־ֻ ֿ֟á

 

( . ֬ ֵָ ,

֮։ ߾ԙ, ֤ )

 

׾פ ָ־Դ ֟- ֟, ߕ֟, ״ן פ ֵ֮ ן ֓߮ ִֻ ֟ | ֟ ָ ֟ ֻ ֵ ֓߮ ָ־Ե ֖֟Ӯ ֿ֟áִ ָ׳ԟ ן | ֣֣ԟ: ԓ߮ ֟ ߕ֟ ֤֮֟ | ƴ ֮ ֵ ָ־ ׾ֻֿ ִ֮ܵ ֻ Ӥ ֟ ֵ֮ , ׾ִֵ ד , ָ־Ե ֮ܵ ,  , ָ֯ ׾ֿ ֮ פ | ׾֮֫ ֟ ָ  ֖֟ӫָ ׻ ֵ ֿ֟á ή ֟ | ׮ֵ ִή ֵ֮ ֿ֟á ׮ֵӴ ֤ | ֣֣ԟ: ֟ ן ׾֪ ֮ ׮ֵ ܵ ֮֬ ִ֗ ֟ |

 

ƴ ׾פ ׮ֵ ֿ֟á ִָ , ή, ־߸ֵ֓- ָ֟, ִֵ ֲֻ ןֵ ֟Ӵ š | ־߸ֵ֓ ֮ ֮ 850 | ή ִ֮ ִָ 㯟, ֵָ߬֓, Ʈ ֖֟ ή ִ֮ ׾ֿ ֟Ӵ ԟ: ׳֮ | Ƹ֣- ָ֟ ο ( problems ) ֵ: ֳ ָ ή ο ׳֮ |

 

֟Դִֻ֮ ֲֻ ֿ֟áӲӬ Ɵ ָָ֬ ƴ ֿá Ɵ֯ ֋ ֙׻֯ (֙), , ֻ־ָ ӳ־֟: ָ֮, ׿ֻ ã֮Ӵ ןֿ߻ | ֲ δ ֯ , ֲ ׮ֵֿ֯ ֆӴ ָïָ ӲӬ |

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1 -־֟ , ֵ֤ ׸ Ɵ, ִꤵ ״ן ָ ׿֟, 1919, 90 | ָ֮֬ ֤, ֱ 1895, ֵ 7, 8, 38.

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ױ ƴ ֢ ֻ֟ ׳֮ ׳֮ ֆ ֵ ή ִ֮ ָ , ׾ßָӲӬ ׾ֿ ֟Ӵ ִ ׾׳֮֟ | ֟ ֻ֟ ׳֮ ׳֮ ֆӴ ֤֮-Τ֮ ӲӬ , ֡ ׾֮֫ ָ ִ ִ֮ , ãִ֮ ֵ ׾ָ ߑ ָ ָ ׾֖ׯ֟ פ ֟ |

 

Οߟ  ִ ִ Γָ ׾׾֬ ׾֖֮ ֆ ֵ֮ ֮ | ִ֮֟: ֳ ָ־Ե ״ Ɵ, ֵܵ֟  Ɵ, ֛ ֛ ܵֆ ׸ | ֛ ܵֆ ε֮ ܵֆ ׻֮ ׻ֵ ָ ־ֿ ֮ , ֿ״ δ ( The place value system of notation ) ׾ָ | ֟ ׮ӿֵֹ ֬ ֿ״ δ ־ָ ִָ ־ ֮ ָӳ ֻ ֳ , ֲ ִ ִ֬ ֮ ָꮮן ָ | ֵ -δ ֛ ׌ֻֿ ֬ , ߮ ֿá ן֯Τ֮ 㻾Ӵ ֯ ֻ߮ ָ׳ ֟ ׾ ֵ֜, ָ״Ƹ ήӴ ֯ ӓ־ ֲ֟ߴ ִ֮ ֿ֟áִ ׸ןԟ פ |

 

֛ Ɵ֯ ֟, ֟ ןָ ™ߴ և, ֪ׯ Ʈ ׮ֵ ִ֮ Ɵ ָ ֲ֟ ֻ֬߮ ִֵ ׾ד , Ο ֲ֟ ή ֲֻ , ֣ׯ ֿ֟áӲӬ Ɵִ ׾֓ , ֣֣ԟ: ֮ 499 ד֟ ֵԳ֙ߵ ֿ֟áӲӬ ֮ ד֟ | ־֤ ܿ׻ ן ( Bakhsali-Manuscript ) ִ ß׻֟ ή ӳ־֟: ָ ָ ֲ֟ ֮ | ֲֻ ß׻֟ ן ƴ ֻ ֟- ֮ ãן ׾ִֵ ׾ß ֮ ״ֻ֟, ֣֣Դ ֵԳ֙ δ㯟 ־ ָ߬ פ ή ֤ ֿ֟á ß | ֟ӲӬ ο ܵ ־ י | ß׻֟ ן ƴ ֻ ֮ ֮ ֿ״δ ֟ӲӬ ֟ εֵ ִֵ ָ ׾פ , ߔ ֖֟ӫָ ׻֟ ָ ֟ ο ( problems ) ֟ |

 

ֵ֟ ֵԳ֙ߵִ ֯ ֿ֟á ׾ֿ ֟ , ִ ƴ ׮ִ ׻֟ ׾ֵ ״ֻ֮ - ֟Դֻ֮߮ ֣״ ֟ ֲ וִ֮ ֟, ׾׮ִֵ ֕ ׮ִֵ ״׻֟ , ֣ ָ ִ, ָ ( indeterminate equations ) ε ߕ֟ | ο ã֟ ֵԳ֙ ֮ ֖֮֟ ׾֤ , ־ ִ ֵԳ֙ߵִ ԟ ֲ ָ־ ׻ ִע ? ֵԳ֙ ׻֟ δ, 飾, Ӧ, , , , ֻ, ïן, ׮ ֡ ִָ ֵԳ֙ ֮ Ԯ ו ֯㸴 ֤ 1 | ֟ ֻ֟ ֮ ׾֤ | ָ ֿ֟á ן ֵ֮ ֮ , ֵԳ֙ߵ ֟ ָ ֟׻֮ ֟ ֜ | ׾֤ ӳ־֮ ָ ָ ο ã֟ ֵԳ֙ ֻ߮ ֿ֟áӲӬ ή ֲֻ ? ׮־ָ ָ | ֿ״δ ׾ָ ־ ֮ ָӳ ֻ ֳ ִֵ | ִ֮ Γָ ֮ ׻ֵ ָ ӓ ֟ײ և | ֿ״δ ε ֻ ֵԳ֙ ή ־ԯΣִ ή Οߟ | ֵԳ֙ ή ήӴ ֮ ܵ֯֬ן ε , ־, ִֵ ָ ָ ֵ | ֟ ™ ֵԳ֙ ׾ß ܵן ָ, ָ֮֟, ֵ֟ Ӯ ־ԯΣִ ή , וִ ֿ״δ ε ֵ | ֵԳ֙ ָ ֮ ß Γ׻֟ ׾ֻ߮ և | ֱ ֟ ֻ ֟ ֮ 499 ֿ֟ ׻ ƴ ֮ ß ״ֻ֟ , ή ֲֻ |

