɣa i tinditin n isnfaln.	null
Very often the right-hand side of an equation is assumed to be zero.	da wala ttgga tsga tayffast n tgdazalt amya.	null
An equation is analogous to a scale into which weights are placed.	tagdazalt tawayt n umsɣal g ttuyagan istaln.	null
This is the starting idea of algebraic geometry	an important area of mathematics.	[ "tadɣ ayd igan tawngimt tadslant n utwal aljibri", " d tga igr istawhmman g tusnakt." ]
To solve equations from either family	one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.	[ "mar ad nfsi tagdazalt illan g ka igat tawja", " da issmras yan tatiqnit n ussiṭn nɣd atwal d igman g ljibr awnɣan", " nɣd afssay awsnak." ]
These equations are difficult in general; one often searches just to find the existence or absence of a solution	and	[ " if they exist", " to count the number of solutions.", "tigdazalin ad cqqant s umata", " da wala ittinig ufgan ad yaf afssay nɣd war afssay", " mk illa", " da issiṭin mcta n ifssayn." ]
In the illustration	x	[ " y and z are all different quantities (in this case real numbers) represented as circular weights", " and each of x", " y", " and z has a different weight.", "g wunuɣ n ussikz", " “k”", " “y” d “z” gan akkʷ anct imzarayn ( g waddad ad uṭṭun n tidt)", " s mdyan s istaln iwrerrayn", " d i ku “k”", " “y”", " d “z” astal istin." ]
Hence	the equation with R unspecified is the general equation for the circle.	[ "sg uya", " tagdazalt d “r” ur iẓliyn", " tga tagdazalt tnmatayt n uwrerry." ]
The process of finding the solutions	or	[ " in case of parameters", " expressing the unknowns in terms of the parameters", " is called solving the equation.", "da as ttinin tiggit n yif n ifssayn nɣd asiwl ɣf ilimsli g tsɣlt", " mk tga tsɣlt tin ufssay n tgdazalt." ]
Multiplying or dividing both sides of an equation by a non-zero quantity.	asfukti nɣd tuṭṭut sin imnaḍn n tgdazalt ɣf tsmkta ur igin amya.	null
An algebraic equation is univariate if it involves only one variable.	da tgga tgdazalt aljibr mm yuwn usnfl ig diks yan usnfl day.	null
In mathematics	the theory of linear systems is the basis and a fundamental part of linear algebra	[ " a subject which is used in most parts of modern mathematics.", "g tusnakt", " tga tmagunt n igrrayn iwnɣann asilan", " d imik adslan g ljibr awnɣan", " ntta d asgum ittusmrasn g kigan ifrdas n tusnakt tatrart." ]
This formalism allows one to determine the positions and the properties of the focuses of a conic.	da ittadja ukurmis ad afgan ad isti imnadn d tẓlayin n iɣisan icɛba.	null
This point of view	outlined by Descartes	[ " enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.", "tannayt ad", " nna iẓli dikart da thyya ar tgadda anaw n tnzgit nna snumln kigan n imusnawn iyunaniyin n tusnakt taqburt." ]
An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.	tagdazalt dyufantin taggadt	[ " tga tagdazalt nna g iɣy ad yili winna iqqarn s tagdazalt mm sin warismawn." ]
Modern algebraic geometry is based on more abstract techniques of abstract algebra	especially commutative algebra	[ " with the language and the problems of geometry.", "tbdda tnzgit tatrart n ljibr ɣf tiqnitin wala inzzɣn ljibr awngim", " numar ljibr ittmnfkan akd tutlayt d tmukrisin n tnzgit." ]
A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation.	da ttumu tnqqiḍt taswirant g izririg n ljibr ig ṭfn izdayn nns ɣr tgdazalt ilan kigan n iwtta.	null
In pure mathematics	differential equations are studied from several different perspectives	[ " mostly concerned with their solutions — the set of functions that satisfy the equation.", "g tusnakt imzarayn", " da ttuɣrant tgdazalin timufayin g wahli n tɣmrin imzarayn", " wahli diks iga win ifssayn nns tarbiɛt n isɣnan itggan tagdazalt." ]
Linear differential equations	which have solutions that can be added and multiplied by coefficients	[ " are well-defined and understood", " and exact closed-form solutions are obtained.", "tigdazalin tizraragin imzaray", " nna ilan ifssayn iɣy ad tturnu ttusfuktu s watig ibddan", " tẓli d tturmas kigan", " d da ttafa ifssayn ilan tlaɣa tunɣidt gin imaqqan." ]
PDEs can be used to describe a wide variety of phenomena such as sound	heat	[ " electrostatics", " electrodynamics", " fluid flow", " elasticity", " or quantum mechanics.", "iɣy ad nsmrs igmamn n “PDE” mar ad nsnumml kigan n wanawn n tumanin zund imsli d tirɣi nɣd iluktrustatik", " iluktrudinamik", " nɣd tanɣla n ibluliwn", " nɣd artutm", " nɣd tamikanikt n idamn." ]
A solution is an assignment of values to the unknown variables that makes the equality in the equation true.	afssay igat ad nadj i tinditin timsnfalin ur ityassn nna ittadjan angiddi g tgdazalt tamddadt.	null
The set of all solutions of an equation is its solution set.	tarbiɛt n ifssayn akkʷ n tgdazalt ayd igan tarbiɛt n ifssayn iẓlin ẓaṛs.	null
Depending on the context	solving an equation may consist to find either any solution (finding a single solution is enough)	[ " all solutions", " or a solution that satisfies further properties", " such as belonging to a given interval.", "sg usatal iɣy ad ittug ufssay n tgdazalt abrid ɣr ifssayn nniḍn (xs ad day tafd yan ufssay)", " nɣd ifssayn maṛṛa", " nɣd afssayn imsasan d tmitar ", " zund tilit sg kan uzmz." ]
In this case	the solutions cannot be listed.	[ "g waddad ad", " ur nzḍar ad ng allas n ifssayn." ]
The variety in types of equations is large	and so are the corresponding methods.	[ "tanawayt g wanawn n tgdazalin imqqur", " ula awd tibridin n umyawaḍ." ]
This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort.	iɣy ad ig uya asrag n tdrsi n tmusna tusnakt	[ " ɛad afssay n itsn imukrisn ḍaṛt tsutin n tzmmar." ]
Polynomials appear in many areas of mathematics and science.	dad tbayant mm iwtta iggudin g kigan n yigran n tusnakt d tmassanin.	null
Many authors use these two words interchangeably.	da issmras kigan n imgayn snat tguriwin ad s umrara.	null
Formally	the name of the polynomial is P	[ " not P(x)", " but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear.", "P ayd igan ism unṣib n kigan n iwtta ur id P(x)", " maka asmrs n untal uɣrif P(x)", " iddad sg uzmz g ur iṣfi usnuḥyu n kigan iwtta d tmskart zaṛs islɣn." ]
However	one may use it over any domain where addition and multiplication are defined (that is	[ " any ring).", "waxxa hakkak", " iɣy ufgan ad tt issmrs g ka igat igr g ityassan usmun d usfukti s (ka igat", " d ka igat taxrst)." ]
Polynomials of small degree have been given specific names.	tid mi ggudin iwtta ilan taskflt mẓẓiyn	[ " ayd ikan ismawn imẓlayn." ]
The polynomial 0	which may be considered to have no terms at all	[ " is called the zero polynomial.", "bu-iwtta 0", " s nɣy ad nini ur akkʷ ili iwtta da itsmma amya mggudy iwtta." ]
