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- 1. ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ ﻭﺍﻟﺼﻼﺓ ﻭﺍﻟﺴﻼﻡ ﻋﻠﻰ ﺃﺷﺮﻑ ﺍﻟﻤﺨﻠﻮﻗﻴﻦ ﻣﺤﻤﺪ ﺳﻴﺪ ﺍﻟﻤﺮﺳﻠﻴﻦ ﻭﻋﻠﻰ ﺁﻟﻪ ﻭﺻﺤﺒﻪ ﺃﺟﻤﻌﻴﻦ ﺃﻣﺎ ﺑﻌﺪ ٬ ﻳﺴﺮﻧﻲ ﺃﻥ ﺃﻗﺪﻡ ﻟﻜﻢ ﻫﺬﺍ ﺍﻟﻌﻤﻞ ﺍﻟﻤﺘﻮﺍﺿﻊ ﻭﻫﻮ ﻋﺒﺎﺭﺓ ﻋﻠﻰ ﺟﻤﻴﻊ ﺩﺭﻭﺱ ﻭﻟﻰ ﺛﺎﻧﻮﻱ ﺇﻋﺪﺍﺩﻱ ﻣﺠﻤﻌﺔ ﻓﻲ ﻛﺘﺎﺏ ﻭﺍﺣﺪ ﻣﻔﻬﺮﺱ ﺍﻟﺮﻳﺎﺿﻴﺎﺕ ﻟﻤﺴﺘﻮﻯ ﺍﻷ ﺟﻤﻌﺖ ﻣﻦ ﻣﻮﻗﻊ www.anissmaths.ift.cx ﻟﻸﺳﺘﺎﺫ ﺍﻟ ﻬﺪﻱ ﻋﻨﻴﺲ ﻤ ﻟﺘﺼﻔﺢ ﺃﻱ ﺩﺭﺱ ﺃﺿﻐﻂ ﻋﻠﻰ ﻋﻨﻮﺍﻧﻪ ﻓﻲ ﺍﻟﻔﻬﺮﺱ ﻭﻛﺬﻟﻚ ﺍﻟﺘﻤﺎﺭﻳﻦ ﻭﻟﻠﺮﺟﻮﻉ ﺇﻟﻰ ﺍﻟﻔﻬﺮﺱ ﺇﺿﻐﻂ ﻋﻠﻰR ﺗﺠﻤﻴﻊ ﻭﺗﺮﺗﻴﺐ ﻭﻓﻬﺮﺳﺖ ALMOHANNAD
- 2. ﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻰ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﻭ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻤﺴﺘﻘﻴﻢ ﻭ ﺃﺟــﺰﺍﺅﻩ ﺴﺮﻳﺔ + ﺍﻟﻜﺘﺎﺑﺎﺕ ﺍﻟﻜﺴﺮﻳﺔ ﻣﻘﺎﺭﻧﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻜ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻰ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻜﺴﺮﻳﺔ ﻭﺍﺳــﻂ ﻗﻄﻌﺔ + ﺍﻟﻤﺘﻔﺎﻭﺗﺔ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻣﻘﺎﺭﻧﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ ﻭ ﺗﻘﺪﻳﻢ ﻣﺜﻠﺜﺎﺕ ﺧــﺎﺻﺔ + ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﻣﺜﻠﺚ ﻃﺮﺡ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ ﻭ ﺟﻤﻊ ﻗﺴﻤﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ ﻭ ﺿﺮﺏ ﺍﻻﺭﺗﻔﺎﻋﺎﺕ ﻓﻲ ﻣﺜﻠﺚ ﻭ ﺍﻟﻤﻨﺼﻔﺎﺕ ﺍﻟـﻘــــﻮﻯ ﺍﻟﺘﻤﺎﺛﻞ ﺍﻟﻤﺮﻛــﺰﻱ ﺍﻟﺘﻌﻤﻴــﻞ ﻭ ﺍﻟﻨﺸــﺮ ﻣﺘــﻮﺍﺯﻱ ﺍﻷﺿــﻼﻉ ﺍﻟﻤـﻌــﺎﺩﻻﺕ ﻭ ﺍﻟﻤﺴــﺎﺋﻞ ﺍﻟﺮﺑﺎﻋﻴــﺎﺕ ﺍﻟﺨــﺎﺻﺔ ﺍﻟﺰﻭﺍﻳﺎ ﺍﻟﻤﻜﻮﻧﺔ ﻣﻦ ﻣﺘﻮﺍﺯﻳﻴﻦ ﻭ ﻗــﺎﻃﻊ ﺍﻟـﺘـﻨــــﺎﺳﺒﻴــﺔ ﺍﻟﻤﻌﻠــﻢ ﻓﻲ ﺍﻟﻤﺴﺘــﻮﻯ + ﺍﻟﻤﺴﺘﻘﻴــﻢ ﺍﻟﻤــﺪﺭﺝ ﺍﻟـــــﺪﺍﺋــﺮﺓ ﺍﻷﺳﻄــﻮﺍﻧﺔ ﺍﻟﻘــﺎﺋﻤﺔ ﻭ ﺍﻟﻤﻮﺷــﻮﺭ ﺍﻟﻘــﺎﺋﻢ ﺍﻟﺤﺠــﻮﻡ ﻭ ﺍﻟﻤﺴــﺎﺣﺎﺕ ﻭ ﺍﻟﻤﺤﻴـﻄــﺎﺕ ﺍﻹﺣﺼــــﺎء
- 3. ﻷﻋﺪﺍﺩ ﺍﻟﺼﺤﻴﺤﺔ ﺍﻟﻄﺒﻴﻌﻴﺔ ﻭ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻰ ﺍ 1( – ﺣﺴﺎﺏ ﺳﻠﺴﻠﺔ ﻣﻦ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺑﺪﻭﻥ ﺃﻗﻮﺍﺱ : ( ﻗﺎﻋﺪﺓ 1 : ﺃ ﻟﺤﺴﺎﺏ ﺗﻌﺒﻴﺮ ﺟﺒﺮﻱ ﻣﻜﻮﻥ ﻣﻦ ﺳﻠﺴﻠﺔ ﻣﻦ ﻋﻤﻠﻴﺘﻲ ﺍﻟﺠﻤﻊ ﻭ ﺍﻟﻄﺮﺡ ﻓﻘﻂ ﺃﻭ ﺍﻟﻀﺮﺏ ﻭ ﺍﻟﻘﺴﻤﺔ ﻓﻘﻂ ﻭ ﺑﺪﻭﻥ ﺃﻗﻮﺍﺱ , ﻧﻨﺠﺰ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻣﻦ ﺍﻟﻴﺴﺎﺭ ﺇﻟﻰ ﺍﻟﻴﻤﻴﻦ ﺣﺴﺐ ﺍﻟﺘﺮﺗﻴﺐ . * ﻣﺜﺎﻝ : 5,1 – 9 – 7,3 + 5,0 + 5,3 – 11 + 5,2 = A 5,1 – 9 – 7,3 + 5,0 + 5,3 – 5,31 = 5,1 – 9 – 7,3 + 5,0 + 5,3 – 01 = 5,1 – 9 – 7,3 + 5,0 + 5,7 = 5,1 – 9 – 7,3 + 8 = 5,1 – 9 – 7,11 = 5,1 – 7,2 = 2,1 = ( ﻗﺎﻋﺪﺓ 2 : ﺏ ﻟﺤﺴﺎﺏ ﺗﻌﺒﻴﺮ ﺟﺒﺮﻱ ﻳﺘﻜﻮﻥ ﻣﻦ ﺳﻠﺴﻠﺔ ﻣﻦ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻭﺑﺪﻭﻥ ﺃﻗﻮﺍﺱ ‘ ﻧﻨﺠﺰ ﻋﻤﻠﻴﺘﻲ ﺍﻟﻀﺮﺏ ﻭ ﺍﻟﻘﺴﻤﺔ ﻗﺒﻞ ﻋﻤﻠﻴﺘﻲ ﺍﻟﺠﻤﻊ ﻭ ﺍﻟﻄﺮﺡ ﺛﻢ ﻧﻄﺒﻖ ﺍﻟﻘﺎﻋﺪﺓ 1 . * ﻣﺜﺎﻝ : 5,1 – 4 : 6,8 + 11 – 2 B = 22 – 2,5 + 7 x 5,1 – 51,2 + 11 – 41 + 5,2 – 22 = 5,1 – 51,2 + 11 – 41 + 5,02 = 5,1 – 51,2 + 11 – 5,43 = 5,1 – 51,2 + 11 – 5,32 = 5,1 – 51,2 + 5,21 = 5,1 – 56,41 = 51,31 =
- 4. 2( – ﺣﺴﺎﺏ ﺳﻠﺴﻠﺔ ﻣﻦ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺑﺄﻗــﻮﺍﺱ : ( ﻗﺎﻋﺪﺓ 3 : ﺝ ﻌﺒﻴﺮ ﺟﺒﺮﻱ ﻣﻜﻮﻥ ﻣﻦ ﺳﻠﺴﻠﺔ ﻣﻦ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺑﺄﻗﻮﺍﺱ ﻟﺤﺴﺎﺏ ﺗ ﻧﺤﺴﺐ ﺃﻭﻻ ﻣﺎ ﺑﻴﻦ ﻗﻮﺳﻴﻦ ﺛﻢ ﻧﻨﺠﺰ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻷﺧﺮﻯ . * ﻣﺜﺎﻝ : 2,3 – ) 4 – 8,5 ( C = 3,5 + [ 14 – ( 1,5 + 3 ) ] x 2 – 0,5 x 2,3 – 8,1 = 3,5 + [ 14 – 4,5 ] x 2 – 0,5 x 2,3 – 8,1 = 3,5 + 9,5 x 2 – 0,5 x 2,3 – 9,0 – 91 + 5,3 = 2,3 – 9,0 – 5,22 = 2,3 – 4,12 = 2,81 = 3( – ﺗﻮﺯﻳﻌﻴﺔ ﺍﻟﻀﺮﺏ ﻋﻠﻰ ﺍﻟﺠﻤﻊ ﻭ ﺍﻟﻄﺮﺡ : ( ﻗﺎﻋﺪﺓ 4 : ﺩ aﻭ bﻭ kﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ . k x ( a + b ) = a x k + b x k ; k x ( a – b ) = a x k – b x k ( a + b ) x k = a x k + b x k ; ( a – b ) x k = a x k – b x k * ﻣﺜﺎﻝ : ) 5,5 – 11 ( D = 2,5 x ( 4 + 7,2 ) E = 3 x 5,5 = 2,5 x 4 + 2,5 x 7,2 = 3 x 11 – 3 x ,61 – 33 = 81 + 01 = 71 = 82 = 5,1 F = ( 6,5 + 1 ) x 5 G = ( 13 – 9,2 ) x 2,9 = 5 x 6,5 + 5 x 1 = 1,5 x 13 – 1,5 x 8,31 – 5,91 = 5 + 5,23 = 5,73 = 7,5 =
