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# Cours1

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### Cours1

1. 1. ‫اﻟﺘﻌﺪاد واﻟﺤﺴﺎب‬ ‫·‬ ‫ﻟﯿﻜﻦ ‪a‬و ‪b‬و ‪ c‬أﻋﺪاد ﺻﺤﯿﺤﺔ ﻃﺒﯿﻌﯿﺔ ﺑﺤﯿﺚ ‪ a‬ﯾﻘﺴﻢ اﻟﺠﺬاء ‪bc‬‬ ‫إذا ﻛﺎن : ‪ a‬و ‪ b‬أوﻟﯿﯿﻦ ﻓﯿﻤﺎ ﺑﯿﻨﮭﻤﺎ ﻓﺈن ‪ a‬ﯾﻘﺴﻢ ‪c‬‬ ‫·‬ ‫ﻟﯿﻜﻦ ‪a‬و ‪b‬و ‪ c‬أﻋﺪاد ﺻﺤﯿﺤﺔ ﻃﺒﯿﻌﯿﺔ إذا ﻛﺎن ‪ a‬ﯾﻘﺴﻢ ‪ c‬و ‪ b‬ﯾﻘﺴﻢ ‪c‬‬ ‫و ‪ a‬و ‪ b‬أوﻟﯿﯿﻦ ﻓﯿﻤﺎ ﺑﯿﻨﮭﻤﺎ ﻓﺈن ‪ ab‬ﯾﻘﺴﻢ ‪c‬‬ ‫·‬ ‫ﯾﻜﻮن ﻋﺪد ﻗﺎﺑﻼ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ 6 إذا ﻛﺎن ھﺬا اﻟﻌﺪد ﻗﺎﺑﻼ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ 2 و 3 .‬ ‫·‬ ‫ﯾﻜﻮن ﻋﺪد ﻗﺎﺑﻼ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ 21 إذا ﻛﺎن ھﺬا اﻟﻌﺪد ﻗﺎﺑﻼ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ 3 و 4 .‬ ‫·‬ ‫ﯾﻜﻮن ﻋﺪد ﻗﺎﺑﻼ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ 51 إذا ﻛﺎن ھﺬا اﻟﻌﺪد ﻗﺎﺑﻼ ﻟﻠﻘﺴﻤﺔ ﻋﻠﻰ 3 و 5 .‬ ‫ﻣﺠﻤﻮﻋﺔ اﻷﻋﺪاد اﻟﺤﻘﻴﻘﻴﺔ ‪R‬‬ ‫·‬ ‫·‬ ‫·‬ ‫·‬ ‫ﻣﺠﻤﻮﻋﺔ اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ ھﻲ اﺗﺤﺎد ﻣﺠﻤﻮﻋﺘﻲ اﻷﻋﺪاد اﻟﻜﺴﺮﯾﺔ اﻟﻨﺴﺒﯿﺔ ‪ Q‬واﻷﻋﺪاد اﻟﺼﻤﺎء ‪I‬‬ ‫ﻟﻜﻞ ﻋﺪد ﻛﺴﺮي ﻧﺴﺒﻲ ﻛﺘﺎﺑﺔ ﻋﺸﺮﯾﺔ دورﯾﺔ ، وﻛﻞ ﻛﺘﺎﺑﺔ ﻋﺸﺮﯾﺔ دورﯾﺔ ﺗﻤﺜﻞ ﻋﺪدا ﻛﺴﺮﯾﺎ وﺣﯿﺪا‬ ‫ﻛﻞ ﻛﺘﺎﺑﺔ ﻋﺸﺮﯾﺔ ﻏﯿﺮ ﻣﺘﻨﺎھﯿﺔ وﻏﯿﺮ دورﯾﺔ ﺗﻤﺜﻞ ﻋﺪدا أﺻﻤﺎ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ اﻟﻌﺪدي ھﻮ ﻣﺴﺘﻘﯿﻢ ﻣﺪرج ﺑﻮاﺳﻄﺔ اﻻﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ ﺣﯿﺚ أن ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ﯾﻤﺜﻞ ﻓﺎﺻﻠﺔ ﻧﻘﻄﺔ ﻣﻦ‬ ‫اﻟﻤﺴﺘﻘﯿﻢ وﻛﻞ ﻧﻘﻄﺔ ﻣﻦ اﻟﻤﺴﺘﻘﯿﻢ ﺗﻤﺜﻞ ﻋﺪدا ﺣﻘﯿﻘﯿﺎ‬ ‫اﻟﻌﻤﻠﻴﺎت ﻓﻲ ‪R‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪدان اﻟﺤﻘﯿﻘﯿﺎن ‪ a‬و ﻻ ﻓﺈن :‬ ‫‪a+b = b+a‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ a‬ﻓﺈن :‬ ‫‪a+0 = 0+a = a‬‬ ‫·‬ ‫اﻟﻔﺮق ﺑﯿﻦ ‪ a‬و ‪ b‬ھﻮ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ d‬ﺣﯿﺚ :‬ ‫‪ a= d+b‬وﻧﻜﺘﺐ ‪d = a – b‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪدان اﻟﺤﻘﯿﻘﯿﺎن ‪ a‬و ‪ b‬ﻓﺈن :‬ ‫‪-(a+b) = -a – b‬‬
2. 2. ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪدان اﻟﺤﻘﯿﻘﯿﺎن ‪ a‬و ‪ b‬ﻓﺈن :‬ ‫‪axb=bxa‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﺗﻜﻦ اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ ‪a‬و ‪ b‬و ‪ c‬ﻓﺈن :‬ ‫‪a(b-c) = ab – ac‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ a‬ﻓﺈن :‬ ‫)‪a x (-1) = (-1) x a = (-a‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪدان اﻟﺤﻘﯿﻘﯿﺎن ‪ a‬و ‪ b‬ﻓﺈن :‬ ‫0 = ‪ ab‬ﯾﻌﻨﻲ 0 = ‪ a‬أو 0 = ‪b‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﺗﻜﻦ اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ ‪a‬و ‪ b‬و ‪ c‬ﻓﺈن :‬ ‫‪a + ( b+c) = ( a+ b) + c = a+ b +c‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ a‬ﻓﺈن :‬ ‫0 = )‪a + (-a‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﺗﻜﻦ اﻷﻋﺪاد اﻟﺤﻘﯿﻘﯿﺔ ‪a‬و ‪ b‬و ‪ c‬ﻓﺈن :‬ ‫‪a – ( b – c) = a – b + c‬‬ ‫‪a – ( b + c) = a – b – c‬‬ ‫·‬ ‫ﻛﻞ ﻋﺪد ﺣﻘﯿﻘﻲ ‪ a‬ﻣﺨﺎﻟﻒ ﻟﻠﺼﻔﺮ ﻟﮫ ﻣﻘﻠﻮب ‪1/a‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪد اﻟﺤﻘﯿﻘﻲ ‪ a‬ﻣﺨﺎﻟﻒ ﻟﻠﺼﻔﺮ ﻓﺈن :‬ ‫1 = ‪a x 1/a‬‬ ‫·‬ ‫‪M‬ﻧﻘﻄﺔ ﻣﻦ اﻟﻤﺴﺘﻘﯿﻢ اﻟﻤﺪرج )‪ (oi‬ﻓﺎﺻﻠﻨﮭﺎ ‪x‬اﻟﻘﯿﻤﺔ اﻟﻤﻄﻠﻘﺔ ﻟـ ‪x‬‬ ‫ھﻲ اﻟﺒﻌﺪ ‪|x| = OM : OM‬‬ ‫·‬ ‫‪ |x| = X‬إذا ﻛﺎن ‪ X‬ﻋﺪد ﻣﻮﺟﺒﺎ‬ ‫·‬ ‫‪ |x| = - X‬إذا ﻛﺎن ‪ X‬ﻋﺪد ا ﺳﺎﻟﺒﺎ‬ ‫·‬ ‫0 = |‪ |x‬ﯾﻌﻨﻲ 0 = ‪X‬‬ ‫·‬ ‫ﻣﮭﻤﺎ ﯾﻜﻦ اﻟﻌﺪدان اﻟﺤﻘﯿﻘﯿﺎن ‪ a‬و ‪ b‬ﻓﺈن :‬ ‫|‪|ab| = |a| .| b‬‬
3. 3. ‫اﻟﻘﻮى ﻓﻲ ‪R‬‬ ‫·‬ ‫إذا ﻛﺎن ‪ a‬و ﻻ ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ ﻣﺨﺎﻟﻔﯿﻦ ﻟﻠﺼﻔﺮ و ‪n‬و ‪ p‬ﻋﺪدﯾﻦ ﺻﺤﯿﺤﯿﻦ ﻓﺈن :‬ ‫‪(a x b ) = an x bn‬‬ ‫‪(an) = anp‬‬ ‫‪an x ap = an+p‬‬ ‫‪( a/b)² = an / bn‬‬ ‫اﻟﺘﺮﺗﻴﺐ واﻟﻤﻘﺎرﻧﺔ ﻓﻲ ‪R‬‬ ‫·‬ ‫ﻟﯿﻜﻦ ‪ a‬و ‪ b‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ‬ ‫0≤ ‪ a-b‬ﯾﻌﻨﻲ ‪a ≤b‬‬ ‫0≥ ‪ a-b‬ﯾﻌﻨﻲ ‪a ≥b‬‬ ‫·‬ ‫ﻟﺘﻜﻦ ‪ x‬و ‪ y‬و ‪ z‬أﻋﺪاد ﺣﻘﯿﻘﯿﺔ‬ ‫‪ a ≤ b‬ﯾﻌﻨﻲ ‪a + c ≤ b +c‬‬ ‫·‬ ‫إذا ﻛﺎن ‪ a‬و ‪ b‬و ‪ c‬و ‪ d‬أﻋﺪاد ﺣﻘﯿﻘﯿﺔ‬ ‫·‬ ‫‪ a ≤ b‬ﯾﻌﻨﻲ ‪a + c ≤ b +c‬‬ ‫·‬ ‫‪ c ≤ d a ≤ b‬ﯾﻌﻨﻲ ‪a + c ≤ b +d‬‬ ‫·‬ ‫ﻧﻌﺘﺒﺮ ‪ a‬و ‪ b‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ‬ ‫1- إذا ﻛﺎن ‪ c‬ﻋﺪدا ﻣﻮﺟﺒﺎ ﻗﻄﻌﺎ ﻓﺈن :‬ ‫‪ a ≤ b‬ﯾﻌﻨﻲ ‪a c ≤ b c‬‬ ‫2- إذا ﻛﺎن ‪ c‬ﻋﺪدا ﺳﺎﻟﺒﺎ ﻗﻄﻌﺎ ﻓﺈن :‬ ‫‪ a ≤ b‬ﯾﻌﻨﻲ ‪a c ≥ b c‬‬ ‫·‬ ‫إذا ﻛﺎن ‪a‬و ‪ b‬و ‪ c‬و ‪ d‬أﻋﺪاد ﺣﻘﯿﻘﯿﺔ ﻣﻮﺟﺒﺔ :‬ ‫‪ A ≤b‬و ‪ c≤d‬إذن ‪ac ≤bd‬‬ ‫·‬ ‫إذا ﻛﺎن ‪ a‬و ‪ b‬و ‪ c‬و ‪ d‬أﻋﺎد ﺣﻘﯿﻘﯿﺔ ﺳﺎﻟﺒﺔ :‬ ‫‪ A ≤b‬و ‪ c ≤d‬إذن ‪ac ≥bd‬‬ ‫·‬ ‫ﻧﻌﺘﺒﺮ ‪ x‬و ‪ y‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ ﻣﻮﺟﺒﯿﻦ‬ ‫‪ X ≤ y‬ﯾﻌﻨﻲ‬ ‫·‬ ‫²‪x² ≤y‬‬ ‫ﻧﻌﺘﺒﺮ ‪ x‬و ‪ y‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ ﺳﺎﻟﺒﯿﻦ‬ ‫‪ X ≤ y‬ﯾﻌﻨﻲ‬ ‫²‪x² ≥y‬‬
