??????????

?????????? ?????? ??????????? ( Algebra , ???? ???????? al-jabr ^[1]) ?????????? ??? ?????????? ?????? ?????. ??? ???????? , ????????? , ???????? ???? ???????? ????????????. ??????? ?????????? ?????? ???? ??????????? ?????????, ?????? ????? ????????? ????????? ????????????. ^[2]^[3] ??????, ????????, ????????? ???????????? ??????????, ????????? ???????????????? ???????? ?????????? ?????????????????. ?????????????? ???????????? ???-?????????? (780 ? 850) ??????? ???? ??????? (1048?1131) ???????? ?????????????????. ^[4]

?????????? ?????? ??????? ??????????????????? ??????????? ??????????????? ?????? ?????????. ??????? ?????????? ????????? ???????????????????. ??????? ????????? ????????? ?????????? ????????????????. ????????? ??????? ?????????? ??????????????? ??????? ????????, ?????????????? ???????????? ????????? ???? ????????????? ??????????? ??????????? . ??????? ?????????? ???????? (abstract) ?????????? ???????????? ???????????? ???????? ?????????? ???????????. ????????????? ?????? ??????? ???????? ?????? ???????????? ?????????? ???????? ???????????????????. ???????? ?????????????? ???????????? ?????? x , y ????? ????????? , a , b ????? ??????????? ????????????????? ?????? ???????? ????????????????????. ^[5] ????????????????, $E=mc^{2}$ ???? ????????????, $E$ and $m$ ?????????? ???????????, $c$ ?????? ?????. ?????? ?????????? ?????????? ??????????????????? ??? ??????????? ??????? ?????????? ???????? ????????????.

?????????????? ???????? ???????????? ???????????? ???????????????? ( algebraist ) ?????????????.

????? ???? ??????????? ???? [ ???? ]

????? ?????? ?????? ??????? ??????????????? ?????? ?? ???????????? ???????? ???????????? ???????. ????? ?????????????????? ???????????? ????? ?????? ??????? ??????? ???????? - ????? - ???????????? - ?????????? ????????????. ????????? ????????? ??????????? ???????????? ??? ??? ?????? - ????? ?????? ?????????? ??????????? ???????. ??????? ???????????? ?????? ????? ??????????????? ??????? ??????, ???????, ???????? ???????????? ?????????????????. ?????? - ?????? ?????.

?????????.		????? ??????
5 + 5 + 5 = 15.		5 + 5 + 5 = 3 x 5 (?????? ???? ????? )
14 + 14 + 14 + 14 = 56		14 + 14 + 14 + 14 = 4 x 14 (???? ???? ????? ????)

5 + 5 + 5 = 3 x 5 (?????? ???? ????? )

5 + 5 + 5 = 3 x 5 (?????? ???? ????? ). ??? ????????? ??????? ???????? ?????????? ???? ???? 3 x 5 = 15 ????? ??????? ????????. ????? ????? ?????????? ??? ?????? ???? ??? ?????????? ??? ???????????????? ????????????. ??????? 5 + 5 + 5 ?????? a + a + a ?? ????????. ???? ?????? ??? ??? ?????? ???? ????????????????. ???? ?????? ?????????????? ????? ?????????? ???? a ? ?????????????????? ???????? ??? ???????? ????? ?????? - ?????????? ????????. ????????? ??? ???????? + ??? ???????? + ??? ???????? ?? ????????. ???? ?????? ???????????. ????? ?????????? ??? ???????? ?????? ???????? ????? ??? ???? ???? ???????????? ??????? ????????? ???????????. ????????? ?????? ??????????? ???????? ?????? ???????? ????? ??????? ????????? ???????????. ?????? ?????????? ???????? ????? ??????? ????????? ????????? ?????? ?????????? ????? ???? ?????? ???????? ????????? ?????? 1 ????? ???????????????? . ???? ????? ?????????? ????????? ???????? ???? ???????? ?????????????????. ?????????? ????? ?????????????????????. ???? ??? ???????? + ??? ???????? + ??? ???????? ??????

   ????????  + ???????? + ???????? ?? ????????.              
   =  ?????? ????????. (????)