 

ָ ֮ 500 ־ ָߵ ֿ֟á ׾ ן ד֡ ׻ֵ ß־ִ ֮֬ ƴָ | ãִ ֵԳ֙ ָߵ ֖֮֟ ֻ֮ ή ׾ֿ Ƣ֯ ֵ ֟ | ֿ֟áӲӬ ή ™ ֮ ָ ֮ 500 ֻ߮ ָߵ ֿ֟á ן : ׮ִ ׻ֵ ƴ Ƥ, ׮ֵ Ɵ, ׾ֿ֟: ״ Ɵ, ֲ֮߮ ֛ | Ӵ ƴ ӛ ״ֻ֟ וִ֮ ֿ֟á ן׾֪ Ԯ ֵ ֟ | ָ ׮ֵ ׬ӿ ִήӴ ֿ֟á ן׾֪ ִ ״ֻ֟ | ִ ָߵ ָָ֟ ֟ , ή וִ ֳԟ , ֵ: ߮ ָ ֟ײ ֮ | : פ ƴ ֮ 400 800 ״ ׮ ן ָ ֿ֟áߵ ׾־ָ ־ ָӳ ֮ 400 ֮ |

 

ֵԌ ׮ֹ ִֿ ƴ ֲ֟ ָӳ ֮ ֙ޛִ ־ֻ ֮ Ɵ֯ ִ֗֟ | ߵ ߸ֻֻ ֮ ή ִ֤ ֿ֮ ׾֮֫ ãֵ߹ ֖֟ ֮ ׻ֵ

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1 δ׿ֲֳ׾ִ֮ |

ֵԳ֙ß ׮֤ן ֯sדԟ ִ֮ || ֵԳ֙ߵ. 2,1.

 

δƳ״ִ֮֮֡ ֯ ֯ܵsô֮ דԟ ֮ ֯㸾״: ו֟ ןֳ֖֮֮֬ ִֵ֮Գ֙ ׮֤ן | ( ָ꿾ֵָ֓ )

--------------------------------

 

ֿ֟á ֿ

 

֮ 1912 ֵ֓ԫָ ָ֟ ֿ֮ ִֵ ׾֮֫ ֳ ֿ֟á ԟ: ׾֮֫ӫָ ֻև ֟ | ֻߴ ִ ή ֵ֮ ֖֟ ֟ήִ֮ ֟ ֻ 2 | ׮ֵ ״ Ɵ ָ Ӵ ׾ֳו֟ , (ִ֬) ֢ ï™, ƻ֟ | ִ ִ ֮ ֮֟, ԟ ֿ֟áӲ֮ ֢ ï™, | ֟ ֻ֟ ִ֬ ֤Ԯִ ֿ֟á ֮ ֤ פ ֵ |

 

֪ׯ ֖֟ ִ ֟ , ָӟ ןֵ 㯟 և | ִ ֲ ֓߮ ֦ 278 þ ָ֬ | ן ׾֪ ή ֮ ֟ (1) ԯΖׯ ; (2) ֦ƾ Ɵ ִ ׻ ή | ֵָֻ (ֳ 1150 .) ֮ ԯΖׯ ִ , ֙˙꟯ֻ3 (965) ή־ָ֟ פ | ֨ ִ ָ ן ή־ָ֟ ָ״Ƹ (505) ֙˙꟯ֻ ָ פ ֵ |

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1 --- ֵ֓ ָ ִפ ָ֟ ß־֮, . . ôָ֣֫ ׻֟, ֦, 1912.

2 : ֿ֟áߵ (, ߮ ֵ֟֙, וֻ 21 (1919), š 115 145.

3 ƟƟ, ׫ָ߲ ִפ, ָ֮, 1895, . 226.

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Դ֬ ִ ׻ ִ֮֟ ή ֵ ֟ | ־ִֻ ָ֯ ܵ ָ֟ ׾֪֮ | ָָ֟ ֣ã֮ ׾ָ֓ ֵ | ֟ ֮ߵ ָ֟ ׮:ӿֵֹ ֨ ׾֮֫ӫָ ׻ ֵ ֟ή 㯟 ֵ 1 | ִ ֳ־֮ ִ ׾ָ֮֫ ׻֟ ή ֿ֟áִ֮ | ָ ƴ ή ֯ | ƴָ ֿ֟áִ֮ ֮ כ ֮ ã֮ , þן ֣֟בֳִµ, ԯΖׯ, ָ֫, סֻΖׯ, סָֻ פ ָ֟ ή ׻֟ ή ־ֻ ִ |

 

 

־ֻ Ɵ

 

־ֻ ֤ ָӳִ ߸ ָ ׻ և | ߸ ֢֖֮ ״ פ֯ | ß: ֖֟ | : ֿ֟áߵִ ־ֻ ԟ , Ծ֟ ן , ֵܵ֟ ֟ ָ, וִ֮ ӓ ֮߮ ֮ ־ָ֟ | ָ , ִ, Ӳ, ֳִ֦֮ ֤֯ , וִ֮ Σִ ֳ ֮ 200 ׮ִ ֮ 600 ֳ | : ־ֻ ׬ӿ ֿ֟áߵִ ֮ 200 600 ߓ ִֵ ֮ | ָ ָ־Ե ֿ֟á ןָ ׻ֵ ־ֻ Σִ Ɵ֯ ή ֟ , ִ ƴ ָߵ ֿ֟á ן ֲ ׬ Ӭָ ִֵ, ԟ ӓ־ ֲ֟ ֟ ״ֻ֟ | ׾ֿ ֵ֮ ֟ ™ ֟ ־ֻ ֿ֟áߵ ִ ֮ 500 | Ƹ֣- ־ִֻ ԟ εֵ ֟ ήִ և ֟, ֣ ִ ãֻ֟ ֳ ו ֻ ֿ֟ ָߵ ֿ֟á ׸ד֟ ׾֮֫ ָ֟ ״ֻ | ־ֻ ֳִ֟ ׸ԟ ׸ָ ֵԳ֙˙ ֿ֟ ήӴ |

 

 

־ֻ֮ԟ ֿ֟á

 

ܵ֋ --- ־ָֻ ֿ״δ ԟ: ׸ד֟ | δ ־ԡ ֲֻ | ƴ ־ֻ ԟ ָ֟ և ܵֆ ֌ ֨ןֵ ã֟ ---

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1 ߻ ,ֵֵ֮֬ ָ֫, 28 ָ ֮ ִ Ӳ֬ ( regarding permutation and combination ) ߮ ׮ִֵ | ׮ִֵ ֟ Σִ ׻ֵ ֵ ֮ ֛ |

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(1)             79999998 ܵ ו פ 7, ִ 8 ִ֬ ָ 9 ָ־ע 1 |

(2)             46666664 ֌ ֵ -- ֚, , ֚ ƕָ, ֚ , ָ 2 |

(3)             22799498 ֌ ֵ -- , ֢և, ׮֮֮־ ƕָ,ָ ֮־3 |

 