Because the degree of a non-zero polynomial is the largest degree of any one term	this polynomial has degree two.	[ "macku lant tskfal tar-amya mm iwtta iggudin", " tga taskflt taxatart g yuwt tguri", " da ttili mm iwtta iggudin g tskflt tiss snat." ]
Polynomials can be classified by the number of terms with nonzero coefficients	so that a one-term polynomial is called a monomial	[ " a two-term polynomial is called a binomial", " and a three-term polynomial is called a trinomial.", "nɣy ad ng mm iwtta iggudin sg mnck n tguriwin ila iggitn ur igin amya", " imk ittyanna i bu-iwtta iggudin s yuwt tguri ism bu-yuwn uwttu", " ar ityanna i bu-iwtta iggudin s snat tguriwin bu-sin iwtta", " ar ityanna i bu mnnaw n iwtta; bu-kṛaḍ iwtta." ]
When it is used to define a function	the domain is not so restricted.	[ "ig da ittusmras mar ad isti tamrst", " ur da ittukraf yigr." ]
A polynomial in one indeterminate is called a univariate polynomial	a polynomial in more than one indeterminate is called a multivariate polynomial.	[ "bu iwtta iggudin g yuwn uwttu ur ityassan da as nttini bu iwtta iggudin ilan yuwn usnfl", " bu iwtta iggudin g kigan ur iẓliyn da as nttini amggudy n iwtta", " amgguwdy n isnfaln." ]
In the case of the field of complex numbers	the irreducible factors are linear.	[ "g waddad n yigr n imḍan uddisn", " da tggan imskarn ityakkasn; izririg." ]
If the degree is higher than one	the graph does not have any asymptote.	[ "ig tkka tskflt nnig yan", " ur da ittili i wunuɣ n usmmal awd yan izririd inmalan." ]
In elementary algebra	methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable.	[ "g ljibr amzwaru", " da ttusɣrant tbridin zund talɣa tamkkuẓt i ufssay n maṛṛa n tgdazalin ilan kigan n iwtta illan g tskflt tamzwarut ula tiss snat g yuwn usnfl." ]
However	root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.	[ "waxxa hakkak nɣy ad nssmrs alguritm n urzzu xf uẓuṛ", " mar ad naf anmala uṭṭun i iẓɣṛan n talɣa ilan iwtta iggudin g ka igat taskflt." ]
Since the 16th century	similar formulas (using cube roots in addition to square roots)	[ " but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation).", "sg usati wiss 16 ttuyassnt talɣiwin n usmsksl ( s ussmrs n izɣṛan igntrn d izɣṛan imkkuẓn)", " maka wanna wala icqqan igat tigdazalin n tskflt tiss kṛaḍt d tiss kkuẓt ( ẓṛ tagdazalt tagntrt d tgdazalt tamkkuẓt)." ]
In 1830	Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals	[ " and showed that for each equation", " one may decide whether it is solvable by radicals", " and", " if it is", " solve it.", "g usggʷas n 1830", " iswr ifarist galu n wisd kigan n tgdazalt n tskflt tamafllat sg kkuẓ", " ur nzḍaṛ at nfsi s uẓuṛ", " issfrud id ka igat tagdazalt iɣy ufgan ad yini is tla afssay s tbridt tzɣṛant", " mk iɣy", " ifsi tt." ]
Nevertheless	formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).	[ "waxxa hakkak", " tyafsarnt talɣiwin n tgdazalin ilan afssay s tskflt tiss 5 d 6 (ẓṛ tamrst tamsmmust d tgdazalt tamṣḍiḍt)." ]
The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1	000 (see Root-finding algorithm).	[ "da ttadjant alguritm tunṣibin ad ttyafsaynt tgdazalin id mm iwtta s umnhal ( g umssuds)", " s tskflt ikkan nnig 1", "000 ( ẓṛ alguritm n yif n izɣṛan)." ]
For a set of polynomial equations in several unknowns	there are algorithms to decide whether they have a finite number of complex solutions	[ " and", " if this number is finite", " for computing the solutions.", "g kigan n tgdazalin id mm iwtta iggudin g kigan n id war-ism", " llant alguritm ittinin is ɣarsn ifssayn mẓlay icqqan", " d mk iga wuṭṭun ad amẓlay i ussiṭn n ifssayn." ]
A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation.	tagdazalt da tgga mm iwtta iggudin nna mi yakka ufgan taɣdft s ifssayn igan imḍan imddadn s tgdazalt dyufantin.	null
The coefficients may be taken as real numbers	for real-valued functions.	[ "nɣy ad ng imggutn d uṭṭun iɣaran n tiggitin ilan atig aɣaran." ]
This equivalence explains why linear combinations are called polynomials.	da issfru ussksl ad amntil n yism n trubba tizririgin ilan kigan n iwtta.	null
In the case of coefficients in a ring	non-constant must be replaced by non-constant or non-unit (both definitions agree in the case of coefficients in a field).	[ "g waddad n imggutn g txrst iqnen ad nsnfl unna ur izzgan ɣr unna ur iwirn", " nɣd unna ur igzimn (isussn ssin ttmsasan g waddad nwaggitn g yigr)." ]
When the coefficients belong to integers	rational numbers or a finite field	[ " there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials).", "ig da ttuɣuln waggitn ɣr imḍan imddadn nɣd uṭṭun umginen nɣd igr iẓlin", " tlla alguritm mar ad tg irm n gar tazmrt n uzgzl d ussiṭn d yiggitn n tinna mi gudin iwtta ur igin tin uzgzl (ẓṛ asfsi mi ggudin iwtta s imskirn)." ]
The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues.	illa g imggudin n iwtta s tẓli tdrast	[ " nɣd amswuri azririg ɣf inɣmisn n tinditin n tnkkint n umswuri." ]
However	the elegant and practical notation we use today only developed beginning in the 15th century.	[ "waxxa hakkak asdukm anwuri iẓiln nssmras assa ittusbuɣlla g tizwuri n usatu wiss 15." ]
This completes the square	converting the left side into a perfect square.	[ "ɣaya “itsmad amkkuẓ”", " d ar itrar tasga tazlmaḍt ad tg amkkuẓ." ]
Descartes' theorem states that for every four kissing (mutually tangent) circles	their radii satisfy a particular quadratic equation.	[ "tamagunt n dikar da ttini llant ɣur kkuẓt n twrerray tugdut (amalu ugdu)", " gan izgn n wagm nns ijj n tagdazalt tamkkuẓt." ]
Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.	ssmrsn imusnawn ibabiliyn n tusnakt; ɣur wattayn n 400 n usggʷas dat tlalit n lmasiḥ	[ " d umusnawn iṣiniyn n tusnakt", " attayn n 200 dat tlalit n lmasiḥ", " tibridin n jyumitik i usfsy n tgdazalin timkkuzin ilan iẓɣṛan umnign." ]
Euclid	the Greek mathematician	[ " produced a more abstract geometrical method around 300 BC.", "isskr uklid amusnaw ayunaniy n tusnakt", " tabridt tajyumitrikt attayn n 300 n usggʷas dat tlalit n lmasiḥ." ]
Al-Khwarizmi goes further in providing a full solution to the general quadratic equation	accepting one or two numerical answers for every quadratic equation	[ " while providing geometric proofs in the process.", "ddan lxawarizm ɣr uggar n uya g ussnkd n ufssay akkʷ isman tagdazalt tamkkuẓt tamattayt", " ittirin yat tmrarut nɣd snat tmrarutin i ku tagdazalt tamkkuẓt", " d tikki n wanẓiwn ajyumitrikn g tiggi ad." ]
Abū Kāmil Shujā ibn Aslam (Egypt	10th century) in particular was the first to accept irrational numbers (often in the form of a square root	[ " cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.", "abu kamil cujaɛ ibn aslam (miṣṛ", " asatu wiss 10); s uẓlay", " d amzwaru innan ah i wuṭṭun imsɣan ( g uẓuṛ amkkuẓ nɣd agntr", " nɣd wiss kkuẓ) zund ifssayn n tgdazalin timkkuẓin nɣd imsɣal g tgdazalt." ]
