ilyasaqit/en-tzm · Datasets at Hugging Face

en stringlengths 1 347	ber stringlengths 0 314	null sequencelengths 1 43 ⌀
By the 17th century	mathematicians generally used decimal fractions with modern notation.	[ "g usatu wiss 17", " ssmrsn imusnawn n tusnakt s umata imtwaln imrawn s uzmmem atrar." ]
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.", "g usggʷas n 1872 ttufsarnt tmagunin n kaṛl wyyirstras ( sg ɣur unlmad nns iy kusak)", " d idwaṛd hayna d juṛj kantur d ritcard didikind." ]
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.", "ar iskan wiyistras d kantur d hin timgunin nnsn ɣf tgffurt tartmi", " isbdd didkin tawngimt n tubuyt (scnit) g ungraw n imḍan n tidt", " ibḍa imḍan n tidt kul xf snat trubba mi ɣur llan kan ifṛḍiṣn imẓlayn." ]
Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations).	sg uya iqnen uxzzr g trabbut tamuzzut n imḍan n ljibr ( ifssayn maṛṛa n tgdazalin mi ggudin iwtta).	null
Aristotle defined the traditional Western notion of mathematical infinity.	isnml arisṭu armmus aɣrbi azayku ɣf tartmi tusnakt.	null
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.", "maka azzigz ad mqquṛn n tmagunt iskrt juṛj kantur; g 1895 ifsr adlis ɣf tmagunt nns tamaynut i trabbut", " issnkd akd tɣawsiwin nniḍn", " imḍan izrrin iwtta", " d isskr turda tasult." ]
A modern geometrical version of infinity is given by projective geometry	which introduces ideal points at infinity	[ " one for each spatial direction.", "tunɣilt tanzgant tatrart sg wartmi d ittufkan sg tanzgant tastuttit", " nnad yakkan tinqqaḍ yattuyn sg ɣur wartmi", " tanqqiḍt i ka igat tanila tadɣarant." ]
The idea of the graphic representation of complex numbers had appeared	however	[ " as early as 1685", " in Wallis's De algebra tractatus.", "tbaynd twngimt n usmdya awnɣan n imḍan uddisn sg 1685 g udlis “walis” “isakatn n ljibr”." ]
In 240 BC	Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers.	[ "g 240 dat tlalit n lmasiḥ", " issmrs iratustans arkkut n iratustans n uẓlay n imḍan imzwura s zzrabit." ]
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.", "tiyafutin nniḍn n ubṭṭu n imḍan imzwura", " diks tuzgit n ulr ittinin masd da ttmɛraqnt trbiɛin n tmggitin ittmfkan imḍan imzwura", " d uswingm n guldbac nna ittini masd ka igat imiḍ axatar igan sin g tlla tugdut iga tamunt n sin imḍan sg imḍan imzwura." ]
Traditionally	the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.)	[ "s tzaykut", " tssnti tsnslt n imḍan iɣaran s wuṭṭun 1 (0 ur djun igi uṭṭun n lyunan iqburn)." ]
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.", "g ungraw n tsila ad 10", " da ittili i wuṭṭun illan g tsga n uyffas wala n wuṭṭun aɣaran atig adɣaran igan 1", " d i wuṭṭun yaḍn atig adɣaran mraw n tikkal ɣf watig adɣaran n wuṭṭun illan g uyffas nns." ]
Negative numbers are usually written with a negative sign (a minus sign).	da wala tyaran wuṭṭun uzdirn s tmatart tuzdirt ( tamatart tanaktamt).	null
Here the letter Z comes .	dadɣ dad itddu uskkil ẓ.	null
Fractions can be greater than	less than	[ " or equal to 1 and can also be positive", " negative", " or 0.", "iɣy ad gin imtwaln; nnig n", " nɣd ddaw n", " nɣ aksuln d 1", " d iɣy ad gin umnign nɣd uzdirn nɣd 0." ]
