Amnni ad icqqa ad t tfhmt ! Amnni ad gis kra n iwalwn 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 [ 1] tanfalit n usnful nɣ tanfalit n Nyuṭun .
Iɣ gan
x
{\displaystyle x}
y
{\displaystyle y}
[ 2]
x
y
=
y
x
{\displaystyle xy=yx}
y
{\displaystyle y}
n
{\displaystyle n}
(
x
+
y
)
n
=
∑
k
=
0
n
(
n
k
)
x
k
y
n
−
k
=
∑
k
=
0
n
(
n
k
)
x
n
−
k
y
k
{\displaystyle (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
k
)
=
n
!
k
!
(
n
−
k
)
!
{\displaystyle {n \choose k}={\frac {n!}{k!\,(n-k)!}}}
(kra n twal ar tt nttara ula
C
n
k
{\displaystyle C_{n}^{k}}
x
0
{\displaystyle x^{0}}
S usnfl ɣ tanfalit
y
{\displaystyle y}
−
y
{\displaystyle -y}
(
x
−
y
)
n
=
(
x
+
(
−
y
)
)
n
=
∑
k
=
0
n
(
n
k
)
x
n
−
k
(
−
y
)
k
{\displaystyle (x-y)^{n}=\left(x+(-y)\right)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}(-y)^{k}}
Imdyatn:
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
.
{\displaystyle {\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}}}
Ha asnful ad ittuyskar i id
n
{\displaystyle n}
(
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
.
{\displaystyle {\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}}}
Nẓḍar ad nsml tanfalit ad s tinawt n ullus .[ 3]
Tummla nɣ taflalit igan tamakazt uggar[ 4]
(
n
k
)
{\textstyle {n \choose k}}
k
{\displaystyle k}
n
{\displaystyle n}
(
x
+
y
)
n
=
(
x
+
y
)
(
x
+
y
)
⋯
(
x
+
y
)
(
n
twal
)
{\displaystyle (x+y)^{n}=(x+y)(x+y)\cdots (x+y)\qquad (n{\text{ twal}})}
ar nttafa yat timrnit n iynfuln s talɣa n
x
j
y
k
{\textstyle x^{j}y^{k}}
j
{\displaystyle j}
k
{\displaystyle k}
x
{\displaystyle x}
y
{\displaystyle y}
j
=
n
−
k
{\textstyle j=n-k}
y
{\displaystyle y}
x
{\displaystyle x}
(
n
k
)
{\textstyle {n \choose k}}
k
{\displaystyle k}
y
{\displaystyle y}
n
{\displaystyle n}
(
x
+
y
)
{\displaystyle (x+y)}
x
n
−
k
y
k
{\textstyle x^{n-k}y^{k}}
(
n
k
)
{\textstyle {n \choose k}}
Ar nsiggil ad nml masd tanfalit ad n
(
a
+
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
k
b
n
−
k
{\textstyle (a+b)^{n}=\sum _{k=0}^{n}{n \choose k}a^{k}b^{n-k}}
ullus
∀
n
∈
N
{\displaystyle \forall n\in \mathbb {N} }
∀
{
a
,
b
}
∈
A
2
{\textstyle \forall \{a,b\}\in \mathbb {A} ^{2}}
a
b
=
b
a
{\displaystyle ab=ba}
A
{\displaystyle \mathbb {A} }
Iɣ
n
=
0
{\displaystyle n=0}
(
a
+
b
)
0
=
1
{\displaystyle (a+b)^{0}=1}
(
0
0
)
a
0
b
0
−
0
=
1
{\displaystyle \textstyle {0 \choose 0}a^{0}b^{0-0}=1}
Tazwart n tanfalit tṣḥa
Rad nɣal mas tanfalit ad tṣḥa ar twala n
n
{\displaystyle n}
(
a
+
b
)
n
+
1
=
(
a
+
b
)
.
(
a
+
b
)
n
{\displaystyle (a+b)^{n+1}=(a+b).(a+b)^{n}}
⇒
(
a
+
b
)
.