- 5. ﺍﻟﻤﺴﺘﻘﻴــﻢ ﻭ ﺃﺟــﺰﺍﺅﻩ – Iﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻨﻘــﻂ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﺔ . 1( – ﺗﻌﺮﻳﻒ : ﻭﺩ , ﻭ ﻫﻮ ﻏﻴﺮ ﻣﺤﺪ ﺍﻟﻤﺴﺘﻘﻴﻢ ﻫﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﻧﻘﻂ ﺍﻟﻤﺴﺘﻮﻯ * ﻣﺜﺎﻝ : ﺍﻟﺸﻜﻞ ﺍﻟﺘﺎﻟﻲ ﻳﻤﺜﻞ ﻣﺴﺘﻘﻴﻤﺎ ﻭ ﻗﺪ ﺭﻣﺰﻧﺎ ﻟﻪ ﺑﺎﻟﺮﻣﺰ : ). (D )(D 2( – ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻤﺎﺭ ﻣﻦ ﻧﻘﻄﺘﻴﻦ : * ﺧﺎﺻﻴﺔ : ﻣﻦ ﻧﻘﻄﺘﻴﻦ ﻣﺨﺘﻠﻔﺘﻴﻦ ﻳﻤﺮ ﻣﺴﺘﻘﻴﻢ ﻭﺣﻴـــﺪ * ﻣﺜﺎﻝ : ﻧﺮﻣﺰﻟﻬﺬﺍ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺑﺎﻟﺮﻣﺰ : ). (AB * ﻣﻼﺣــﻈـﺔ ﻫﺎﻣــﺔ : ﻣﻦ ﻧﻘﻄﺔ ﻭﺍﺣﺪﺓ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ ﺗﻤﺮ ﻋــﺪﺓ ﻣﺴﺘﻘﻴﻤﺎﺕ
- 6. 3( – ﺍﻟﻨﻘﻂ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﺔ : ﺗﻜﻮﻥ ﻧﻘﻂ ﻣﺴﺘﻘﻴﻤﻴﺔ ﺇﺫﺍ ﻛﺎﻧﺖ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻧﻔﺲ ﺍﻟﻤﺴﺘﻘﻴﻢ * ﺗﻌﺮﻳﻒ : * ﻣﺜﺎﻝ : ﻧﻘﻮﻝ ﺃﻥ ﺍﻟﻨﻘﻂ Aﻭ Bﻭ Cﻭ Dﻣﺴﺘﻘﻴﻤﻴﺔ . ﻧﻘﻮﻝ ﺃﻥ ﺍﻟﻨﻘﻂ Eﻭ Fﻭ Gﻏﻴﺮ ﻣﺴﺘﻘﻴﻤﻴﺔ . _ IIﺍﻷﻭﺿﺎﻉ ﺍﻟﻨﺴﺒﻴﺔ ﻟﻤﺴﺘﻘﻴﻤﻴﻦ ﻓﻲ ﺍﻟﻤﺴﺘﻮﻯ : 1( – ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺍﻟﻤﺘﻘﺎﻃﻌﺎﻥ : * ﺗﻌﺮﻳﻒ : ﻳﻜﻮﻥ ﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻘﺎﻃﻌﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺎ ﻳﺸﺘﺮﻛﺎﻥ ﻓﻲ ﻧﻘﻄﺔ ﻭﺍﺣﺪﺓ * ﻣﺜﺎﻝ : ﻧﻘﻮﻝ ﺃﻥ ) (Dﻭ ) (Lﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻘﺎﻃﻌﺎﻥ . 2( ﺴﺘﻘﻴﻤﺎﻥ ﺍﻟﻤﻨﻄﺒﻘﺎﻥ : ﺍﻟﻤ * ﺗﻌﺮﻳﻒ : ﻳﻜﻮﻥ ﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﻨﻄﺒﻘﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺎ ﻳﺸﺘﺮﻛﺎﻥ ﻓﻲ ﺃﻛﺜﺮ ﻣﻦ ﻧﻘﻄﺔ ﻭﺍﺣﺪﺓ . * ﻣﺜﺎﻝ : ﻧﻘﻮﻝ ﺃﻥ ) (Lﻭ ) (Kﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﻨﻄﺒﻘﺎﻥ .
- 7. 3( – ﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺍﻟﻤﺘﻮﺍﺯﻳﺎﻥ ﻗﻄﻌﺎ : ﻳﻜﻮﻥ ﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻮﺍﺯﻳﻴﻦ ﻗﻄﻌﺎ ﺇﺫﺍ ﻛﺎﻧﺎ ﻻ ﻳﺸﺘﺮﻛﺎﻥ ﻓﻲ ﺃﻳﺔ ﻧﻘﻄﺔ * ﺗﻌﺮﻳﻒ : * ﻣﺜﺎﻝ : ﻧﻘﻮﻝ ﺃﻥ ) (Dﻭ ) (Lﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻮﺍﺯﻳﺎﻥ ﻗﻄﻌﺎ ﻭ ﻧﻜـــﺘﺐ : )(D) // (L ﻭ ﻧﻘﺮﺃ : ) (Dﻳﻮﺍﺯﻱ ) (Lﻭ ) (Lﻳﻮﺍﺯﻱ . ﺃ _ IIIﺍﻟﻤﺴﺘﻘﻴﻤﺎﻥ ﺍﻟﻤﺘﻌﺎﻣﺪﺍﻥ : 1( – ﺗﻌﺮﻳﻒ : ﻳﻜﻮﻥ ﻣﺴﺘﻘﻴﻤﺎﻥ ﻣﺘﻌﺎﻣﺪﻳﻦ ﺇﺫﺍ ﻛﺎﻧﺎ ﻳﺤﺪﺩﺍﻥ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ * ﻣﺜﺎﻝ : ﻧﻘﻮﻝ ﺃﻥ ﺍﻟﻤﺴﺘﻘﻴﻢ ) (Dﻋﻤﻮﺩﻱ ﻋﻠﻰ ﺍﻟﻤﺴﺘﻘﻴﻢ ) ( Rﻭ ﻧﻜــﺘﺐ : )( R ) ^ (D ﻭ ﻧﻘﺮﺃ : ) (Dﻋﻤﻮﺩﻱ ﻋﻠﻰ ) ( Rﺃﻭ ) ( Rﻋﻤﻮﺩﻱ ﻋﻠﻰ )(D 2( – ﺧـﺎﺻﻴﺔ : ﻣﻦ ﻧﻘﻄﺔ ﻣﻌﻠﻮﻣﺔ ﻳﻤﺮ ﻣﺴﺘﻘﻴﻢ ﻭﺣﻴــﺪ ﻋﻤﻮﺩﻱ ﻋﻠﻰ ﻣﺴﺘﻨﻘﻴﻢ ﻣﻌﻠﻮﻡ _ IVﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ : 1( – ﻣﺜﺎﻝ : ﺟﺰء ﺍﻟﻤﺴﺘﻘﻴﻢ ) (Dﺍﻟﻤﻠﻮﻥ ﺑﺎﻷﺣﻤﺮ ﻳﺴﻤﻰ : ﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ ﺃﺻﻠﻪ Aﻭ ﻳﻤﺮ ﻣﻦ . B ﻭ ﻳﺮﻣﺰ ﻟﻪ ﺑﺎﻟﺮﻣﺰ : ). [AB ﻧﺴﻤﻲ ﺍﻟﻤﺴﺘﻘﻴﻢ ) : (Dﺣــﺎﻣﻞ ﻧﺼﻒ ﺍﻟﻤﺴﺘﻘﻴﻢ ). [AB
- 8. 2( – ﻧﺼﻔﺎ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻤﺘﻘﺎﺑﻼﻥ : * ﺗﻌﺮﻳﻒ : ﻳﻜﻮﻥ ﻧﺼﻔﺎ ﻣﺴﺘﻘﻴﻢ ﻣﺘﻘﺎﺑﻠﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺎ ﻣﺨﺘﻠﻔﻴﻦ ﻭ ﻛﺎﻥ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻷﺻﻞ ﻭ ﻧﻔﺲ ﺍﻟﺤــﺎﻣﻞ . * ﻣﺜﺎﻝ : ﻧﻼﺣﻆ ﺃﻥ ﻧﺼﻔﻲ ﺍﻟﻤﺘﻘﻴﻢ ) [ABﻭ ) [ACﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﺮﺃﺱ Aﻭ ﻧﻔﺲ ﺍﻟﺤﺎﻣﻞ ). (D ﻧﻘﻮﻝ ﺃﻥ ) [ABﻭ ) [ACﻧﺼﻔﺎ ﻣﺴﺘﻘﻴﻢ ﻣﺘﻘﺎﺑﻠﻴﻦ . 3( – ﺍﻟﻤﺴﻘﻂ ﺍﻟﻌﻤﻮﺩﻱ ﻟﻨﻘﻄﺔ ﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ : * ﺗﻌﺮﻳﻒ : ﺍﻟﻤﺴﻘﻂ ﺍﻟﻌﻤﻮﺩﻱ ﻟﻨﻘﻄﺔ Eﻋﻠﻰ ﻣﺴﺘﻘﻴﻢ ) (Dﻫﻲ Hﻧﻘﻄﺔ ﺗﻘﺎﻃﻊ )(D ﻭ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻌﻤﻮﺩﻱ ﻋﻠﻴﻪ ﻓﻲ . H * ﻣﺜﺎﻝ : ﺍﻟﻤﺴﺎﻓﺔ EHﺗﺴﻤﻰ : ﺍﻟﻤﺴﺎﻓﺔ ﺑﻴﻦ ﺍﻟﻨﻘﻄﺔ Eﻭ ﺍﻟﻤﺴﺘﻘﻴﻢ )(D _ Vﺍﻟﻘﻄﻌﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﺔ : 1( – ﻣﺜﺎﻝ : ﻧﺴﻤﻲ ﻫﺬﺍ ﺍﻟﺸﻜﻞ : ﻗـﻄــﻌـﺔ ﻣﺴﺘﻘﻴـﻤﻴــﺔ . ﻭ ﻧﺮﻣﺰ ﻟﻬﺎ ﺑﺎﻟﺮﻣﺰ : ]. [AB Aﻭ Bﻳﺴﻤﻴﺎﻥ : ﻃﺮﻓﻲ ﻄﻌﺔ ]. [AB ﺍﻟﻘ ﺍﻟﻤﺴﺘﻘﻴﻢ ) (ABﻳﺴﻤﻰ ﺣﺎﻣﻞ ﺍﻟﻘﻄﻌﺔ ][AB 2( – ﻣﻨﺘﺼﻒ ﻗﻄﻌﺔ : * ﺗﻌﺮﻳﻒ : ﻣﻨﺘﺼﻒ ﻗﻄﻌﺔ ﻫﻮ ﻧﻘﻄﺔ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﺍﻟﻘﻄﻌﺔ ﻭ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﻃﺮﻓﻲ ﻫﺬﻩ ﺍﻟﻘﻄﻌــﺔ .
