Asnful n Nyuṭun

Tga tanfalit n usnful n Nyuṭun yat tanfalit tusnakt iskr tt umusnak amqqran Isḥaq Nyuṭun^[1] fad ad yaf usbuɣlu n kraygat taẓḍurt n kra n usnful. Tṭṭaf ismawn yaḍna zun d tanfalit n usnful nɣ tanfalit n Nyuṭun.

Mad tettini

Iɣ gan $x$ d $y$ sin ifrdisn n kra n uzbg (s umdya sin imḍanen ilawn nɣ ismlaln, sin igtfuln, sin isiruwn imkkuẓn nna dar illa nafs tiddi, atg.) nna ɣ tlla tasunflt^[2] (ad t-igan mas-d $x y = y x$ — s umdya i isiruwn : $y$ iga-tt isiruw n tulut) s ɣikk ad, i kraygat $n$ amḍan ummid agaman, nṭṭaf :

(x + y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{k} y^{n - k} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{n - k} y^{k}

,

Ma ɣ imḍann

$(\binom{n}{k}) = \frac{n!}{k! (n - k)!}$

(kra n twal ar tt nttara ula $C_{n}^{k}$ ) gan imuskirn isnfal, « ! » ɛnan sis uskir d $x^{0}$ afrdis-tiggt n uzbg.

S usnfl ɣ tanfalit $y$ s $- y$ , ar nttafa:

$(x - y)^{n} = {(x + (- y))}^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) x^{n - k} (- y)^{k}$

Imdyatn:

\begin{matrix} n = 2, & (x + y)^{2} & = x^{2} + 2 x y + y^{2}, & (x - y)^{2} & = x^{2} - 2 x y + y^{2}, \\ n = 3, & (x + y)^{3} & = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}, & (x - y)^{3} & = x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3}, \\ n = 4, & (x + y)^{4} & = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}, \\ n = 7, & (x + y)^{7} & = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7} . \end{matrix}

Amdya

Ha asnful ad ittuyskar i id $n$ mẓẓiynin:

$\begin{matrix} (x + y)^{0} & = 1, \\ (x + y)^{1} & = x + y, \\ (x + y)^{2} & = x^{2} + 2 x y + y^{2}, \\ (x + y)^{3} & = x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}, \\ (x + y)^{4} & = x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}, \\ (x + y)^{5} & = x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5}, \\ (x + y)^{6} & = x^{6} + 6 x^{5} y + 15 x^{4} y^{2} + 20 x^{3} y^{3} + 15 x^{2} y^{4} + 6 x y^{5} + y^{6}, \\ (x + y)^{7} & = x^{7} + 7 x^{6} y + 21 x^{5} y^{2} + 35 x^{4} y^{3} + 35 x^{3} y^{4} + 21 x^{2} y^{5} + 7 x y^{6} + y^{7}, \\ (x + y)^{8} & = x^{8} + 8 x^{7} y + 28 x^{6} y^{2} + 56 x^{5} y^{3} + 70 x^{4} y^{4} + 56 x^{3} y^{5} + 28 x^{2} y^{6} + 8 x y^{7} + y^{8} . \end{matrix}$

Tummla

Nẓḍar ad nsml tanfalit ad s tinawt n ullus.^[3]

Tummla nɣ taflalit igan tamakazt uggar^[4] ar tswurri s usfki n imuskirn isnfal $(\binom{n}{k})$ mas iga umḍan n tifulin n $k$ ifrdisn ɣ yat tagrumma nna ɣ illa $n$ ifrdisn. Kudnna nsbuɣla tanfalit

(x + y)^{n} = (x + y) (x + y) \dots (x + y) (n twal)

ar nttafa yat timrnit n iynfuln s talɣa n $x^{j} y^{k}$ ma ɣ $j$ d $k$ ar mmaln s trtib ad nit uṭṭun n twal nna ɣ nxtar $x$ nɣ $y$ lliɣ tn nkka art nsbuɣlu. Illa darnɣ s bzziz $j = n - k$ , acku kraygat twal ur ar gis ntxtar $y$ , ar ntxtar $x$ . S tɣarast yaḍna, maḥd darɣ $(\binom{n}{k})$ n tɣarasin ur mrwasnin n lixtiyyar n $k$ twal atig n $y$ zɣ $n$ tinfulin $(x + y)$ isggt ddaw as, aynful $x^{n - k} y^{k}$ ixṣṣa ad ibayn ɣ usbuɣlu d umuskir $(\binom{n}{k})$ .