His solution was largely based on Al-Khwarizmi's work.	afssay nns da ittuɣul kigan ɣf twuri n lxawarizmi.	null
However	at some point the quadratic formula begins to lose accuracy because of round off error	[ " while the approximate method continues to improve.", "waxxa hakkak", " g yat tfrkt da ttusntay talɣa tamkkuzt mi ittaccka unɣad; s uzgl n usnmila", " tssudu tɣarast n usnmila g tixxitrt." ]
Methods of numerical approximation existed	called prosthaphaeresis	[ " that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.", "llant tbridin n usmila amiḍan da as ttinin taɣart timkisit yakkan isanfn ɣf tiggitin ittamẓn tizi kigan", " zund asfukti d yisy n tzmrt d uzɣṛan." ]
Computational algorithms for finding the solutions are an important part of numerical linear algebra	and play a prominent role in engineering	[ " physics", " chemistry", " computer science", " and economics.", "ifssayn gan agzzum istawhmman g ljibr izririg amiḍan", " d ɣurs tawila mqqurn g tnzgit d fizik d cimi d tusnamssudst d tdamsa." ]
For solutions in an integral domain like the ring of the integers	or in other algebraic structures	[ " other theories have been developed", " see Linear equation over a ring.", "ismd amm txṛst n imḍan imddadn", " nɣd g tanɣiwin nniḍn n ljibr", " ttusbuɣllant tmagunin yaḍn", " ẓṛ tagdazalt tazrirgt aflla n txṛst." ]
This allows all the language and theory of vector spaces (or more generally	modules) to be brought to bear.	[ "wad ar ittadja ad ittug ka igat tutlayt d tmagunt n istumn n imnidn ( s umata ifrdasn ursiln)." ]
Such a system is known as an underdetermined system.	ityassn ungraw ad s yism n angraw ur iẓliyn.	null
The second system has a single unique solution	namely the intersection of the two lines.	[ "angraw wiss sin ɣars afssay iẓlin igan amrjal n izrirign." ]
Any two of these equations have a common solution.	ka igat snat tgdazalin g tgadazalin ad ɣarsnt afssay iccarn.	null
A system of equations whose left-hand sides are linearly independent is always consistent.	aha da itgga ungraw n tgdazalin nna mi tggant tsggin nns tiẓlmaḍ timẓlay; azririg imzgi.	null
This yields a system of equations with one fewer equation and one fewer unknown.	dad yakka maya angraw n tgdazalin diks tagdazalt yazdurn	[ " d tayḍ ur tyassn." ]
Type 3: Add to one row a scalar multiple of another.	anaw 3: rnu as i yan wadur aslag amiḍan n wadur nniḍn.	null
For instance	systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition.	[ "s umdya", " iɣy ad nfsi igrrayn ilan adurn imẓlay umnign inawayn s zzrabit tasftayt s tslṭ ntculiski." ]
A completely different approach is often taken for very large systems	which would otherwise take too much time or memory.	[ "da wala ittuḍfuṛ usara amziray i igrrayn mqqurn kigan", " nna ittettrn kigan n tizi nɣd timktit." ]
This leads to the class of iterative methods.	aya ar ittawy ɣr tgrruma n tbridin yulsn.	null
In mathematics	a series is	[ " roughly speaking", " a description of the operation of adding infinitely many quantities", " one after the other", " to a given starting quantity.", "g tusnakt", " tga tgffurt s talɣa n unmili; asnuml i tiggit n turnut iwudiyn wartmi iggudin", " yat ḍaṛt yat ar agudi n kan ussnti." ]
In addition to their ubiquity in mathematics	infinite series are also widely used in other quantitative disciplines such as physics	[ " computer science", " statistics and finance.", "d turnut xf unɣal nns g tusnakt", " da ttusmrasnt tsnslin wartmi g ufaɣul abaraw g tẓlayin n ugudiy nnidn zund fizik d tussnamssuds taddadant d ussẓṛf." ]
Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise	but when he reaches the position of the tortoise at the beginning of the race	[ " the tortoise has reached a second position; when he reaches this second position", " the tortoise is at a third position", " and so on.", "da tssfru tmzarayt n zinu gr axil d kfrun taẓlayt ad inmn ɣf watign wartmi", " axil ittazla ḍaṛt kfrun", " maka adday yawḍ ansa n kfrun g tizwiri n uḥmmzwur", " dan ittawḍ kfrun aswir wiss sin", " ign yuwḍ wiss sin da ttili g wiss kṛaḍ", " ar tddu." ]
This argument does not prove that the sum is equal to 2 (although it is)	but it does prove that it is at most 2.	[ "ur da izzga wanẓa is tag tmuttrt nns 2 (waxxa iga imkinnaɣ)", " maca da tzzga ddaw 2." ]
Tests for uniform convergence include the Weierstrass' M-test	Abel's uniform convergence test	[ " Dini's test", " and the Cauchy criterion.", "smunn yirimn n usnmala n ussuds", " irim n “wayrstris m Weierstrass' M-tes”", " d yirim n usnmala amyiwn izlin s “abil Abel's” d “ dini Dini's” d anaway “kuci Cauchy ”." ]
The convergence is uniform on closed and bounded (that is	compact) subsets of the interior of the disc of convergence: to wit	[ " it is uniformly convergent on compact sets.", "asnmala da ittusuds g trabbutin tayyawin iqnn gint timẓaly ( ig gant tuddizin)", " g tgzzumt tagnsut n disk n usnmala: i tiɣist", " inmala s talɣa iẓlin g trabbutin yudrn." ]
The Hilbert–Poincaré series is a formal power series used to study graded algebras.	tagffurt “ilbir-bwankari Hilbert–Poincaré” tga tagffurt tunsibt ittusmrasn g tɣuri n ljibr amḍfuṛ.	null
In the 17th century	James Gregory worked in the new decimal system on infinite series and published several Maclaurin series.	[ "g usatu wiss 17 iswuri “gims griguri James Gregory” g ungraw amrawan amaynu; ɣf tgffurt tartmi", " d ifsr kigan n tgffurin n “maklurin Maclaurin”." ]
Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so	and with him begins the discovery of effective criteria.	[ "isddid “kuci” (1821) ɣf yirimn inmalanen uqjirn", " issakz ig mmlmalant snat tgffurin amẓaraw nns ur id ccil as ad mayan", " ar ids ttusntay tufayt n inawayn imṛwitn." ]
A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence.	taɣarast n usmun tga ikz awttu i trabbut tayyawt n trabbut n tgffurin immɛraqn	[ " ittanfn s talɣa tamddadt armmus aklasiki n tnmila." ]
Indian scholars have been using factorial formulas since at least the 12th century.	da ssmrasn imassn ihindiyn talɣiwin n imggutn sg usatu wiss 12.	null
In functional languages	the recursive definition is often implemented directly to illustrate recursive functions.	[ "g tutlayin tisɣnan", " da wala ittuga s wusrid usissn amaɣul", " mar ad issfru tisɣna n tmaɣult." ]
Other implementations (such as computer software such as spreadsheet programs) can often handle larger values.	ɣint tsnsitin yaḍn (zund iɣawasn n umssuds zund iɣawasn n ismyallayn n ismmaln) asmkd n tinditin tixatarin g kigan n waddadn.	null
Compared to the Pickover definition of the superfactorial	the hyperfactorial grows relatively slowly.	[ "s uzmzazal n usissn bikufr n umswur amagur", " da ittugm umswur n usigz n tilal s timmisɣt." ]

