ملزمة تكييف 246 صفحة صادق حسين (2) PDF

‫ﺍﳌﻘﺪﻣﺔ‬ ‫ﻭﻇﻔﻨﺎ ﺍﳌﻠﺰﻣﺔ ﻟﺘﻘﺪﻳﻢ ﺍﳊﻠﻮﻝ ﺍﻟﺘﻔﺼﻴﻠﻴﺔ ﻟﻠﻄﺎﻟﺐ ﻭﻟﻴﺲ ﺍﳊﻠﻮﻝ ﺍﻟﺘﻘﻠﻴﺪﻳﺔ‪ ،‬ﻟﺬﻟﻚ ﺳﻴﺠﺪ ﺍﻟﻄﺎﻟﺐ‬ ‫ﺍﳌﻼﺣﻈﺎﺕ ﻭﺍﻟﺘﻔﺼﻴﻞ ﺿﻤﻦ ﺍﳊﻞ ﻛﺬﻟﻚ ﺳﻨﺘﻮﻗﻒ ﻟﻠﺘﻮﺿﻴﺢ ﻋﻨﺪ ﻣﻮﺍﺟﻬﺔ ﺍﳊﺎﻻﺕ ﺍﳋﺎﺻﺔ‬ ‫‪ ‬ﺇﺳﺘﺨﺪﻣﻨﺎ ﺍﻟﻠﻮﻥ ﺍﻟﺬﻫﱯ ﻟﺒﻴﺎﻥ ﺍﻻﺧﺘﺼﺎﺭ ﺍﻟﺘﺎﻡ ﺧﻼﻝ ﺍﳊﻞ‬ ‫‪ ‬ﺇﺳﺘﺨﺪﻣﻨﺎ ﺍﻟﻠﻮﻥ ﺍﻷﺧﻀﺮ ﻟﺒﻴﺎ...

‫ﺍﳌﻘﺪﻣﺔ‬ ‫ﻭﻇﻔﻨﺎ ﺍﳌﻠﺰﻣﺔ ﻟﺘﻘﺪﻳﻢ ﺍﳊﻠﻮﻝ ﺍﻟﺘﻔﺼﻴﻠﻴﺔ ﻟﻠﻄﺎﻟﺐ ﻭﻟﻴﺲ ﺍﳊﻠﻮﻝ ﺍﻟﺘﻘﻠﻴﺪﻳﺔ‪ ،‬ﻟﺬﻟﻚ ﺳﻴﺠﺪ ﺍﻟﻄﺎﻟﺐ‬ ‫ﺍﳌﻼﺣﻈﺎﺕ ﻭﺍﻟﺘﻔﺼﻴﻞ ﺿﻤﻦ ﺍﳊﻞ ﻛﺬﻟﻚ ﺳﻨﺘﻮﻗﻒ ﻟﻠﺘﻮﺿﻴﺢ ﻋﻨﺪ ﻣﻮﺍﺟﻬﺔ ﺍﳊﺎﻻﺕ ﺍﳋﺎﺻﺔ‬ ‫‪ ‬ﺇﺳﺘﺨﺪﻣﻨﺎ ﺍﻟﻠﻮﻥ ﺍﻟﺬﻫﱯ ﻟﺒﻴﺎﻥ ﺍﻻﺧﺘﺼﺎﺭ ﺍﻟﺘﺎﻡ ﺧﻼﻝ ﺍﳊﻞ‬ ‫‪ ‬ﺇﺳﺘﺨﺪﻣﻨﺎ ﺍﻟﻠﻮﻥ ﺍﻷﺧﻀﺮ ﻟﺒﻴﺎﻥ ﺇﺿﺎﻓﺔ ﻣﻘﺪﺍﺭ ﺑﺼﻮﺭﺓ ﻣﻌﻴﻨﺔ ﻟﺘﻮﺣﻴﺪ ﻣﻘﺎﻣﺎﺕ ﺃﻭ ﻟﻠﻀﺮﺏ ﺑﺎﻟﻌﺪﺩ ﺍﳌﺮﺍﻓﻖ ﺃﻭ‬ ‫ﻟﺘﻮﻓﲑ ﻣﻘﺪﺍﺭ ﺩﺍﺧﻞ ﺍﻟﺘﻜﺎﻣﻞ‬ ‫ﺍﳌﻠﺰﻣﺔ ﺗﻘﺪﻡ ﺣﻠﻮﻝ ﺍﻷﻣﺜﻠﺔ ﻭﺍﻟﺘﻤﺎﺭﻳﻦ ﻭﺍﻷﺳﺌﻠﺔ ﺍﻟﻮﺯﺍﺭﻳﺔ‪ ،‬ﺃﻏﻠﺐ ﺃﺳﺌﻠﺔ ﺍﻟﻜﺘﺎﺏ ﻭﺭﺩﺕ ﺿﻤﻦ ﺍﻷﺳﺌﻠﺔ‬ ‫ﺍﻟﻮﺯﺍﺭﻳﺔ ﻭﺭﻏﻢ ﺫﻟﻚ ﱂ ﻳﺘﻢ ﺍﻹﺷﺎﺭﺓ ﺇﱃ ﺫﻟﻚ‪ ،‬ﻷﻧﻪ ﺑﺎﳋﺘﺎﻡ ﻋﻠﻰ ﺍﻟﻄﺎﻟﺐ ﺍﻻﳌﺎﻡ ﲜﻤﻴﻊ ﺍﻷﻓﻜﺎﺭ ﺍﳌﻄﺮﻭﺣﺔ‬ ‫ﺿﻤﻦ ﺍﳌﻨﻬﺞ ﺑﻐﺾ ﺍﻟﻨﻈﺮ ﻋﻦ ﺗﻀﻤﻴﻨﻬﺎ ﰲ ﺍﻷﺳﺌﻠﺔ ﺍﻟﻮﺯﺍﺭﻳﺔ ﺃﻡ ﻻ‪.‬‬ ‫ ﺳﻴﺠﺪ ﺍﻟﻄﺎﻟﺐ ﺇﻥ ﺍﻷﺳﺌﻠﺔ ﻣﺮﺗﺒﺔ ﻭﻣﺼﻨﻔﺔ ﺣﺴﺐ ﻧﻮﻉ ﺍﻟﺴﺆﺍﻝ ﻭﻟﻴﺲ ﺣﺴﺐ ﺗﺴﻠﺴﻠﻪ ﰲ ﺍﻟﻜﺘﺎﺏ‪ ،‬ﻭﻫﺬﺍ‬ ‫ﻳﻔﻴﺪ ﺍﻟﻄﺎﻟﺐ ﺑﻮﺟﻮﺩ ﺍﻷﺳﺌﻠﺔ ﺍﳌﺘﺸﺎﲠﺔ ﰲ ﻣﻜﺎﻥ ﻭﺍﺣﺪ‪.‬‬ ‫ﻣﺪﺭﺱ ﺍﳌﺎﺩﺓ‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫ﻣﺎﺫﺍ ﻳﻌﲏ ﺗﺴﺎﻭﻱ ﻋﺪﺩﻳﻦ ﻣﺮﻛﺒﲔ‬ ‫ﺍﳌﻮﺿﻮﻉ ﺍﻻﻭﻝ‪ :‬ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫ ‬ ‫ﻤجموﻋﺔ اﻻﻋداد اﻟمر ﺔ )‪ (ℂ‬ﺘضم أﻋداداً ﺠدﯿدة ﻟم سبق ﻟنﺎ دراﺴتﻬﺎ‬ ‫اﻟتخیﻠﻲ = اﻟتخیﻠﻲ‬ ‫اﻟحق ﻘﻲ = اﻟحق ﻘﻲ‬ ‫ﺴﺎ ﻘﺎً‪ ،‬وﻋﻠ ﻪ ﺨﻼل ﻫذا اﻟﻔصﻞ ﺴنﻌرف اﻟﻌدد اﻟمر ب وﻨﻌرف اﻟﻌمﻠ ﺎت‬ ‫ﻃﺮﻳﻘﺔ ﲨﻊ ﺍﻷﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ‬ ‫اﻟحسﺎﺒ ﺔ ﻋﻠ ﻪ‪ ،‬وﻨتﻌﻠم طر ﻘﺔ رﻓﻌﻪ اﻟﻰ أس وطر ﻘﺔ إ جﺎد ﺠذرﻩ‪ ،‬إﺴتﻌد‬ ‫ ‬ ‫اﻟتخیﻠﻲ ‪ +‬اﻟتخیﻠﻲ‬ ‫اﻟحق ﻘﻲ ‪ +‬اﻟحق ﻘﻲ‬ ‫ﻤﻌنﺎ ﻟتوﻀ ﺢ ﻫذﻩ اﻟمﻔﺎه م اﻟجدﯿدة‪.‬‬ ‫ﻃﺮﻳﻘﺔ ﺿﺮﺏ ﺍﻷﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ‬ ‫‪ ‬اﻟﻌدد اﻟمر ب ﻫو ﻋدد ﻤتر ب ﻤن ﺠزﺌین‪ :‬ﺠزء ﺤق ﻘﻲ وﺠزء ﺘخیﻠﻲ‬ ‫ﻨضرب اﻟﻌدد اﻷول ﺎﻟﻌدد اﻟثﺎﻨﻲ ﺤسب ﺨﺎﺼ ﺔ اﻟتوز ﻊ‬ ‫ﻤضرو ﺎً ﺒـ𝒊𝒊 ﺘر ط ﺒینﻬمﺎ ﻋﻼﻤﺔ اﻟجمﻊ‬ ‫ﻧﻈﲑ ﻭﻣﺮﺍﻓﻖ ﺍﻟﻌﺪﺩ ﺍﳌﺮﻛﺐ‬ ‫𝒊𝒊 ﻋدد ﺤق ﻘﻲ ‪ +‬ﻋدد ﺤق ﻘﻲ = اﻟﻌدد اﻟمر ب‬ ‫① اﻟنظیر اﻟجمﻌﻲ ﻟﻠﻌدد اﻟمر ب 𝒃𝒃𝒃𝒃 ‪ ، 𝒄𝒄 = 𝒂𝒂 +‬ﻫو ﻋكس إﺸﺎرة‬ ‫ ‬ ‫ ‬ ‫ ‬ ‫اﻟجزء اﻟﺤﻘﯿﻘﻲ‬ ‫اﻟﺘﺨﯿﻠﻲ‬ ‫اﻟﺠﺰء‬ ‫‪ ‬ﻻﺤظ إن اﻟجزء اﻟتخیﻠﻲ ﻫو ﻋدد ﺤق ﻘﻲ ﻤضرو ﺎ ﺎﻟوﺤدة اﻟتخیﻠ ﺔ اﻟﻌدد‬ ‫ً‬ ‫𝒃𝒃𝒃𝒃 ‪⟹ −𝒄𝒄 = −𝒂𝒂 −‬‬ ‫𝒊𝒊 ‪ ،‬ﺤیث‬ ‫② اﻟنظیر اﻟضر ﻲ ﻟﻠﻌدد اﻟمر ب 𝒃𝒃𝒃𝒃 ‪ ، 𝒄𝒄 = 𝒂𝒂 +‬ﻫو ﻤﻘﻠوب اﻟﻌدد‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‪𝒊𝒊 = √−‬‬ ‫= ⟹‬ ‫𝒃𝒃𝒃𝒃 ‪𝒄𝒄 𝒂𝒂 +‬‬ ‫‪ ‬ﺘسمﻰ اﻟص ﻐﺔ اﻟتﺎﻟ ﺔ اﻟص ﻐﺔ اﻟجبر ﺔ أو اﻟص ﻐﺔ اﻟﻌﺎد ﺔ ﻟﻠﻌدد‬ ‫③ ﻤراﻓق اﻟﻌدد اﻟمر ب 𝒃𝒃𝒃𝒃 ‪ ، 𝒄𝒄 = 𝒂𝒂 +‬ﻨﻌكس إﺸﺎرة اﻟجزء اﻟتخیﻠﻲ‬ ‫اﻟمر ب‬ ‫ﻓﻘط‬ ‫𝒃𝒃𝒃𝒃 ‪𝒄𝒄 = 