The following paragraph will focus primarily on positive real numbers.	ttwan n tsddart ad s talɣa tadslant ɣf imḍan n tidt ign umnign.	null
Thus	for example	[ " one half is 0.5", " one fifth is 0.2", " one-tenth is 0.1", " and one fiftieth is 0.02.", "imki", " s umdya", " azgn iga 0.5", " tiss smmus twal tga 0.2", " tiss mrawt twal tga 0.1", " d yan xf smmus mraw iga 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.	ur id ɣas imdyatn ad ayd ityassn day	[ " maka maṛṛa uṭṭun n tidt ur wala gin umgin", " aya aɣf ur lin talɣiwin ittyalsn", " aɣf ur illi wuṭṭun amraw amaksal." ]
Since not even the second digit after the decimal place is preserved	the following digits are not significant.	[ "macku ur illi uḥṭṭu n wiss sin ḍaṛt tiskrt tamrawt", " aɣf ur yad stawhmman wuṭṭun d ikkan dat." ]
For example	0.999...	[ " 1.0", " 1.00", " 1.000", " ...", " all represent the natural number 1.", "s umdya", " 0.999...", " 1.0", " 1.00", " 1.000", " ...", " kullutn smdyan s wuṭṭun aɣaran 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.", "tiyira", " ig llan wuṭṭun maṛṛa g wuṭṭun 0", " da itgga wuṭṭun 0", " d mk gan wuṭṭun kul g wuṭṭun n tgffurt ur itfukkun g 9", " tɣid ad tsugzzd tẓa g uyffas g udɣar amraw", " trnud yan i tgffurt g llan 9 n tisnatin g uẓlmaḍ n wansa amraw." ]
Thus the real numbers are a subset of the complex numbers.	d gin imḍan n tidt; tarbiɛt tayyawt n imḍan uddisn	[ "" ]
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.	[ "tslkan tmagunt tadslant n ljibr ɣf imḍan uddisn ittggan igr n ljibr ign amaɣun", " aɣ ittinin isd amggudy n iwtta akd irwin ɣurs aẓuṛ g imḍan uddisn." ]
The primes have been widely studied for more than 2000 years and have led to many questions	only some of which have been answered.	[ "ttuga tzrawt ɣf imḍan imzwura g ufuɣal amiriw uggar n 2000 n usggʷas", " yuwin ɣr kigan n isqsitn", " ttufka tmrarut i itsn day." ]
Real numbers that are not rational numbers are called irrational numbers.	da tsmman imḍan n tidt	[ " nna ur igin uṭṭunen umginen; imḍan ur igin umginen." ]
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.", "uṭṭun ittusiṭn; wrn ɣf tiggitin n ussiṭn ittuwalfn", " g illa ussiṭn n uẓuṛ n bu iwtta iggudin", " aya aɣf tskar igr amaqqan n tidt g llan wuṭṭun n ljibr n tidt." ]
One reason is that there is no algorithm for testing the equality of two computable numbers.	yan g imntiln igat ur tlli alguritm n yirm n tngiddit ingr sin wuṭṭun ign win ussiṭn.	null
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.	[ "da iskuttu ungraw n wuṭṭun d ittufkan s usila ittusmrasn i wuṭṭun", " ka igat talgamt iɣin ad yili", " maka talgamt n imḍan imzwura", " da takka imẓlay iɣudan n tusnakt." ]
The former gives the ordering of the set	while the latter gives its size.	[ "da yakka umzwaru assuds n trbiɛt", " ar yakka umggaru aksay nns." ]
This standard basis makes the complex numbers a Cartesian plane	called the complex plane.	[ "asila ad amsɣal da itgga imḍan uddisn g uswir adikartiy da itsmma aswur uddis." ]
The complex numbers of absolute value one form the unit circle.	da tggan imḍan uddisn n watig amggaru	[ " yat tzgunt ign yuwt." ]
In domain coloring the output dimensions are represented by color and brightness	respectively.	[ "g yigr n usuɣn", " da ttusmdyan wuggugn n ussufɣ s ukʷli d uzznẓṛ amzday." ]