ִ (1) ו ֨ן ֵ Ɵִ ã֮ ֵ ֟ , ָ֟ƴ4 ã֮Ӵ | ֿ״δ ׸ֵ ֨ | (2) ܵ֋ ƻ ֌ և | Ɵִ Γ׻֟ ָ֬ ן ָ | ָ -δ , ָ֬֟: Ɵִ ֵ ֟ 5 | ֻ ִ δ ֵ: ִֵ ֵ ֵ | (3) ֲ ֛ ܵ ƻ ֌ և | ָָ (2) (3) ï™: ׳֮ ã֮ ׻ֵ ֵ |

 

֛ ֵܵ --- ׾פ Ɵִ ֛ ֵܵ ֵ֟ ִֵ և | ־ִֻ ָ ߾ָ׿ֵ ( ֯δ ) פ ָ ׾֟ | ׮׿ֹ֟ ׻ և ֲ ֛ ܵ ֵԯ ֮µ | ܵ ־ִֻ6 ֟־ ߓ, ־ ׮׿֟, י-י-י י-י-י-י ߓ և | ֮ ---

 

226 227 ߓ | ־, ׬ ׮ֵ֟- (1,00,00,000)3 (1,00,00,000)4 ߓ | ־, ־ԣ ׮׿֟- 225 x 226 | ߾ ܵ ָ֮֟ 79228162514264337593543950336 |

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1. . 3, š 98, ֣ 51 | 괴ָ֙, ߾ӛ š 633.

2. . 3, š 99, ֣ 52 | 3. . 3. . 100, ֣ 53.

4. -ָ֟ 1, 27 Ʈ ֿ֟á ן, וֻ 1, 1935, 16 5. , Ծ֟ . 14. 6 . 3, 253 7 괴ָ֙,߾ӛ, (. . . ߸ߕ) . 104

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ܵ | ִ ֮ ã֮ ו֮֟ ( 1,00,00,000 )4 , ָ ֛ ܵ | ֟ ־ָֻ ֟ Ӯ ֮µ ֱ ׮ֻ ֨ ܵ ֮µ ֮µִ ִ , ׻ֵ ܵ־ֻ ֟ |

 

׻ εֵ

 

־ִֻ , , , , Դ ִ֮ ׮ֻ֮, ֣ ܵֆ ֟ ׮ֻ֮ ( The raising of numbers to given powers ) פ ׻ εֆ ֮ ֲֻ | ε֋ ׳֮, ӲӬִ և | ־ִֻ ԟ ֟ ֨ӟ ( theory of indices ) ָ ֟ ή ׳֮ | ׮ֵֿ֟: ֮֨ ֣״ , ֮ 500 | ִ֮֨Ӭ ׻ ׾ָ֓ ׮ִ׻֟ εֆ ָָ֬ Οߟ :--- (1) , (2) ֮, (3) ָָ , (4) ָָ ֮ (5) ܵ ܵ֟㻵 ֟ ׮ֻ֮ ( The raising of numbers to their own power ), (6) Դ, (7) ִ֮, (8) ָָ Դ, (9) ָָ ִ֮, פ ֲ ֟ Ӵ ֙ ֵ |

 

Ƹ֣--- 3/2 ֮ Σִ Դ | 9 ֮ ֮ | 6

֮ , ֮ , פ1 | ָָ ִ֮ ߓ ׻ ָ -

 

Σִ ֮ () 2 = 2

,, ߵ ֮ (2) 2 = 4 = 226

,, ֮ 23

,, ,, 2

ָ --- Σִ Դ ֮ 1/2

׫ߵ ,, ,, 1/2 2

,, ,, ߵ ,, ,, 1/2 3

2

,, ,, ,, 1/2

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1 ־ֻ, 3 š 53

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ԟ---Ӿԟ

 

׸ ֲ ԟ-Ӿԟ ε ܵ ܵ֟㻵 ֟ Դ ֵ |

 

Ƹ֣ --- ԟӾԟ |

 

ִִ֮ ־ִֻ ׾ָ֮- ֮ ִ ε ֵ | ܵ ׾ָ֮ ֮ ԟ ܵ | , ׾ָ֮ ---

11111...... ָ

ֵԌ Ο ã֮ ָ ( ׾־֟ ܵ ) | ױ ׾ָ֮- ֲֻ ܵֆ ָïָ ܵ ԟ-Ӿԟ ֯ ֟ , ܵ Σִ ԟ-Ӿԟ ƻ֟ | , ԟ-Ӿԟ |

׾ָ֮- ָ : ε , ԟ ׾֮֬ ױ ,

׫ߵ ԟӾԟ () ֯ | ׾֮֬ : ָ ߵ ԟ-Ӿԟ

 

() ֯ |

()

 

־ִֻ ε ε ߮ ָ ׬ ֟ | ߵ ԟӾԟ ָ1 ֛ ܵֆ ܵ֟ ֮ ִӬִ ֵ | ε ֮ ܵ ֯ , ֮ ֟ 2 ߵ־ָ ԟӾԟ 256 256 ֟ |

 

֟ ֮֨

ֵԌ ֮ ï™ ־ָֻ ֟ ֮֨ ԟ: ׸ד֟ | ---

 

(1) = +

(2) / = -

(3) () = ֮

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1 ־ֻ, 3, 20 פ.

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֮֨ εӲӬ Ƹ ־ִֻ | Ƹ ׮ִ ָ 1- ֵ 2 7 Դ 2 2 ֲ ֟ | ԟ-

 

2 2 7/2 2 6 = 2 2 6

 

ֲ ֿ״δ ֮ ֵ ֲ ׫δ δ ε֋ ( The operations of duplation and mediation ) Ɵ֯ ִ֗ ֟ | ָߵ ֿ֟á ήӴ εֆ ד֮ ״ֻ֟ | εֆ ״ ֮ ׮־ Ɵ֯ ֮֟ , ֟ӲӬ ή ָ֤ þָ ֟ | ־ִֻ εֆ ד֮ ״ֻ֟ | ܵֆ ָָ ׾ָ֓ ׮ֵֿ֟: ׫δ ε ׸ñי , ׫δ ε ֿ״δ Γָ ָ־Դ ֿ Γ׻֟ | ָ δ ֨ן ֢ ֻ֟ | ־ִֻ ε ƴ 2, 3, 4 פ ָֻ֬ ֑׸ ִ֮֨ ָ֬ ֟ |

 

֑׸ ( Logarithm)

־ִֻ ׮ִ ׸ ֲ ֵ ֟ 2 ___

(1)   ԓ__ ו֮֟ ָ ܵ ָָ ֬ ֬ , ֮ ܵ ԓ ֟ - 2 ԓ =

ԓ ֮ ƴ ֬׮ ֨ןִ ָ _ ( ) = ׸ ֑׸ ָ֬ 2 |

(2)   Կֻ__ ܵ ԓ ܵ Կֻ | _ Կֻ = ֿ = ׸ ׸ ֑׸ ָ֬ 2 |

(3)   ס 3___ו֮֟ ָ ܵ ָָ 3 ׾ֳו֟ ֟ , ֮ ܵ ס | _ ס = ס֔ = ׸ 3 | ֑׸ ָ֬ 3 | 1 ־ֻ 3, š.253 פ. 2. ־ֻ 3, . 21 פ. 3. ־ֻ 3, . 56.