𝒂𝒂 +‬‬ ‫𝒃𝒃𝒃𝒃 ‪⟹ 𝒄𝒄 = 𝒂𝒂 −‬‬ ‫ﻣﻼﺣﻈﺎﺕ‬ ‫④ ﻻ ﺘوﺠد ف ﺔ ﻟﻘسمﺔ اﻻﻋداد اﻟمر ﺔ‪ ،‬وﻋﻠ ﻪ ﻟتحو ﻞ اﻟكسور اﻟﻰ‬ ‫① جب أن ﻨﻌﻠم إن‬ ‫اﻟص ﻐﺔ اﻟﻌﺎد ﺔ ﻟﻠﻌدد اﻟمر ب‪ ،‬ﻨضرب سط وﻤﻘﺎم اﻟﻌدد اﻟمر ب ﺎﻟﻌدد‬ ‫ ‬ ‫𝟏𝟏‪𝒊𝒊𝟐𝟐 = −‬‬ ‫𝒊𝒊‪𝒊𝒊𝟑𝟑 = −‬‬ ‫𝟏𝟏 = 𝟒𝟒𝒊𝒊‬ ‫اﻟمراﻓق ﻟﻠمﻘﺎم‬ ‫و صورة ﻋﺎﻤﺔ‪ :‬مكن اﻟتﻌﺎﻤﻞ ﻤﻊ ﻗوى 𝒊𝒊 ﺤسب اﻟﻘﺎﻋدة‬ ‫⑤ اﻟﻌددان اﻟمتراﻓﻘﺎن‪ :‬اﻟﻌدد اﻻول = ﻤراﻓق اﻟﻌدد اﻻﺨر‬ ‫𝟑𝟑 ‪𝒊𝒊𝟒𝟒𝟒𝟒+𝒓𝒓 = 𝒊𝒊𝒓𝒓 , 𝒓𝒓 = 𝟎𝟎, 𝟏𝟏, 𝟐𝟐,‬‬ ‫وﻫنﺎ 𝟒𝟒𝟒𝟒 ﻨﻘصد ﺒﻬﺎ أﺤد ﻤضﺎﻋﻔﺎت اﻟﻌدد ‪ ، 4‬أي إن‬ ‫ﺧﻮﺍﺹ ﻣﺮﺍﻓﻖ ﺍﻟﻌﺪﺩ ﺍﳌﺮﻛﺐ‬ ‫اﻟﻌدد 𝟒𝟒 ‬ ‫اﻟ ﺎﻗﻲ ‪ +‬ﻤضﺎﻋﻒ‬ ‫ ‬ ‫ اﻟ ﺎﻗﻲ ‬ ‫① اﺸﺎرة اﻟمراﻓق ﺘتوزع ﻋﻠﻰ ﻋمﻠ ﺎت اﻟجمﻊ واﻟطرح واﻟضرب واﻟﻘسمﺔ‬ ‫𝒊𝒊‬ ‫𝒊𝒊 =‬ ‫ ‬ ‫ = ) 𝟏𝟏𝒄𝒄 ‪( 𝒄𝒄 +‬‬ ‫𝒄𝒄 ‪𝒄𝒄𝟏𝟏 +‬‬‫𝟐𝟐 ‬ ‫▼ ﻤضﺎﻋﻔﺎت اﻟﻌدد ‪ 4‬ﺘشمﻞ ﻤضﺎﻋﻔﺎت ﻤضﺎﻋﻔﺎﺘﻬﺎ‪ ،‬أي إﻨﻬﺎ ﺘشمﻞ‬ ‫𝟏𝟏 ⎧‬ ‫ ⎪‬ ‫ = ) 𝟏𝟏𝒄𝒄 ‪( 𝒄𝒄𝟏𝟏 −‬‬ ‫𝒄𝒄 ‪𝒄𝒄𝟏𝟏 −‬‬‫𝟐𝟐 ‬ ‫ﻤضﺎﻋﻔﺎت اﻻﻋداد 𝟖𝟖 و 𝟏𝟏𝟏𝟏 و 𝟏𝟏𝟏𝟏 و‪...‬اﻟﺦ‬ ‫ ⟹‬ ‫ = ) 𝟐𝟐𝒄𝒄 ∙ 𝟏𝟏𝒄𝒄 (‬ ‫ ∙ 𝟏𝟏𝒄𝒄‬ ‫𝟐𝟐𝒄𝒄‬ ‫② ﯿرﻤز ﻟمجموﻋﺔ اﻷﻋداد اﻟمر ﺔ ﺎﻟرﻤز ‪ ،ℂ‬وﻫﻲ أوﺴﻊ ﻤن ﻤجموﻋﺔ‬ ‫ ⎨‬ ‫𝒄𝒄‬ ‫𝒄𝒄 = 𝟏𝟏 ⎪‬ ‫𝟏𝟏 ‬ ‫اﻷﻋداد اﻟحق ق ﺔ‪ ،‬أي إن ‪𝑹𝑹 ⊆ ℂ‬‬ ‫ ‬ ‫𝟐𝟐𝒄𝒄‬ ‫𝟐𝟐𝒄𝒄 ⎩‬ ‫② اﻟمراﻓق ﻟﻠمراﻓق = اﻟﻌدد اﻷﺼﻠﻲ‬ ‫ﻣﻼﺣﻈﺎﺕ ﻣﻬﻤﺔ‬ ‫𝒄𝒄 = ) 𝒄𝒄 (‬ ‫① ﻓﻲ ﺤﺎﻟﺔ اﻟﻘوة اﻟسﺎﻟ ﺔ ﻨستخدم اﻟص ﻐﺔ‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟏𝟏‬ ‫ اﻟتخیﻠﻲ ‪ +‬اﻟحق ﻘﻲ = ﻤراﻓﻘﻪ × اﻟﻌدد ③‬ ‫𝒏𝒏 = 𝒏𝒏‪𝒊𝒊−‬‬ ‫𝒊𝒊‬ ‫𝟐𝟐𝒃𝒃 ‪𝒄𝒄 ∙ 𝒄𝒄 = 𝒂𝒂𝟐𝟐 +‬‬ ‫② ﺨﻼل ﻫذا اﻟﻔصﻞ‪ :‬مكن إ جﺎد ق مﺔ اﻟجذر اﻟتر ﻌﻲ ﻟﻠﻌدد اﻟسﺎﻟب‬ ‫④ ﻞ ﻋدد ﺤق ﻘﻲ = ﻤراﻓﻘﻪ‬ ‫ﺤسب اﻟﻘﺎﻋدة‬ ‫𝑹𝑹 ∈ 𝒄𝒄 ∀ ‪𝒄𝒄 = 𝒄𝒄 ,‬‬ ‫𝒊𝒊𝒂𝒂√ = 𝒂𝒂‪√−‬‬ ‫ﺃﺳﺌﻠﺔ "ﺿﻊ ﺑﺎﻟﺼﻴﻐﺔ ﺍﻟﻌﺎﺩﻳﺔ ‪ /‬ﺑﺴﻂ ‪ /‬ﺍﺛﺒﺖ ﺍﻥ"‬ ‫③ ﺨﻼل ﻫذا اﻟﻔصﻞ‪ :‬أي ﻋدد ﺤق ﻘﻲ مكن تﺎﺒتﻪ حﺎﺼﻞ ﺠمﻊ‬ ‫ﻤر ﻌین‪ ،‬مكن ﺘحﻠیﻠﻪ ﻀمن اﻷﻋداد اﻟمر ﺔ ﺤسب اﻟص ﻐﺔ اﻟتﺎﻟ ﺔ‪:‬‬ ‫اﻤثﻠﺔ ص‪ 6‬اﻤثﻠﺔ ﺤول ﺤسﺎب ﻗوى 𝒊𝒊‪:‬‬ ‫)𝒚𝒚𝒚𝒚 ‪𝒙𝒙𝟐𝟐 + 𝒚𝒚𝟐𝟐 = (𝒙𝒙 + 𝒚𝒚𝒚𝒚)(𝒙𝒙 −‬‬ ‫𝟏𝟏‪𝒊𝒊𝟐𝟐 = −‬‬ ‫④ ﻟحﻞ اﻟمﻌﺎدﻻت و جﺎد ق م 𝒙𝒙 و 𝒚𝒚 جب تﺎ ﺔ اﻟطرﻓین ﺎﻟص ﻐﺔ‬ ‫𝒊𝒊‪𝒊𝒊𝟑𝟑 = 𝒊𝒊𝟐𝟐 ∙ 𝒊𝒊 = −𝟏𝟏 ∙ 𝒊𝒊 = −‬‬ ‫𝟏𝟏 = )𝟏𝟏‪𝒊𝒊𝟒𝟒 = 𝒊𝒊𝟐𝟐 ∙ 𝒊𝒊𝟐𝟐 = (−𝟏𝟏)(−‬‬ ‫اﻟﻌﺎد ﺔ ﻟﻠﻌدد اﻟمر ب‬ ‫𝒊𝒊‪𝒊𝒊𝟐𝟐𝟐𝟐 = 𝒊𝒊𝟐𝟐𝟐𝟐+𝟑𝟑 = 𝒊𝒊𝟐𝟐𝟐𝟐 ∙ 𝒊𝒊𝟑𝟑 = 𝟏𝟏 ∙ 𝒊𝒊𝟑𝟑 = 𝒊𝒊𝟑𝟑 = −‬‬ ‫𝒊𝒊 = 𝟏𝟏𝒊𝒊 ∙ 𝟏𝟏 = 𝟏𝟏𝒊𝒊 ∙ 𝟖𝟖𝟖𝟖𝒊𝒊 = 𝟏𝟏‪𝒊𝒊𝟖𝟖𝟖𝟖 = 𝒊𝒊𝟖𝟖𝟖𝟖+‬‬ ‫‪ 1‬ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍ ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫𝟎𝟎 = 𝟏𝟏 ‪= 𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏 + 𝟏𝟏 − 𝟐𝟐𝟐𝟐 −‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝒊𝒊‬ ‫= 𝟕𝟕‪𝒊𝒊−‬‬ ‫=‬ ‫=‬ ‫=‬ ‫=‬ ‫𝒊𝒊 = 𝟑𝟑‪= 𝒊𝒊𝟒𝟒−‬‬ ‫ﻤثﺎل ‪ -9-‬ﺠد اﻟنظیر اﻟضر ﻲ ﻟﻠﻌدد 𝟐𝟐𝟐𝟐 ‪ 𝒄𝒄 = 𝟐𝟐 −‬وﻀﻌﻪ ﺎﻟص ﻐﺔ‬ ‫𝟑𝟑𝒊𝒊 𝟑𝟑𝒊𝒊 ∙ 𝟏𝟏 𝟑𝟑𝒊𝒊 ∙ 𝟒𝟒𝒊𝒊 𝟑𝟑‪𝒊𝒊𝟕𝟕 𝒊𝒊𝟒𝟒+‬‬ ‫𝟏𝟏𝟏𝟏‪−‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝒊𝒊‬ ‫𝒊𝒊 اﻟﻌﺎد ﺔ ﻟﻠﻌدد اﻟمر ب‪.