???????? ?????????????????? ?????? ????? ?????????? ???????????????. ????? ?????????? ??????? ??????????????? ?? a ?????? ???????? ?? ?????? ????? ????????. ????????? ????? ?????? ?????????? ????????????? ??????? ????? ???????????? ?? ???????? ?????? a ?? ?????? ????? ?????????????.)

?????? ???????? = ?????? a (???????? ?????? a ?? ?????? ????? ????????????????? )

                     =
 3 
 a     (?????? ?????? 3 ?? ?????? ????? ????????????????? )

3 a = 3 x a ??? ??? ??? ????. ????? ????????????. 3 a ?????? ??????????. ????? ????????? ??????????? ?? ???????? ????????????????. ?????? ???????? ???????? ?????? ?? ???????? ???????????????????? ??????? ????? ?????????? ??????????? ????????????? ??????? ?? ???????? ?????????????????????? . .

?????? ???? ???????? ?????? ?????? ???????? ????? ???? - ?? ?????? ????????? ????????????. ???????? ?????? ????? ???????? ???????????. ?????? ???????? ???????? ??????? ?????? ???? ??????? ??????? ???????? ???? ???????? ?????? ???????????? ????? ?????????? ??????? ??????? ??????? ????????? ???????????????? . 3a ?????? a3 ????? ???????? ?????? ??????? ???????? ????????. ?????????? a x 3 (a ?? 3)????? ????? ??????????????.

???? ??????. [ ???? ]

????? ?????????? ?????? ??? ??????? ?????? ??????????????? . ??? ??????????? ??????????????? ??????????????? ???????, ??????? ?????? ???? ??? ???? ?????????????????. ?????? ???? ???????????????? 1, 2, 3, 4, .... ????????????? ?????????? ????????? ????????? ( ???????, ???????, ????? ) ?????????. ?????? ???????? ??????????? ????????? ???????????????? ?????? ??????????? ??????, ???? ???????????????? ?? ????? ???????????? ????????. ???? "+" ?? ???? ???????? ??????????? ??????? ???????????? ????????????. ?????? ???????? ????????? ?????? ??????? ?????????. ??????? ??????????????? ???????, ??????? ?????? ??? ??? ???????. '??? ?????? ????????? " - " ?? ???? ?????????????? ???????????????? .' ?????? ?????? ??????? ??????????? " - " ?? ???? ???????? ??????? ?????????. ?????? ?????? ?????? ??????????? ?????? ???????? ????????? ???? ????????? ???? ???????????? ??????????????????? ????????????????. ???????? ??????????? ??????, ???? ???????????????? ???????????. ???????? ??????? ??????????? ??????? ?? ???? ????????????? ???? ????????? . ??? ??????????? ?????? ???????????? ????? ?????????????? ????????????? ?????????? ?????? ????? ????????? ???????????? ???? ?????????? ??????? ??????. ????????? ???????????? ???????? .

?????? [ ???? ]

??????????????? ??????? ???????? ??????????????????? 1, 2, 3, ... ??????????? ?????? ??????? ???????? ???? ????????. ??????????????? ????? ?????????? ?????????? ???? ???????? ????????. ???????? ?????? ??? ?????????? ??????? ???????? ?????????? ??????????? ????????????? ???????????? ???????. ??????? ?????????????? ???????? ???????????? ?????????? ???? ??????????? ?????????? ???? ???? ?????????? 5?? ???????????? )????????????. ??? ????????? ????? ????????????. ????????????? 4??? ???????????? ??????????????? 9??? ???????????? ???-?????????? ??????? Hisab al-dschabr wa-l-muqabala ???? ?????? ???? ?????????. ????????????? 13?? ???????????? "al-jabr" ???? ????????? ????? ?????? ???? ?????? ?????? ???????????? ?????????????????? ????? ?????? ??????????????????. ???? ????? ?????? ?????? ?????? ???????? ????????? ???? ????? ?????????. 17?? ???????????? ??????? ???????? ????????????? Ludus algebrae et almucgrabalaeque ???? ??????? ??????????. ???????? ????? ???????? ???????? ???????? ???????? ?????????.