- 9. * ﻣﺜﺎﻝ : ﻧﺴﻤﻲ ﺍﻟﻨﻘﻄﺔ Mﻣﻨﺘﺼﻒ ﺍﻟﻘﻄﻌــﺔ ]. [AB ﻭ MA = MB * ﺑﺘﻌﺒﻴﺮ ﺁﺧﺮ : Mﻣﻨﺼﻒ ﺍﻟﻘﻄﻌﺔ ] [ABﻳﻌﻨﻲ ﺃﻥ : ]M Î [AB 3( – ﺍﻟﻘﻄﻌﺘﺎﻥ ﺍﻟﻤﺘﻘﺎﻳﺴﺘﺎﻥ : * ﺗﻌﺮﻳﻒ : ﺗﻜﻮﻥ ﻗﻄﻌﺘﺎﻥ ﻣﺘﻘﺎﻳﺴﺘﻴﻦ ﺇﺫﺍ ﻛﺎﻥ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻄـــﻮﻝ * ﻣﺜﺎﻝ : ﻧﻘﻮﻝ ﺃﻥ ] [ABﻭ ] [CDﻗﻄﻌﺘﺎﻥ ﻣﺘﻘﺎﻳﺴﺘﺎﻥ , ﻭ ﻧﻜــﺘﺐ : AB = CD
- 10. ﺍﻟﻜﺘﺎﺑﺎﺕ ﺍﻟﻜﺴﺮﻳﺔ ﻭ ﻣﻘﺎﺭﻧﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻜﺴﺮﻳﺔ 1( – ﺍﻟﻜﺘﺎﺑﺎﺕ ﺍﻟﻜﺴﺮﻳﺔ ﻟﻌﺪﺩ ﻛﺴﺮﻱ : * ﻗﺎﻋﺪﺓ 1 : aﻭ bﻋﺪﺩﺍﻥ ﻋﺸﺮﻳﺎﻥ ﻏﻴﺮ ﻣﻨﻌﺪﻣﻴﻦ . ﻳﻤﻜﻦ ﺇﻳﺠﺎﺩ ﻛﺘﺎﺑﺎﺕ ﻛﺴﺮﻳﺔ ﺍﻋﺪﺩ ﻛﺴﺮﻱ ﻭ ﺫﻟﻚ ﺑﻀﺮﺏ ﺃﻭ ﻌﺪﺩ ﺍﻟﻜﺴﺮﻱ ﻋﻠﻰ ﻧﻔﺲ ﺍﻟﻌﺪﺩ ﺍﻟﻐﻴﺮ ﺍﻟﻤﻨﻌﺪﻡ . ﻗﺴﻤﺔ ﺣﺪﻱ ﻫﺬﺍ ﺍﻟ aﻭ bﻭ mﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ ﺑﺤﻴﺚ : m aﻏﻴﺮ ﻣﻨﻌﺪﻣﻴﻦ . ﻭ ﺑﺘﻌﺒﻴﺮ ﺁﺧﺮ : a ´ m a a : m a ;; = = b ´ m b b : m b 6 2 : 21 21 51 3 ´ 5 5 = ;; = = = * ﺃﻣﺜﻠﺔ : 7 2 : 41 41 72 3 ´ 9 9 2( – ﺟﻌﻞ ﻣﻘﺎﻡ ﻋﺸﺮﻱ ﻟﻜﺘﺎﺑﺔ ﻛﺴﺮﻳﺔ ﻋﺪﺩﺍ ﺻﺤﻴﺤﺎ : * ﻗﺎﻋﺪﺓ 2 : ﻟﺠﻌﻞ ﻣﻘﺎﻡ ﻋﺪﺩ ﻛﺴﺮﻱ ﻋﺪﺩﺍ ﺻﺤﻴﺤﺎ , ﻧﻀﺮﺏ ﺣﺪﻱ ﻫﺬﺍ ﺍﻟﻌﺪﺩ ﺍﻟﻜﺴﺮﻱ ﻓﻲ : ﺃﻭ 001 0001 ﺃﻭ ....... ﺃﻭ 01 31 00031 0001 ´ 31 7 007 001´ 7 011 01 11 11 ´ = = ;; = = ;; = = * ﺃﻣﺜﻠﺔ : 2101 0001 ´ 210 1 210 1 , , 2 001´ 20 0 20 0 , , 53 01´ 5 3 5 3 , , 3( – ﻣﻘﺎﺭﻧﺔ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻤﻘﺎﻡ : * ﻗﺎﻋﺪﺓ 3 : , ﻓﺈﻥ ﺃﻛﺒﺮﻫﻤﺎ ﻫﻮ ﺍﻟﺬﻱ ﻟﻪ ﺃﻛﺒﺮ ﺑﺴﻂ ﺇﺫﺍ ﻛﺎﻥ ﻟﻌﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻧﻔﺲ ﺍﻟﻤﻘﺎﻡ
- 11. 71 15 17 31 7 3 > 15 ﻷﻥ > ;; ﻷﻥ 17 < 31 < ;; ﻷﻥ 3 > 7 > * ﺃﻣﺜﻠﺔ : 2 2 9 9 11 11 71 4( – ﺭﻧﺔ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﺒﺴﻂ : ﻣﻘﺎ * ﻗﺎﻋﺪﺓ 4 : , ﻓﺈﻥ ﺃﻛﺒﺮﻫﻤﺎ ﻫﻮ ﺍﻟﺬﻱ ﻟﻪ ﺃﺻﻐﺮ ﻣﻘﺎﻡ ﺇﺫﺍ ﻛﺎﻥ ﻟﻌﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻧﻔﺲ ﺍﻟﺒﺴﻂ 71 71 7 7 3 3 ﻷﻥ 22 < 9 > ﻷﻥ 31 > 14 ;; > ﻷﻥ 13 < 11 ;; > * ﺃﻣﺜﻠﺔ : 22 9 31 14 13 11 5( – ﺭﻧﺔ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻣﻘﺎﻡ ﺃﺣﺪﻫﻤﺎ ﻣﻀﺎﻋﻒ ﻟﻤﻘﺎﻡ ﺍﻵﺧﺮ : ﻣﻘﺎ * ﻗﺎﻋﺪﺓ 4 : ﻟﻤﻘﺎﺭﻧﺔ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻣﻘﺎﻡ ﺃﺣﺪﻫﻤﺎ ﻣﻀﺎﻋﻒ ﻟﻤﻘﺎﻡ ﺍﻵﺧﺮ , ﻧﻮﺣﺪ ﻣﻘﺎﻣﻴﻬﻤﺎ ﺛﻢ ﻧﻄﺒﻖ ﺍﻟﻘﺎﻋﺪﺓ 3 * ﻣﺜﺎﻝ : 7 5 ﻭ ﻟﻨﻘﺎﺭﻥ ﺍﻟﻌﺪﺩﻳﻦ : 4 61 82 4 ´ 7 7 5 5 = = ﻭ = ﻟﺪﻳﻨﺎ : 61 4 ´ 4 4 61 61 82 5 ﻷﻥ 82 < 5 < ﻭﺑﻤﺎ ﺃﻥ 61 61 7 5 < ﻓﺈﻥ 4 61 6( – ﺭﻧﺔ ﻋﺪﺩ ﻛﺴﺮﻱ ﻭ 1 : ﻣﻘﺎ 5 : * ﻗﺎﻋﺪﺓ ﻳﻜﻮﻥ ﻋﺪﺩ ﻛﺴﺮﻱ ﺃﻛﺒﺮ ﻣﻦ 1 ﺇﺫﺍ ﻛﺎﻥ ﺑﺴﻄﻪ ﺃﻛﺒﺮ ﻣﻦ ﻣﻘﺎﻣﻪ , ﻭ ﻳﻜﻮﻥ ﺃﺻﻐﺮ ﻣﻦ 1 ﺇﺫﺍ ﻛﺎﻥ ﺑﺴﻄﻪ ﺃﺻﻐﺮ ﻣﻦ ﻣﻘﺎﻣﻪ . 5 17 ﻷﻥ 3 < 5 1 < ;; 25 > 17 ﻷﻥ 1 > * ﻣﺜﺎﻝ : 73 25
- 12. ﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻰ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻜﺴــﺮﻳﺔ 1( – ﺟﺪﺍء ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ : * ﻗﺎﻋﺪﺓ 1 : ﻘﺎﻡ ﻓﻲ ﺍﻟﻤﻘﺎﻡ . ﻟﺤﺴﺎﺏ ﺟﺪﺍء ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻧﻀﺮﺏ ﺍﻟﺒﺴﻂ ﻓﻲ ﺍﻟﺒﺴﻂ ﻭ ﺍﻟﻤ a c a ´ c c a = ´ ﻋﺪﺩﺍﻥ ﻛﺴﺮﻳﺎﻥ : ﻭ b d b ´ d d b 54 3 ´ 51 3 51 3 31 711 9 ´ 31 77 7 ´ 11 7 11 = ´ = ´ 5,1 = ;; = 9 ´ = ;; = ´ = * ﺃﻣﺜﻠﺔ : 07 7 ´ 01 7 01 7 22 22 1 ´ 22 01 2 ´ 5 2 5 ﺎﻡ : 2( – ﻣﺠﻤﻮﻉ ﻭ ﻓﺮﻕ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻤﻘ * ﻗﺎﻋﺪﺓ 2 : ﻭ ﻓﺮﻕ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻤﻘﺎﻡ : ﻧﺤﺘﻔﻆ ﺑﻨﻔﺲ ﻟﺤﺴﺎﺏ ﻣﺠﻤﻮﻉ ﺃ . ﺍﻟﻤﻘﺎﻡ ﺛﻢ ﻧﺤﺴﺐ ﻣﺠﻤﻮﻉ ﺃﻭ ﻓﺮﻕ ﺍﻟﺒﺴﻄﻴﻦ a c a - c a c a + c c a = - = + ﻭ ( > c ) a ﻭ ﻋﺪﺩﺍﻥ ﻛﺴﺮﻳﺎﻥ : b b b b b b b b 8 91 - 72 91 72 81 7 + 11 7 11 = - = ;; = + = * ﺃﻣﺜﻠﺔ : 9 9 9 9 5 5 5 5 3( – ﻣﺠﻤﻮﻉ ﻭ ﻓﺮﻕ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻣﻘﺎﻡ ﺃﺣﺪﻫﻤﺎ ﻣﻀﺎﻋﻒ ﻣﻘﺎﻡ ﺍﻵﺧﺮ : * ﻗﺎﻋﺪﺓ 3 : ﻟﺤﺴﺎﺏ ﻣﺠﻤﻮﻉ ﺃﻭ ﻓﺮﻕ ﻋﺪﺩﻳﻦ ﻛﺴﺮﻳﻴﻦ ﻣﻘﺎﻡ ﺃﺣﺪﻫﻤﺎ ﻣﻀﺎﻋﻒ ﻟﻤﻘﺎﻡ ﺍﻵﺧﺮ , ﻧﻮﺣﺪ ﻣﻘﺎﻣﻴﻬﻤﺎ ﺛﻢ ﻧﻄﺒﻖ ﺍﻟﻘﺎﻋﺪﺓ 2 . 23 7 - 93 7 93 7 31 62 11 + 51 11 51 11 5 - = - = = ;; = + + = = * ﺃﻣﺜﻠﺔ : 3 9 9 9 9 9 12 12 12 7 12 12
- 13. ﺣﺎﻻﺕ ﺧـــﺎﺻﺔ : 8 83 2 - 04 2 04 1 93 12 + 81 12 81 7 9 = - - = = ;; = + + = = 02 02 01 4 02 02 42 42 8 21 42 42 93 33 27 3 21 59 53 + 06 53 06 5 51 = - - = ;; = + + = = 66 66 66 6 11 82 82 4 7 82 82 ﺗﻘﻨﻴﺎﺕ ﻭ ﻣﻬﺎﺭﺍﺕ 7 1 1 31 7 5 + + ,1 = B 5 ;; + 11 = A + + + ﻟﻨﺤﺴﺐ ﺍﻟﻤﺠﻤﻮﻋﻴﻦ Aﻭ Bﺑﺄﺑﺴﻂ ﻃﺮﻳﻘﺔ : 02 5 9 6 3 6 7 1 1 7 31 5 + + ,1 = B 5 ) + ( + ) + ( + 11 = A 02 5 6 6 9 3 7 1 51 1 12 31 + 5 = B + + + 11 = A ) + ( + 6 9 9 02 5 02 22 81 1 7 51 + + 11 = A + ) + ( = B 9 6 5 02 02 22 1 22 + 3 + 11 = A = B + 9 5 02 22 + 41 = A 1 11 9 + = B 5 01 22 621 = A + 2 11 9 9 + = B 841 01 01 = A 31 9 = B 01
- 14. ﺍﻟﻤﺘﻔﺎﻭﺗﺔ ﺍﻟﻤﺜﻠﺜﻴﺔ ﻭ ﻭﺍﺳــﻂ ﻗﻄﻌﺔ 1( – ﺍﻟﻤﺘﻔﺎﻭﺗﺔ ﺍﻟﻤﺜﻠﺜﻴﺔ : * ﺧﺎﺻﻴﺔ 1 : Aﻭ Bﻭ Cﺛﻼﺙ ﻧﻘﻂ ﻣﺨﺘﻠﻔــﺔ ﺇﺫﺍ ﻛﺎﻧﺖ Cﺗﻨﺘﻤﻲ ﺇﻟﻰ ﺍﻟﻘﻄﻌﺔ ] [ABﻓﺈﻥ : AB = AC + BC ﺇﺫﺍ ﻛﺎﻧﺖ Cﻻ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﺍﻟﻘﻄﻌﺔ ] [ABﻓﺈﻥ : AB < AC + BC * ﻣﺜﺎﻝ : AB = AC + BC AB < AC + BCﻭ ﻛﺬﻟﻚ : AC < AB + BCﻭ BC < AB + AC ﻭ ﻣﻨﻪ ﻧﺴﺘﻨﺘﺞ ﻣﺎ ﻳﻠﻲ : ﻲ ﻣﺜﻠﺚ ﻃﻮﻝ ﺃﻱ ﺿﻠﻊ ﻣﻦ ﺃﺿﻼﻋﻪ ﺃﺻﻐﺮ ﻣﻦ ﻣﺠﻤﻮﻉ ﻃﻮﻟﻲ ﻓ ﺍﻟﻀﻠﻌﻴﻦ ﺍﻵﺧــــﺮﻳﻦ . ﺗﻄﺒﻴﻖ : ﻫﻞ ﻳﻤﻜﻦ ﺭﺳﻢ ﺍﻟﻤﺜﻠﺚ ABCﺑﺤﻴﺚ : AB = 7cmﻭ AC = 17cmﻭ BC = 5 cm؟ ﻧﻼﺣﻆ ﺃﻥ : 21 = 7 + 5 ﻭ ﺃﻥ 21 > 71 ﺃﻱ ﺃﻥ AC > AB + BC ﺇﺫﻥ : ﻻ ﻳﻤﻜﻦ ﺭﺳﻢ ﺍﻟﻤﺜﻠﺚ . ABC 2( – ﻭﺍﺳـــﻂ ﻗـﻄــﻌــﺔ : * ﺗﻌــﺮﻳﻒ : ﻭﺍﺳﻂ ﻗﻄﻌﺔ ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮ ﻣﻦ ﻣﻨﺘﺼﻒ ﺍﻟﻘﻄﻌــﺔ ﻭ ﻋﻤﻮﺩﻱ ﻋﻠﻰ ﺣﺎﻣﻠﻬﺎ
