More Related Content
More from tarek1961moussa (15)
W1ac+cours
- 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