Stone age cultures

including ancient indigenous American groups

[ " used tallies for gambling", " personal services", " and trade-goods.", "ssmrsnt tdlsiwin n uzmz-aggun g amunt trubba timirikanin tiqbuṛin", " imḍan i tqbbaḍt d twuriwin tinmɣurin d tɣawsiwin n mnziwt." ]

Beginning about 3500 BC

clay tokens were gradually replaced by number signs impressed with a round stylus at different angles in clay tablets (originally containers for tokens) which were then baked.

[ "sg wattayn n 3500 dat tlalit n lmasiḥ ttusnflt tmatarin n waluḍ imik s imik ɣr tmatarin n wuṭṭun imssiggzn s uɣanib awrerray ilan tiɣmrin imzarayn g tfilitin n waluḍ ( ikkan gant tin ttuknat) nna yad inwan." ]

These cuneiform number signs resembled the round number signs they replaced and retained the additive sign-value notation of the round number signs.

aɣnt tmatarin n wuṭṭun akmam ad

[ " timitar n wuṭṭun awrerray nna illan g udɣar nns tḥḍu atig n uzmmem n tmitar n turnut i tmitar n wuṭṭun aqlallay." ]

Sexagesimal numerals were a mixed radix system that retained the alternating base 10 and base 6 in a sequence of cuneiform vertical wedges and chevrons.

kkant wuṭṭun amṣḍiṣ-mraw gan angraw n ljidr iccarn iḥṭṭun talgamt 10 tawalit d talgamt 6 g tgffurt d tgusin tikmamin ibddan d cifrun.

null

Unique numbers of troops and measures of rice appear as unique combinations of these tallies.

dad tffɣn imḍan iẓlin n taggalin d imsɣaln n idgl am trubba istin g imḍan ad.

null

Conventional tallies are quite difficult to multiply and divide.

icqqa kigan usfukti d bṭṭu n imḍan izaykutn.

null

Jews began using a similar system (Hebrew numerals)

with the oldest examples known being coins from around 100 BC.

[ "ssntin wudayn assmrs n ungraw amm wa (uṭṭun n lɛibriya)", " acku kkant imdyatn iqburn ityassn adrim azaɣur attayn n 100 dat tlalit n lmasiḥ." ]

The Maya of Central America used a mixed base 18 and base 20 system

possibly inherited from the Olmec

[ " including advanced features such as positional notation and a zero.", "tssmrs lmaya g amrika n wammas; angraw iccarn gr tlgamt 18 d tlgamt 20", " iɣy is asntid udjan ulmk", " g tlla tbɣurt yattuyn zund azmmem asursan d umya." ]

Knowledge of the encodings of the knots and colors was suppressed by the Spanish conquistadors in the 16th century

and has not survived although simple quipu-like recording devices are still used in the Andean region.