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.", "yuw uswuri ɣf tmukrisin n tmggudin n iwtta timattutin g tyira ɣr tmagunt tasilant n ljibr", " nna issfrun akd imḍan uddisn", " illa ufssay n ka igat tagdazalt ila iwtta iggudin; sg tskʷflt tamzwarut nɣd nnig as." ]
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.	baynd tsmktitin n wisl g tiwin tikadimitin zund “kubnhagn”	[ " maka tzri ur tt yannay awd yan kigan." ]
Later classical writers on the general theory include Richard Dedekind	Otto Hölder	[ " Felix Klein", " Henri Poincaré", " Hermann Schwarz", " Karl Weierstrass and many others.", "sg imarratn iklasikn g tizi yaḍn ɣf tmagunt tamattut n ritcard didikind", " nɣd utu hildr", "d filiks klin", " d hinri bwankir", " d hirman cwartz", " d karl wirstras d wiyyaḍ." ]
The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).	ur ittusyaha ussmrs n imḍan iwngimn g uɣufal amiriw	[ " ar tawuri n lyunard ulr (1707-1783)", " d kaṛl fridritc gaws (1777-1855)." ]
The integers form the smallest group and the smallest ring containing the natural numbers.	da tggan imḍan imddadn; tarbiɛt tamẓẓant	[ " d txrst tamẓẓant g llan imḍan iɣaranen." ]
It is the prototype of all objects of such algebraic structure.	iga amdya amnzu n maṛṛa imɣnawn	[ " zund tuskiwt tajibrit." ]
Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68	C	[ " Java", " Delphi", " etc.).", "da ttubdarn wanawn n inmmaln n usnmili n imiḍ amddad ilan taɣzi iwrn (nɣd tirbiɛin tayyawin)", " igan s tmatart “ int “ nɣd “Integer” g kigan n tutlayin n usɣiws ( zund Algol68", " d C", " d Java", "d Delphi", " d tiyyaḍ)." ]
These are provable properties of rational numbers and positional number systems	and are not used as definitions in mathematics.	[ "imẓlay ad iɣy ad tnt nswr i uṭṭun umginen", " d ingrawn imiḍan imsursn", " d urda ttusmras am isissn." ]
Since the triangle is isosceles	a = b).	[ "mayd iga wamkṛaḍ amsaskl n igalaln", " a = b)." ]
Since c is even	dividing c by 2 yields an integer.	[ "mayd iga c amsin", " da aɣd takka tubḍut n c xf 2; imiḍ amddad." ]
Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.	asnfl n 4y2 ɣr c2	[ " g tgadazt tamzwarut (c2 = 2b2)", " aɣ yakkan 4y2= 2b2." ]
Since b2 is even	b must be even.	[ "mayd iga b2 amsin", " iqnen ad ig b amsin." ]
However this contradicts the assumption that they have no common factors.	maka aya itmgla d umrdu ittinin ur llin imggitn imccurn grasn.	null
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.”", "waxxa hakkak ", " illa yilɣ n hibasus ɣf tzmmar nns", " ayd inna yan wumiy", " yukz", " ikiz nns aynna illa g yil", " grnt imddukal nns ifitaɣursn g tizi yaḍn", "...", " acku snfln ya ufṛḍiṣ g iɣzwr ur irin... taɣalt ittinin iɣy uzgzl n tumanin g iɣzwr ɣr imḍan imddadn d usɣal nnsn." ]
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.", "s umdya", " g twngimt nnk tagzzumt inmn: iɣ ad ttubḍu ɣr snat", " d uzgn n uzgn ɣr sin", " d uzgn n uzgn ɣf azng", " imki." ]
This is just what Zeno sought to prove.	aya ayd itnaɣ zinun ad tt iswr.	null
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.", "g twngimin n liɣriq", " ukus n umddad n kan tannayt ur da iswar amddad n tannayt nniḍn", " xf uya asn iqqn uzrru uggar." ]
