Asnful n Nyuṭun

Amnni ad icqqa ad t tfhmt ! Amnni ad gis kra n iwaliwn nna ur iẓḍaṛ yan ad tn ifhm iɣ ur issin ɣ umawal iẓlin n tutlayt Taclḥiyt tatrart. Mad igan afssay ? Fad ad tfhmt mad gis illan tzḍart ad tawst i yixf nnk s umawal lli illan ɣ izddar akkʷ n tasna. Iɣ t ur tufit, tzḍart ad nn taggʷt ɣ imawaln d isgzawaln.

Tannayt f mamnk asa itgga usɣlu n kra n usnful.

Tga tanfalit n usnful n Nyuṭun yat tanfalit tusnakt iskr tt umusnak amqqran Isḥaq Nyuṭun 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 (ad t-igan mas-d $xy=yx$ — s umdya i isiruwn : $y$ iga-tt isiruw n tulut) s ɣikk ad, i kraygat $n$ amḍan ummid agaman, nṭṭaf :

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}$,

Ma ɣ imḍann

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

(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:

$(x-y)^{n}=\left(x+(-y)\right)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}(-y)^{k}$

Imdyatn:

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

Amdya

Ha asnful ad ittuyskar i id $n$ mẓẓiynin:

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

Tummla

Nẓḍar ad nsml tanfalit ad s tinawt n ullus.

Tummla nɣ taflalit igan tamakazt uggar ar tswurri s usfki n imuskirn isnfal ${\textstyle {n \choose k}}$ mas iga umḍan n tifulin n $k$ ifrdisn ɣ yat tagrumma nna ɣ illa $n$ ifrdisn. Kudnna nsbuɣla tanfalit

$(x+y)^{n}=(x+y)(x+y)\cdots (x+y)\qquad (n{\text{ twal}})$

ar nttafa yat timrnit n iynfuln s talɣa n ${\textstyle x^{j}y^{k}}$ 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 ${\textstyle j=n-k}$, acku kraygat twal ur ar gis ntxtar $y$, ar ntxtar $x$. S tɣarast yaḍna, maḥd darɣ ${\textstyle {n \choose k}}$ n tɣarasin ur mrwasnin n lixtiyyar n $k$ twal atig n $y$ zɣ $n$ tinfulin $(x+y)$ isggt ddaw as, aynful ${\textstyle x^{n-k}y^{k}}$ ixṣṣa ad ibayn ɣ usbuɣlu d umuskir ${\textstyle {n \choose k}}$.

Tummla s ullus

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

Tazwart

Iɣ $n=0$

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

$\textstyle {0 \choose 0}a^{0}b^{0-0}=1$

Tazwart n tanfalit tṣḥa

Taɣaldt

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

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

$\Rightarrow (a+b).\sum _{k=0}^{n}\textstyle {n \choose k}a^{k}b^{n-k}$

$\Rightarrow \left(\sum _{k=0}^{n}\textstyle {n \choose k}a^{k+1}b^{n-k}\right)+\left(\sum _{k=0}^{n}\textstyle {n \choose k}a^{k}b^{n+1-k}\right)$

Ad nsrs $p=k+1$

$\Rightarrow \left(\sum _{p=1}^{n+1}\textstyle {n \choose p-1}a^{p}b^{n+1-p}\right)+\left(\sum _{k=0}^{n}\textstyle {n \choose k}a^{k}b^{n+1-k}\right)$

$\Rightarrow \left(\sum _{k=1}^{n}\left(\textstyle {n \choose k-1}+\textstyle {n \choose k}\right)a^{k}b^{n+1-k}\right)+\textstyle {n \choose 0}a^{0}b^{n+1}+\textstyle {n \choose n}a^{n+1}b^{0}$

$\Rightarrow \left(\sum _{k=1}^{n}\textstyle {n+1 \choose k}a^{k}b^{n+1-k}\right)+\textstyle {n+1 \choose 0}a^{0}b^{n+1}+\textstyle {n+1 \choose n+1}a^{n+1}b^{0}$

$\Rightarrow \left(\sum _{k=0}^{n+1}\textstyle {n+1 \choose k}a^{k}b^{n+1-k}\right)$

Amsgra

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

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

$(X+Y)^{n}=\sum _{k=0}^{n}{n \choose k}X^{n-k}Y^{k}$

s

$\prod _{i=1}^{n}(X+Y_{i})=\sum _{k=0}^{n}\sigma _{k}(Y_{1},\ldots ,Y_{n})X^{n-k}$,

maɣ id $\sigma _{k}$ nɛna sisn igtfuln ujjuṛn ifrdasn.

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

$$\left(\sum _{i=1}^{m}x_{i}\right)^{n}=\sum _{\left|{\vec {k}}\right|=n}{n \choose {\vec {k}}}\prod _{i=1}^{m}x_{i}^{k_{i}}$$d s imksanen ur gin ummidn nɣ gan ummidn izdrn.

Ẓṛ uggar

• Asaḍuf n id kusinus
• Targadda n Cauchy–Schwarz
• Alugaritm agaman
• Asktiɣmr

== Tisaɣulin

==

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]Aggur:Tusnakt/Tin imgradn