??????? ???????? ?????????? [ ???? ]

?????????? ?????? ??? ????. ????? ????????????? ?????? ????????????? ?????? ????? ??? ???????? ????????? ??????? ????????? ??????????? ??????????. ????? ?????????? ?????????? ????????????? ??????????? ?????? ?????????????? (notation) ????????????????. ????????? ??????? ?????? ???????. ??????????? ??????? ???????????? . ?????????? ??????? ????? ?????????????????. ?????? ??? ??????????? ??????????????? ???????. ?????? ???????? ???????? '????' ???? ??????? ?????????? ???????????? ????????????????. $(x+y)^{2},(x-y)^{2}$ ???? ???????????,

x^{2}-y^{2}=(x+y)(x-y)

????? ?????????? ??????? ??????? ???????? ????? ??????????????????. ??????? ???????? ??????? ????? ??????? ????????? ???? ????????? ???????? ?????????? ???????, ?????? ???????????.

?????????????? ?????????? ??????????? ???????????? ??????????? ??????? ???????? ????????. ??????? ??????????? ????????????? ????? ?????? ??????. ??????????? ?? ( $\pi$ ) ??????? ???????????????????????????? ???????????? ???????? ??????????????? ???? (Method of exhaustion) ????? ?????. ?????????? ???????? ??? ??????????? ??????????? ??????????? ??????????????? ?????? ???? ??????? ???????? ??????????? ????????? ??????? ???????? ??????? $\pi$ ???? ?????????? ??????????????? ?????????????? ???????? ???.

(?????????? ???????? / ??????? ) ?????? (???????????? ??????????? ???????? / ???????) ?????? '?????'

???? ??????? '????????' ???? ????????? ?????????????????????????. ???????????? ??????? ???????? ???? ' ????? '????????? ??????? ????????? ??????????? ????????? ??????!

??????? ??????????? ?????????? [ ???? ]

?????? ?????????? ??????????? ????????? ????????? ?????????? ?????? ?????????????????? ?????????? ??????? ???????. ???????????? (3?? ??????????) ?????? '?????????' ???? ?????? ??????????????? ??????? ???????????? ?????????????? ???????????. ???????, ??????? ????????? ??????? ????? ???????? ????? ????????? ???????????? ??????????? ?????????? ??? ?????? ?????? ???????? ??????????????.

???????? ???????????? ????? ??? ??????????????? ??????????????? ??????????? ???????? ????????????????? (70 ????????? ???????) ????? 1881 ??? ??????? ???????????? ???? ?????????????? ????????????????????. ?????? ??? ????????????????????, ??????????????? ??? ??????????, ???????? ??????????????????????????. ?????????? , ?????????????? , ??????? ?????????? ???????? , ????????? ??????????? , ????????????? , ??????????????? ???? ???? ??????????. ??????? ???? ??????, ????????????????? ??????????? ??????????????? '????????? ????' (Golden Rule) ????? ????????? ????? ??????????? ?????????? ???? ???? ?????????? ????????????????. ??????????-????-????? ???? ????? ???????? ??????????. ????????? ???????????? Rule of Three ????? ?????????????. ??? ???? ???????????????,

??????? ?????????? ??????? ?????? ??????????, ??????? ????? ??????? ?????????????????
?????? ???? ??? ????: ?????????? - ???? - ????? ????????????? ???? ????????? ?????. ???????????? ????????? ???????? ???????? ??????? ???????? ???. ??????? ????!
??????????, ????? ???????? ??? ???????????????. ???? ???? ???????????????? ??????????.

?????????? ??????????? (Indeterminate Equations) ???????????? ?????????????????? ????????? ??????????? ?????? ???????? ???????????? ????????? ????. ????????????????????? ????????????? ??????????? 4?? ???????????? ???????? ????????????????, ???????????? ?????????? ????????????? (7?? ??????????), ???????? I ( 600 - 680 ), ???????? II ( 1114 - 1185 ) ?????????? ????????????? ??????? ????? ?????? ????????? ????????????? ?????????????. ???????? II ???? ??????? ???? ??????? ??????????? ???????????????.