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.", "aksay”...ur igi imiḍ", " maka ismdya s inmallatn zund: imzrayn n uzririg", " d tɣmrin", " d tijimma", " d iksayn", " d tizi iɣin ad tmziriy", " imk nttini s talɣa izdin." ]
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.", "acku ur ityakz watig n wanc illan g uksay", " iɣy iduksus ad issiṭn asɣaln igan win usqqul", " d tinna ur igin tin usqqul sg ustay n usɣl sg uksay", " d usɣl zund assiksl n sin isɣal." ]
This incommensurability is dealt with in Euclid's Elements	Book X	[ " Proposition 9.", "da ittuga i winna ur iẓḍaṛn i usqqul g ifṛḍisn n iqlids", " adlis mraw", " asumr 9." ]
In fact	in many cases algebraic conceptions were reformulated into geometric terms.	[ "g tinawt; g kigan n waddadn ittuyalls tihkct i irmmusn n ljibr ɣr irman inzganen." ]
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.	iqnen wusun n itsn irmmusn idslan agnsu n tmagunt n dɣi	[ " imzaray n tilit n tiggi azrru akkʷ ign aɣzuran i tiggiwin d iswingmn illan ḍaṛt tmagunt ad." ]
However	historian Carl Benjamin Boyer writes that such claims are not well substantiated and unlikely to be true.	[ "waxxa hakkak", " yara unmzruy kaṛl bn jamin bwir an; “zund tafakult ad ur ilin iswrn iɣudan", " ur tnni ad tg tamddadt”." ]
Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed.	kan imusnawn n tusnakt zund brahamaguta (asggʷas n 628 ḍaṛt tlalit n lmasiḥ)	[ " d baskara (g 629 ḍaṛt tlalit n lmasiḥ)", " imyiwasn g yigr ad", " mk skrn imusnawn n tusnakt yaḍn iḍfaṛn mayan." ]
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.", "g usggʷas 1872 tyafsarnt tmagunin n kaṛl wiyrstras ( sg ɣur unlmad nns irnst kusak) d idwaṛd hayn ( tasɣnt n kṛil", " 74)", " d juṛj kantur (analin", " 5)", " d ritcaṛd didikind." ]
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.", "iska wiyrstras d kantur d hin; timagunin nnsn ɣf tgffurt tartmi", " isbdd didkin tamagunt nns ɣf twngimt n wubuy (sicnit)", " g ungraw n maṛṛa uṭṭun umginen", " artn yaṭṭu ɣf snat trubba ilan itsn imẓlayn istin." ]
Dirichlet also added to the general theory	as have numerous contributors to the applications of the subject.	[ "irna awd diliclit i tmagunt tamattut", " imk gan kigan n winna yuwsn n tsnsitin n usntl." ]
This asserts that every integer has a unique factorization into primes.	aya islkan n wis d ku imiḍ amddad ɣars imggi amẓlay g imḍan imzwura.	null
To show this	suppose we divide integers n by m (where m is nonzero).	[ "mar ad nsbayn ayad yurda is da naṭṭu imḍan amddadn n xf m ( acku m ur tgi tamyat)." ]
If 0 never occurs	then the algorithm can run at most m − 1 steps without using any remainder more than once.	[ "ig ur akkʷ ittuga 0", " tɣy ad tswuri alguritm ɣf kigan n tsurifin m - 1", " bla nssmrs awdyan umagur wahli n tikkal." ]
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).	[ "“g tusnakt", " uṭṭunen iɣaranen gantn tinna ittusmrasn g ussiṭn ( imk illan g : llant sḍiṣ n idrimn aflla n tdabut)", " d ussuds ( imk illan g : wad aɣrm wiss kṛaḍ g tmurt).”" ]
These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.	da tggant tgffurin n izdawn n wuṭṭu iɣaran g llan ( iẓlin)	[ " g imagrawn n wuṭṭun yaḍn." ]
The first major advance in abstraction was the use of numerals to represent numbers.	azwar amzwaru n tanzɣt	[ " igat ussmrs amaṭṭun i usmdya n wuṭṭun." ]