[ "ttwamsay tussan n isntln n ugllay d ikʷlan g ɣur wazzaɣn n sbanya g uzmz wiss 16", " ur tnjim waxxa llan ingmam unziln n uzmmem yaɣn g “kibbu”", " tsul da ttussmras g wansa n “landiz”." ]

Zero was first used in India in the 7th century CE by Brahmagupta.

ittusmrs umya tiklt izwarn g lhind sg ɣur “brahmaguta”

[ " g uzmz wiss 7 ḍaṛt tlalit n lamsiḥ." ]

Arabic mathematicians extended the system to include decimal fractions

and Muḥammad ibn Mūsā al-Ḵwārizmī wrote an important work about it in the 9th century.

[ "sbirwn imusnawn aɛrabn n tusnakt angraw mar ad yamẓ imtwaln imrawn", " d yaru muḥmmad bn musa lxawarizmi yat twuri istawhmman ɣf uynnaɣ", " g uzmz wiss 9." ]

The binary system (base 2)

was propagated in the 17th century by Gottfried Leibniz.

[ "ityafsar ungraw amsin (talgamt 2)", " g uzmz wiss 17 sg ɣur gutfrid libniz." ]

The variables for which the equation has to be solved are also called unknowns

and the values of the unknowns that satisfy the equality are called solutions of the equation.

[ "imskiln s nn iqn ad as tyafsay tgdazalt", " ttuyassn s warism", " d tinditin tilimslitin nna iskarn asiksl g ifssayn n tgdazalt." ]

A conditional equation is only true for particular values of the variables.

tagdazaltn tfada tga day tamddadt mar ad tg tazlɣa i tinditin n isnfaln.

null

Very often the right-hand side of an equation is assumed to be zero.

da wala ttgga tsga tayffast n tgdazalt amya.

null

An equation is analogous to a scale into which weights are placed.

tagdazalt tawayt n umsɣal g ttuyagan istaln.

null

This is the starting idea of algebraic geometry

an important area of mathematics.

[ "tadɣ ayd igan tawngimt tadslant n utwal aljibri", " d tga igr istawhmman g tusnakt." ]

To solve equations from either family

one uses algorithmic or geometric techniques that originate from linear algebra or mathematical analysis.

[ "mar ad nfsi tagdazalt illan g ka igat tawja", " da issmras yan tatiqnit n ussiṭn nɣd atwal d igman g ljibr awnɣan", " nɣd afssay awsnak." ]

These equations are difficult in general; one often searches just to find the existence or absence of a solution

and

[ " if they exist", " to count the number of solutions.", "tigdazalin ad cqqant s umata", " da wala ittinig ufgan ad yaf afssay nɣd war afssay", " mk illa", " da issiṭin mcta n ifssayn." ]

In the illustration

x

[ " y and z are all different quantities (in this case real numbers) represented as circular weights", " and each of x", " y", " and z has a different weight.", "g wunuɣ n ussikz", " “k”", " “y” d “z” gan akkʷ anct imzarayn ( g waddad ad uṭṭun n tidt)", " s mdyan s istaln iwrerrayn", " d i ku “k”", " “y”", " d “z” astal istin." ]

Hence

the equation with R unspecified is the general equation for the circle.

[ "sg uya", " tagdazalt d “r” ur iẓliyn", " tga tagdazalt tnmatayt n uwrerry." ]

The process of finding the solutions

or

[ " in case of parameters", " expressing the unknowns in terms of the parameters", " is called solving the equation.", "da as ttinin tiggit n yif n ifssayn nɣd asiwl ɣf ilimsli g tsɣlt", " mk tga tsɣlt tin ufssay n tgdazalt." ]

Multiplying or dividing both sides of an equation by a non-zero quantity.

asfukti nɣd tuṭṭut sin imnaḍn n tgdazalt ɣf tsmkta ur igin amya.

null

An algebraic equation is univariate if it involves only one variable.

da tgga tgdazalt aljibr mm yuwn usnfl ig diks yan usnfl day.

null

In mathematics

the theory of linear systems is the basis and a fundamental part of linear algebra

[ " a subject which is used in most parts of modern mathematics.", "g tusnakt", " tga tmagunt n igrrayn iwnɣann asilan", " d imik adslan g ljibr awnɣan", " ntta d asgum ittusmrasn g kigan ifrdas n tusnakt tatrart." ]

This formalism allows one to determine the positions and the properties of the focuses of a conic.

da ittadja ukurmis ad afgan ad isti imnadn d tẓlayin n iɣisan icɛba.

null

This point of view

outlined by Descartes

[ " enriches and modifies the type of geometry conceived of by the ancient Greek mathematicians.", "tannayt ad", " nna iẓli dikart da thyya ar tgadda anaw n tnzgit nna snumln kigan n imusnawn iyunaniyin n tusnakt taqburt." ]

An exponential Diophantine equation is one for which exponents of the terms of the equation can be unknowns.

tagdazalt dyufantin taggadt

[ " tga tagdazalt nna g iɣy ad yili winna iqqarn s tagdazalt mm sin warismawn." ]

Modern algebraic geometry is based on more abstract techniques of abstract algebra

especially commutative algebra

[ " with the language and the problems of geometry.", "tbdda tnzgit tatrart n ljibr ɣf tiqnitin wala inzzɣn ljibr awngim", " numar ljibr ittmnfkan akd tutlayt d tmukrisin n tnzgit." ]

A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation.

da ttumu tnqqiḍt taswirant g izririg n ljibr ig ṭfn izdayn nns ɣr tgdazalt ilan kigan n iwtta.

null

In pure mathematics

differential equations are studied from several different perspectives

[ " mostly concerned with their solutions — the set of functions that satisfy the equation.", "g tusnakt imzarayn", " da ttuɣrant tgdazalin timufayin g wahli n tɣmrin imzarayn", " wahli diks iga win ifssayn nns tarbiɛt n isɣnan itggan tagdazalt." ]

Linear differential equations

which have solutions that can be added and multiplied by coefficients

[ " are well-defined and understood", " and exact closed-form solutions are obtained.", "tigdazalin tizraragin imzaray", " nna ilan ifssayn iɣy ad tturnu ttusfuktu s watig ibddan", " tẓli d tturmas kigan", " d da ttafa ifssayn ilan tlaɣa tunɣidt gin imaqqan." ]