(4)   ֟ԓ 1___ו֮֟ ָ ܵ ָָ 4 ׾ֳו֟ , ֮ ܵ ֟ԓ | _ ֟ԓ = ֔ = ׸ 4 | ֑׸ ָ֬ 4 |

 

־ִֻ ֑׸ӲӬ ׮ִ ׸ִ ֵ ֵ __

1.      2 ׸ ( /) = ׸ _ ׸

2.      ׸ (.) = ׸ + ׸

3.      3 2 ׸ = | ֑׸ ָ֬ 2 |

4.      4 ׸ ( )2 = 2 ׸

5.      5 ׸ ׸ (/)2 = ׸ +1+׸ ׸ ,

( ) ( 2 ׸ )

= ׸ + ׸ 2 + ׸ ׸

= ׸ + 1+ ׸ ׸ |

׸ 2= 1, ֲ ָ֬ 2 |

6. { } =

6 ׸ ׸

7. ֻ֮ ܵ , ---

Σִ ԟ Ӿԟ = ( ֻ֮ )

,, ׫ߵ ,, = = ,,

,, ߵ ,, = = ,,

 

־ִֻ ׮ִ ׸ִ פ ֵ 7 ---

() ׸ =

() ׸ ׸ = ׸ + ׸ ׸

() ׸ = ׸

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1 ־ֻ, 3, . 56. 2 ־ֻ, 3, . 60 3 ־ֻ, 3, . 55. 4. ־ֻ, 3, . 21 פ. 5. Ծ֟. 6. Ծ֟ | ֟ ֮ߵ ήִ ֑׸ ׸״֟ | ܵ ܵ | Σִ ԟӾԟ ׿ ( ) ׫ߵ ԟ Ӿԟ ׿ | 7. ־ֻ, 3, . 21. 24.

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( ) ׸ ׸ = ׸ + ׸ ׸

= ׸ + ׸ ׸ + ׸

( ) ׸ = ׸

( ) ׸ ׸ = ׸ + ׸ ׸ | פ

( 8 ) 1 ׸ ׸ < 2

 

ִ֟ ׮ִ ִ֟ ֟ ---

׸ + ׸ + ׸ ׸ < 2

 

׳֮ --- ִ֟ ׳֮ ׻ εֆ, ו֮ ֮ ־ִֻ ׻ֵ ֵ , ן׸ ƴ ׳֮ӲӬ ֟ ֟Ӳ֬ ֟ ήִ ״ֻ֟ | ִ ׮ִ ׻֟ ֮ߵ ---

 

( 1) 2 2 = +

+ ( / ) +1

(2)3 ֻ֮ ܵ , ֕ פ ֵ δֿ: ֲ ( ׳֮ ) ֮ | ׮ִ ׻֟ ִ + ׸ִ פ ֵ ---

=

+ ( /) + 1

־ =

1 + ( /)

( 3) 4 פ = , = , --- ( - ) + =

 

( 4) 5 פ =, - = -

+ +1

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1 ־ֻ, 3, .24. 2. ־ֻ, 3, . 46.

3 ־ֻ, 3, 46. 4. ־ֻ, 3, . 47, ֣ 27.

5 3, . 46, ֣ 24.

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= +

- -1

( 5) 1 פ = , - =-

+ + 1

= +

- - 1

( 6) 2 פ = , = + , -

= -

+1

 

פ = - , --- = +

- 1

( 7)3 פ - , ָ ׳֮ , ---

- = (-)

( 8) 4 פ = , = - , --- =

+ -

 

( 9) 5 פ =, = + , - =

- +

 

( 10)6 פ = , = , -= -

+ +

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1 3, . 46, ֣ 24. 2. 3, . 46, ֣ 25.

3 3, . 46, ֣ 28. 4. 3, . 48 ֣ 29.

5 3, . 49, ֣ 30. 6. 3, . 49 ֣ 31.

--------------------------------

 

(11)1 פ =, = , - = +

- -

 

ֲ ׸ִ ־ֻ ԟ ָ֟Ӵ ֵ ֟ | ֟ӲӬ ֟ ήִ ״ֻ֟ | ָ֟ Դ֬ ־ ή | ֮ ֲ ֟ӲӬ ή, ־ Ծ֟ ֆ ׻ֵ ֵ | ֟ ָ ε ׮ֹ | ֻ ôָ־ֿ ֲ ך ִ֬ ׾֮֬ ִ֗ ֟ | ׮ִֵ ׮ֵֿ֟: ֻ ֲ ֿ״-δ ֟ εֆӴ ֵ Γ׻֟ |

 

׿ --- ׿ ε ־ִֻ ã֮ ָ ֵ ֵ 2 | εӲӬ ׸ ֲ - , δ - ֟ ήӴ ״ֻ֟ | ֮ ׿ ε ֮ ־ָ ָ־Դ ֿ״ δ ׾ָ ֟Դ֮ |

֮

֛ ܵֆ ε --- ֮ ֲ ׾׾֬ ε ֳ ֓߮ ןֵ Ɵִ ֵ ֟ | ׸ ִָ֤֗ ߔ և | þֳ׾ ֮ ׸ ָ ׾֟ ֛ ܵֆ ε , ֮ Ԯֿáִ ܵֆ ß | ׮ִ ׾־֮ ֵ ָ־Դ ֿ׮ ֮ ӲӬ ֻ֮ ׾׾֬ ־֮ֆ ߲֨ ֣ ֮ӲӬ ֮ ׸ ׮ֻ֮ ֱ |

֛ ܵֆ ֌ ׻ֵ ד֟ ֣ ֮ ֮ ׾ ֳ ֲ ׮ ׾ָ֓ ׾ֿ ָ߯ ֟ | ִ ״֛ߕ֮ ִ ֙ δ Ӥ֕ ֮ ε֟ ֮ ֿ׮ ֮ ߴ ( limit) ׾ִֵ ׾ָ֓ | ֛ ܵֆ ֌ ݵ | ָ־Դ Ʈ, ֿ׮Ӯ ֛ ܵֆ ε ֵ ׻ֵ ד֟ ׾ָ | ׾ֿ֟: ׮ֵ ָ ִß ߾, ֻ-Τ ־ ֿ-Τ פ δ ׮ֹ ε֟ |

֛ ֵܵ ֌ ߮ ָ ִֵ ֵ ֵ ---

( 1 ) ֿ״-δ ( place-value notation ) --- וִ ִ֮ ֵ ֵ | ӲӬִ ֟ ֮ߵ ִ֮1 ָָ֬ 10/140 ֛ ܵֆ ֌ ֻ ִ ׻֟ ֵ |