Tummla s ullus

Ar nsiggil ad nml masd tanfalit ad n $(a + b)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) a^{k} b^{n - k}$ tṣḥa s tinawt n ullus $\forall n \in ℕ$ d $\forall {a, b} \in 𝔸^{2}$ s $a b = b a$ (maɣ $𝔸$ iga uzbg mknna ira igt)

Tazwart

Iɣ $n = 0$

$(a + b)^{0} = 1$

$(\binom{0}{0}) a^{0} b^{0 - 0} = 1$

Tazwart n tanfalit tṣḥa

Taɣaldt

Rad nɣal mas tanfalit ad tṣḥa ar twala n $n$

$(a + b)^{n + 1} = (a + b) . (a + b)^{n}$

$\Rightarrow (a + b) . \sum_{k = 0}^{n} (\binom{n}{k}) a^{k} b^{n - k}$

$\Rightarrow (\sum_{k = 0}^{n} (\binom{n}{k}) a^{k + 1} b^{n - k}) + (\sum_{k = 0}^{n} (\binom{n}{k}) a^{k} b^{n + 1 - k})$

Ad nsrs $p = k + 1$

$\Rightarrow (\sum_{p = 1}^{n + 1} (\binom{n}{p - 1}) a^{p} b^{n + 1 - p}) + (\sum_{k = 0}^{n} (\binom{n}{k}) a^{k} b^{n + 1 - k})$

$\Rightarrow (\sum_{k = 1}^{n} ((\binom{n}{k - 1}) + (\binom{n}{k})) a^{k} b^{n + 1 - k}) + (\binom{n}{0}) a^{0} b^{n + 1} + (\binom{n}{n}) a^{n + 1} b^{0}$

$\Rightarrow (\sum_{k = 1}^{n} (\binom{n + 1}{k}) a^{k} b^{n + 1 - k}) + (\binom{n + 1}{0}) a^{0} b^{n + 1} + (\binom{n + 1}{n + 1}) a^{n + 1} b^{0}$

$\Rightarrow (\sum_{k = 0}^{n + 1} (\binom{n + 1}{k}) a^{k} b^{n + 1 - k})$

Amsgra

S tifadiwin lli nfka i $a$ d $b$ , tanfalit $(a + b)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) a^{k} b^{n - k}$ tṣḥa i $\forall n \in ℕ$ .

Asmata

Nẓḍar ad nals i usbuɣlu s ullus fad ad nml tanfalit n Laybniz i tazllumt tis- $n$ n kra n ufaris.

Taɣarast tamsuddst n tawlawalt ad aɣ yujjan ad nskr asmata n tulut agtful

(X + Y)^{n} = \sum_{k = 0}^{n} (\binom{n}{k}) X^{n - k} Y^{k}

s

\prod_{i = 1}^{n} (X + Y_{i}) = \sum_{k = 0}^{n} σ_{k} (Y_{1}, \dots, Y_{n}) X^{n - k}

,

maɣ id $σ_{k}$ nɛna sisn igtfuln ujjuṛn ifrdasn.

Nẓḍar ula ad nskr asmata n tanfalit ad s timrniyin n $m$ irman ismlaln lan yat tuẓḍurt tummidt $n$

${(\sum_{i = 1}^{m} x_{i})}^{n} = \sum_{| \vec{k} | = n} (\binom{n}{\vec{k}}) \prod_{i = 1}^{m} x_{i}^{k_{i}}$ d s imksanen ur gin ummidn nɣ gan ummidn izdrn.