PDEs can be used to describe a wide variety of phenomena such as sound

heat

[ " electrostatics", " electrodynamics", " fluid flow", " elasticity", " or quantum mechanics.", "iɣy ad nsmrs igmamn n “PDE” mar ad nsnumml kigan n wanawn n tumanin zund imsli d tirɣi nɣd iluktrustatik", " iluktrudinamik", " nɣd tanɣla n ibluliwn", " nɣd artutm", " nɣd tamikanikt n idamn." ]

A solution is an assignment of values to the unknown variables that makes the equality in the equation true.

afssay igat ad nadj i tinditin timsnfalin ur ityassn nna ittadjan angiddi g tgdazalt tamddadt.

null

The set of all solutions of an equation is its solution set.

tarbiɛt n ifssayn akkʷ n tgdazalt ayd igan tarbiɛt n ifssayn iẓlin ẓaṛs.

null

Depending on the context

solving an equation may consist to find either any solution (finding a single solution is enough)

[ " all solutions", " or a solution that satisfies further properties", " such as belonging to a given interval.", "sg usatal iɣy ad ittug ufssay n tgdazalt abrid ɣr ifssayn nniḍn (xs ad day tafd yan ufssay)", " nɣd ifssayn maṛṛa", " nɣd afssayn imsasan d tmitar ", " zund tilit sg kan uzmz." ]

In this case

the solutions cannot be listed.

[ "g waddad ad", " ur nzḍar ad ng allas n ifssayn." ]

The variety in types of equations is large

and so are the corresponding methods.

[ "tanawayt g wanawn n tgdazalin imqqur", " ula awd tibridin n umyawaḍ." ]

This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort.

iɣy ad ig uya asrag n tdrsi n tmusna tusnakt

[ " ɛad afssay n itsn imukrisn ḍaṛt tsutin n tzmmar." ]

Polynomials appear in many areas of mathematics and science.

dad tbayant mm iwtta iggudin g kigan n yigran n tusnakt d tmassanin.

null

Many authors use these two words interchangeably.

da issmras kigan n imgayn snat tguriwin ad s umrara.

null

Formally

the name of the polynomial is P

[ " not P(x)", " but the use of the functional notation P(x) dates from a time when the distinction between a polynomial and the associated function was unclear.", "P ayd igan ism unṣib n kigan n iwtta ur id P(x)", " maka asmrs n untal uɣrif P(x)", " iddad sg uzmz g ur iṣfi usnuḥyu n kigan iwtta d tmskart zaṛs islɣn." ]

However

one may use it over any domain where addition and multiplication are defined (that is

[ " any ring).", "waxxa hakkak", " iɣy ufgan ad tt issmrs g ka igat igr g ityassan usmun d usfukti s (ka igat", " d ka igat taxrst)." ]

Polynomials of small degree have been given specific names.

tid mi ggudin iwtta ilan taskflt mẓẓiyn

[ " ayd ikan ismawn imẓlayn." ]

The polynomial 0

which may be considered to have no terms at all

[ " is called the zero polynomial.", "bu-iwtta 0", " s nɣy ad nini ur akkʷ ili iwtta da itsmma amya mggudy iwtta." ]

Because the degree of a non-zero polynomial is the largest degree of any one term

this polynomial has degree two.

[ "macku lant tskfal tar-amya mm iwtta iggudin", " tga taskflt taxatart g yuwt tguri", " da ttili mm iwtta iggudin g tskflt tiss snat." ]

Polynomials can be classified by the number of terms with nonzero coefficients

so that a one-term polynomial is called a monomial

[ " a two-term polynomial is called a binomial", " and a three-term polynomial is called a trinomial.", "nɣy ad ng mm iwtta iggudin sg mnck n tguriwin ila iggitn ur igin amya", " imk ittyanna i bu-iwtta iggudin s yuwt tguri ism bu-yuwn uwttu", " ar ityanna i bu-iwtta iggudin s snat tguriwin bu-sin iwtta", " ar ityanna i bu mnnaw n iwtta; bu-kṛaḍ iwtta." ]

When it is used to define a function

the domain is not so restricted.

[ "ig da ittusmras mar ad isti tamrst", " ur da ittukraf yigr." ]

A polynomial in one indeterminate is called a univariate polynomial

a polynomial in more than one indeterminate is called a multivariate polynomial.

[ "bu iwtta iggudin g yuwn uwttu ur ityassan da as nttini bu iwtta iggudin ilan yuwn usnfl", " bu iwtta iggudin g kigan ur iẓliyn da as nttini amggudy n iwtta", " amgguwdy n isnfaln." ]

In the case of the field of complex numbers

the irreducible factors are linear.

[ "g waddad n yigr n imḍan uddisn", " da tggan imskarn ityakkasn; izririg." ]

If the degree is higher than one

the graph does not have any asymptote.

[ "ig tkka tskflt nnig yan", " ur da ittili i wunuɣ n usmmal awd yan izririd inmalan." ]

In elementary algebra

methods such as the quadratic formula are taught for solving all first degree and second degree polynomial equations in one variable.

[ "g ljibr amzwaru", " da ttusɣrant tbridin zund talɣa tamkkuẓt i ufssay n maṛṛa n tgdazalin ilan kigan n iwtta illan g tskflt tamzwarut ula tiss snat g yuwn usnfl." ]

However

root-finding algorithms may be used to find numerical approximations of the roots of a polynomial expression of any degree.

[ "waxxa hakkak nɣy ad nssmrs alguritm n urzzu xf uẓuṛ", " mar ad naf anmala uṭṭun i iẓɣṛan n talɣa ilan iwtta iggudin g ka igat taskflt." ]

Since the 16th century

similar formulas (using cube roots in addition to square roots)

[ " but much more complicated are known for equations of degree three and four (see cubic equation and quartic equation).", "sg usati wiss 16 ttuyassnt talɣiwin n usmsksl ( s ussmrs n izɣṛan igntrn d izɣṛan imkkuẓn)", " maka wanna wala icqqan igat tigdazalin n tskflt tiss kṛaḍt d tiss kkuẓt ( ẓṛ tagdazalt tagntrt d tgdazalt tamkkuẓt)." ]

In 1830

Évariste Galois proved that most equations of degree higher than four cannot be solved by radicals

[ " and showed that for each equation", " one may decide whether it is solvable by radicals", " and", " if it is", " solve it.", "g usggʷas n 1830", " iswr ifarist galu n wisd kigan n tgdazalt n tskflt tamafllat sg kkuẓ", " ur nzḍaṛ at nfsi s uẓuṛ", " issfrud id ka igat tagdazalt iɣy ufgan ad yini is tla afssay s tbridt tzɣṛant", " mk iɣy", " ifsi tt." ]

Nevertheless

formulas for solvable equations of degrees 5 and 6 have been published (see quintic function and sextic equation).