(2) ֟ ׮ִֵ ( Law of indices Ӿ ) ֵ ֛ ܵֆ ֟ ֌ ׻ֵ ֵ | ---

( ) 22 = 4

( ) (22) 22 =44=256

( ) { (22)22 } { (22)22 } = 256256

ו 2 ߵ ԟ-־ԟ | ܵ ִß ( universe ) ׾֪ ( protons electrons ) ܵ ֛ |

( 3) ֑׸ ( ԓ ) ־ ֑׸ ֬׸ ( ԓꤿֻ ) ֵ ֛ ܵֆ ׾ָ֓ ܵֆ ׾ָ֓ ָ ׻ֵ ֵ | ---

( ) ׸2 22 = 2

( ) ׸2 ׸2 44 =3

( ) ׸2 ׸2 256256 =11

ִ ֵֿ ֕ ܵ ֌ ׻ֵ ƴ ֵԌ ߮ ָ ָ ֵ | ֿ״δ ִß ָ֬ ִע ֮ և | ֛ ܵֆ ֟ ֛ , ֑׸ ֵ ֟ | ֬׮ ֤֣׾֖֮ ׸ִ ( magnitudes ) ֌ ׻ֵ ֟ ׮ִֵ ֵ ־ָ֬ | 1 ֛ ܵֆ ֣ ܵ-ִ ӲӬִ ׾ֿ ֮֮ ׻ֵ ֵ Ʈ ֿ֟á ן ( History of Hindu Mathematic ), ߻ֻ ָ֮ߤ, , ָ ׿֟, 1, 11 פ. Ƹ֣--- ׾ֳָ ׾֪1 ֮ ׌ ָ և --- 136. 2 256 ֣, ܵֆ ׾֮ (distribution of primes) ד֟ ֻ Ì ܵ (skewes number) ׮ִ ָ ֌ ֟ -

34

10

10

10

ܵֆ ֌ ֻ ֵԌ ִß ָ ֵ ־ִֻ ֵ | ï™ ָ־Դ ָ ֮ ֟־ ֟ײ ־-ָ֬ ֵ |

֮

־ִֻ ֮ ֵ ֟ | Ɵִ ֮ ֲ ֵ Դ | ִ ֲ ֮ ֵ ָ ֮ ݵָ ָ | ---

( 1 ) ִ֮֮ 2 ___ ִ ֮ | ß-ִֵ ֣֣ԟ: ֮ ׾ָ֓ ׾֮ ֻ 㟾 ֙ ׻ֵ ָ֬ ִֻ֓ ־ ֮µ ָ ׻ֵ, ־ Ɵִ, ֮ פ ֟ | ãִ `֮` ֲ ִִ֡ ֮ | ִ֮֮ Ɵ |

1 ܵ 136.2/256 ֿ״-δ ֌ ָ ָ -

15,747,724,136,275,002,577,605,653,961,181,555,468,044,717,914,572,116,709,336,231,425,076,185,631,031,296,

2 ߵ ԟ-־ԟ ԟ 256256 ׾ֳָ ִß ׾֪ ܵ ׬ | פ ƴ ִß ׾ ָ֟ӕ ֮ ׾֪ י, ׾֪ ׸ע ׾ `ֻ`֮ , ִß ֻ ܵ---

34

10

10

10 |

ܵ ܵֆ ( primes ) ׾ֳ ( distribution ) ӲӬ ֟ | 2 ߾֕߾״֤ ָָ ӟ | ־ֻ 3, 11 .

 

( 2) ã֮֮֮֯1 - ָׯ֟ ֮, ãׯ֟ ֮ | ֣֣ ֮ | ß ֮ ָ ׻ֵ ֟ ֲ ε ֟ |

( 3) ֮֮2 ___ ֻ֟ ִֵ ֟ ֮ ֮ | Ӗ ֵ ׻ֵ ֟ ו֮ ֮-׾ֵ á ֮ , ו ֟Դִ֮ ֵ |

( 4 ) ֮֮֮֟__ ܵ֟ ֮ | Ӗ ֿ֟áִ ε挟 ß׾ ֮ Դ և |

( 5 ) Τ׿֮ ׸ִ߮ ԟ ֮ ָ |

( 6 )֮֮__ פ֟ ֮ | ֮ פִ ߬ ֹ ֮ Οߟ |

( 7 ) ׾ßָ֮֮___ ׫׾ßָ֟ ־ šߵ ֮ | Οָ֟ ֿ֮ |

( 8 ) ֵ֮֮__ ׫פ֟ ֮ | Ƹ ߬ פֆ ֮ ֟

( 9 ) ־Ԯ֮ __ ֿ֟ ֮ | ס֬-׾ß ֮ ԟ ָ֮ ֿ֮ |

(10) ־֮֮__ ֻ֟ ִֵ ֟ ֮ ֮ | Ӗ ֵ ׻ֵ ֟ ו ֮-׾ֵ á ֮ ו ֵ |

(11)֮֟__ ׮֟ãֵ ׮־ֿ ֮ |

 

쌟 ֯ וִ ֲ ִ־ ֵ ו֮ Դ `֮` Ӗ ε Ɵִ |

1 ӟ ִ ד֢ ---- ָ֛ ֋ ׾֤ ӟ״פ ־ ֟ ִ | .3, .11 12 . 2. ӟ Ӥ׾ ִ֤ ִ֤ | .3, 12.

 

֮֮֮ ( Numerical infinite )

 

־ִֻ ï™ פ ֵ ִ ֮ Ӗ ε1 ֮֮֮ Դ ֵ , ֮ Դ , ֮ ָ δ ι ֵ ֟2 ֵ ֮֮֮ Ԯߵ ִ 3 ֮ ӳ־֟: -Ɵִ ֮ ԟ ֮֮֮ ׸ ׬ ׾ֿ֤ ׳֮ ׳֮ ָ և , ֣ ε ֮ Γ׻֟ ֵ ־ִֻ ֮ ׸ և ֮ӲӬ ε֋ ܵ֟ ܵ֟ ִ δ ֣ ֣ ָ ׻֟

 

ܵ֟, ܵ֟ ֮ δ ֵ Ɵִ ִ֓߮֟ ֻ֟ ֵ Οߟ ׳ֵ֯ ֤ ָ֓߮֟ ήӴ ֮ ִ֓ ֮ Դ ε㌟ ו Դ ƴ ׸ ߔ ήӴ ã֮ ֮֮֮֮ ׻ֵ Ƹ֣-- ״֓Ӧ ָ ־ ֟ײ ׻֟ ή סָֻ ָ ָߟ֮֮, ㌟֮֮ ֑֮ ֮֮֮ ֛ ָ ܵ , ֮ ή ָ ܵֆ ߮ ܵ ---

 

( 1 ) ܵ֟--- ו ƴ ֮

( 2) ܵ֟--- ו ƴ ֮

( 3) ֮--- ו ƴ ֮

 

ֵԌ ߮ ָ ܵ-δ : ߮ ߮ γ ֵ ׮ִ ָ ---

 

( 1 ) ܵ֟ --- ( ֮ߵ ) ܵֆ ߮ ---

( ) ֑֮-ܵ֟ ( ִ֟ ܵ ) ו ƴ ֮

( ) ֬-ܵ֟ ( ߓ ܵ ) ו ƴ ֮

--------------------------------

1 ־ֻ 3, . 16.

2 ֆӟ ִֹ֯, ֟ ֤֤֬ . 3, . 17.

3 ӟ ߵ ִ . 3, . 16.