- 15. * ﻣﺜﺎﻝ : ﻟﻨﺮﺳﻢ ﻗﻄﻌﺔ ] [ABﻗﻄﻌﺔ ﻭ ) (Dﻭﺍﺳﻄﻬﺎ * ﺧﺎﺻﻴﺔ 2 : ﻛﻞ ﻧﻘﻄﺔ ﺗﻨﺘﻤﻲ ﺇﺍﻟﻰ ﻭﺍﺳﻂ ﻗﻄﻌﺔ ﺗﻜﻮﻥ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﻃﺮﻓﻴﻬﺎ * ﺑﺘﻌﺒﻴﺮ ﺁﺧــﺮ : ] [ABﻗﻄﻌﺔ ﻭ )∆( ﻭﺍﺳﻄﻬﺎ ﻭ Mﻧﻘﻄﺔ ﻣﻦ ﺍﻣﺴﺘﻮﻯ . ) M Î (Dﻳﻌﻨﻲ ﺃﻥ MA = MB * ﺧﺎﺻﻴﺔ 3 : ﻛﻞ ﻧﻘﻄﺔ ﻣﺘﺴﺎﻭﻳﺔ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﻃﺮﻓﻲ ﻗﻄﻌﺔ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻭﺍﺳﻂ ﻫﺬﻩ ﺍﻟﻘﻄﻌﺔ * ﺑﺘﻌﺒﻴﺮ ﺁﺧــﺮ : ] [ABﻗﻄﻌﺔ ﻭ )∆( ﻭﺍﺳﻄﻬﺎ ﻭ Mﻧﻘﻄﺔ ﻣﻦ ﺍﻣﺴﺘﻮﻯ . MA = MBﻳﻌﻨﻲ ﺃﻥ )M Î (D
- 16. 3( – ﻭﺍﺳﻄﺎﺕ ﻣﺜﻠﺚ : * ﺗﻌﺮﻳﻒ 2 : ﻭﺍﺳﻂ ﻣﺜﻠﺚ ﻫﻮ ﻭﺍﺳﻂ ﻛﻞ ﺿـــﻠﻊ ﻣﻦ ﺃﺿــــﻼﻋــﻪ ﻣﺜﺎﻝ : ABCﻣﺜﻠﺚ ﻭ (Dﻭﺍﺳﻂ ﺍﻟﻀﻠﻊ ]. [BC ) ﻧﺴﻤﻲ ﺍﻟﻤﺴﺘﻘﻴﻢ (Dﻭﺍﺳﻂ ﺍﻟﻤﺜﻠﺚ ABC ) ﺧﺎﺻﻴﺔ 4 : * ﻭﺍﺳﻄﺎﺕ ﻣﺜﻠﺚ ﺗﺘﻼﻗﻰ ﻓﻲ ﻧﻘﻄﺔ ﻭﺍﺣﺪﺓ ﺗﺴﻤﻰ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﻴﻄﺔ ﺑﻬﺬﺍ ﺍﻟﻤﺜﻠﺚ ﻣﺜﺎﻝ :
- 17. ﺗﻘﺪﻳﻢ ﻭ ﻣﻘﺎﺭﻧﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ _Iﺗ ﻘ ﺪﻳﻢ . ـ ـــ 1( – ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻤﻮﺟﺒﺔ ﻭ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﺴﺎﻟﺒﺔ : * ﺗﻌﺮﻳﻒ 1 : ﺍﻷﻋﺪﺍﺩ ﻣﺜﻞ : 0 ; 1 ; 2 , 41 ; 41,3 ; 11 ; 5,2 ﺗﺴﻤﻰ ﺃﻋﺪﺍﺩﺍ ﻋﺸﺮﺑﺔ ﻣﻮﺟﺒﺔ . ﺍﻷﻋﺪﺍﺩ ﻣﺜﻞ : 0 ; 2 ; 1 ; 44,0 ; 21 ; 5,2 ﺗﺴﻤﻰ ﺃﻋﺪﺍﺩﺍ ﻋﺸﺮﻳﺔ ﺳﺎﻟﺒﺔ . ﺍﻟﻌﺪﺩ 0 ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻣﻮﺟﺐ ﻭ ﺳﺎﻟﺐ ﻓﻲ ﺁﻥ ﻭﺍﺣﺪ . * ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﺒﻴﺔ : 2( – ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴ * ﺗﻌﺮﻳﻒ 2 : ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻤﻮﺟﺒﺔ ﻭ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﺴﺎﻟﺒﺔ ﺗﻜﻮﻥ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ * ﻣﻼﺣﻈﺔ ﺭﻫﺎﻣﺔ : ﺍﻷﻋﺪﺍﺩ ﻣﺜﻞ : 0 ; 1 ; 8 , 2 ; 41 ; 1 ; 5 ; 15 ; 11 .... ﺗﺴﻤﻰ ﺃﻋﺪﺍﺩﺍ ﺻﺤﻴﺤﺔ ﻧﺴﺒﻴﺔ . ﻛﻞ ﻋﺪﺩ ﺻﺤﻴﺢ ﻧﺴﺒﻲ ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ . ﺍﻟﻌﺪﺩ ﻣﺜﻞ : 21,41 ﺃﻭ 5,2 ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﻭ ﻟﻴﺲ ﺑﻌﺪﺩ ﺻﺤﻴﺢ ﻧﺴﺒﻲ . 3( – ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻤﺪﺭﺝ : ﻧﻌﺘﺒﺮ ) (Dﻣﺴﺘﻘﻴﻤﺎ ﻭ Oﻭ Iﻧﻘﻄﺘﻴﻦ ﻣﺨﺘﻠﻔﺘﻴﻦ ﻣﻦ ) . (Dﻟﻨﺪﺭﺝ ﺍﻟﻤﺴﺘﻘﻴﻢ ) (Dﺑﻮﺍﺳﻄﺔ ﺍﻟﻘﻄﻌ ﺔ ][OI ) ﺃﻇﺮ ﺍﻟﺸﻜــﻞ ﺃﺳﻔﻠﻪ ( . )(D E F O I A B , , , , , , , , , , , , , , , 0 1 ﺍﻷﻋﺪﺍﺩ ﺍﻟﺴﺎﻟﺒﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﻮﺟﺒﺔ ﻛﻞ ﻧﻘﻄﺔ ﻣﻦ ﺍﻟﻤﺴﺘﻘﻴﻢ ) (Dﻣﺮﺗﺒﻄﺔ ﺑﻌﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﻳﺴﻤﻰ ﺃﻓﺼﻮﻝ ﻫﺬﻩ ﺍﻟﻨﻘﻄﺔ . ﺍﻟﻨﻘﻄﺔ Oﺗﺴﻤﻰ ﺃﺻﻞ ﺍﻟﻤﺴﺘﻘﻴﻢ ﺍﻟﻤﺪﺭﺝ ). (D ﻃﻮﻝ ﺍﻟﻘﻄﻌﺔ ] [OIﻳﺴﻤﻰ ﻭﺣــﺪﺓ ﺍﻟﺘﺪﺭﻳﺞ . ﺍﻟﻨﻘﻄﺔ Eﺃﻓﺼﻮﻟﻬﺎ 4 ﺍﻟﻨﻘﻄﺔ Aﺃﻓﺼﻮﻟﻬﺎ 3 ﺍﻟﻨﻘﻄﺔ Oﺃﻓﺼﻮﻟﻬﺎ 0 ﺍﻟﻨﻘﻄﺔ Fﺃﻓﺼﻮﻟﻬﺎ 5,3 ﺍﻟﻨﻘﻄﺔ Bﺃﻓﺼﻮﻟﻬﺎ 5,3 Iﺃﻓﺼﻮﻟﻬﺎ 1 ﺍﻟﻨﻘﻄﺔ 4( – ﻣﺴﺎﻓﺔ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﻋﻦ ﺍﻟﺼﻔﺮ : * ﺗﻌﺮﻳﻒ 3 : ﺪﺭﺟﺎ ﺃﺻﻠﻪ Oﻭ Mﻧﻘﻄﺔ ﻣﻦ ) (Dﺃﻓﺼﻮﻟﻬﺎ ﺍﻟﻌﺪﺩ . a ﻧﻌﺘﺒﺮ ) (Dﻣﺴﺘﻘﻴﻤﺎ ﻣ ﻣﺴﺎﻓﺔ ﺍﻟﻌﺪﺩ aﻋﻦ ﺍﻟﺼﻔﺮ ﻫﻮ ﻃﻮﻝ ﺍﻟﻘﻄﻌﺔ ]. [OM
- 18. 5( – ﻣﻘﺎﺑﻞ ﻋﺪﺩ ﻋﺸﺮﻱ : * ﺗﻌﺮﻳﻒ 4 : ﻳﻜﻮﻥ ﻋﺪﺩﺍﻥ ﻣﺘﻘﺎﺑﻠﻴﻦ ﺇﺫﺍ ﻛﺎﻧﺖ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﺍﻟﺼﻔﺮ ﻭ ﺇﺷﺎﺭﺗﺎﻫﻤﺎ ﻣﺨﺘﻠﻔﺘﻴﻦ . 11 ﻭ 11 ﻋﺪﺩﺍﻥ ﻣﺘﻘﺎﺑﻼﻥ ;; 2,1 ﻭ 2,1 ﻋﺪﺩﺍﻥ ﻣﺘﻘﺎﺑﻼﻥ * ﺃﻣﺜﻠﺔ : 3 ﻭ 3 ﻋﺪﺩﺍﻥ ﻣﺘﻘﺎﺑﻼﻥ 23,0 ﻭ 23,0 ﻋﺪﺩﺍﻥ ﻣﺘﻘﺎﺑﻼﻥ ;; ﻣﻘﺎﺑﻞ ﺍﻟﻌﺪﺩ 0 ﻫﻮ ﺍﻟﻌﺪﺩ 0 _ IIﻟﻤﻘ ﺎﺭﻧــﺔ : ﺍ ـــــ 1( – ﻣﻘﺎﺭﻧﺔ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ : * ﻗﺎﻋﺪﺓ 1 : ﻛﻞ ﻋﺪﺩ ﻋﺸﺮﻱ ﻣﻮﺟﺐ ﺃﻛﺒﺮ ﻣﻦ ﻛﻞ ﻋﺪﺩ ﻋﺸﺮﻱ ﺳﺎﻟﺐ ﻏﻴﺮ ﻣﻨﻌﺪﻡ ;; 7,41 > 22 0 < 21,33 ;; 0 > 44,52 ;; 5,1 < 54,0 * ﺃﻣﺜﻠﺔ : 2( – ﻣﻘﺎﺭﻧﺔ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﺳﺎﻟﺒﻴﻦ : * ﻗﺎﻋﺪﺓ 2 : ﺇﺫﺍ ﻛﺎﻥ ﻋﺪﺩﺍﻥ ﻋﺸﺮﻳﺎﻥ ﺳﺎﻟﺒﺎﻳﻦ ﻓﺈﻥ ﺃﻛﺒﺮﻫﻤﺎ ﻫﻮ ﺍﻷﻗﺮﺏ ﻣﻦ ﺍﻟﺼﻔﺮ ;; 1 < 5,2 3522 > 0 ;; 63 > 1,0 * ﺃﻣﺜﻠﺔ : ﺍﻟﻌﺪﺩ 0 ﻫﻮ ﺃﻛﺒﺮ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺴﺎﻟﺒﺔ ﻭ ﺃﺻﻐﺮ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﻮﺟﺒﺔ * ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﻭ £ . 3( – ﺍﻟﺮﻣﺰﺍﻥ : ³ ﻭ 33 ³ 33 ﺍﻟﺮﻣﺰ ³ ﻳﻘﺮﺃ : ﺃﻛﺒﺮ ﻣﻦ ﺃﻭ ﻳﺴﺎﻭﻱ ﻭ ﻳﺴﺘﻌﻤﻞ ﻓﻲ ﺣﺎﻟﺘﻴﻦ ﻣﺜﻞ : 32 ³ 3,11 ﺴﺎﻭﻱ ﻭ ﻳﺴﺘﻌﻤﻞ ﻓﻲ ﺣﺎﻟﺘﻴﻦ ﻣﺜﻞ : 5,1 5,73 £ ﻭ 6,7 – 6,7 £ ﺍﻟﺮﻣﺰ £ ﻳﻘﺮﺃ : ﺃﺻﻐﺮ ﻣﻦ ﺃﻭ ﻳ ﺗﻘﻨﻴﺎﺕ : ﻟﺘﺮﺗﻴﺐ ﻋﺪﺓ ﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ ﻧﺴﺒﻴﺔ ﻧﺮﺗﺐ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺴﺎﻟﺒﺔ ﻓﻴﻤﺎ ﺑﻴﻨﻬﺎ ﺛﻢ ﻧﺮﺗﺐ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﻮﺟﺒﺔ ﻓﻴﻤﺎ ﺑﻴﻨﻬﺎ ﺛﻢ ﻧﺮﺗﺐ ﺍﻟﻜﻞ ﻣﺜﺎﻝ : ﻟﻨﺮﺗﺐ ﺍﻷﻋﺪﺍﺩ : 6,41 ;; 11 ;; 55,8 ;; 9,5 ;; 6 ;; 5,1 ;; 52 ;; 0 ﻟﺪﻳﻨﺎ : 0 < 5,1 < 6 < 55,8 < 6,41 ﻭ 52 < 11 < 9,5 < 0 52 < 11 < 9,5 < 0 < 5,1 < 6 < 55,8 < 6,41 ﺇﺫﻥ