‬‬ ‫= 𝟑𝟑 𝟏𝟏𝟏𝟏 = 𝟑𝟑‪= 𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏+‬‬ ‫=‬ ‫𝟑𝟑‪= 𝒊𝒊𝟒𝟒−‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝒊𝒊 ∙ 𝒊𝒊‬ ‫𝟑𝟑𝒊𝒊 𝟑𝟑𝒊𝒊 ∙ 𝟏𝟏‬ ‫= اﻟنظیر اﻟضر ﻲ‬ ‫𝟏𝟏‬ ‫𝒊𝒊 =‬ ‫𝟐𝟐𝟐𝟐‪𝟐𝟐−‬‬ ‫𝟏𝟏‬ ‫𝟐𝟐𝟐𝟐 ‪𝟐𝟐 +‬‬ ‫𝟐𝟐𝟐𝟐 ‪𝟐𝟐 +‬‬ ‫𝟐𝟐𝟐𝟐 ‪𝟐𝟐 +‬‬ ‫ﻤثﺎل ‪ -1-‬اﻛتب ﻤﺎ ﯿﻠﻲ ﺄ سط ﺼورة‪:‬‬ ‫=‬ ‫ ‬ ‫= ‬ ‫=‬ ‫𝟏𝟏𝟏𝟏‬ ‫𝟎𝟎‪𝟏𝟏𝟏𝟏+‬‬ ‫𝟐𝟐𝟐𝟐 ‪𝟐𝟐 − 𝟐𝟐𝟐𝟐 𝟐𝟐 +‬‬ ‫)𝟐𝟐( ‪(𝟐𝟐) +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟒𝟒 ‪𝟒𝟒 +‬‬ ‫𝒊𝒊 = 𝒊𝒊 )𝒂𝒂‬ ‫𝟏𝟏 = )𝟏𝟏()𝟏𝟏( = 𝟎𝟎𝒊𝒊 ∙ 𝒊𝒊 =‬ ‫𝟏𝟏𝟏𝟏‬ ‫𝟏𝟏 𝟏𝟏 𝟐𝟐𝟐𝟐 𝟐𝟐 𝟐𝟐𝟐𝟐 ‪𝟐𝟐 +‬‬ ‫𝟏𝟏‪𝒃𝒃) 𝒊𝒊𝟓𝟓𝟓𝟓 = 𝒊𝒊𝟓𝟓𝟓𝟓+𝟐𝟐 = 𝒊𝒊𝟓𝟓𝟓𝟓 ∙ 𝒊𝒊𝟐𝟐 = (𝟏𝟏)(−𝟏𝟏) = −‬‬ ‫=‬ ‫𝒊𝒊 ‪= + = +‬‬ ‫𝟖𝟖‬ ‫𝟒𝟒 𝟒𝟒 𝟖𝟖 𝟖𝟖‬ ‫𝟗𝟗𝟗𝟗𝒊𝒊 ∙ )𝟏𝟏( = 𝟗𝟗𝟗𝟗𝒊𝒊 ∙ )𝟒𝟒𝟒𝟒(𝟑𝟑𝒊𝒊 = 𝟗𝟗𝟗𝟗‪𝒄𝒄) 𝒊𝒊𝟏𝟏𝟏𝟏𝟏𝟏+𝟗𝟗𝟗𝟗 = 𝒊𝒊𝟑𝟑(𝟒𝟒𝟒𝟒)+‬‬ ‫ﻤثﺎل ‪ -12-‬ﻀﻊ ﻼً ﻤمﺎ ﺄﺘﻲ ﺎﻟصورة 𝒃𝒃𝒃𝒃 ‪: 𝒂𝒂 +‬‬ ‫𝒊𝒊 = 𝒊𝒊)𝟏𝟏( = 𝟏𝟏𝒊𝒊 ∙ 𝟗𝟗𝟗𝟗𝒊𝒊 = 𝟏𝟏‪= 𝒊𝒊𝟗𝟗𝟗𝟗 = 𝒊𝒊𝟗𝟗𝟗𝟗+‬‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 +‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟏𝟏 + 𝟐𝟐𝟐𝟐 +‬‬ ‫𝟏𝟏𝟏𝟏‪−‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝒊𝒊‬ ‫)𝒂𝒂‬ ‫ =‬ ‫ ‬ ‫= ‬ ‫)𝒅𝒅‬ ‫𝒊𝒊‬ ‫=‬ ‫𝟏𝟏𝟏𝟏‬ ‫=‬ ‫𝟏𝟏‪𝟏𝟏𝟏𝟏+‬‬ ‫=‬ ‫𝟏𝟏𝒊𝒊 ∙ 𝟏𝟏𝟏𝟏‬ ‫=‬ ‫=‬ ‫𝟏𝟏‪= 𝒊𝒊𝟒𝟒−‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝟏𝟏 − 𝒊𝒊 𝟏𝟏 +‬‬ ‫)𝟏𝟏( ‪(𝟏𝟏) +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝟏𝟏‬ ‫∙‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝒊𝒊𝟐𝟐 𝟏𝟏 ‪𝟏𝟏 + 𝟐𝟐𝟐𝟐 −‬‬ ‫𝒊𝒊‪= 𝒊𝒊𝟑𝟑 = −‬‬ ‫=‬ ‫=‬ ‫𝒊𝒊 ‪= 𝒊𝒊 = 𝟎𝟎 +‬‬ ‫ﻤثﺎل ‪ -2-‬اﻛتب اﻻﻋداد اﻻﺘ ﺔ ﻋﻠﻰ ﺼورة 𝒃𝒃𝒃𝒃 ‪: 𝒂𝒂 +‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫𝟐𝟐‬ ‫𝒊𝒊 ‪𝟐𝟐 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟐𝟐 − 𝒊𝒊 𝟑𝟑 −‬‬ ‫𝟎𝟎𝟎𝟎 ‪𝒂𝒂) − 𝟓𝟓 = −𝟓𝟓 +‬‬ ‫)𝒃𝒃‬ ‫ =‬ ‫ ‬ ‫ ‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 + 𝟒𝟒𝟒𝟒 𝟑𝟑 −‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪𝒃𝒃) √−𝟏𝟏𝟏𝟏𝟏𝟏 = √𝟏𝟏𝟏𝟏𝟏𝟏 √−𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟎𝟎 +‬‬ ‫𝟒𝟒 ‪𝟔𝟔 − 𝟖𝟖𝟖𝟖 − 𝟑𝟑𝟑𝟑 + 𝟒𝟒𝒊𝒊𝟐𝟐 𝟔𝟔 − 𝟏𝟏𝟏𝟏𝟏𝟏 −‬‬ ‫=‬ ‫=‬ ‫𝒊𝒊𝟑𝟑√ ‪𝒄𝒄) − 𝟏𝟏 − √−𝟑𝟑 = −𝟏𝟏 −‬‬ ‫𝟐𝟐)𝟒𝟒( ‪(𝟑𝟑)𝟐𝟐 +‬‬ ‫𝟏𝟏𝟏𝟏 ‪𝟗𝟗 +‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪𝟐𝟐 −‬‬ ‫𝟐𝟐‬ ‫𝟏𝟏𝟏𝟏‬ ‫𝟓𝟓 𝟏𝟏 𝒊𝒊𝟐𝟐𝟐𝟐√ 𝟏𝟏 𝒊𝒊𝟐𝟐𝟐𝟐√ ‪𝟏𝟏 + √−𝟐𝟐𝟐𝟐 𝟏𝟏 +‬‬ ‫=‬ ‫=‬ ‫‪−‬‬ ‫𝒊𝒊‬ ‫)𝒅𝒅‬ ‫=‬ ‫‪= +‬‬ ‫𝒊𝒊 ‪= +‬‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐‬ ‫𝟒𝟒‬ ‫𝟒𝟒‬ ‫𝟒𝟒‬ ‫𝟒𝟒‬ ‫𝟒𝟒 𝟒𝟒‬ ‫𝟐𝟐𝟐𝟐 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝟐𝟐𝟐𝟐 −𝟐𝟐 −‬‬ ‫ﻤثﺎل ‪ -4-‬ﺠد ﻤجموع اﻟﻌددﯿن اﻟمر بین ﻓﻲ ﻞ ﻤمﺎ ﺄﺘﻲ‪:‬‬ ‫)𝒄𝒄‬ ‫ =‬ ‫ ‬ ‫ ‬ ‫𝒊𝒊 ‪−𝟐𝟐 +‬‬ ‫𝒊𝒊 ‪−𝟐𝟐 + 𝒊𝒊 −𝟐𝟐 −‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝒂𝒂) 𝟑𝟑 + 𝟒𝟒√𝟐𝟐𝒊𝒊 , 𝟓𝟓 −‬‬ ‫𝟐𝟐𝒊𝒊𝟐𝟐 ‪−𝟐𝟐 − 𝒊𝒊 − 𝟒𝟒𝟒𝟒 −‬‬ ‫=‬ ‫𝒊𝒊 𝟐𝟐√𝟐𝟐 ‪𝟑𝟑 + 𝟒𝟒√𝟐𝟐𝒊𝒊 + 𝟓𝟓 − 𝟐𝟐√𝟐𝟐𝒊𝒊 = (𝟑𝟑 + 𝟓𝟓) + 𝟒𝟒√𝟐𝟐 −‬‬ ‫𝟐𝟐)𝟏𝟏( ‪(−𝟐𝟐)𝟐𝟐 +‬‬ ‫𝒊𝒊𝟓𝟓‪−𝟐𝟐 − 𝒊𝒊 − 𝟒𝟒𝟒𝟒 + 𝟐𝟐 −‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪= 𝟖𝟖 +‬‬ ‫=‬ ‫=‬ ‫𝒊𝒊‪= −‬‬ ‫𝟓𝟓𝟓𝟓 ‪𝒃𝒃) 𝟑𝟑 , 𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝟒𝟒 +‬‬ ‫𝟓𝟓‬ ‫𝒊𝒊 ‪= 𝟎𝟎 −‬‬ ‫𝟓𝟓𝟓𝟓 ‪𝟑𝟑 + 𝟐𝟐 − 𝟓𝟓𝟓𝟓 = 𝟓𝟓 −‬‬ ‫𝟑𝟑𝟑𝟑 ‪𝒄𝒄) 𝟏𝟏 − 𝒊𝒊 ,‬‬ ‫𝟐𝟐𝟐𝟐 ‪ 𝟏𝟏 − 𝒊𝒊 + 𝟑𝟑𝟑𝟑 = 𝟏𝟏 + (−𝟏𝟏 + 𝟑𝟑)𝒊𝒊 = 𝟏𝟏 +‬ﺘمر ن ‪ -1-‬ﻀﻊ ﻼً ﻤمﺎ ﺄﺘﻲ ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ ﻟﻠﻌدد اﻟمر ب‪:‬‬ ‫𝒊𝒊 ‪𝒊𝒊𝟓𝟓 = 𝒊𝒊𝟒𝟒+𝟏𝟏 = 𝒊𝒊𝟏𝟏 = 𝒊𝒊 = 𝟎𝟎 +‬‬ ‫ﻤثﺎل ‪ -5-‬ﺠد ﻨﺎﺘﺞ‪(𝟕𝟕 − 𝟏𝟏𝟏𝟏𝟏𝟏) − (𝟗𝟗 + 𝟒𝟒𝟒𝟒) :‬‬ ‫𝟎𝟎𝟎𝟎 ‪𝒊𝒊𝟔𝟔 = 𝒊𝒊𝟒𝟒+𝟐𝟐 = 𝒊𝒊𝟐𝟐 = −𝟏𝟏 = −𝟏𝟏 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪(𝟕𝟕 − 𝟏𝟏𝟏𝟏𝟏𝟏) − (𝟗𝟗 + 𝟒𝟒𝟒𝟒) = 𝟕𝟕 − 𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟗𝟗 −‬‬ ‫𝟎𝟎𝟎𝟎 ‪𝒊𝒊𝟏𝟏𝟏𝟏𝟏𝟏 = 𝒊𝒊𝟏𝟏𝟏𝟏𝟏𝟏+𝟎𝟎 = 𝒊𝒊𝟎𝟎 = 𝟏𝟏 = 𝟏𝟏 +‬‬ ‫𝒊𝒊)𝟒𝟒 ‪= (𝟕𝟕 − 𝟗𝟗) + (−𝟏𝟏𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝒊𝒊𝟗𝟗𝟗𝟗𝟗𝟗 = 𝒊𝒊𝟗𝟗𝟗𝟗𝟗𝟗+𝟑𝟑 = 𝒊𝒊𝟑𝟑 = −𝒊𝒊 = 𝟎𝟎 −‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= −𝟐𝟐 −‬‬ ‫} ⋯ ‪𝒊𝒊𝟒𝟒𝟒𝟒+𝟏𝟏 , ∀𝒏𝒏 ∈ 𝒘𝒘 = {𝟎𝟎, 𝟏𝟏, 𝟐𝟐, 𝟑𝟑, 𝟒𝟒,‬‬ ‫𝒊𝒊 ‪𝒊𝒊𝟒𝟒𝟒𝟒+𝟏𝟏 = 𝒊𝒊𝟏𝟏 = 𝒊𝒊 = 𝟎𝟎 +‬‬ ‫ﻤثـــﺎل ‪ -6-‬ﺤـــﻞ اﻟمﻌـــﺎدﻟـــﺔ‪ (𝟐𝟐 − 𝟒𝟒𝟒𝟒) + 𝒙𝒙 = −𝟓𝟓 + 𝒊𝒊 :‬ﺤیـــث‬ ‫)𝟐𝟐𝟐𝟐 ‪(𝟐𝟐 + 𝟑𝟑𝟑𝟑) 𝟐𝟐 + (𝟏𝟏𝟏𝟏 +‬‬ ‫‪𝒙𝒙 ∈ ℂ‬‬ ‫𝟐𝟐‬ ‫)𝟐𝟐𝟐𝟐 ‪= 𝟒𝟒 + 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟗𝟗𝒊𝒊 + (𝟏𝟏𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪(𝟐𝟐 − 𝟒𝟒𝟒𝟒) + 𝒙𝒙 = −𝟓𝟓 +‬‬ ‫)𝟐𝟐𝟐𝟐 ‪= (𝟒𝟒 + 𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟗𝟗) + (𝟏𝟏𝟏𝟏 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝒙𝒙 = −𝟓𝟓 + 𝒊𝒊 − 𝟐𝟐 +‬‬ ‫)𝟐𝟐𝟐𝟐 ‪= (−𝟓𝟓 + 𝟏𝟏𝟏𝟏𝟏𝟏) + (𝟏𝟏𝟏𝟏 +‬‬ ‫𝒙𝒙‬ ‫=‬ ‫𝒊𝒊)𝟒𝟒 ‪(−𝟓𝟓 − 𝟐𝟐) + (𝟏𝟏 +‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟕𝟕 +‬‬ ‫𝟓𝟓𝟓𝟓 ‪𝒙𝒙 = −𝟕𝟕 +‬‬ ‫𝟐𝟐‬ ‫𝒊𝒊𝟏𝟏𝟏𝟏 ‪(𝟏𝟏𝟏𝟏 + 𝟑𝟑𝟑𝟑)(𝟎𝟎 + 𝟔𝟔𝟔𝟔) = 𝟎𝟎 + 𝟔𝟔𝟔𝟔𝟔𝟔 + 𝟎𝟎 +‬‬ ‫ﻤثﺎل ‪ -7-‬ﺠد ﻨﺎﺘﺞ ﻼً ﻤمﺎ ﺄﺘﻲ‪:‬‬ ‫𝟔𝟔𝟔𝟔𝟔𝟔 ‪= −𝟏𝟏𝟏𝟏 +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐𝒊𝒊𝟏𝟏𝟏𝟏 ‪𝒂𝒂) (𝟐𝟐 − 𝟑𝟑𝟑𝟑)(𝟑𝟑 − 𝟓𝟓𝟓𝟓) = 𝟔𝟔 − 𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟗𝟗𝟗𝟗 +‬‬ ‫ 𝟐𝟐)𝒊𝒊 ‪(𝟏𝟏 + 𝒊𝒊)𝟒𝟒 − (𝟏𝟏 − 𝒊𝒊)𝟒𝟒 = (𝟏𝟏 + 𝒊𝒊)𝟐𝟐 − (𝟏𝟏 −‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟔𝟔 − 𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟏𝟏𝟏𝟏 = −𝟗𝟗 −‬‬ ‫𝟐𝟐 𝟐𝟐‬ ‫𝟐𝟐 𝟐𝟐‬ ‫ 𝒊𝒊 ‪= 𝟏𝟏 + 𝟐𝟐𝟐𝟐 + 𝒊𝒊 − 𝟏𝟏 − 𝟐𝟐𝟐𝟐 +‬‬ ‫𝟐𝟐)𝟒𝟒𝟒𝟒( ‪𝒃𝒃) (𝟑𝟑 + 𝟒𝟒𝟒𝟒) = (𝟑𝟑)𝟐𝟐 + 