???????? ????????? ?????? ?????????? [ ???? ]

1619??? ???????? ???????? ?????????? ?????????? ???????????? ??????? ???????????? ??????????????. ???????? ?????????? ?????????? ???????????????????, ??????????? ??????? ???????????????, ????????????? ???? ???????? ???????????? ???????????, ?????????????? ?????????? ????????? ???????????. ??????????????????? ???????????? ????????? ?????????? ?????????????? ??????. ??????? ??? ????? ??????? ????? ??????????? ?????????????????? ????????? ???? ?????????? ????????????? ?????????. ?????? ????? ?????????? ???????? ????????? ????????. ????????????? ????????????? ?????? ??????????? ?????????? ??????????? ?????, ????? ?????????? ?????????????? ?????? ?????????? ?????????????, ??????? ????????????? ????????? ?????????? ?????????????? ????? ????????. ?????????????? ?? ???? ???????? ???????????? ????????? ??????? ???????.

?????????????? ??????? ????????? ??????? ???????????? ????????? ????????????? ?????? ????. ???????? ?????????? ??????????? ??????????????? ?????? ???????????? ??????????????? ????????? ???????? ?????????? ??????????? ????????. ???????????????? ??? ???????? ??????????? 19??? ???????????? ???? ?????????. ????? ???? ?????? ??????????????? ?????????? ??????? ????????????? ??????????? ??????? ?????????????????? ???????? ????? ??????? ???????? ???????.

???????? [ ???? ]

???? ???????????? ???????????????? ???????? ????? ??????. ???????????????? (??? ??????????):

???????? ( 1501 - 1576 ), ( ??????? ) 1545 ??? ???????, ???????????????????????? ????????? ?????????????.
?????? ( 1540 - 1603 ), ( ???????? ) $\pi$ ???? ??? ?????????? ?????????? ???????????????. ?? ???? ????????????? ??????????????? ????? ?????? ?????????????? ??????? ?????? ?????????????? ????? ????? ??????????????.
??????? (????????) ( 1601 - 1665 ), ??? ???????????? ?? ?????????? ???????????????. ???? ??????? ?????? ???????? ??? ??????? ?????? ?????????????? ???????? ??????? ?????? ????? ??????????????, ??? ?????????????? ??????? ???????????????? ??? ??????? ????????????? ????????? ????????.
???? ????????? ( 1642 - 1727 ) ( ??????????? ) ???? ???????????? ??????? ????????????????????? ??????? ??????. ??????? ?????????? , ????????? ???????? , ????? ???? ?????????????????? ????, ?????????????? ???????? ??????: ?????????????????? ; ????????? ????? ?????? ????????.
?????????? ( ??????? )( 1646 - 1716 ), ?????? ?????????????????? ??????????? ????? ??????? ????????????????? ??????. ?????? ??????????? ???????????????? ?????. ???????????? ?????????????? ????????????? ?????? ???????????????? ???????????????????.
?????? ?????????? ( 1654 - 1705 )( ??????????????? ) ????????????? ??????????? ???????????? ?? ???????????????? ??????????. ???? ????? ???????????? ?????????? ?????? ????? ????????????????.
?????? ( 1707 - 1783 ), (???????????????, ????? , ???????) ??????? ??????????????? ???????????? ????????. ?????? ?????? ?? ?????? ??????????????????? ???????? ??????. ??? ???????????? ???????? ??????? ???????????????????. ????????? ????????????? ???????????? ???? ??????? ???????????. ??????? ?????????? , ?????????????? ???????? ????? ?? ??????????? ??????????. ??? ??? ??????? ????? ????????????? ?????????? ???????? ???????????????, ??????? ???? ??????? ?????????? ???????????. ????? ???????????? ?????. ??????? ???????????? ????????????? ????????????? ????????????? ?? ??????????? ???????????.
???????????? ( 1736 - 1813 ), ( ??????? ) ??? ???????????????? ??????????? ???????????????? ??????? ????????????? ?????????.
???? ( 1777 - 1855 ), (???????) ????????????????? ?????????? ??????????. ?????????????? ????????????? ???????? ??????????????? ??? ???????? ???????????????? ???? ???????? ?????????? ??????? ?????????.
???? ( 1789 - 1857 ), (????????). ??????? ????????????????????????, ????????????????, ???, ?????? ?????????? 20?? ???????????? ?????? ????????????? ????????? ???????????????? (Theory of Substitutions) ????? ?????????????????. ????????????? ????? ????? 20??? ???????????? ???????????? ??? ??????? ?????????????? ??????? ???????. ?????????? ????????? ????????? ????????? ???????????? ???????? ??????????. ?????? ????????????? ???? ??????? ?????????????? ?????? ???????? ???? ?????????? ?????????, ???? ???????????????? ?????????? ?????????,???? ??? ??????????? ????? ??????????????? ???????? ????????????? ???????????? ????????? ???????????????? ????? ???? ??????? ??????????. ?????? ??? ????????? ???????????? ????.
?????? ( 1811 - 1832 ),(????????) ?????? ?????????? ?????????? ????? ??????????????? ?????? ???????? ????? ??????? ??????????????? ????????????? ???????????????? ???????????? ?????? ?????????. ??????? ?? ?????????????? ????? ????????????????? ????????????? ?????????????????????????????? ????????????? ?????????.
????? (1802-1829),
?????? (1804-1851),
????????? (1805-1865),
??????????? (1814-1897),
?????? (1821-1895)