--------------------------------

() ™ ܵ֟ ( ֲ ֛ ܵ ) ו ƴ ֮

 

(2) ܵ֟ ( ֮ߵ ) ߮ ---

 

( ) ָߟ-ܵ֟ ( Σִ ܵ ) ו ƴ ֮

( ) ㌟-ܵ֟ ( ߓ ܵ ) ו ƴ ֮

( ) ܵ֟ܵ֟ ( ܵ-ܵ ) ו ƴ ֮

 

Ԍ ߮ Ο : ߮ ߮ γ , ֑֮ ( ֲ ), ִ֬ ( ߓ ) ™ ( ֲ ֛ ) ָ ܵ֟ ߟָ ׮ִ ܵ֋ ׾™ ֟ ---

 

1 ֑֮-ָߟ-ܵ֟ ------------

2 ִ֬-ָߟ-ܵ֟ ------------

3 ™-ָߟ-ܵ֟ ------------

1 ֑֮-㌟-ܵ֟ ------------

2 ִ֬-㌟-ܵ֟ ------------

3 ™-㌟-ܵ֟ ------------

1 ֑֮-ܵ֟ܵ֟ ------------

2 ִ֬-ܵ֟ܵ֟ ------------

3 ™-ܵ֟ܵ֟ -----------

 

( 3 ) ֮ --- ו ƴ ֮ ߮ ---

( ) ָߟ-֮ ( Σִ ֮֮ ) ו ƴ ֮

( ) ㌟-֮ ( ߓ ֮ ) ו ƴ ֮

( ) ֮֮֮ ( : ߴ ֮ ) ו ƴ ֮

 

ܵ֟ ִ֮ ߮ Ο : ߮ ߮ γ ֑֮, ִ֬ ™ : ֮ ƴ ׮ִ ܵ֋ ֯ ---

 

1 ֑֮-ָߟ֮֮ ---------

2 ִ֬-ָߟ֮֮ ---------

3 ™-ָߟ֮֮ ---------

1 ֑֮-㌟֮֮ ---------

2 ִ֬-㌟֮֮ ---------

3 ™-㌟֮֮ ---------

1 ֑֮-֮֮֮ ---------

2 ִ֬-֮֮֮ ---------

3 ™-֮֮֟ ---------

 

ܵ֟ ܵ֟ ׸--- ֳ ή ָ ֑֮ ܵ֟ 2 , , ή ֟ ׳֮֟ ֲ ܵ ִܵ֟ ״׻֟ ִ֬ ִܵ֟ 2 ™ ܵ֟ ߓ ִß ֮ ֟ , ֣ ™-ܵ֟ ֑֮-ָߟܵ֟ Ծ֟ ԟ ֮ ִ ԟ = - 1 סָֻ ׮ִ ָ ִֵ֗ 1 ---

 

ָ֮ ׾ֿ, ԟ ֻ֬, ״ ֻ δ־ָ ֵֻ ֮ ߴ֋ ָָ ֜ ס֕ֆӾֻ ִꮦߵ ֹ ״ ִֵֻ ֵֻ ׾ßָ Ծ֟ ֵֻ ׾ßָ ꮦ֟ ( ֲ Σִ ߓ ) ( 100,000 ) ֮ ־ֻ , ִ߯ ƻ֟

 

֮ ָ ָ ֛˜ ֮ וֵ Ο ֮ ־ֻ ƕָ ֮ Ƹ 1, 1, 1 1 Ƶ ֮ וֵ 1 ָ ߕ ָ פ ֵ ױ ָ ָ ֻ ֵ ֲ ׿ ָ ֵ, וִ ֲ ָ ָ ߕ ε ׻ֵ ו֮֟ ָ ߕ ־ֿ ܵ ָ ---

 

ָ֮ ֛˜ ׻ֵ __ 19791209299968.10 3 1 ָ ӌ ׿ ֵ - 1799200845451636363636363636363636363636363636 ӯ ָ δ - 1997112938451316363636363636363636363636363636.

--------------------------------

1 סָֻ. ֣ 35.

--------------------------------

Ԍ ε ƴ ָ֮ ֛ ָ ߕ ׿ֵ㌟ ֵԌ ׿ֵ㌟ ׸ ֛˜ ߕ ׮׻ֵ ִ߯ ָӳ Ο ߯ ִ ֵֻ ߕ ׻ֵ ߕ ܵ ִ , ׻ֵ ׮ִ ߕ ִ㦾ֵֻ ָ ֛ ߕ 1 ִ ֛˜ ֻ וֵ, ֻ֮֟׻ֵ ε ָ և

 

֮ ֮ וֵ ו ִ ߴֵ֯Ԯ ֲָָ וִ ׮ִ ָ ߕ ֻ ֮ 2 Ƶ 2 Ԍ ָ ָ ׿ֵ㌟ ָ ֮ וֵ ױ ߕ ֯ ׮ִ ִ㦾ֵֻ ߯-ִ㦹 ֵֻӴ Ԍ ָ δֿ: ߕ ׻ֵ ׫ߵ ָ ׾ִָ֮ ׮ִ ָ ִ㦾ֵֻ ָ ֛ 1 ָ ֻ , ֻ֮֟ ׻ֵ ε ׫ߵ ָ

 

ױ ֮ ֮ וֵ ו־ ׮ִ ֯ ִ㦾ֵֻ ֲָָ ֣ ƕָ ֮ Ƹ ֮ 3 Ƶ 3 ָ ׿ֵ㌟ ָ Ƶ ױ ߕ ִ߯Ӵ Ԍ ָ ֻ֮ Ƶ ִ ָ 1 ֻ Ƶ

 

֮ וֵ ε ֲ ֻ և ֲ 1 ׿ֵ㌟ ָ ֵ εִ ƴ ָָ ֜ ָ ֮ ֛ __

1, 2,................ ,...................

 

֮ ׻וֵ 1 ׿ֵ ָ ׮ִ ֮ ֯

 

Σִ ׿ֵ㌟ ָ ֛˜ ֮ ֻ־ֵֻ ֤ וִ ׯ֔ ε ָ ׮ִ ߕ ֻ ֵ , ָ Ο ֻ ãֻ ִֵֻ ߕ ꛮ ε ֜և ֲ 1 ߕ כ ε ֲ ֻ ֵ ֲ 1 ׿ֵ㌟ ָ ֵ ֮ וֵ ε ƴ ׮ִ ֮ ֯ ֲ ױ ε ָ וֵ 1 ׿ֵ㌟ ָ ֮ ֻ ֵ ֮ ׻וֵ ε ִ ƴ ֯ ֋ ָ֑֮֯ߟܵ֟ δ ִֻ֮ ָ ߕ ܵ ֲָָ ™ ܵ֟ = = - 1

 

ֵԻ֮ --- ܵֆ ߮ Ӵ ׾ֳ֌ ܵ ׳ֵ֯ Οߟ - ܵ֟ ԟ ֮ ִ ܵ-ִ ֻײ ־ ܵ־׌ ֵ ׯ ָ ֻ״֟ ֋ ִ ֮ ֮֜ ׻ֵ ָ־Դ ά֮֟: -֮ ָָ֬ ܵ-ִ ִ ֮և և Ʈ 1017 ֮ ִ ֌ ֻ ָ ִ ӟ™ ֵ 1017 ָ ܵ֋ ִ ָ־ע ָ ֌ , ƴ - ( million millon ) פ ֟ ־ ֵ ָ־ע ָ ( cumbersome ) ׮ֵ ֮ Ԯ ׾ָֿ֮ ӲӬ ׾ָ֓ ׻ֵ 1017 ֛ ܵֆ ־ֿ ֛ ֋ Ӯ ֛ ֛ ܵֆ ִ ׻֟ ׻ֵ ׮ֵ ִ֮ܵ ƴ ֟ 1 ӫָ ׻֟ ܵ-