- 19. ﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﻣﺜﻠﺚ / ﻣﺜﻠﺜﺎﺕ ﺧﺎﺻﺔ ﻣﺠ _Iﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﻣﺜﻠﺚ . 1( – ﺍﻟﺰﻭﺍﻳﺎ : ﺗﻌﺎﺭﻳﻒ ﻭ ﻣﻔﺮﺩﺍﺕ : Tﺍﻟﺸﻜﻞ ﺟﺎﻧﺒﻪ ﻳﺴﻤﻰ : ﺯﺍﻭﻳﺔ . ﻳﺮﻣﺰ ﻟﻬﺬﻩ ﺍﻟﺰﺍﻭﻳﺔ ﺑﺎﻟﺮﻣﺰ : A ˆ B O ﺍﻟﻨﻘﻄﺔ Oﺗﺴﻤﻰ ﺭﺃﺱ ﻫﺬﻩ ﺍﻟﺰﺍﻭﻳﺔ . ﻧﺼﻔﺎ ﺍﻟﻤﺴﺘﻘﻴﻢ ) [OAﻭ ) [OBﻳﺴﻤﻴﺎﻥ : ﺿﻠﻌﻲ ﻫﺬﻩ ﺍﻟﺰﺍﻭﻳﺔ . Tﺯﻭﺍﻳﺎ ﺧﺎﺻﺔ : ± ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻨﻌﺪﻣﺔ : ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻨﻌﺪﻣﺔ ﻫﻲ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ °0 . ± ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺤﺎﺩﺓ : ﺎ ﻣﺤﺼﻮﺭ ﺑﻴﻦ °0 ﻭ °09 . ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﺤﺎﺩﺓ ﻫﻲ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬ ± ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ : ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻘﺎﺋﻤﺔ ﻫﻲ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ °09 .
- 20. ± ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻨﻔﺮﺟﺔ : ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻨﻔﺮﺟﺔ ﻫﻲ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ ﻣﺤﺼﻮﺭ ﺑﻴﻦ °09 ﻭ °081 . ± ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﺔ : ﺍﻭﻳﺔ ﺍﻟﻤﺴﺘﻘﻴﻤﻴﺔ ﻫﻲ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳﻬﺎ °081 ﺍﻟﺰ ± ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻠﻴــﺌﺔ : ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﻠﻴﺌﺔ ﻫﻲ ﺯﺍﻭﻳﺔ ﻗﻴﺎﺳــﻬﺎ °063 . Tﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﻘﺎﻳﺴﺘﺎﻥ : ﺗﻜﻮﻥ ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﺎﻳﺴﺘﻴﻦ ﺇﺫﺍ ﻛﺎﻥ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﻘﻴﺎﺱ . Tﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﺤﺎﺫﻳﺘﺎﻥ : ﺗﻜﻮﻥ ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﺤﺎﺫﻳﺘﻴﻦ ﺇﺫﺍ ﻛﺎﻥ : ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻟﺮﺃﺱ . ﻟﻬﻤﺎ ﺿﻠﻊ ﻣﺸﺘﺮﻙ . ﻭ ﻳﺘﻘﺎﻃﻌﺎﻥ ﻓﻲ ﺍﻟﻀﻠﻊ ﺍﻟﻤﺸﺘﺮﻙ . Tﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﺘﺎﻣﺘﺎﻥ : ﺗﻜﻮﻥ ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﻴﻦ ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻬﻤﺎ ﻳﺴﺎﻭﻱ °09 Tﺍﻟﺰﺍﻭﻳﺘﺎﻥ ﺍﻟﻤﺘﻜﺎﻣﻠﺘﺎﻥ : ﺗﻜﻮﻥ ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﻜﺎﻣﻠﺘﻴﻦ ﺇﺫﺍ ﻛﺎﻥ ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﻬﻤﺎ ﻳﺴﺎﻭﻱ °081 2( – ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﻣﺜﻠﺚ : * ﺧﺎﺻﻴﺔ 1 : ﻣﺠﻤﻮﻉ ﻗﻴﺎﺳﺎﺕ ﺯﻭﺍﻳﺎ ﻣﺜﻠﺚ ﻳﺴﺎﻭﻱ °081 ABCﻣﺜﻠﺚ
- 21. 3( – ﻣﺜﻠﺜﺎﺕ ﺧـــﺎﺻﺔ : ± ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ : * ﺗﻌﺮﻳﻒ 1 : ﻛﻞ ﻣﺜﻠﺚ ﻟﻪ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ ﻳﺴﻤﻰ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻫﻮ ﻣﺜﻠﺚ ﻟﻪ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ * ﻣﺜﺎﻝ : ABCﻣﺜﺎﺙ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻲ . A * ﺧﺎﺻﻴﺔ 2 : ﺇﺫﺍ ﻛﺎﻥ ﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﺯﺍﻭﻳﺔ ﻓﺈﻥ ﺯﺍﻭﻳﺘﺎﻩ ﺍﻟﺤﺎﺩﺗﻴﻦ ﻣﺘﺘﺎﻣﺘﻴﻦ * ﺧﺎﺻﻴﺔ 3 : ﺇﺫﺍ ﻛﺎﻥ ﻟﻤﺜﻠﺚ ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﺘﺎﻣﺘﺎﻥ ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ± ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ : * ﺗﻌﺮﻳﻒ 2 : ﻳﻜﻮﻥ ﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﺇﺫﺍ ﻛﺎﻥ ﻟﻪ ﺿﻠﻌﺎﻥ ﻣﺘﻘﺎﻳﺴﺎﻥ * ﻣﺜﺎﻝ : ABCﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﺭﺃﺳﻪ A
- 22. * ﺧﺎﺻﻴﺔ : 4 ﺇﺫﺍ ﻛﺎﻥ ﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴ ﻴﻦ ﻓﺈﻥ ﺯﺍﻭﺗﻲ ﺍﻟﻘﺎﻋﺪﺓ ﻣﺘﻘﺎﻳﺴﺘﺎﻥ ﺎﻗ ˆ ˆ ﺑﺘﻌﺒﻴﺮ ﺁﺧﺮ : ABCﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﺭﺃﺳﻪ Aﻳﻌﻨﻲ ﺃﻥ : B = C * ﺧﺎﺻﻴﺔ : 5 ﺇﺫﺍ ﻛﺎﻥ ﻟﻤﺜﻠﺚ ﺯﺍﻭﻳﺘﺎﻥ ﻣﺘﻘﻠﻴﺴﺘﺎﻥ ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ˆ ˆ ﺑﺘﻌﺒﻴﺮ ﺁﺧﺮ : ABCﻣﺜﻠﺚ ﺑﺤﻴﺚ B = Cﻳﻌﻨﻲ ﺃﻥ : ABCﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﺭﺃﺳﻪ . A ± ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﻭ ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ : * ﺗﻌﺮﻳﻒ 3 : ﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﻭ ﺍﻟﻘﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻫﻮ ﻣﺜﻠﺚ ﻟﻪ ﺿﻠﻌﺎﻥ ﻣﺘﻘﺎﻳﺴﺎﻥ ﻭ ﺯﺍﻭﻳﺔ ﻗﺎﺋﻤﺔ ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴ C ﻦ ﻭ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﻲ . A * ﻣﺜﺎﻝ : ABCﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴ A B * ﺧﺎﺻﻴﺔ : 6 ﺇﺫﺍ ﻛﺎﻥ ﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﻭ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻓﺈﻥ ﺯﺍﻭﻳﺘﻲ ﺍﻟﻘﺎﻋﺪﺓ ﻣﺘﻘﺎ ﻳﺴﺘﺎﻥ ﻭ ﻗﻴﺎﺳﻬﻤﺎ °54 ˆ ˆ ° 54 = ABC = ACB * ﻣﺜﺎﻝ : ABCﻣﺜﻠﺚ ﻗﺎﺋﻢ ﺍﻟﺰﺍﻭﻳﺔ ﻭ ﻣﺘﺴﺎﻭﻱ ﺍﻟﺴﺎﻗﻴﻦ ﻓﻲ Aﺇﺫﻥ :
- 23. ± ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴﺎﻭﻱ ﺍﻷﺿﻼﻉ : * ﺗﻌﺮﻳﻒ 4 : ﺍﻟﻤﺜﻠﺚ ﺍﻟﻤﺘﺴﺎﻭﻱ ﺍﻷﺿﻼﻉ ﻫﻮ ﻣﺜﻠﺚ ﺟﻤﻴﻊ ﺃﺿﻼﻋﻪ ﻣﺘﻘﺎﻳﺴﺔ * ﻣﺜﺎﻝ : ABCﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻷﺿﻼﻉ . * ﺧﺎﺻﻴﺔ : 7 ﺇﺫﺍ ﻛﺎﻥ ﻣﺜﻠﺚ ﻣﺘﺴﺎﻭﻱ ﺍﻷﺿﻼﻉ ﻓﺈﻥ ﺟﻤﻴﻊ ﺯﻭﺍﻳﺎﻩ ﻣﺘﻘﺎﻳﺴﺔ ﻭ ﻗﻴﺎﺱ ﻛﻞ ﻣﻨﻬﺎ °06 * ﺧﺎﺻﻴﺔ : 8 ﺇﺫﺍ ﻛﺎﻧﺖ ﺯﻭﺍﻳﺎ ﻣﺜﻠﺚ ﻣﺘﻘﺎﻳﺴﺔ ﻓﺈﻧﻪ ﻳﻜﻮﻥ ﻣﺘﺴﺎﻭﻱ ﺍﻷﺿﻼﻉ