[ "waxxa hakkak", " tyafsarnt talɣiwin n tgdazalin ilan afssay s tskflt tiss 5 d 6 (ẓṛ tamrst tamsmmust d tgdazalt tamṣḍiḍt)." ]

The most efficient algorithms allow solving easily (on a computer) polynomial equations of degree higher than 1

000 (see Root-finding algorithm).

[ "da ttadjant alguritm tunṣibin ad ttyafsaynt tgdazalin id mm iwtta s umnhal ( g umssuds)", " s tskflt ikkan nnig 1", "000 ( ẓṛ alguritm n yif n izɣṛan)." ]

For a set of polynomial equations in several unknowns

there are algorithms to decide whether they have a finite number of complex solutions

[ " and", " if this number is finite", " for computing the solutions.", "g kigan n tgdazalin id mm iwtta iggudin g kigan n id war-ism", " llant alguritm ittinin is ɣarsn ifssayn mẓlay icqqan", " d mk iga wuṭṭun ad amẓlay i ussiṭn n ifssayn." ]

A polynomial equation for which one is interested only in the solutions which are integers is called a Diophantine equation.

tagdazalt da tgga mm iwtta iggudin nna mi yakka ufgan taɣdft s ifssayn igan imḍan imddadn s tgdazalt dyufantin.

null

The coefficients may be taken as real numbers

for real-valued functions.

[ "nɣy ad ng imggutn d uṭṭun iɣaran n tiggitin ilan atig aɣaran." ]

This equivalence explains why linear combinations are called polynomials.

da issfru ussksl ad amntil n yism n trubba tizririgin ilan kigan n iwtta.

null

In the case of coefficients in a ring

non-constant must be replaced by non-constant or non-unit (both definitions agree in the case of coefficients in a field).

[ "g waddad n imggutn g txrst iqnen ad nsnfl unna ur izzgan ɣr unna ur iwirn", " nɣd unna ur igzimn (isussn ssin ttmsasan g waddad nwaggitn g yigr)." ]

When the coefficients belong to integers

rational numbers or a finite field

[ " there are algorithms to test irreducibility and to compute the factorization into irreducible polynomials (see Factorization of polynomials).", "ig da ttuɣuln waggitn ɣr imḍan imddadn nɣd uṭṭun umginen nɣd igr iẓlin", " tlla alguritm mar ad tg irm n gar tazmrt n uzgzl d ussiṭn d yiggitn n tinna mi gudin iwtta ur igin tin uzgzl (ẓṛ asfsi mi ggudin iwtta s imskirn)." ]

The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues.

illa g imggudin n iwtta s tẓli tdrast

[ " nɣd amswuri azririg ɣf inɣmisn n tinditin n tnkkint n umswuri." ]

However

the elegant and practical notation we use today only developed beginning in the 15th century.

[ "waxxa hakkak asdukm anwuri iẓiln nssmras assa ittusbuɣlla g tizwuri n usatu wiss 15." ]

This completes the square

converting the left side into a perfect square.

[ "ɣaya “itsmad amkkuẓ”", " d ar itrar tasga tazlmaḍt ad tg amkkuẓ." ]

Descartes' theorem states that for every four kissing (mutually tangent) circles

their radii satisfy a particular quadratic equation.

[ "tamagunt n dikar da ttini llant ɣur kkuẓt n twrerray tugdut (amalu ugdu)", " gan izgn n wagm nns ijj n tagdazalt tamkkuẓt." ]

Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.

ssmrsn imusnawn ibabiliyn n tusnakt; ɣur wattayn n 400 n usggʷas dat tlalit n lmasiḥ

[ " d umusnawn iṣiniyn n tusnakt", " attayn n 200 dat tlalit n lmasiḥ", " tibridin n jyumitik i usfsy n tgdazalin timkkuzin ilan iẓɣṛan umnign." ]

Euclid

the Greek mathematician

[ " produced a more abstract geometrical method around 300 BC.", "isskr uklid amusnaw ayunaniy n tusnakt", " tabridt tajyumitrikt attayn n 300 n usggʷas dat tlalit n lmasiḥ." ]

Al-Khwarizmi goes further in providing a full solution to the general quadratic equation

accepting one or two numerical answers for every quadratic equation

[ " while providing geometric proofs in the process.", "ddan lxawarizm ɣr uggar n uya g ussnkd n ufssay akkʷ isman tagdazalt tamkkuẓt tamattayt", " ittirin yat tmrarut nɣd snat tmrarutin i ku tagdazalt tamkkuẓt", " d tikki n wanẓiwn ajyumitrikn g tiggi ad." ]

Abū Kāmil Shujā ibn Aslam (Egypt

10th century) in particular was the first to accept irrational numbers (often in the form of a square root

[ " cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.", "abu kamil cujaɛ ibn aslam (miṣṛ", " asatu wiss 10); s uẓlay", " d amzwaru innan ah i wuṭṭun imsɣan ( g uẓuṛ amkkuẓ nɣd agntr", " nɣd wiss kkuẓ) zund ifssayn n tgdazalin timkkuẓin nɣd imsɣal g tgdazalt." ]

His solution was largely based on Al-Khwarizmi's work.

afssay nns da ittuɣul kigan ɣf twuri n lxawarizmi.

null

However

at some point the quadratic formula begins to lose accuracy because of round off error

[ " while the approximate method continues to improve.", "waxxa hakkak", " g yat tfrkt da ttusntay talɣa tamkkuzt mi ittaccka unɣad; s uzgl n usnmila", " tssudu tɣarast n usnmila g tixxitrt." ]

Methods of numerical approximation existed

called prosthaphaeresis

[ " that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.", "llant tbridin n usmila amiḍan da as ttinin taɣart timkisit yakkan isanfn ɣf tiggitin ittamẓn tizi kigan", " zund asfukti d yisy n tzmrt d uzɣṛan." ]

Computational algorithms for finding the solutions are an important part of numerical linear algebra

and play a prominent role in engineering

[ " physics", " chemistry", " computer science", " and economics.", "ifssayn gan agzzum istawhmman g ljibr izririg amiḍan", " d ɣurs tawila mqqurn g tnzgit d fizik d cimi d tusnamssudst d tdamsa." ]

For solutions in an integral domain like the ring of the integers

or in other algebraic structures

[ " other theories have been developed", " see Linear equation over a ring.", "ismd amm txṛst n imḍan imddadn", " nɣd g tanɣiwin nniḍn n ljibr", " ttusbuɣllant tmagunin yaḍn", " ẓṛ tagdazalt tazrirgt aflla n txṛst." ]

This allows all the language and theory of vector spaces (or more generally

modules) to be brought to bear.