Ẓṛ uggar

Isaɣuln

↑ S ṣṣaḥt, tanfalit ad tettussan yad zɣ tasut tis-X, slawann akk dar id bu-tusnakt ihindiyn, iɛrabn d ifarisiyn (Al-Karaji) d ɣ tasut tis-XIII, amusnak acinwi Yang Hui imlatt. Ɣ 1665, Nyuṭun iskr as asmata nns s tuẓḍurin ur gin ummidn.
↑ Tafada ad tga ḍaṛuṛiyya, d tga tagdazalt i tidt n tanfalit i n = 2.
↑ Tummla n Asnful n Nyuṭun : s tinawt n ullus - AVIDYU YUTUB
↑ Tummla n Asnful n Nyuṭun : s tinawt n uddun - AVIDYU YUTUB

== Isaɣuln

==

Tinidlisn d izdayn

(en) J. L. Coolidge, « The Story of the Binomial Theorem », Amer. Math. Monthly, vol. 56, n^o 3,‎ 1949, p. 147-157 (JSTOR 2305028, Ɣr ɣ wanṭirnit)

(en) Graham, Ronald; Knuth, Donald; Patashnik, Oren (1994). "(5) Binomial Coefficients". Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153–256. ISBN 978-0-201-55802-9. OCLC 17649857.

(en) Bag, Amulya Kumar (1966). "Binomial theorem in ancient India". Indian J. History Sci. 1 (1): 68–74.

(en) Solomentsev, E.D. (2001) [1994], " Newton binomial ", , Encyclopedia of Mathematics, EMS Press

Amawal

Tiguriwin

Tanfalit = formule
Asnful = binome
Amusnak = mathematicien
Asbuɣlu = developpement
Taẓḍurt = puissance
Azbg = anneau
Afrdis = element
Amḍan = nombre
Ilaw = reel
Asmlal = complexe
Agtful = polynome
Isiruw = matrice
Amkkuẓ = carre
Tasunflt = commutativite
Tulut = identite
Ummid = entier
Agaman = naturel
Amuskir = coeficient
Uskir = factorielle
Afrdis-tiggt = element unite
Tinawt = énoncé
Ullus = recurence
Asfki = definion
Amakaz = intuitif
Tafult = partie
Tagrumma = ensemble
Timrnit = somme
Iynfuln = monômes
Tazllumt = derive
Tamsuddst = derive
Awlawal = variant
Agtful = polynimiale
Ujjuṛ = symetrique
Afrdas = elementaire
Irm = terme
Amksan = exposant
Uzdir = negatif

Imawaln

Amawal umniḍ DGLAi n Asinag Agldan n Tussna Tamaziɣt, s kraḍ tutlayin (tamaziɣt tatrart tamɣribit - tafransist - taɛrabt) [Alink nns]

Amawal Tafsut n tusnakt - MCB Alger, Tiwi Uzzu s sin tutlayin (tamaziɣt taqbaylit - tafransist) [Alink nns]Talɣa:Aggur:Tusnakt/Tin imgradn

[1] S ṣṣaḥt, tanfalit ad tettussan yad zɣ tasut tis-X, slawann akk dar id bu-tusnakt ihindiyn, iɛrabn d ifarisiyn (Al-Karaji) d ɣ tasut tis-XIII, amusnak acinwi Yang Hui imlatt. Ɣ 1665, Nyuṭun iskr as asmata nns s tuẓḍurin ur gin ummidn.

[2] Tafada ad tga ḍaṛuṛiyya, d tga tagdazalt i tidt n tanfalit i n = 2.

[3] Tummla n Asnful n Nyuṭun : s tinawt n ullus - AVIDYU YUTUB

[4] Tummla n Asnful n Nyuṭun : s tinawt n uddun - AVIDYU YUTUB

[1]

[2]

[3]

[4]

Asnful n Nyuṭun

Tumayin

Mad tettini

Amdya

Tummla

Tummla s ullus

Tazwart

Taɣaldt

Amsgra

Asmata

Ẓṛ uggar

Isaɣuln

== Isaɣuln

Tinidlisn d izdayn

Amawal

Tiguriwin

Imawaln

Talgamt n ustara

Asnful n Nyuṭun

Mad tettini

Amdya

Tummla

Tummla s ullus

Tazwart

Taɣaldt

Amsgra

Asmata

Ẓṛ uggar

Isaɣuln

== Isaɣuln

Tinidlisn d izdayn

Amawal

Tiguriwin

Imawaln

Talgamt n ustara

Siggl