By the 17th century

mathematicians generally used decimal fractions with modern notation.

[ "g usatu wiss 17", " ssmrsn imusnawn n tusnakt s umata imtwaln imrawn s uzmmem atrar." ]

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.", "g usggʷas n 1872 ttufsarnt tmagunin n kaṛl wyyirstras ( sg ɣur unlmad nns iy kusak)", " d idwaṛd hayna d juṛj kantur d ritcard didikind." ]

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.", "ar iskan wiyistras d kantur d hin timgunin nnsn ɣf tgffurt tartmi", " isbdd didkin tawngimt n tubuyt (scnit) g ungraw n imḍan n tidt", " ibḍa imḍan n tidt kul xf snat trubba mi ɣur llan kan ifṛḍiṣn imẓlayn." ]

Hence it was necessary to consider the wider set of algebraic numbers (all solutions to polynomial equations).

sg uya iqnen uxzzr g trabbut tamuzzut n imḍan n ljibr ( ifssayn maṛṛa n tgdazalin mi ggudin iwtta).

null

Aristotle defined the traditional Western notion of mathematical infinity.

isnml arisṭu armmus aɣrbi azayku ɣf tartmi tusnakt.

null

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.", "maka azzigz ad mqquṛn n tmagunt iskrt juṛj kantur; g 1895 ifsr adlis ɣf tmagunt nns tamaynut i trabbut", " issnkd akd tɣawsiwin nniḍn", " imḍan izrrin iwtta", " d isskr turda tasult." ]

A modern geometrical version of infinity is given by projective geometry

which introduces ideal points at infinity

[ " one for each spatial direction.", "tunɣilt tanzgant tatrart sg wartmi d ittufkan sg tanzgant tastuttit", " nnad yakkan tinqqaḍ yattuyn sg ɣur wartmi", " tanqqiḍt i ka igat tanila tadɣarant." ]

The idea of the graphic representation of complex numbers had appeared

however

[ " as early as 1685", " in Wallis's De algebra tractatus.", "tbaynd twngimt n usmdya awnɣan n imḍan uddisn sg 1685 g udlis “walis” “isakatn n ljibr”." ]

In 240 BC

Eratosthenes used the Sieve of Eratosthenes to quickly isolate prime numbers.

[ "g 240 dat tlalit n lmasiḥ", " issmrs iratustans arkkut n iratustans n uẓlay n imḍan imzwura s zzrabit." ]

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.", "tiyafutin nniḍn n ubṭṭu n imḍan imzwura", " diks tuzgit n ulr ittinin masd da ttmɛraqnt trbiɛin n tmggitin ittmfkan imḍan imzwura", " d uswingm n guldbac nna ittini masd ka igat imiḍ axatar igan sin g tlla tugdut iga tamunt n sin imḍan sg imḍan imzwura." ]

Traditionally

the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.)

[ "s tzaykut", " tssnti tsnslt n imḍan iɣaran s wuṭṭun 1 (0 ur djun igi uṭṭun n lyunan iqburn)." ]

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.", "g ungraw n tsila ad 10", " da ittili i wuṭṭun illan g tsga n uyffas wala n wuṭṭun aɣaran atig adɣaran igan 1", " d i wuṭṭun yaḍn atig adɣaran mraw n tikkal ɣf watig adɣaran n wuṭṭun illan g uyffas nns." ]

Negative numbers are usually written with a negative sign (a minus sign).

da wala tyaran wuṭṭun uzdirn s tmatart tuzdirt ( tamatart tanaktamt).

null

Here the letter Z comes .

dadɣ dad itddu uskkil ẓ.

null

Fractions can be greater than

less than

[ " or equal to 1 and can also be positive", " negative", " or 0.", "iɣy ad gin imtwaln; nnig n", " nɣd ddaw n", " nɣ aksuln d 1", " d iɣy ad gin umnign nɣd uzdirn nɣd 0." ]

The following paragraph will focus primarily on positive real numbers.

ttwan n tsddart ad s talɣa tadslant ɣf imḍan n tidt ign umnign.

null

Thus

for example

[ " one half is 0.5", " one fifth is 0.2", " one-tenth is 0.1", " and one fiftieth is 0.02.", "imki", " s umdya", " azgn iga 0.5", " tiss smmus twal tga 0.2", " tiss mrawt twal tga 0.1", " d yan xf smmus mraw iga 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.

ur id ɣas imdyatn ad ayd ityassn day

[ " maka maṛṛa uṭṭun n tidt ur wala gin umgin", " aya aɣf ur lin talɣiwin ittyalsn", " aɣf ur illi wuṭṭun amraw amaksal." ]

Since not even the second digit after the decimal place is preserved

the following digits are not significant.