𝟐𝟐(𝟑𝟑)(𝟒𝟒𝟒𝟒) +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫]𝟏𝟏 ‪= [𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏] − [𝟏𝟏 − 𝟐𝟐𝟐𝟐 −‬‬ ‫𝟐𝟐‬ ‫𝟏𝟏𝟏𝟏 ‪= 𝟗𝟗 + 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟏𝟏𝟏𝟏𝒊𝒊𝟐𝟐 = 𝟗𝟗 + 𝟐𝟐𝟐𝟐𝟐𝟐 −‬‬ ‫𝟐𝟐𝒊𝒊𝟒𝟒 ‪= (𝟐𝟐𝟐𝟐)𝟐𝟐 − (−𝟐𝟐𝟐𝟐)𝟐𝟐 = 𝟒𝟒𝒊𝒊𝟐𝟐 −‬‬ ‫𝟐𝟐𝟐𝟐𝟐𝟐 ‪= −𝟕𝟕 +‬‬ ‫𝟎𝟎𝟎𝟎 ‪= −𝟒𝟒 + 𝟒𝟒 = 𝟎𝟎 = 𝟎𝟎 +‬‬ ‫𝒊𝒊 ‪𝒄𝒄) 𝒊𝒊(𝟏𝟏 + 𝒊𝒊) = 𝒊𝒊 + 𝒊𝒊𝟐𝟐 = 𝒊𝒊 − 𝟏𝟏 = −𝟏𝟏 +‬‬ ‫𝟐𝟐‬ ‫𝒊𝒊‪𝟏𝟏𝟏𝟏 + 𝒊𝒊 𝟏𝟏𝟏𝟏 + 𝒊𝒊 −‬‬ ‫𝒊𝒊 ‪−𝟏𝟏𝟏𝟏𝟏𝟏 −‬‬ ‫𝟏𝟏 ‪−𝟏𝟏𝟏𝟏𝟏𝟏 +‬‬ ‫𝟓𝟓‬ ‫𝟏𝟏𝟏𝟏‬ ‫=‬ ‫= ‬ ‫𝟐𝟐‬ ‫=‬ ‫‪𝒅𝒅) − (𝟒𝟒 + 𝟑𝟑𝟑𝟑) = −𝟏𝟏𝟏𝟏 −‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝒊𝒊‪−‬‬ ‫𝒊𝒊‪−‬‬ ‫𝟏𝟏‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟏𝟏 −‬‬ ‫𝟐𝟐)𝒊𝒊 ‪𝒆𝒆) (𝟏𝟏 + 𝒊𝒊)𝟐𝟐 + (𝟏𝟏 −‬‬ ‫) 𝟐𝟐𝒊𝒊 ‪= (𝟏𝟏 + 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 ) + (𝟏𝟏 − 𝟐𝟐𝟐𝟐 +‬‬ ‫‪2‬‬ ‫‪set by: Sadiq Hussain‬‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 +‬‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 + 𝟒𝟒𝟒𝟒 𝟑𝟑 + 𝟒𝟒𝟒𝟒 𝟑𝟑 +‬‬ ‫𝟐𝟐𝒊𝒊𝟏𝟏𝟏𝟏 ‪𝟗𝟗 + 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟏𝟏𝟏𝟏𝟏𝟏 +‬‬ ‫=‬ ‫ ‬ ‫‪ −‬‬ ‫ ‬ ‫ ‬ ‫=‬ ‫ ‬ ‫= ‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 − 𝟒𝟒𝟒𝟒 𝟑𝟑 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 + 𝟒𝟒𝟒𝟒 𝟑𝟑 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 − 𝟒𝟒𝟒𝟒 𝟑𝟑 − 𝟒𝟒𝟒𝟒 𝟑𝟑 +‬‬ ‫𝟐𝟐)𝟒𝟒( ‪(𝟑𝟑)𝟐𝟐 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 −‬‬ ‫𝟐𝟐𝟐𝟐𝟐𝟐 ‪𝟗𝟗 + 𝟐𝟐𝟒𝟒𝟒𝟒 − 𝟏𝟏𝟏𝟏 −𝟕𝟕 +‬‬ ‫=‬ ‫‪−‬‬ ‫=‬ ‫=‬ ‫𝟐𝟐)𝟒𝟒( ‪(𝟑𝟑)𝟐𝟐 + (𝟒𝟒)𝟐𝟐 (𝟑𝟑)𝟐𝟐 +‬‬ ‫𝟏𝟏𝟏𝟏 ‪𝟗𝟗 +‬‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 + 𝟒𝟒𝟒𝟒 𝟑𝟑 −‬‬ ‫𝟐𝟐𝟐𝟐 𝟕𝟕‪−‬‬ ‫=‬ ‫‪−‬‬ ‫=‬ ‫‪+‬‬ ‫𝒊𝒊‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐 𝟐𝟐𝟐𝟐‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 + 𝟒𝟒𝟒𝟒 − 𝟑𝟑 +‬‬ ‫𝟖𝟖‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝟑𝟑𝟑𝟑 ‪𝟐𝟐 −‬‬ ‫𝟐𝟐𝒊𝒊𝟑𝟑 ‪𝟐𝟐𝟐𝟐 −‬‬ ‫𝟑𝟑 ‪𝟐𝟐𝟐𝟐 +‬‬ ‫=‬ ‫=‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 = 𝒊𝒊‬ ‫=‬ ‫ ‬ ‫= ‬ ‫=‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟑𝟑𝟑𝟑 ‪𝟐𝟐 + 𝟑𝟑𝟑𝟑 𝟐𝟐 + 𝟑𝟑𝟑𝟑 𝟐𝟐 −‬‬ ‫)𝟑𝟑( ‪(𝟐𝟐) +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟗𝟗 ‪𝟒𝟒 +‬‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟏𝟏 − 𝒊𝒊)𝟐𝟐 (𝟏𝟏 +‬‬ ‫𝟑𝟑‬ ‫𝟐𝟐‬ ‫)𝒃𝒃‬ ‫‪+‬‬ ‫𝟐𝟐‪= −‬‬ ‫=‬ ‫‪+‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟏𝟏 − 𝒊𝒊)𝟐𝟐 (𝟏𝟏 +‬‬ ‫𝟑𝟑‬ ‫= 𝑳𝑳𝑳𝑳𝑳𝑳‬ ‫‪+‬‬ ‫𝟑𝟑 𝒊𝒊 ‪𝟑𝟑 +‬‬ ‫𝟑𝟑 𝒊𝒊 ‪𝟑𝟑 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟑𝟑 − 𝟑𝟑𝟑𝟑 + 𝒊𝒊 −‬‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫ ‬ ‫ = ‬ ‫ ‬ ‫ = ‬ ‫ ‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝟐𝟐)𝟏𝟏( ‪(𝟏𝟏)𝟐𝟐 +‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟏𝟏 − 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 𝟏𝟏 + 𝟐𝟐𝟐𝟐 +‬‬ ‫=‬ ‫‪+‬‬ ‫𝟑𝟑 𝟏𝟏 ‪𝟑𝟑 − 𝟑𝟑𝟑𝟑 + 𝒊𝒊 +‬‬ ‫𝟑𝟑 𝟐𝟐𝟐𝟐 ‪𝟒𝟒 −‬‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫ =‬ ‫ = ‬ ‫ ‬ ‫𝟏𝟏 ‪𝟏𝟏 − 𝟐𝟐𝟐𝟐 − 𝟏𝟏 𝟏𝟏 + 𝟐𝟐𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫𝟐𝟐‬ ‫=‬ ‫‪+‬‬ ‫𝟑𝟑 𝟐𝟐𝟐𝟐 𝟒𝟒‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫𝟑𝟑)𝒊𝒊 ‪= − = (𝟐𝟐 −‬‬ ‫𝒊𝒊 ‪−𝟐𝟐𝟐𝟐 𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝟐𝟐𝟐𝟐 𝟏𝟏 +‬‬ ‫𝟐𝟐 𝟐𝟐‬ ‫=‬ ‫ ‬ ‫‪ +‬‬ ‫ ‬ ‫ ‬ ‫)𝒊𝒊 ‪= (𝟐𝟐 − 𝒊𝒊)𝟐𝟐 (𝟐𝟐 −‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝟏𝟏 − 𝒊𝒊 𝟏𝟏 +‬‬ ‫𝟐𝟐𝒊𝒊𝟐𝟐 ‪−𝟐𝟐𝟐𝟐 +‬‬ ‫𝟐𝟐𝒊𝒊𝟐𝟐 ‪𝟐𝟐𝟐𝟐 +‬‬ ‫)𝒊𝒊 ‪= 𝟒𝟒 − 𝟒𝟒𝟒𝟒 + 𝒊𝒊𝟐𝟐 (𝟐𝟐 −‬‬ ‫=‬ ‫‪+‬‬ ‫)𝒊𝒊 ‪= (𝟒𝟒 − 𝟒𝟒𝟒𝟒 − 𝟏𝟏)(𝟐𝟐 −‬‬ ‫𝟐𝟐)𝟏𝟏( ‪(𝟏𝟏)𝟐𝟐 + (𝟏𝟏)𝟐𝟐 (𝟏𝟏)𝟐𝟐 +‬‬ ‫𝟐𝟐𝟐𝟐 ‪−𝟐𝟐 − 𝟐𝟐𝟐𝟐 −𝟐𝟐 +‬‬ ‫)𝒊𝒊 ‪= (𝟑𝟑 − 𝟒𝟒𝟒𝟒)(𝟐𝟐 −‬‬ ‫=‬ ‫‪+‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟔𝟔 − 𝟑𝟑𝟑𝟑 − 𝟖𝟖𝟖𝟖 + 𝟒𝟒𝒊𝒊𝟐𝟐 = 𝟐𝟐 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐 𝟐𝟐‪−‬‬ ‫𝟐𝟐 𝟐𝟐‪−‬‬ ‫𝟐𝟐𝒊𝒊𝟏𝟏𝟏𝟏 ‪𝟐𝟐 + 𝟑𝟑𝟑𝟑 𝟏𝟏 + 𝟒𝟒𝟒𝟒 𝟐𝟐 + 𝟖𝟖𝟖𝟖 + 𝟑𝟑𝟑𝟑 +‬‬ ‫=‬ ‫‪− 𝒊𝒊 +‬‬ ‫𝒊𝒊 ‪+‬‬ ‫×‬ ‫=‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝟒𝟒 +‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟒𝟒 + 𝒊𝒊 − 𝟒𝟒𝟒𝟒 −‬‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 = 𝟐𝟐‪= −𝟏𝟏 − 𝒊𝒊 − 𝟏𝟏 + 𝒊𝒊 = −‬‬ ‫𝟏𝟏𝟏𝟏 ‪𝟐𝟐 + 𝟏𝟏𝟏𝟏𝟏𝟏 −‬‬ ‫=‬ ‫𝟒𝟒 = 𝟑𝟑𝒊𝒊 ‪𝒄𝒄) (𝟏𝟏 − 𝒊𝒊) 𝟏𝟏 − 𝒊𝒊𝟐𝟐 𝟏𝟏 −‬‬ ‫𝟏𝟏 ‪𝟒𝟒 − 𝟑𝟑𝟑𝟑 +‬‬ ‫𝟐𝟐‬ ‫𝟑𝟑‬ ‫𝟑𝟑𝟑𝟑 ‪−𝟏𝟏𝟏𝟏 + 𝟏𝟏𝟏𝟏𝟏𝟏 𝟓𝟓 +‬‬ ‫ 𝒊𝒊 ‪𝑳𝑳𝑳𝑳𝑳𝑳 = (𝟏𝟏 − 𝒊𝒊) 𝟏𝟏 − 𝒊𝒊 𝟏𝟏 −‬‬ ‫=‬ ‫ ‬ ‫ ‬ ‫𝟑𝟑𝟑𝟑 ‪𝟓𝟓 −‬‬ ‫𝟑𝟑𝟑𝟑 ‪𝟓𝟓 +‬‬ ‫])𝒊𝒊‪= (𝟏𝟏 − 𝒊𝒊)[𝟏𝟏 − (−𝟏𝟏)][𝟏𝟏 − (−‬‬ ‫𝟐𝟐𝒊𝒊𝟑𝟑𝟑𝟑 ‪−𝟓𝟓𝟓𝟓 − 𝟑𝟑𝟑𝟑𝟑𝟑 + 𝟓𝟓𝟓𝟓𝟓𝟓 +‬‬ ‫)𝒊𝒊 ‪= (𝟏𝟏 − 𝒊𝒊)(𝟏𝟏 + 𝟏𝟏)(𝟏𝟏 +‬‬ ‫=‬ ‫𝟐𝟐)𝟑𝟑( ‪(𝟓𝟓)𝟐𝟐 +‬‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 = 𝟒𝟒 = )𝟐𝟐(𝟐𝟐 = )𝟏𝟏 ‪= 𝟐𝟐(𝟏𝟏 − 𝒊𝒊)(𝟏𝟏 + 𝒊𝒊) = 𝟐𝟐(𝟏𝟏 +‬‬ ‫𝟑𝟑𝟑𝟑 ‪−𝟓𝟓𝟎𝟎 − 𝟑𝟑𝟑𝟑𝟑𝟑 + 𝟓𝟓𝟓𝟓𝟓𝟓 −‬‬ ‫ﻤثﺎل ‪ -13-‬ﺤﻠﻞ ﻼً ﻤن اﻟﻌددﯿن 𝟏𝟏𝟏𝟏 ‪ 𝟓𝟓𝟓𝟓,‬اﻟﻰ ﺤﺎﺼﻞ ﻀرب ﻋﺎﻤﻠین‬ ‫=‬ ‫𝟗𝟗 ‪𝟐𝟐𝟐𝟐 +‬‬ ‫𝟐𝟐𝟐𝟐 𝟖𝟖𝟖𝟖‪−𝟖𝟖𝟖𝟖 + 𝟐𝟐𝟐𝟐𝟐𝟐 −‬‬ ‫ﻤن ﺼورة 𝒃𝒃𝒃𝒃 ‪ 𝒂𝒂 +‬ﺤیث 𝒃𝒃 ‪ 𝒂𝒂,‬ﻋددﯿن ﻨسبیین‬ ‫=‬ ‫=‬ ‫‪+‬‬ ‫𝒊𝒊‬ ‫𝟑𝟑𝟑𝟑‬ ‫𝟑𝟑𝟑𝟑‬ ‫𝟑𝟑𝟑𝟑‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫)𝟏𝟏𝟏𝟏 ‪𝟏𝟏𝟏𝟏 = 𝟗𝟗 + 𝟏𝟏 = (𝟑𝟑) + (𝟏𝟏) = (𝟑𝟑 − 𝟏𝟏𝟏𝟏)(𝟑𝟑 +‬‬ ‫𝟑𝟑)𝒊𝒊 ‪(𝟏𝟏 + 𝒊𝒊)𝟑𝟑 + (𝟏𝟏 −‬‬ ‫)𝒊𝒊 ‪= (𝟑𝟑 − 𝒊𝒊)(𝟑𝟑 +‬‬ ‫)𝒊𝒊 ‪= (𝟏𝟏 + 𝒊𝒊)𝟐𝟐 (𝟏𝟏 + 𝒊𝒊) + (𝟏𝟏 − 𝒊𝒊)𝟐𝟐 (𝟏𝟏 −‬‬ ‫)𝟐𝟐𝟐𝟐 ‪𝟓𝟓𝟓𝟓 = 𝟒𝟒𝟒𝟒 + 𝟒𝟒 = (𝟕𝟕)𝟐𝟐 + (𝟐𝟐)𝟐𝟐 = (𝟕𝟕 − 𝟐𝟐𝟐𝟐)(𝟕𝟕 +‬‬ ‫)𝒊𝒊 ‪= 𝟏𝟏 + 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 (𝟏𝟏 + 𝒊𝒊) + 𝟏𝟏 − 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 (𝟏𝟏 −‬‬ ‫)𝒊𝒊 ‪ = (𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏)(𝟏𝟏 + 𝒊𝒊) + (𝟏𝟏 − 𝟐𝟐𝟐𝟐 − 𝟏𝟏)(𝟏𝟏 −‬طر ﻘﺔ أﺨرى‪:‬‬ ‫)𝟑𝟑𝟑𝟑 ‪𝟏𝟏𝟏𝟏 = 𝟏𝟏 + 𝟗𝟗 = (𝟏𝟏)𝟐𝟐 + (𝟑𝟑)𝟐𝟐 = (𝟏𝟏 − 𝟑𝟑𝟑𝟑)(𝟏𝟏 +‬‬ ‫)𝒊𝒊 ‪= 𝟐𝟐𝟐𝟐(𝟏𝟏 + 𝒊𝒊) + (−𝟐𝟐𝟐𝟐)(𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫)𝟕𝟕𝟕𝟕 ‪𝟓𝟓𝟓𝟓 = 𝟒𝟒 + 𝟒𝟒𝟒𝟒 = (𝟐𝟐) + (𝟕𝟕) = (𝟐𝟐 − 𝟕𝟕𝟕𝟕)(𝟐𝟐 +‬‬ ‫=‬ ‫ 𝟐𝟐𝒊𝒊𝟐𝟐 ‪ 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝒊𝒊𝟐𝟐 + −𝟐𝟐𝟐𝟐 +‬‬ ‫𝟎𝟎𝟎𝟎 ‪= 𝟐𝟐𝟐𝟐 − 𝟐𝟐 − 𝟐𝟐𝟐𝟐 − 𝟐𝟐 = −𝟒𝟒 = −𝟒𝟒 +‬‬ ‫ﺘمر ن ‪ -4-‬ﺤﻠﻞ ﻼً ﻤن اﻻﻋداد 𝟖𝟖𝟖𝟖 ‪ 𝟐𝟐𝟐𝟐, 𝟏𝟏𝟏𝟏𝟏𝟏, 𝟒𝟒𝟒𝟒,‬اﻟﻰ ﺤﺎﺼﻞ‬ ‫ﺘمر ن ‪ -3-‬اﺜبت ان‪:‬‬ ‫ﻀرب ﻋﺎﻤﻠین ﻤن ﺼورة 𝒃𝒃𝒃𝒃 ‪ 𝒂𝒂 +‬ﺤیث 𝒃𝒃 ‪ 𝒂𝒂,‬ﻋددﯿن ﻨسبیین‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟖𝟖‬ ‫)𝟕𝟕𝟕𝟕 ‪𝟖𝟖𝟖𝟖 = 𝟑𝟑𝟑𝟑 + 𝟒𝟒𝟒𝟒 = (𝟔𝟔)𝟐𝟐 + (𝟕𝟕)𝟐𝟐 = (𝟔𝟔 − 𝟕𝟕𝟕𝟕)(𝟔𝟔 +‬‬ ‫)𝒂𝒂‬ ‫‪−‬‬ ‫=‬ ‫𝒊𝒊‬ ‫)𝒊𝒊 ‪(𝟐𝟐 −‬‬‫𝟐𝟐‬ ‫)𝒊𝒊 ‪(𝟐𝟐 +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‬ ‫)𝟒𝟒𝟒𝟒 ‪𝟒𝟒𝟒𝟒 = 𝟐𝟐𝟐𝟐 + 𝟏𝟏𝟏𝟏 = (𝟓𝟓)𝟐𝟐 + (𝟒𝟒)𝟐𝟐 = (𝟓𝟓 − 𝟒𝟒𝟒𝟒)(𝟓𝟓 +‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟐𝟐‬ ‫)𝟐𝟐( ‪𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟒𝟒 = (𝟏𝟏𝟏𝟏) +‬‬ ‫𝟐𝟐‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳‬ ‫=‬ ‫‪−‬‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟐𝟐 − 𝒊𝒊)𝟐𝟐 (𝟐𝟐 +‬‬ ‫)𝟐𝟐𝟐𝟐 ‪= (𝟏𝟏𝟏𝟏 − 𝟐𝟐𝟐𝟐)(𝟏𝟏𝟏𝟏 +‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫)𝟐𝟐𝟐𝟐 ‪𝟐𝟐𝟐𝟐 = 𝟐𝟐𝟐𝟐 + 𝟒𝟒 = (𝟓𝟓)𝟐𝟐 + (𝟐𝟐)𝟐𝟐 = (𝟓𝟓 − 𝟐𝟐𝟐𝟐)(𝟓𝟓 +‬‬ ‫=‬ ‫𝟐𝟐‬ ‫‪−‬‬ ‫𝒊𝒊 ‪𝟒𝟒 − 𝟒𝟒𝟒𝟒 +‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟒𝟒 + 𝟒𝟒𝟒𝟒 +‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫طر ﻘﺔ أﺨرى‪:‬‬ ‫=‬ ‫‪−‬‬ ‫𝟏𝟏 ‪𝟒𝟒 − 𝟒𝟒𝟒𝟒 − 𝟏𝟏 𝟒𝟒 + 𝟒𝟒𝟒𝟒 −‬‬ ‫)𝟔𝟔𝟔𝟔 ‪𝟖𝟖𝟖𝟖 = 𝟒𝟒𝟒𝟒 + 𝟑𝟑𝟑𝟑 = (𝟕𝟕)𝟐𝟐 + (𝟔𝟔)𝟐𝟐 = (𝟕𝟕 − 𝟔𝟔𝟔𝟔)(𝟕𝟕 +‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫=‬ ‫‪−‬‬ ‫)𝟓𝟓𝟓𝟓 ‪𝟒𝟒𝟒𝟒 = 𝟏𝟏𝟏𝟏 + 𝟐𝟐𝟐𝟐 = (𝟒𝟒)𝟐𝟐 + (𝟓𝟓)𝟐𝟐 = (𝟒𝟒 − 𝟓𝟓𝟓𝟓)(𝟒𝟒 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 − 𝟒𝟒𝟒𝟒 𝟑𝟑 +‬‬ ‫‪ 3‬ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍ ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫وزاري ‪ 2020‬دور ﺜﺎﻨﻲ ﺘكمیﻠﻲ ﻀﻊ ﺎﻟص ﻐﺔ اﻟﻌدد ﺔ‬ ‫𝟐𝟐)𝟏𝟏𝟏𝟏( ‪𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟒𝟒 + 𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟐𝟐)𝟐𝟐 +‬‬ ‫)𝒊𝒊 ‪(𝟑𝟑 + 𝟒𝟒𝟒𝟒)𝟐𝟐 + (𝟓𝟓 − 𝟑𝟑𝟑𝟑)(𝟏𝟏 −‬‬ ‫)𝟏𝟏𝟏𝟏𝟏𝟏 ‪= (𝟐𝟐 − 𝟏𝟏𝟏𝟏𝟏𝟏)(𝟐𝟐 +‬‬ ‫)𝟓𝟓𝟓𝟓 ‪𝟐𝟐𝟐𝟐 = 𝟒𝟒 + 𝟐𝟐𝟐𝟐 = (𝟐𝟐)𝟐𝟐 + (𝟓𝟓)𝟐𝟐 = (𝟐𝟐 − 𝟓𝟓𝟓𝟓)(𝟐𝟐 +‬‬ ‫)𝒊𝒊 ‪(𝟑𝟑 + 𝟒𝟒𝟒𝟒)𝟐𝟐 + (𝟓𝟓 − 𝟑𝟑𝟑𝟑)(𝟏𝟏 −‬‬ ‫𝟐𝟐𝒊𝒊𝟑𝟑 ‪= 𝟗𝟗 + 