- 24. ﺟﻤﻊ ﻭ ﻃﺮﺡ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ 1( – ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ : ( ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻹﺷﺎﺭﺓ : ﺃ * ﻗﺎﻋﺪﺓ 1 : ﻟﺤﺴﺎﺏ ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻹﺷﺎﺭﺓ ﻧﺤﺘﻔﻆ ﺑﺎﻹﺷﺎﺭﺓ ﺛﻢ ﻧﺠﻤﻊ ﻣﺴﺎﻓﺘﻴﻬﻤﺎ ﻋﻦ ﺍﻟﺼﻔﺮ . ;; 9,32 = 5,1 + 4,22 * ﺃﻣﺜﻠﺔ : 5,21 = ) 7 + 5,5( – = ) 7 –( + 5,5 – ;; 51,071 = 51,85 + 211 522,175 – = ) 75 + 522,,415 ( – = ) 75 –( + 522,415 – ( ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ : ﺏ * ﻗﺎﻋﺪﺓ 2 : ﻟﺤﺴﺎﺏ ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ ﻧﺄﺧﺬ ﺇﺷﺎﺭﺓ ﺍﻟﻌﺪﺩ ﺍﻷﺑﻌﺪ ﻋﻦ ﺍﻟﺼﻔﺮ ﺛﻢ ﻧﺤﺴﺐ ﻓﺮﻕ ﻣﺴﺎﻓﺘﻴﻬﻤﺎ ﻋﻦ ﺍﻟﺼﻔﺮ . 62,31 – = ) 41,21 – 4,52( – = ) 4,52 –( + 41,21 * ﺃﻣﺜﻠﺔ : 98,12 = ) 11,41 – 63 ( + = 63 + 11,41 – 5,97 = ) 5,54 – 521 ( + = ) 5,54 –( + 521 51,02 – = ) 5,11 – 56,13 ( – = 5,11 + 56,13 – ﺎﺑﻠﻴﻦ : ( ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻣﺘﻘ ﺝ * ﻗﺎﻋﺪﺓ 3 : ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻣﺘﻘﺎﺑﻠﻴﻦ ﻳﻜﻮ ﺩﺍﺋﻤﺎ ﻣﻨﻌﺪﻣﺎ ) ﺃﻱ ﻳﺴﺎﻭﻱ ﺻﻔﺮ ( . 0 = ) a + ( aﻭ 0 = a + a aﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ . ;; 0 = ) 88,521 –( + 88,521 * ﺃﻣﺜﻠﺔ : 0 = 7633 + 7633 – 0 = ) 85211 –( + 85211 ;; 0 = 7,953 + 7,953 – 2( – ﻓﺮﻕ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ : * ﻗﺎﻋﺪﺓ 4 : ﻟﺤﺴﺎﺏ ﻓﺮﻕ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻧﻀﻴﻒ ﺇﻟﻰ ﺍﻟﺤﺪ ﺍﻷﻭﻝ ﻣﻘﺎﺑﻞ ﺍﻟﺤﺪ ﺍﻟﺜﺎﻧﻲ . aﻭ bﻋﺪﺩﺍﻥ ﻋﺸﺮﻳﺎﻥ ﻧﺴﺒﻴﺎﻥ : ) a – b = a + ( b * ﺃﻣﺜﻠﺔ : 57,9 = ) 5,11 – 52,12 ( + = ) 5,11 –( + 52,12 = 5,11 – 52,12 55,52 = 21 + 55,31 = ) 21 ( – 55,31 05 = ) 61 + 43( – = ) 61 –( + 43 – = 61 – 43 – 41,54 = ) 02 – 41,56 ( – = 02 + 41,56 – = ) 02 –( – 41,56 –
- 25. ﺗﻘﻨﻴﺎﺕ 1( ﻹﺯﺍﻟﺔ ﺍﻷﻗﻮﺍﺱ ﺍﻟﻤﺴﺒﻮﻗﺔ ﺑﻌﻼﻣﺔ + : ﻧﺰﻳﻞ ﻋﻼﻣﺔ + ﻭ ﻧﺤﺪﻑ ﺍﻷﻗﻮﺍﻕ ﺑﺪﻭﻥ ﺗﻐﻴﻴﺮ ﺇﺷﺎﺭﺓ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺘﻲ ﺑﺪﺍﺧﻠﻬﺎ . ﻹﺯﺍﻟﺔ ﺍﻷﻗﻮﺍﺱ ﺍﻟﻤﺴﺒﻮﻗﺔ ﺑﻌﻼﻣﺔ – : ﻧﺰﻳﻞ ﻋﻼﻣﺔ – ﻭ ﻧﺤﺪﻑ ﺍﻷﻗﻤﺎﺱ ﻣﻊ ﺗﻐﻴﻴﺮ ﺇﺷﺎﺭﺓ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺘﻲ ﺑﺪﺍﺧﻠﻬﺎ . )2 + 11 – 45 ( + )5,1 – 33 + 5,2 –( + 11 = A * ﺃﻣﺜﻠﺔ : 2 + 11 – 45 + 5,1 – 33 + 5,2 – 11 = ) 66,42 + 5,1 – 25 ( – ) 1+ 85 – 44,21 + 55 –( – 6,2 = B 66,42 – 5,1 + 25 – 1 – 85 + 44,21 – 55 + 6,2 = 2( ﺣﺴﺎﺏ ﺗﻌﺒﻴﺮ ﺟﺒﺮﻱ ﻳﺤﺘﻮﻱ ﻋﻠﻰ ﺃﻗﻮﺍﺱ ﻭ ﻣﻌﻘﻮﻓﺎﺕ ﺑﺎﺳﺘﻌﻤﺎﻝ ﺍﻟﻘﺎﻋﺪﺓ ﺃﻋﻼﻩ . 1( – ﻧﺰﻳﻞ ﺍﻷﻗﻮﺍﺱ ﻭ ﺍﻟﻤﻌﻘﻮﻓﺎﺕ ﺑﺪﺃ ﺑﺎﻷﻗﻮﺍﺱ ﺍﻟﺪﺍﺧﻠﻴﺔ ﻣﻊ ﺗﻄﺒﻴﻖ ﺍﻟﻘﺎﻋﺪﺓ ﺃﻋﻼﻩ . 2( – ﻧﺠﻤﻊ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﺘﻘﺎﺑﻠﺔ ﻓﻴﻤﺎ ﺑﻴﻨﻬﺎ ﺛﻢ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻤﻮﺟﺒﺔ ﻭ ﺍﻷﻋﺪﺍﺩ ﺍﻟﺴﺎﻟﺒﺔ 7 – ) 5,2 + 41 –( – ) 1+ 5,11 –( + 5,2 = A * ﻠﺔ : ﺃﻣﺜ 7 – 5,2 – 41 + 1 + 5,11 – 5,2 = 7 – 5,11 – 41 + 1 + 5,2 – 5,2 = 5,71 – 51 + 0 = ) 51 – 5,71 ( – = 5,2 – = ) 3 + 5,5–( – 22 + ] 1 – ) 7 – 5,3 ( + 5,11 –[ – ) 1 – 5,3 ( = B 3 – 5,5 + 22 + ] 1 – 7 – 5,3 + 5,11–[ – 1 – 5,3 = 3 – 5,5 + 22 + 1 – 7 – 5,3 – 5,11 + 1 – 5,3 = 3 – 7 – 5,5 + 22 + 5,11 + 1 – 1 + 5,3 – 5,3 = 01 – 93 + 0 + 0 = 01 – 93 = 92 =
- 26. ﺿﺮﺏ ﻭ ﻗﺴﻤﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ 1( – ﺿﺮﺏ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ : ( ﺟﺪﺍء ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻹﺷﺎﺭﺓ : ﺃ * ﻗﺎﻋﺪﺓ 1 : ﺟﺪﺍء ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻹﺷﺎﺭﺓ ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻣﻮﺟﺐ 5,0 = ) 01–( – 21 x (–5 ) = 105 ;; 0,05 x * ﺃﻣﺜﻠﺔ : 0 = ) 621–( –125,89 x 0 = 0 ;; 0 x ( ﺟﺪﺍء ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ : ﺏ * ﻗﺎﻋﺪﺓ 2 : ﺟﺪﺍء ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﺳﺎﻟﺐ ;; 15– = ) 2–( 25,5 x * ﺃﻣﺜﻠﺔ : 575– = 05 –11,5 x ;; 011– = ) 5–( 22 x 057 = 01 –75 x ( ﺟﺪﺍء ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﻓﻲ : 1 ﻭ 1 : ﺝ * ﻗﺎﻋﺪﺓ 3 : ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻣﺘﻘﺎﺑﻠﻴﻦ ﻳﻜﻮ ﺩﺍﺋﻤﺎ ﻣﻨﻌﺪﻣﺎ ) ﺃﻱ ﻳﺴﺎﻭﻱ ﺻﻔﺮ ( . aﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ . a + ( 1 ) = aﻭ 1 + a = a 0 = ) a + ( aﻭ ﻭ 1 x a = a a + a = 0 a x 1 = a aﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ . * ﺃﻣﺜﻠﺔ : 7633 = 1 1 x (– 125,88 ) = –125,88 ;; 3367 x 35211– = 85211 – 359,7 x (–1 ) = 359,7 ;; – 1 x ( ﺟﺪﺍء ﻋﺪﺓ ﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ ﻧﺴﺒﻴﺔ : ﺩ * ﻗﺎﻋﺪﺓ 4 : ﺮﻳﺔ ﻧﺴﺒﻴﺔ ﻳﻜﻮﻥ : ﺟﺪﺍء ﻋﺪﺓ ﺃﻋﺪﺍﺩ ﻋﺸ . ﻣﻮﺟﺒﺎ : ﺇﺫﺍ ﻋﺪﺩ ﻋﻮﺍﻣﻠﻪ ﺍﻟﺴﺎﻟﺒﺔ ﺯﻭﺟﻴﺎ ﺳﺎﻟﺒﺎ : ﺇﺫﺍ ﻛﺎﻥ ﻋﺪﺩ ﻋﻮﺍﻣﻠﻪ ﺍﻟﺴﺎﻟﺒﺔ ﻓﺮﺩﻳﺎ . ) 5–( A = –5 x 1,3 x (–7 ) x (–25 ) x 1 x * ﺃﻣﺜﻠﺔ : 7,1 B = 11 x (–25,4 ) x 14 x (–1 ) x (–0,5 ) x * ﻟﺪﻳﻨﺎ ﺍﻟﺠﺪﺍء Aﻋﺪﺩ ﻋﻮﺍﻣﻠﻪ ﺍﻟﺴﺎﻟﺒﺔ ﻫﻮ 4 ﻭ ﻫﻮ ﻋﺪﺩ ﺯﻭﺟﻲ , ﺇﺫﻥ Aﻋﺪﺩ ﻣﻮﺟﺐ . * ﻟﺪﻳﻨﺎ ﺍﻟﺠﺪﺍء Bﻋﺪﺩ ﻋﻮﺍﻣﻠﻪ ﺍﻟﺴﺎﻟﺒﺔ ﻫﻮ 3 ﻭ ﻫﻮ ﻋﺪﺩ ﻓﺮﺩﻱ , ﺇﺫﻥ Bﻋﺪﺩ ﺳﺎﻟﺐ .