[ "macku ur illi uḥṭṭu n wiss sin ḍaṛt tiskrt tamrawt", " aɣf ur yad stawhmman wuṭṭun d ikkan dat." ]

For example

0.999...

[ " 1.0", " 1.00", " 1.000", " ...", " all represent the natural number 1.", "s umdya", " 0.999...", " 1.0", " 1.00", " 1.000", " ...", " kullutn smdyan s wuṭṭun aɣaran 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.", "tiyira", " ig llan wuṭṭun maṛṛa g wuṭṭun 0", " da itgga wuṭṭun 0", " d mk gan wuṭṭun kul g wuṭṭun n tgffurt ur itfukkun g 9", " tɣid ad tsugzzd tẓa g uyffas g udɣar amraw", " trnud yan i tgffurt g llan 9 n tisnatin g uẓlmaḍ n wansa amraw." ]

Thus the real numbers are a subset of the complex numbers.

d gin imḍan n tidt; tarbiɛt tayyawt n imḍan uddisn

[ "" ]

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.

[ "tslkan tmagunt tadslant n ljibr ɣf imḍan uddisn ittggan igr n ljibr ign amaɣun", " aɣ ittinin isd amggudy n iwtta akd irwin ɣurs aẓuṛ g imḍan uddisn." ]

The primes have been widely studied for more than 2000 years and have led to many questions

only some of which have been answered.

[ "ttuga tzrawt ɣf imḍan imzwura g ufuɣal amiriw uggar n 2000 n usggʷas", " yuwin ɣr kigan n isqsitn", " ttufka tmrarut i itsn day." ]

Real numbers that are not rational numbers are called irrational numbers.

da tsmman imḍan n tidt

[ " nna ur igin uṭṭunen umginen; imḍan ur igin umginen." ]

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.", "uṭṭun ittusiṭn; wrn ɣf tiggitin n ussiṭn ittuwalfn", " g illa ussiṭn n uẓuṛ n bu iwtta iggudin", " aya aɣf tskar igr amaqqan n tidt g llan wuṭṭun n ljibr n tidt." ]

One reason is that there is no algorithm for testing the equality of two computable numbers.

yan g imntiln igat ur tlli alguritm n yirm n tngiddit ingr sin wuṭṭun ign win ussiṭn.

null

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.

[ "da iskuttu ungraw n wuṭṭun d ittufkan s usila ittusmrasn i wuṭṭun", " ka igat talgamt iɣin ad yili", " maka talgamt n imḍan imzwura", " da takka imẓlay iɣudan n tusnakt." ]

The former gives the ordering of the set

while the latter gives its size.

[ "da yakka umzwaru assuds n trbiɛt", " ar yakka umggaru aksay nns." ]

This standard basis makes the complex numbers a Cartesian plane

called the complex plane.

[ "asila ad amsɣal da itgga imḍan uddisn g uswir adikartiy da itsmma aswur uddis." ]

The complex numbers of absolute value one form the unit circle.

da tggan imḍan uddisn n watig amggaru

[ " yat tzgunt ign yuwt." ]

In domain coloring the output dimensions are represented by color and brightness

respectively.