?????????????? ??? ???????? [ ???? ]

?????????????? ??????? ????? ??? ???????? . ???????????? ??????????????? ???????? ?????? ????? ????? ???????????? ?????????? ????????????? ??????????? ???????. ????????????? ???????? ??????????????? ?????? ???? ??????? ????????????? ?????? ????????, ?????????? ?????????? ????????, ??????????? ???????????? ?????? ?????. ???????????? ??? ???????? ?????????? ????? ????? ??????????? ??????? ????????? ??????????? ???????????????? ???????????.

??????? ?????? ????? ?????????? ?????? ??? ????? ??????????. ??????????? ??? ?????? ??????????? ????? ?????????, ????????? ???????????? ??????? ???????? ??????? ???????? ???????????? ????????? ???????.

??????????? [ ???? ]

↑ "algebra" . Online Etymology Dictionary .
↑ I. N. Herstein, Topics in Algebra , "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964
↑ I. N. Herstein, Topics in Algebra , "...it also serves as the unifying thread which interlaces almost all of mathematics." p. 1, Ginn and Company, 1964
↑ "Omar Khayyam" . Encyclopedia Britannica . ????????????? ???? 5 October 2014 .
↑ See Boyer 1991 , Europe in the Middle Ages , p. 258: "In the arithmetical theorems in Euclid's Elements VII?IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."

??????????? [ ???? ]

Eli Maor. e : The story of a number. 1994. Princeton University Press. Princeton, NJ ??????????? ?????????? ??? 0-691-03390-0 .
Paul J. Nahin. An Imaginary tale. The Story of $\surd {-1}$ .1998. Princeton University Press. Princeton, NJ. ??????????? ?????????? ??? 0-691-02795-1 .
E.T. Bell. Men of Mathematics.1937. Simon & Schuster, New York, NY . ??????????? ?????????? ??? 0-671-46401-9

[1] "algebra" . Online Etymology Dictionary .

[2] I. N. Herstein, Topics in Algebra , "An algebraic system can be described as a set of objects together with some operations for combining them." p. 1, Ginn and Company, 1964

[3] I. N. Herstein, Topics in Algebra , "...it also serves as the unifying thread which interlaces almost all of mathematics." p. 1, Ginn and Company, 1964

[4] "Omar Khayyam" . Encyclopedia Britannica . ????????????? ???? 5 October 2014 .

[citeboyer-5] See Boyer 1991 , Europe in the Middle Ages , p. 258: "In the arithmetical theorems in Euclid's Elements VII?IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's Algebra made use of lettered diagrams; but all coefficients in the equations used in the Algebra are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."

[1]

[2]

[3]

[4]

[5]

?? ? ?? ?????????? ?????????
???????	?????????? ???????? ??????? ??????? ?????????? multilinear abstract ????????? / ??? ???????? ?????????? / ???????? Category theory ????????? Computation ???????????????? ??????????? ??????????? / Dynamical systems Functional analysis ?????? ???????? ????????? ( discrete ?????????? differential finite ) Graph theory ????? ???????? Lie theory Mathematical logic Mathematical physics Mathematical statistics ??????? ?????????? Optimization ????????? ???????? Representation theory ???? ???????? ??????? ???????????? ??????? ???????????
?????????	Pure ???????? ?????????? Computational
?????? ???????? /