[ "wad ar ittadja ad ittug ka igat tutlayt d tmagunt n istumn n imnidn ( s umata ifrdasn ursiln)." ]

Such a system is known as an underdetermined system.

ityassn ungraw ad s yism n angraw ur iẓliyn.

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The second system has a single unique solution

namely the intersection of the two lines.

[ "angraw wiss sin ɣars afssay iẓlin igan amrjal n izrirign." ]

Any two of these equations have a common solution.

ka igat snat tgdazalin g tgadazalin ad ɣarsnt afssay iccarn.

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A system of equations whose left-hand sides are linearly independent is always consistent.

aha da itgga ungraw n tgdazalin nna mi tggant tsggin nns tiẓlmaḍ timẓlay; azririg imzgi.

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This yields a system of equations with one fewer equation and one fewer unknown.

dad yakka maya angraw n tgdazalin diks tagdazalt yazdurn

[ " d tayḍ ur tyassn." ]

Type 3: Add to one row a scalar multiple of another.

anaw 3: rnu as i yan wadur aslag amiḍan n wadur nniḍn.

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For instance

systems with a symmetric positive definite matrix can be solved twice as fast with the Cholesky decomposition.

[ "s umdya", " iɣy ad nfsi igrrayn ilan adurn imẓlay umnign inawayn s zzrabit tasftayt s tslṭ ntculiski." ]

A completely different approach is often taken for very large systems

which would otherwise take too much time or memory.

[ "da wala ittuḍfuṛ usara amziray i igrrayn mqqurn kigan", " nna ittettrn kigan n tizi nɣd timktit." ]

This leads to the class of iterative methods.

aya ar ittawy ɣr tgrruma n tbridin yulsn.

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In mathematics

a series is

[ " roughly speaking", " a description of the operation of adding infinitely many quantities", " one after the other", " to a given starting quantity.", "g tusnakt", " tga tgffurt s talɣa n unmili; asnuml i tiggit n turnut iwudiyn wartmi iggudin", " yat ḍaṛt yat ar agudi n kan ussnti." ]

In addition to their ubiquity in mathematics

infinite series are also widely used in other quantitative disciplines such as physics

[ " computer science", " statistics and finance.", "d turnut xf unɣal nns g tusnakt", " da ttusmrasnt tsnslin wartmi g ufaɣul abaraw g tẓlayin n ugudiy nnidn zund fizik d tussnamssuds taddadant d ussẓṛf." ]

Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise

but when he reaches the position of the tortoise at the beginning of the race

[ " the tortoise has reached a second position; when he reaches this second position", " the tortoise is at a third position", " and so on.", "da tssfru tmzarayt n zinu gr axil d kfrun taẓlayt ad inmn ɣf watign wartmi", " axil ittazla ḍaṛt kfrun", " maka adday yawḍ ansa n kfrun g tizwiri n uḥmmzwur", " dan ittawḍ kfrun aswir wiss sin", " ign yuwḍ wiss sin da ttili g wiss kṛaḍ", " ar tddu." ]

This argument does not prove that the sum is equal to 2 (although it is)

but it does prove that it is at most 2.

[ "ur da izzga wanẓa is tag tmuttrt nns 2 (waxxa iga imkinnaɣ)", " maca da tzzga ddaw 2." ]

Tests for uniform convergence include the Weierstrass' M-test

Abel's uniform convergence test

[ " Dini's test", " and the Cauchy criterion.", "smunn yirimn n usnmala n ussuds", " irim n “wayrstris m Weierstrass' M-tes”", " d yirim n usnmala amyiwn izlin s “abil Abel's” d “ dini Dini's” d anaway “kuci Cauchy ”." ]

The convergence is uniform on closed and bounded (that is

compact) subsets of the interior of the disc of convergence: to wit

[ " it is uniformly convergent on compact sets.", "asnmala da ittusuds g trabbutin tayyawin iqnn gint timẓaly ( ig gant tuddizin)", " g tgzzumt tagnsut n disk n usnmala: i tiɣist", " inmala s talɣa iẓlin g trabbutin yudrn." ]

The Hilbert–Poincaré series is a formal power series used to study graded algebras.

tagffurt “ilbir-bwankari Hilbert–Poincaré” tga tagffurt tunsibt ittusmrasn g tɣuri n ljibr amḍfuṛ.

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In the 17th century

James Gregory worked in the new decimal system on infinite series and published several Maclaurin series.

[ "g usatu wiss 17 iswuri “gims griguri James Gregory” g ungraw amrawan amaynu; ɣf tgffurt tartmi", " d ifsr kigan n tgffurin n “maklurin Maclaurin”." ]

Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so

and with him begins the discovery of effective criteria.

[ "isddid “kuci” (1821) ɣf yirimn inmalanen uqjirn", " issakz ig mmlmalant snat tgffurin amẓaraw nns ur id ccil as ad mayan", " ar ids ttusntay tufayt n inawayn imṛwitn." ]

A summability method is such an assignment of a limit to a subset of the set of divergent series which properly extends the classical notion of convergence.

taɣarast n usmun tga ikz awttu i trabbut tayyawt n trabbut n tgffurin immɛraqn

[ " ittanfn s talɣa tamddadt armmus aklasiki n tnmila." ]

Indian scholars have been using factorial formulas since at least the 12th century.

da ssmrasn imassn ihindiyn talɣiwin n imggutn sg usatu wiss 12.

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In functional languages

the recursive definition is often implemented directly to illustrate recursive functions.

[ "g tutlayin tisɣnan", " da wala ittuga s wusrid usissn amaɣul", " mar ad issfru tisɣna n tmaɣult." ]

Other implementations (such as computer software such as spreadsheet programs) can often handle larger values.

ɣint tsnsitin yaḍn (zund iɣawasn n umssuds zund iɣawasn n ismyallayn n ismmaln) asmkd n tinditin tixatarin g kigan n waddadn.

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Compared to the Pickover definition of the superfactorial

the hyperfactorial grows relatively slowly.

[ "s uzmzazal n usissn bikufr n umswur amagur", " da ittugm umswur n usigz n tilal s timmisɣt." ]