--------------------------------

1 ׮ֵ ֓߮ Ɵִ ߑ ֻ δ ִ ׻ և ֟ δ ָ ִ־ֻ ָ ---

1 17 = 84 י

2 = 5 18 = ,,

3 = 84 19 ִ = ,,

4 = ,, 20 ִ = ,, ִ

5 ֵ = ,, 21 = ,, ִ

6 ֵ = ,, ֵ 22 = ,,

7 = ,, ֵ 23 = ,,

8 = ,, 24 = ,,

9 ֩ = ,, 25 ֟ = ,,

10 ֩ = ,, ֩ 26 ֟ = ,, ֟

11 ׻֮ = ,, ֩ 27 ֻ֟ = ,, ֟

12 ׻֮ = ,, ׻֮ 28 ֻ֟ = ,, ֻ֟

13 ֻ = ,, ׻֮ 29 = ,, ֻ֟

14 ֻ = ,, ֻ 30 ß֯׻֟ = ,,

15 י = ,, ֻ 31 ֻ֯ = ,, ß֯׻֟

16 י = ,, י

ִ־ֻ סֻΖׯ ( 4--6 ֟ײ ) ׸ӿ֯ ( 8 ֟ײ ) ֕־ן ( 8 ֟ײ ) ֳִ ֣ և ֟ סֻΖׯ ָ֮ ֻ֯ δ 84 31 ָ ָïָ ֯ - ֻ֯- 8431 ֣ ܵ 90 δ ֑׸ ׻ ( Logarithmic tables ) ָ 8431 ܵ 60 δ ֯ ֵ ־ֻ, 3, ß־֮ , . 34 --- ִ֤.

ִ ׮ִ ד֢ ---

 

1 = 1 15 = ( 10,000,000)8

2 = 10 16 ׮ָ = ( 10,000,000)9

3 ֟ = 100 17 = ( 10,000,000)10

4 = 1,000 18 ֲ = ( 10,000,000)11

5 = 10,000 19 = ( 10,000,000)12

6 ֟ = 100,000 20 ׮ = ( 10,000,000)13

7 ֟= 1,000,000 21 ֻ = ( 10,000,000)14

8 י = 10,000,000 22 = ( 10,000,000)15

9 = (10,000,000)2 23 ӛ = ( 10,000,000)16

10 יי =(10,000,000)3 24 ֤ = ( 10,000,000)17

11 = (10,000,000)4 25 ֮ = ( 10,000,000)18

12 ׮֮ = (10,000,000)5 26 ֮ = ( 10,000,000)19

13 ׳֮ = (10,000,000)6 27 ܵ = ( 10,000,000)20

14 ײ֮ = (10,000,000)7

֟ ִ ׮ִ ִ ܵ ׳ֵ֯ Οߟ ܵ ָ ܵ֋ ֮֟ߟ

 

ܵ ׸ִ ִֵ ִֵ ָ ֿ ֤֟ ״֓Ӧ ܵ֟ ֵԌ ܵ, ו֟ δ 10140 , ׮ֵֿ֟: ׳֮

 

ܵ֟ --- ָ ܵ֟ ߮ ܵ ִ Ο ߮ ; ָ ׮פ™ ε ƴ ״֓Ӧ ָ ׮ִ δ ֯ ---

֑֮-ָߟ-ܵ֟ ( ) = + 1

ִ֬-ָߟ-ܵ֟ ( ) > , < .

™-ָߟ-ܵ֟ ( ) = - 1

---

֑֮-㌟-ܵ֟ ( ) = ( )

ִ֬- ㌟-ܵ֟ ( ) > , < .

™-㌟-ܵ֟ ( = - 1 )

---

֑֮-ܵ֟ܵ֟ ( ) = ( )2

ִ֬-ܵ֟ܵ֟ ( ) > , <

™-ܵ֟ܵ֟ ( ) = - 1 )

---

֑֮-ָߟ-֮

֮ --- ֮ ܵ֋ ׮ִ ָ ---

()

() ()

() ()

() ()

= ()

 

ֻ֮ = + 1

 

()

ֻ֮ = () + 4 ׿ֵ2

 

ֲ ---

()

֑֮-ָߟ-֮ ( ) = ()

 

ִ֬-ָߟ-֮ ( ) > , ӟ <

™-ָߟ-֮ ( = - 1,

--------------------------------

1 - (1) ִ, (2) ִ, (3) ߾, (4) ֿ, (5) ןš (֮ïן ߾), (6) ןš (֮ïן ߾).

2. ָ ִֵ - (1) ֻ ִֵ, (2) ֿ Τ, (3) ֲӬֆ־ֵã֮, (4) ׾ֳ-ן֓.

--------------------------------

---

( )

֑֮ ㌟-֮ ( ) = ( )

ִ֬-㌟-֮ ( ) > , ӟ <

™-㌟-֮ ( )= - 1

 

---

֑֮-֮֮֮ ( ) = ( )2

ִ֬-֮֮֮ ( ) > , ӟ <

 

---

 

™ ֮֮֮ ׻ֵ ε㌟ , ״֓Ӯ ָ ׮ִ ָ ֯ ---

 

= [ ֮֕ ֮֕] [ ֮֕ ֮֕ ]

( ֮֕) ( ֮֕) ( ֮֕) ( ֮֕) + ׿ֵ 1

 

=

( ) ( ) + ׿ֵ2

 

= ֡ ֡

( ֡) ( ֡)

 

, ֻ֖֮ ׿ ֛ ---

 

= ֻ֖֮ - + = ֻ֖֮,

 

ֵԻ֮ --- ֵԌ ׾־ָ ׾ ׮֟ ---

(1) ֑֮-ָߟ-֮ ( ) ֮ ֲ ִ ֯ ֵ ָ ׿ֵӴ ׬ ֮ ֮ ׻ֵ ֵ

--------------------------------

1. ׿ֵ - (1) ֨, (2) ָ֬ ֮ïן ׮, (3) ֮ïן, (4) 㮤ֻ (5) ־ָֻ (6) ֿ.