- 27. * ﻗﺎﻋﺪﺓ 5 : ﻻ ﻳﺘﻐﻴﺮ ﺟﺪﺍء ﻋﺪﺓ ﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ ﻧﺴﺒﻴﺔ ﺇﺫﺍ ﻏﻴﺮﻧﺎ ﺗﺮﺗﻴﺐ ﻋﻮﺍﻣﻠﻪ ﺃﻭ ﻋﻮﺿﻨﺎ ﺑﻌﻀﺎ ﻣﻨﻬﺎ ﺑﺠﺪﺍﺋﻬﺎ . ) 5,1–( A = (–2 ) x 5,5 x 50 x * ﻣﺜﺎﻝ : ) ) 5,1–( = ( –2 x 50 ) x ( 5,5 x ) 52,8–( = –100 x 528 = ﺗﻘﻨﻴﺎﺕ ﻟﺤﺴﺎﺏ ﺟﺪﺍء ﻋﺪﺓ ﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ ﻧﺴﺒﻴﺔ ﻧﺤﺪﺩ ﺃﻭﻻ ﺇﺷﺎﺭﺓ ﻫﺬﺍ ﺍﻟﺠﺪﺍء ﺛﻢ ﻧﻄﺒﻖ ﺍﻟﻘﺎﻋﺪﺓ 4 . ﺃﻣﺜﻠﺔ : 5,6 A = (–7,5 ) x 25 x –4 ) x ) 5,6 = + ( 7,5 x 25 x 4 x ) 5,6 = + ( ( 25 x 5 ) x ( 7,5 x 57,84 = 100 x 5784 = 5,7 B = –6 x 5 x (–1,5 ) x (–1 ) x ) 5,7 = – ( 6 x 5 x 1 x ) 5,7 = – ( (6 x 5 x 1 ) x ( 1,5 x ) 52,11 = – ( 30 x 5,733– = 2( – ﻗﺴﻤﺔ ﺍﻷﻋﺪﺍﺩ ﺍﻟﻌﺸﺮﻳﺔ ﺍﻟﻨﺴﺒﻴﺔ : ( ﺧﺎﺭﺝ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻹﺷﺎﺭﺓ : ﺃ * ﻗﺎﻋﺪﺓ 6 : ﺧﺎﺭﺝ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻟﻬﻤﺎ ﻧﻔﺲ ﺍﻹﺷﺎﺭﺓ ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﻣﻮﺟﺐ ;; 51,26 = ) 31 –( : 59,708 – 011 = 1,7 : 187 * ﺃﻣﺜﻠﺔ : ( ﺧﺎﺭﺝ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ : ﺏ * ﻗﺎﻋﺪﺓ 7 : ﺧﺎﺭﺝ ﻋﺪﺩﻳﻦ ﻋﺸﺮﻳﻴﻦ ﻧﺴﺒﻴﻴﻦ ﻣﺨﺘﻠﻔﻴﻦ ﻓﻲ ﺍﻹﺷﺎﺭﺓ ﻫﻮ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﺳﺎﻟﺐ ;; 51,26 – = ) 31–( : 59,708 011 – = 1,7 : 187 – * ﺃﻣﺜﻠﺔ :
- 28. - a a - a a a = ﻭ = -= * ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : - b b b - b b ( ﺍﻟﺨﺎﺭﺝ ﺍﻟﻤﻘﺮﺏ ﻭ ﺍﻟﺘﺄﻃﻴﺮ : ﺝ 1( – ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺨﺎﺭﺝ ﻣﻮﺟﺒﺎ : 22 22 7 ﻧﻌﺘﺒﺮ ﺍﻟﺨﺎﺭﺝ * ﻣﺜﺎﻝ : 7 01 41,3 03 02 22 ﺇﻟﻰ 1 ﻧﺘﻔﺮﻳﻂ ﻫﻲ : 3 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 ﺇﻟﻰ 1 ﺑﺈﻓﺮﺍﻁ ﻫﻲ : 4 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 22 < 3 ﺇﻟﻰ 1 ﻫﻮ : 4 < ﺇﺫﻥ ﺗﺄﻃﻴﺮ ﺍﻟﻌﺪﺩ 7 7 22 ﺇﻟﻰ 1,0 ﻧﺘﻔﺮﻳﻂ ﻫﻲ : 1,3 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 ﺇﻟﻰ 1,0 ﺑﺈﻓﺮﺍﻁ ﻫﻲ : 2,3 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 22 < 1,3 2,3 < ﺇﻟﻰ 1,0 ﻫﻮ : ﺇﺫﻥ ﺗﺄﻃﻴﺮ ﺍﻟﻌﺪﺩ 7 7 2( – ﺇﺫﺍ ﻛﺎﻥ ﺍﻟﺨﺎﺭﺝ ﺳﺎﻟﺒﺎ : 22 - * ﻣﺜﺎﻝ : ﻧﻌﺘﺒﺮ ﺍﻟﺨﺎﺭﺝ 7 22 - ﺇﻟﻰ 1 ﻧﺘﻔﺮﻳﻂ ﻫﻲ : 4 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 - ﺇﻟﻰ 1 ﺑﺈﻓﺮﺍﻁ ﻫﻲ : 3 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 22 - ﺇﻟﻰ 1 ﻫﻮ : 3 < - < 4 ﺇﺫﻥ ﺗﺄﻃﻴﺮ ﺍﻟﻌﺪﺩ 7 7 22 - ﺇﻟﻰ 1,0 ﻧﺘﻔﺮﻳﻂ ﻫﻲ : 2,3 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 - ﺇﻟﻰ 1,0 ﺑﺈﻓﺮﺍﻁ ﻫﻲ : 1,3 . * ﺍﻟﻘﻴﻤﺔ ﺍﻟﻤﻘﺮﺑﺔ ﻟﻠﻌﺪﺩ 7 22 22 - ﺇﻟﻰ 1,0 ﻫﻮ : 1,3 < - < 2,3 ﺇﺫﻥ ﺗﺄﻃﻴﺮ ﺍﻟﻌﺪﺩ 7 7
- 29. ﺍﻟﻤﻨﺼﻔــﺎﺕ ﻭ ﺍﻻﺭﺗﻔــﺎﻋــﺎﺕ ﻓﻲ ﻣﺜﻠﺚ 1( – ﺍﻟﻤﻨﺼﻔﺎﺕ ﻓﻲ ﻣﺜﻠﺚ : ( ﻣﻨﺼﻒ ﺯﺍﻭﻳﺔ : ﺃ * ﺗﻌﺮﻳﻒ 1 : ﻣﻨﺼﻒ ﺯﺍﻭﻳﺔ ﻫﻮ ﻧﺼﻒ ﻣﺴﺘﻘﻴﻢ ﺃﺻﻠﻪ ﺭﺃﺱ ﺍﻟﺰﺍﻭﻳﺔ , ﻳﻮﺟﺪ ﺑﺪﺍﺧﻠﻬﺎ ﻭ ﻳﻘﺴﻤﻬﺎ ﺇﻟﻰ ﻴﻦ ﺯﺍﻭﻳﺘﻴﻦ ﻣﺘﻘﺎﻳﺴﺘ * ﻣﺜﺎﻝ : ﻧﻌﺘﺒﺮ A ˆ Bﺯﺍﻭﻳﺔ ﻭ ) [OMﻣﻨﺼﻔﻬﺎ . O ( ﺍﻟﺨﺎﺻﻴﺔ ﺍﻟﻤﻤﻴﺰﺓ ﻟﻤﻨﺼﻒ ﺯﺍﻭﻳﺔ : ﺏ * ﺍﻟﺨﺎﺻﻴﺔ ﺍﻟﻤﺒﺎﺷــﺮﺓ : ﻛﻞ ﻧﻘﻄﺔ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻣﻨﺼﻒ ﺯﺍﻭﻳﺔ ﺗﺒﻌﺪ ﺑﻨﻔﺲ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﺿﻠﻌﻲ ﻫﺬﻩ ﺍﻟﺰﺍﻭﻳﺔ ﺎ : EK = EL ﺳﻴﻜﻮﻥ ﻟﺪﻳﻨ * ﺍﻟﺨﺎﺻﻴﺔ ﺍﻟﻌﻜﺴﻴﺔ : ﺔ ﺗﻨﺘﻤﻲ ﺇﻟﻰ ﻣﻨﺼﻒ ﻫﺬﻩ ﺍﻟﺰﺍﻭﻳﺔ ﻛﻞ ﻧﻘﻄﺔ ﺗﺒﻌﺪ ﺑﻨﻔﺲ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﺿﻠﻌﻲ ﺯﺍﻭﻳ * ﺍﻟﺨﺎﺻﻴﺔ ﺍﻟﻤﻤﻴﺰﺓ : ﻣﻨﺼﻒ ﺯﺍﻭﻳﺔ ﻫﻮ ﻣﺠﻤﻮﻋﺔ ﻣﻦ ﻧﻘﻂ ﺍﻟﺰﺍﻭﻳﺔ ﺍﻟﻤﺘﺴﺎﻭﻳﺔ ﺍﻟﻤﺴﺎﻓﺔ ﻋﻦ ﺿﻠﻌﻴﻬﺎ ( ﻣﻨﺼﻔﺎﺕ ﻣﺜﻠﺚ : ﺝ * ﺗﻌﺮﻳﻒ 2 : ﻣﻨﺼﻒ ﻣﺜﻠﺚ ﻫﻮ ﻣﻨﺼﻒ ﺇﺣﺪﻯ ﺯﻭﺍﻳﺎﻩ