∑
k
=
0
n
(
n
k
)
a
k
b
n
−
k
{\displaystyle \Rightarrow (a+b).\sum _{k=0}^{n}\textstyle {n \choose k}a^{k}b^{n-k}}
⇒
(
∑
k
=
0
n
(
n
k
)
a
k
+
1
b
n
−
k
)
+
(
∑
k
=
0
n
(
n
k
)
a
k
b
n
+
1
−
k
)
{\displaystyle \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
{\displaystyle p=k+1}
⇒
(
∑
p
=
1
n
+
1
(
n
p
−
1
)
a
p
b
n
+
1
−
p
)
+
(
∑
k
=
0
n
(
n
k
)
a
k
b
n
+
1
−
k
)
{\displaystyle \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)}
⇒
(
∑
k
=
1
n
(
(
n
k
−
1
)
+
(
n
k
)
)
a
k
b
n
+
1
−
k
)
+
(
n
0
)
a
0
b
n
+
1
+
(
n
n
)
a
n
+
1
b
0
{\displaystyle \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}}
⇒
(
∑
k
=
1
n
(
n
+
1
k
)
a
k
b
n
+
1
−
k
)
+
(
n
+
1
0
)
a
0
b
n
+
1
+
(
n
+
1
n
+
1
)
a
n
+
1
b
0
{\displaystyle \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}}
⇒
(
∑
k
=
0
n
+
1
(
n
+
1
k
)
a
k
b
n
+
1
−
k
)
{\displaystyle \Rightarrow \left(\sum _{k=0}^{n+1}\textstyle {n+1 \choose k}a^{k}b^{n+1-k}\right)}
S tifadiwin lli nfka i
a
{\displaystyle a}
b
{\displaystyle b}
(
a
+
b
)
n
=
∑
k
=
0
n
(
n
k
)
a
k
b
n
−
k
{\displaystyle (a+b)^{n}=\sum _{k=0}^{n}\textstyle {n \choose k}a^{k}b^{n-k}}
∀
n
∈
N
{\displaystyle \forall n\in \mathbb {N} }
Nẓḍar ad nals i usbuɣlu s ullus fad ad nml tanfalit n Laybniz i tazllumt tis-
n
{\displaystyle n}
Taɣarast tamsuddst n tawlawalt ad aɣ yujjan ad nskr asmata n tulut agtful
(
X
+
Y
)
n
=
∑
k
=
0
n
(
n
k
)
X
n
−
k
Y
k
{\displaystyle (X+Y)^{n}=\sum _{k=0}^{n}{n \choose k}X^{n-k}Y^{k}}
s
∏
i
=
1
n
(
X
+
Y
i
)
=
∑
k
=
0
n
σ
k
(
Y
1
,
…
,
Y
n
)
X
n
−
k
{\displaystyle \prod _{i=1}^{n}(X+Y_{i})=\sum _{k=0}^{n}\sigma _{k}(Y_{1},\ldots ,Y_{n})X^{n-k}}
maɣ id
σ
k
{\displaystyle \sigma _{k}}
Nẓḍar ula ad nskr asmata n tanfalit ad s timrniyin n
m
{\displaystyle m}
n
{\displaystyle n}
(
∑
i
=
1
m
x
i
)
n
=
∑
|
k
→
|
=
n
(
n
k
→
)
∏
i
=
1
m
x
i
k
i
{\displaystyle \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}}}
↑ 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
(en) J. L. Coolidge , « The Story of the Binomial Theorem », Amer. Math. Monthly , vol. 56, no 3, 1949, p. 147-157 (JSTOR 2305028, Ɣr ɣ wanṭirnit )
(en) Concrete Mathematics (2nd ed.). Addison Wesley. pp. 153–256. ISBN 978-0-201-55802-9 . OCLC 17649857.
(en) Indian J. History Sci . 1 (1): 68–74.
(en) " Newton binomial " , , Encyclopedia of Mathematics , EMS Press
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
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
^{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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88704e95bcdc63d9e4aa97ef94e2bd7bdf2f28fb)