4. 4. ‫·‬ ‫ﻟﯿﻜﻦ ‪ x‬و ‪ y‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ‬ ‫|‪|x|≤|y‬‬ ‫²‪ x² ≤y‬ﯾﻌﻨﻲ‬ ‫‪ X‬و ‪ y‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ ﻣﺨﺎﻟﻔﯿﻦ ﻟﻠﺼﻔﺮ وﻟﮭﻤﺎ ﻧﻔﺲ اﻟﻌﻼﻣﺔ‬ ‫·‬ ‫‪ X ≤ y‬ﯾﻌﻨﻲ‬ ‫·‬ ‫إذا ﻛﺎن ‪ a‬و ‪ b‬و ‪ c‬و ‪ d‬أﻋﺪاد ﺣﻘﯿﻘﯿﺔ ﻓﺈن :‬ ‫‪(a+b)(c+d) = ac + ad + bc + bd‬‬ ‫‪(a+b)(c-d) = ac – ad + bc - bd‬‬ ‫‪(a-b)(c-d) = ac – ad - bc - bd‬‬ ‫‪(a-b)(c+d) = ac + ad - bc - bd‬‬ ‫·‬ ‫إذا ﻛﺎن ‪ a‬و ‪ b‬ﻋﺪدﯾﻦ ﺣﻘﯿﻘﯿﻦ :‬ ‫²‪( a +b) ² = a² + 2ab + b‬‬ ‫²‪(a -b) ² = a² - 2ab + b‬‬ ‫²‪( a + b) ( a – b)= a² - b‬‬ ‫·‬ ‫ﺣﺼﺮ ﻋﺪﺩ ﺣﻘﻴﻘﻲ‬ ‫ﺍﻟﻜﺘﺎﺑﺔ ‪b‬‬ ‫‪ a‬ﺃﻭ ‪b‬‬ ‫‪x‬‬ ‫‪x‬‬ ‫‪ a‬ﺗﺴﻤﻰ ﺣﺼﺮ ﻟﻠﻌﺪﺩ ‪. x‬‬ ‫ﺍﻟﻔﺮﻕ ‪ b – a‬ﻳﺴﻤﻰ ﻣﺪﻯ ﺍﻟﺤﺼﺮ‬ ‫·‬ ‫ﺣﺼﺮ ﻣﺠﻤﻮﻉ ﻋﺪﺩﻳﻦ :‬ ‫‪ a‬ﻭ ‪ b‬ﻭ ‪ c‬ﻭ ‪ d‬ﻭ ‪ x‬ﻭ ‪ y‬ﺃﻋﺪﺍﺩ ﺣﻘﻴﻘﻴﺔ.‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪b‬‬ ‫‪x‬‬ ‫ﻓﺈﻥ ‪b+d‬‬ ‫‪x+y‬‬ ‫·‬ ‫‪a‬ﻭ ‪d‬‬ ‫‪y‬‬ ‫‪a+ c‬‬ ‫ﺣﺼﺮ ﺟﺬﺍﺀ ﻋﺪﺩﻳﻦ ﻣﻮﺟﺒﻴﻦ‬ ‫‪c‬‬
5. 5. ‫‪ a‬ﻭ ‪ b‬ﻭ ‪ c‬ﻭ ‪ d‬ﻭ ‪ x‬ﻭ ‪ y‬ﺃﻋﺪﺍﺩ ﺣﻘﻴﻘﻴﺔ ﻣﻮﺟﺒﺔ‬ ‫ﺇﺫﺍ ﻛﺎﻥ ‪b‬‬ ‫‪x‬‬ ‫ﻓﺈﻥ : ‪bd‬‬ ‫‪xy‬‬ ‫‪b‬‬ ‫‪ac‬‬ ‫ﺍﻟﻤﺠﺎﻻﺕ ﺍﻟﻤﺤﺪﻭﺩﺓ ﻓﻲ‬ ‫·‬ ‫‪b‬‬ ‫‪a‬ﻭ ‪d‬‬ ‫‪y‬‬ ‫‪c‬‬ ‫‪a‬‬ ‫‪x‬‬ ‫‪x‬‬ ‫[ ‪]a ; b‬‬ ‫‪a‬‬ ‫‪a‬‬ ‫‪b‬‬ ‫‪x‬‬ ‫‪b‬‬ ‫‪a>x‬‬ ‫]‪]a;b‬‬ ‫ﺍﻟﻤﺠﺎﻻﺕ ﻏﻴﺮ ﺍﻟﻤﺤﺪﻭﺩﺓ ﻓﻲ‬ ‫·‬ ‫‪X≥a‬‬ ‫[‬ ‫+ ; ‪[a‬‬ ‫‪X‬‬ ‫[‬ ‫+ ; ‪]a‬‬ ‫‪a‬‬ ‫‪X≤a‬‬ ‫‪a‬‬ ‫]‬‫‪X‬‬ ‫·‬ ‫]‬‫ﺍﻟﻤﺠﺎﻻﺕ ﺍﻟﺨﺎﺻﺔ‬ ‫|‪ |x‬ﺗﺴﻤﻰ ﺍﻟﻤﺠﺎﻝ‬ ‫|‪ |x‬ﺗﺴﻤﻰ ﺍﻟﻤﺠﺎﻝ‬ ‫|‪ |x‬ﻫﻲ [‬ ‫+ ; ‪[a‬‬ ‫-]‬ ‫|‪ |x‬ﻫﻲ [‬ ‫+ ; ‪]a‬‬ ‫-]‬