- 30. * ﻣﺜﺎﻝ : ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﻟﻠﻤﺜﻠﺚ ﺛﻼﺙ ﻣﻨﺼﻔﺎﺕ . * ﺧﺎﺻﻴـــﺔ : ﻣﻨﺼﻔﺎﺕ ﻣﺜﻠﺚ ﺗﺘﻼﻗﻰ ﻓﻲ ﻧﻘﻄﺔ ﻭﺍﺣﺪﺓ ﺗﺴﻤﻰ ﻣﺮﻛﺰ ﺍﻟﺪﺍﺋﺮﺓ ﺍﻟﻤﺤﺎﻃﺔ ﺑﻬﺬﺍ ﺍﻟﻤﺜﻠﺚ * ﻣﺜﺎﻝ : ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﻹﻳﺠﺎﺩ ﻣﺮﻛﺰ ﺩﺍﺋﺮﺓ ﻣﺤﺎﻃﺔ ﺑﻤﺜﻠﺚ ﻳﻜﻔﻲ ﺭﺳﻢ ﻣﻨﺼﻔﻴﻦ ﻓﻘﻂ ﻣﻦ ﻣﻨﺼﻔﺎﺕ ﻫﺬﺍ ﺍﻟﻤﺜﻠﺚ . 2( – ﺍﻻﺭﺗﻔﺎﻋﺎﺕ ﻓﻲ ﻣﺜﻠﺚ : ( ﺟﺪﺍء ﻋﺪﺓ ﺃﻋﺪﺍﺩ ﻋﺸﺮﻳﺔ ﻧﺴﺒﻴﺔ : ﺃ * ﺗﻌﺮﻳﻒ 3 : ﺍﺭﺗﻔﺎﻉ ﻣﺜﻠﺚ ﻫﻮ ﻣﺴﺘﻘﻴﻢ ﻳﻤﺮ ﻣﻦ ﺃﺣﺪ ﺭﺅﻭﺱ ﺍﻟﻤﺜﻠﺚ ﻭ ﻋﻤﻮﺩﻱ ﻋﻠﻰ ﺣﺎﻣﻞ ﺍﻟﻀﻠﻊ ﺍﻟﻤﻘﺎﺑﻞ ﻟﻬﺬﺍ ﺍﻟﺮﺃﺱ . * ﻣﺜﺎﻝ : ABCﻣﺜﻠﺚ ﻭ ) (AHﺍﻻﺭﺗﻔﺎﻉ ﺍﻟﻤﻮﺍﻓﻖ ﻟﻠﻀﻠﻊ ]. [BC
- 31. · ﺣﺎﻟﺔ ﺧﺎﺻ ﺔ : ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﻟﻠﻤﺜﻠﺚ ﺛﻼﺙ ﺍﺭﺗﻔﺎﻋﺎﺕ . * ﺧﺎﺻﻴﺔ : ﺍﺭﺗﻔﺎﻋﺎﺕ ﻣﺜﻠﺚ ﺗﺘﻼﻗﻰ ﻓﻲ ﻧﻘﻄﺔ ﻭﺍﺣﺪﺓ ﺗﺴﻤﻰ ﻣﺮﻛﺰ ﺗﻌﺎﻣــﺪ ﻫﺬﺍ ﺍﻟﻤﺜﻠﺚ . * ﻣﺜﺎﻝ : 02 ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﻟﺮﺳﻢ ﻣﺮﻛﺰ ﺗﻌﺎﻣـــﺪ ﻣﺜﻠﺚ ﻳﻜﻔﻲ ﺭﺳﻢ ﺍﺭﺗﻔﺎﻋﻴﻦ ﻓﻘﻂ ﻣﻦ ﺍﺭﺗﻔﺎﻋﺎﺕ ﻫﺬﺍ ﺍﻟﻤﺜﻠﺚ .
- 32. ﺍﻟـﻘــــــــــــــــﻮﻯ 1( – ﻗﻮﺓ ﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ : ( ﻣﺜﺎﻝ : ﺃ ﻧﻌﺘﺒﺮ ﺍﻟﺠﺪﺍء ﺍﻵﺗﻲ : 5,2 A = 2,5 x 2,5 x 2,5 x 2,5 x ﻳﺘﻜﻮﻥ ﻫﺬﺍﺍﻟﺠﺪﺍء ﻣﻦ ﺧﻤﺴﺔ ﻋﻮﺍﻣﻞ ﻣﺴﺎﻭﻳﺔ ﻟﻠﻌﺪﺩ 5,2 . ﻧﺴﻤﻲ ﺇﺫﻥ ﻫﺬﺍ ﺍﻟﺠﺪﺍء : ﺍﻟﻘﻮﺓ ﺍﻟﺨﺎﻣﺴﺔ ﻟﻠﻌﺪﺩ 5,2 . 5 .. ﺘﺐ : )5,2( ﻭ ﻧﻘــﺮﺃ : ﺇﺛﻨﺎﻥ ﺃﺱ ﺧﻤﺴــﺔ ﻭ ﻧﻜ 5 5 ﺍﻟﻌﺪﺩ 5,2 ﻳﺴﻤﻰ : ﺃﺳﺎﺱ ﺍﻟﻘﻮﺓ )5,2( ﻭ ﺍﻟﻌﺪﺩ 5 ﻳﺴﻤﻰ : ﺃﺱ ﺍﻟﻘﻮﺓ )5,2( . ( ﺗﻌﺮﻳﻒ : ﺏ . aﻋﺪﺩ ﻋﺸﺮﻱ ﻧﺴﺒﻲ ﺃ ﺒﺮ ﻣﻦ 1 ﻭ nﻋﺪﺩ ﺻﺤﻴﺢ ﻃﺒﻴﻌﻲ ﻏﻴﺮ ﻣﻨﻌﺪﻡ ﻛ a n = a ´ a ´ a ´ a ´ a ´ ...... ´ a 44442 44414 3 ) nﻣﻦ ﺍﻟﻌﻮﺍﻣﻞ ( ﻣﻼﺣﻈﺎﺕ ﻫﺎﻣﺔ : 0 0 1 ﺓ 0 ﻻ ﻣﻌﻨﻰ ﻟﻬﺎ . ﺍﻟﻘﻮ , , 1 = ( 0 ¹ a ) a a = a ﻣﻔــــﺮﺩﺍﺕ : n · ﻧﺴﻤﻲ aﺃﺳــﺎﺱ ﺍﻟﻘﻮﺓ . a n · ﻧﺴﻤﻲ nﺃﺱ ﺍﻟﻘﻮﺓ . a ( ﺇﺷﺎﺭﺓ ﻗـــﻮﺓ ﺃﺳﺎﺳﻬﺎ ﺳـــﺎﻟﺐ : ﺝ * ﺧــﺎﺻﻴﺔ 1 : ﺗﻜﻮﻥ ﻗــﻮﺓ ﺃﺳﺎﺳﻬﺎ ﺳﺎﻟﺐ : · ﻣﻮﺟﺒﺔ : ﺇﺫﺍ ﻛﺎﻥ ﺃﺳﻬﺎ ﻋﺪﺩﺍ ﺯﻭﺟﻴﺎ . · ﺳﺎﻟﺒﺔ : ﺇﺫﺍ ﻛﺎﻥ ﺃﺳﻬﺎ ﻋﺪﺩﺍ ﻓﺮﺩﻳﺎ . 61 )11 ( ﻋﺪﺩ ﻣﻮﺟﺐ , ﻷﻥ ﺃﺳﻬﺎ ﻫﻮ 61 ﻭﻫﻮ ﻋﺪﺩ ﺯﻭﺟﻲ . ﺍﻟﻘﻮﺓ * ﻣﺜﺎﻝ : 12 ﺍﻟﻘﻮﺓ )9,5 ( ﻋﺪﺩ ﺳﺎﻟﺐ , ﻷﻥ ﺃﺳﻬﺎ ﻫﻮ 12 ﻭ ﻫﻮ ﻋﺪﺩ ﻓﺮﺩﻱ . 8 8 * ﻣﻼﺣﻈﺔ ﻫﺎﻣﺔ : ﺍﻟﻘﻮﺓ )5 ( ﺗﺨﺘﻠﻒ ﻋﻦ ﺍﻟﻘﻮﺓ 5 ﻷﻥ : 8 ) 5 ( ﺃﺳﺎﺳﻬﺎ ﻫﻮ ) 5 ( ﻭﺣﺴﺐ ﺍﻟﺨﺎﺻﻴﺔ 1 ﻓﻬﻲ ﻣﻮﺟﺒﺔ . 8 5 ﺃﺳﺎﺳﻬﺎ ﻫﻮ 5 ﻭ ﻫﻲ ﺳﺎﻟﺒﺔ ﻷﻧﻬﺎ ﻻﺗﺨﻀﻊ ﻟﻠﺨﺎﺻﻴﺔ 1 .
- 33. 2( – ﺧـﺼـــﺎﺋـــﺺ ﺍﻟﻘــﻮﻯ : aﻭ bﻋﺪﺩﺍﻥ ﻋﺸﺮﻳﺎﻥ ﻧﺴﺒﻴﺎﻥ ﻏﻴﺮ ﻣﻨﻌﺪﻣﻴــﻦ . mﻭ nﻋﺪﺩﺍﻥ ﺻﺤﻴﺤﺎﻥ ﻃﺒﻴﻴﻌﻴﺎﻥ . a m ´ a n = a m + n m - n a m = æ a ö ) (m > n ÷ ç a n è a ø n ) ( a m = a m ´n m ) ´a m ´ b m = ( a b m a m = æ a ö ÷ ç b m è b ø * ﺃﻣﺜﻠــﺔ : 62 a ´ a = a + 14 = a 21 41 21 42 a 5 ´ a ´ a 7 ´ a = a 5 + 11 + 7 + 1 = a 11 32 a 23 ´ b 23 = (a ´ b ) 72 = 51 - 24 = 24 a 51 a a a 54 (a 9 ) 5 = a 9 ´ 5 = a 11a = æ a ö 11 ÷ ç ÷ 11 ç a è b ø 3( – ﻗــــﻮﻯ ﺍﻟـﻌـــﺪﺩ 01 : * ﺧــﺎﺻﻴﺔ 2 : nﻋﺪﺩ ﺻﺤﻴﺢ ﻃﺒﻴﻌﻲ ﻏﻴﺮ ﻣﻨﻌﺪﻡ : 0.............0000001 = 10 n 3442441 ) nﻣﻦ ﺍﻷﺻﻔﺎﺭ ( * ﺃﻣﺜﻠــﺔ : 5 000001 = 01 11 000000000001 = 01 22 00000000000000000000001 = 01