2. ׿ֵ - (1) ִԦ, (2) ִԦ, ( ֑ ׾ֳ-ן֓ )

--------------------------------

(2) ™-֮-֮ ( ֈ) ָֻ֖֮׿ ִ֯δ ֵԌ ׾־ָ ׳ֵ֯ ׮֟ ™ ֮֮֮ ֟ εָ֫ ֯ , ε ֮ ֵ ֵ ֣֣ԟ: ָ֟֫ ֯ ܵ ׬ : Οߟ ֻ֖֮ ֮ ׻ֵ ™ -֮֮֮ ֮

 

ָ סָֻ֮ԟ ׾־ָ ƴ ӿִֵ ָߟ֮֮ ㌟֮֮ ߮ ߮ ָ ֣ ֑֮ ֮֮֮ ִ֓ ֮ , ֲ ܵ֟ ֱ֮ ֵ , ׿ֵ ִ և ִܵ֟֡ ־ֻ ֮ ִ֓ ֮ , ï™: פ ֵ ֵ ׿ ™ ֮ 1 ־ִֻ פ ֵ ֮֮֮ ־ԡ ֵ֟ ִ֬-֮֮֮ : ־ָֻ֮ ִ֬-֮֮֮ ֮ ־ִֻ ׻֟ ׿ֵ ״ֻ֮ ׮ִ ן ֛ 2---

 

ֻ֟ ִß ׯ ׯ ԟ ֻ ִֵ ( time-instants) ãׯ֟ ( ִ פ-֮֟ ֮֟ ) ָ ״֣֤™ ߾ָ׿ ׿ֵӴ ֲָָ - ֆ ָ ֮ ָֻ׿ ™ ֟ , , ߾ָ׿ ָ 3 ־ִֻ ָ ׮ ׮ֻ ֵ ״֣֤™ ׿ ߟ ִֵ ׬

 

ֵԌ ן ֻ - ן ( one-to-one correspondence ) ָ ֬׮ ֮ ֮ ֮֨ ( Theory of infinite cardinals ) ָ֬ ן ׸״֟ ֮ ״ִֻ֮ ֵ㌟ , ׻ֵ ִֻ֮ ֛ ׸״֟ ׿ֵ ״ֻ֮ ׻ֵ ׻ֵ ֵ - ֮ ֛ ׿ֵ ו֮ ( elements ) ֮ ܵ֟ Ӗ ָ ™ ֟

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1. ӟ ֋ ּӟ ӟ֢׾ָ֤ . 3, . 25.

2. ־ֻ 3, . 28.

3. ӟӟ ׯ-ׯ Ƹן ֻ . 3, . 28 3 . 28 ֻ ״֕ӟ ״֓և ߾? פ

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™ -ήӴ ִֵ ֮ ׮ֵֿ פ ֵ , ׻ֵ (ׯ-ׯ) ֻ֯Τ ׸״֟ Ƶ, , þֵ ֮ ִֻ֮ ׮ִ ֟ ָ ֑֮-ָߟ-֮, ׸ָ֮ ֻ֯Τ ׿ ׬ , ׸״֟

 

ָ ֵ֟ , - ן ן ֮ ֮ ֵ֮ ׻ֵ ֲ βֻ ֮֬ ֨ , ֮֨ ֣ ־-Σִ ε ׮ֵ

 

ܵֆ ֵԌ ִ ֮ ֮ ֮֨ ׾֟ ֣״ ε֟ פև ִ֮֨ ӳ߸ ׾ָ ֮ ִ - 1 ܵ ֮ , ֮ ߴ ׮ִֵ ׾ָ֯ߟ ׮ֵ ֮֨ ܵ ԟ-Ӿԟ ԟ ־߮ ܵ ֮ , ㌟֯ פ ֓ ֓߮ Ɵ ™-ܵ֟ ֮ ֟ , ֮ ܵֆ עִ ֬׮ ֮ ֮ ֮֨ ( Theory of infinite cardinals ) ߴ ׮ֹ ֵ ֿ֟áߵ ׾ ֮ ֓߮ ֻ ָ״ ãןִ ָ ε֟ ֱ֟ ֿӳ־ ֵֿ ε֟

 

֮ ָ ֢ ֕ ꮙ ־ ֟ײ ֻ֬ ֳ ε-֨ פֵ Ӯ ߴ֟ߟ ( transfinite ) ܵֆ ֨ӟ ãׯ֟ ֮ ׿ֵ ( domain ) ׾ִֵ  ֿ֟á ׻ֵ ™ ָ֬, ֮ ׻ֵ βֻ ֮֬ ֟ӲӬ ֮ ׾ָ֓ ֌ ׻ֵ ״ֻ և ߴ֟ߟ ܵֆ ֨ӟ ֮ ֣״ ãִ ܵֆ ֮ ( calculus ) ֯ ֵ , ׻ֵ ƴ βֻ֟ ֿ֟áߵ ׾ִֿ ָ

 

 

 

ֲ-

 

־ֻ ֿ֟á ִ ֟ ִ֮ ֻ֮ ֿ Ʈ ֲ ֵ ֵ ִֹ ֲ ׮ִ ָ ---

 

֮ - Infinite. ִ֮ - Cube root

֮ ֮ ֮֨ - Theory ֟ ׮ֻ֮, 0 - Raising of

of infinite cardinals numbers to given powers.

֯ - Proportion. ֟ - Powers.

δ - Operation of mediation. ֟ ֮֨ - Theory of indices.

ԓ - Number of times a ֟ԓ - Number of times that a number

number is halved; mediation; can be divided by 4.

Logarithm.

ܵ֟ - Innumerable. ד֮ - Trace.

ִ֟ - Inequality. - Addition.

- Notational place. ן׾֪ - Astronomy.

֟ - Arithmetic. י - Notes.

-Element ס - Number of times that a number

ָ֬ - Base ( of logarithm ) can be divided by 3.

׾ָ - Discovery; invention. ס֕ - Radius.

ָָ - Successive ׿ - Rule of three.

פ֟ - One directional. ִ֮ - Scale of ten.

ן _ One-to-one ֿ״δ - Decimal place-value

correspondence notation.

- Art. ׫δ - Operation of duplation.

ֻ֯Τ - Time-instant. ׫׾ßָ֟ - Two-dimensional;

- Indeterminte equation superficial

ꮦ֟ - Initial circle;central ׮期 - Abstract reasoning.

Core.

ε - Operation. ׮ִֵ - Rule.

֯Τ - Locations; points or ֨ן - Method.

Places.

״ן - Mensuration. ׸ִ - Result.

֟, 0á - Mathematics. ׸ - Magnitude.

֖֟ - Mathematician. ׸߮ - Dimensionless.

- Multiplication. ׸״֟ ֮ - Finite cardinals.

- Integer. ׾֖֮ - Science.

ε - process; operation. ׾֪ - protons and electrons.

Οָ֟ ֮ ֿ - Infinite ׾׮ִֵ - Barter and exchange.

Plane area.

ο - problem. ׾ָ֮ - Distribution; spreading.

֣״ - Elementary; primitive. ׾ָ֮- - Spread and give.

- Subtraction. ׾ֿ - Analysis.

ߕ֟ - Algebra. ׾ßָ - Details.

ָ֮ - Cylindrical. - Circle.

- Division ֕ - Interest.

֕ - Divisor. - Diameter.

׳֮ - fraction. ӌָ ׿ - Super incumbent

cone.

, 0׻ ε - Fundamental - School.

Operation. ߲֨ - Classify.

׿ - Aggregate. ִꮦߵ - Concentric.

ܵ - prime. ָ ִ - Simple equation.

ָ - General outline. - Symbol, notation.

֑׸ - Logarithm. δ - Scale of notation.

ֲ - Quotient. ܵ - Number.

- Square. ܵ֟ - Numberable.

Դ - Square root. ܵ֟㻵 ֟ - Raising of a number to

Կֻ-Logarithm of logarithm. its own power.

ִ - Quadratic equation. ֟֟ - Continuum.

ԟ-Ӿԟ - Raising a number ָ֬ - Generalised.

to its own power (ܵ֟㻵 ֟) ߴ - Boundary.

ֵֻ - Ring. ߴ֟ߟ ܵ - Transfinite number.

׾֮ - Distribution. - Formula.

 

 

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