[ "g yigr n usuɣn", " da ttusmdyan wuggugn n ussufɣ s ukʷli d uzznẓṛ amzday." ]

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.", "yuw uswuri ɣf tmukrisin n tmggudin n iwtta timattutin g tyira ɣr tmagunt tasilant n ljibr", " nna issfrun akd imḍan uddisn", " illa ufssay n ka igat tagdazalt ila iwtta iggudin; sg tskʷflt tamzwarut nɣd nnig as." ]

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed.

baynd tsmktitin n wisl g tiwin tikadimitin zund “kubnhagn”

[ " maka tzri ur tt yannay awd yan kigan." ]

Later classical writers on the general theory include Richard Dedekind

Otto Hölder

[ " Felix Klein", " Henri Poincaré", " Hermann Schwarz", " Karl Weierstrass and many others.", "sg imarratn iklasikn g tizi yaḍn ɣf tmagunt tamattut n ritcard didikind", " nɣd utu hildr", "d filiks klin", " d hinri bwankir", " d hirman cwartz", " d karl wirstras d wiyyaḍ." ]

The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855).

ur ittusyaha ussmrs n imḍan iwngimn g uɣufal amiriw

[ " ar tawuri n lyunard ulr (1707-1783)", " d kaṛl fridritc gaws (1777-1855)." ]

The integers form the smallest group and the smallest ring containing the natural numbers.

da tggan imḍan imddadn; tarbiɛt tamẓẓant

[ " d txrst tamẓẓant g llan imḍan iɣaranen." ]

It is the prototype of all objects of such algebraic structure.

iga amdya amnzu n maṛṛa imɣnawn

[ " zund tuskiwt tajibrit." ]

Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68

C

[ " Java", " Delphi", " etc.).", "da ttubdarn wanawn n inmmaln n usnmili n imiḍ amddad ilan taɣzi iwrn (nɣd tirbiɛin tayyawin)", " igan s tmatart “ int “ nɣd “Integer” g kigan n tutlayin n usɣiws ( zund Algol68", " d C", " d Java", "d Delphi", " d tiyyaḍ)." ]

These are provable properties of rational numbers and positional number systems

and are not used as definitions in mathematics.

[ "imẓlay ad iɣy ad tnt nswr i uṭṭun umginen", " d ingrawn imiḍan imsursn", " d urda ttusmras am isissn." ]

Since the triangle is isosceles

a = b).

[ "mayd iga wamkṛaḍ amsaskl n igalaln", " a = b)." ]

Since c is even

dividing c by 2 yields an integer.

[ "mayd iga c amsin", " da aɣd takka tubḍut n c xf 2; imiḍ amddad." ]

Substituting 4y2 for c2 in the first equation (c2 = 2b2) gives us 4y2= 2b2.

asnfl n 4y2 ɣr c2

[ " g tgadazt tamzwarut (c2 = 2b2)", " aɣ yakkan 4y2= 2b2." ]

Since b2 is even

b must be even.

[ "mayd iga b2 amsin", " iqnen ad ig b amsin." ]

However this contradicts the assumption that they have no common factors.

maka aya itmgla d umrdu ittinin ur llin imggitn imccurn grasn.

null

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.”", "waxxa hakkak ", " illa yilɣ n hibasus ɣf tzmmar nns", " ayd inna yan wumiy", " yukz", " ikiz nns aynna illa g yil", " grnt imddukal nns ifitaɣursn g tizi yaḍn", "...", " acku snfln ya ufṛḍiṣ g iɣzwr ur irin... taɣalt ittinin iɣy uzgzl n tumanin g iɣzwr ɣr imḍan imddadn d usɣal nnsn." ]

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.", "s umdya", " g twngimt nnk tagzzumt inmn: iɣ ad ttubḍu ɣr snat", " d uzgn n uzgn ɣr sin", " d uzgn n uzgn ɣf azng", " imki." ]

This is just what Zeno sought to prove.

aya ayd itnaɣ zinun ad tt iswr.

null

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.", "g twngimin n liɣriq", " ukus n umddad n kan tannayt ur da iswar amddad n tannayt nniḍn", " xf uya asn iqqn uzrru uggar." ]

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.", "aksay”...ur igi imiḍ", " maka ismdya s inmallatn zund: imzrayn n uzririg", " d tɣmrin", " d tijimma", " d iksayn", " d tizi iɣin ad tmziriy", " imk nttini s talɣa izdin." ]

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.", "acku ur ityakz watig n wanc illan g uksay", " iɣy iduksus ad issiṭn asɣaln igan win usqqul", " d tinna ur igin tin usqqul sg ustay n usɣl sg uksay", " d usɣl zund assiksl n sin isɣal." ]

This incommensurability is dealt with in Euclid's Elements

Book X

[ " Proposition 9.", "da ittuga i winna ur iẓḍaṛn i usqqul g ifṛḍisn n iqlids", " adlis mraw", " asumr 9." ]

In fact

in many cases algebraic conceptions were reformulated into geometric terms.

[ "g tinawt; g kigan n waddadn ittuyalls tihkct i irmmusn n ljibr ɣr irman inzganen." ]

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.

iqnen wusun n itsn irmmusn idslan agnsu n tmagunt n dɣi

[ " imzaray n tilit n tiggi azrru akkʷ ign aɣzuran i tiggiwin d iswingmn illan ḍaṛt tmagunt ad." ]

However

historian Carl Benjamin Boyer writes that such claims are not well substantiated and unlikely to be true.

[ "waxxa hakkak", " yara unmzruy kaṛl bn jamin bwir an; “zund tafakult ad ur ilin iswrn iɣudan", " ur tnni ad tg tamddadt”." ]

Mathematicians like Brahmagupta (in 628 AD) and Bhāskara I (in 629 AD) made contributions in this area as did other mathematicians who followed.

kan imusnawn n tusnakt zund brahamaguta (asggʷas n 628 ḍaṛt tlalit n lmasiḥ)

[ " d baskara (g 629 ḍaṛt tlalit n lmasiḥ)", " imyiwasn g yigr ad", " mk skrn imusnawn n tusnakt yaḍn iḍfaṛn mayan." ]

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.", "g usggʷas 1872 tyafsarnt tmagunin n kaṛl wiyrstras ( sg ɣur unlmad nns irnst kusak) d idwaṛd hayn ( tasɣnt n kṛil", " 74)", " d juṛj kantur (analin", " 5)", " d ritcaṛd didikind." ]

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.", "iska wiyrstras d kantur d hin; timagunin nnsn ɣf tgffurt tartmi", " isbdd didkin tamagunt nns ɣf twngimt n wubuy (sicnit)", " g ungraw n maṛṛa uṭṭun umginen", " artn yaṭṭu ɣf snat trubba ilan itsn imẓlayn istin." ]

Dirichlet also added to the general theory

as have numerous contributors to the applications of the subject.

[ "irna awd diliclit i tmagunt tamattut", " imk gan kigan n winna yuwsn n tsnsitin n usntl." ]

This asserts that every integer has a unique factorization into primes.

aya islkan n wis d ku imiḍ amddad ɣars imggi amẓlay g imḍan imzwura.

null

To show this

suppose we divide integers n by m (where m is nonzero).

[ "mar ad nsbayn ayad yurda is da naṭṭu imḍan amddadn n xf m ( acku m ur tgi tamyat)." ]

If 0 never occurs

then the algorithm can run at most m − 1 steps without using any remainder more than once.

[ "ig ur akkʷ ittuga 0", " tɣy ad tswuri alguritm ɣf kigan n tsurifin m - 1", " bla nssmrs awdyan umagur wahli n tikkal." ]

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).

[ "“g tusnakt", " uṭṭunen iɣaranen gantn tinna ittusmrasn g ussiṭn ( imk illan g : llant sḍiṣ n idrimn aflla n tdabut)", " d ussuds ( imk illan g : wad aɣrm wiss kṛaḍ g tmurt).”" ]

These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.

da tggant tgffurin n izdawn n wuṭṭu iɣaran g llan ( iẓlin)

[ " g imagrawn n wuṭṭun yaḍn." ]

The first major advance in abstraction was the use of numerals to represent numbers.

azwar amzwaru n tanzɣt

[ " igat ussmrs amaṭṭun i usmdya n wuṭṭun." ]