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟏𝟏𝟏𝟏𝒊𝒊𝟐𝟐 + 𝟓𝟓 − 𝟓𝟓𝟓𝟓 − 𝟑𝟑𝟑𝟑 +‬‬ ‫وزاري ‪ 2012‬دور ﺜﺎﻨﻲ ﻀﻊ ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ ﻟﻠﻌدد اﻟمر ب اﻟمﻘدار‪:‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟏𝟏𝟏𝟏 + 𝟏𝟏𝟏𝟏𝟏𝟏 − 𝟏𝟏𝟏𝟏 − 𝟑𝟑 = −𝟓𝟓 +‬‬ ‫𝟓𝟓)𝒊𝒊 ‪(𝟏𝟏 + 𝒊𝒊)𝟓𝟓 − (𝟏𝟏 −‬‬ ‫وزاري ‪ 2021‬دور ﺜﺎﻟث اﺤ ﺎﺌﻲ اﺜبت ان‪:‬‬ ‫𝟓𝟓)𝒊𝒊 ‪(𝟏𝟏 + 𝒊𝒊)𝟓𝟓 − (𝟏𝟏 −‬‬ ‫𝟏𝟏𝟏𝟏 = 𝟑𝟑𝒊𝒊 ‪(𝟐𝟐 − 𝒊𝒊) 𝟏𝟏 − 𝒊𝒊𝟐𝟐 𝟐𝟐 −‬‬ ‫)𝒊𝒊 ‪= (𝟏𝟏 + 𝒊𝒊)𝟒𝟒 (𝟏𝟏 + 𝒊𝒊) − (𝟏𝟏 − 𝒊𝒊)𝟒𝟒 (𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫ 𝟑𝟑𝒊𝒊 ‪𝑳𝑳𝑳𝑳𝑳𝑳 = (𝟐𝟐 − 𝒊𝒊) 𝟏𝟏 − 𝒊𝒊𝟐𝟐 𝟐𝟐 −‬‬ ‫)𝒊𝒊 ‪= (𝟏𝟏 + 𝒊𝒊)𝟐𝟐 (𝟏𝟏 + 𝒊𝒊) − (𝟏𝟏 − 𝒊𝒊)𝟐𝟐 (𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫)]𝒊𝒊‪= (𝟐𝟐 − 𝒊𝒊)(𝟏𝟏 − [−𝟏𝟏])(𝟐𝟐 − [−‬‬ ‫)𝒊𝒊 ‪= 𝟏𝟏 + 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 (𝟏𝟏 + 𝒊𝒊) − 𝟏𝟏 − 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 (𝟏𝟏 −‬‬ ‫)𝒊𝒊 ‪= (𝟐𝟐 − 𝒊𝒊)(𝟏𝟏 + 𝟏𝟏)(𝟐𝟐 +‬‬ ‫)𝒊𝒊 ‪= [𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏]𝟐𝟐 (𝟏𝟏 + 𝒊𝒊) − [𝟏𝟏 − 𝟐𝟐𝟐𝟐 − 𝟏𝟏]𝟐𝟐 (𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫ )𝟏𝟏( ‪= 𝟐𝟐(𝟐𝟐 − 𝒊𝒊)(𝟐𝟐 + 𝒊𝒊) = 𝟐𝟐 (𝟐𝟐) +‬‬ ‫)𝒊𝒊 ‪= [𝟐𝟐𝟐𝟐]𝟐𝟐 (𝟏𝟏 + 𝒊𝒊) − [−𝟐𝟐𝟐𝟐]𝟐𝟐 (𝟏𝟏 −‬‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 = 𝟏𝟏𝟏𝟏 = ]𝟓𝟓[𝟐𝟐 =‬ ‫)𝒊𝒊 ‪= −𝟒𝟒(𝟏𝟏 + 𝒊𝒊) − (−𝟒𝟒)(𝟏𝟏 −‬‬ ‫ﺘمﻬیدي ‪ 2021‬اﺤ ﺎﺌﻲ ﻀﻊ ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ ﻨﺎﺘﺞ‪:‬‬ ‫)𝒊𝒊 ‪= −𝟒𝟒(𝟏𝟏 + 𝒊𝒊) + 𝟒𝟒(𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟖𝟖𝟖𝟖‪= −𝟒𝟒 − 𝟒𝟒𝟒𝟒 + 𝟒𝟒 − 𝟒𝟒𝟒𝟒 = −‬‬ ‫ 𝒊𝒊𝟐𝟐√ ‪ 𝟏𝟏 − √𝟐𝟐𝒊𝒊 − 𝟐𝟐 −‬‬ ‫وزاري ‪ 2017‬دور اول اﺤ ﺎﺌﻲ اﺜبت ان‪:‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫ 𝒊𝒊𝟐𝟐√ ‪ 𝟏𝟏 − √𝟐𝟐𝒊𝒊 − 𝟐𝟐 −‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟔𝟔‪−‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫‪+‬‬ ‫𝟐𝟐‬ ‫=‬ ‫ 𝒊𝒊𝟐𝟐 ‪= 𝟏𝟏 − 𝟐𝟐√𝟐𝟐𝒊𝒊 + 𝟐𝟐𝒊𝒊 − 𝟒𝟒 − 𝟒𝟒√𝟐𝟐𝒊𝒊 +‬‬ ‫)𝟐𝟐𝟐𝟐 ‪(𝟏𝟏 +‬‬ ‫)𝟐𝟐𝟐𝟐 ‪(𝟏𝟏 −‬‬ ‫𝟐𝟐𝟐𝟐‬ ‫= 𝑳𝑳𝑳𝑳𝑳𝑳 𝟐𝟐 ‪= 𝟏𝟏 − 𝟐𝟐√𝟐𝟐𝒊𝒊 − 𝟐𝟐 − 𝟒𝟒 − 𝟒𝟒√𝟐𝟐𝒊𝒊 −‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫‪+‬‬ ‫ 𝒊𝒊𝟐𝟐√𝟒𝟒 ‪= −𝟏𝟏 − 𝟐𝟐√𝟐𝟐𝒊𝒊 − 𝟐𝟐 −‬‬ ‫𝟐𝟐)𝟐𝟐𝟐𝟐 ‪(𝟏𝟏 + 𝟐𝟐𝟐𝟐)𝟐𝟐 (𝟏𝟏 −‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝒊𝒊𝟐𝟐√𝟒𝟒 ‪= −𝟏𝟏 − 𝟐𝟐√𝟐𝟐𝒊𝒊 − 𝟐𝟐 +‬‬ ‫=‬ ‫𝟐𝟐‬ ‫‪+‬‬ ‫𝒊𝒊𝟒𝟒 ‪𝟏𝟏 + 𝟒𝟒𝟒𝟒 +‬‬ ‫𝟐𝟐𝒊𝒊𝟒𝟒 ‪𝟏𝟏 − 𝟒𝟒𝟒𝟒 +‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪= −𝟑𝟑 +‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫=‬ ‫‪+‬‬ ‫وزاري ‪ 2021‬دور اول اﺤ ﺎﺌﻲ اﺜبت ان‪:‬‬ ‫𝟒𝟒 ‪𝟏𝟏 + 𝟒𝟒𝟒𝟒 − 𝟒𝟒 𝟏𝟏 − 𝟒𝟒𝟒𝟒 −‬‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 −‬‬ ‫𝟏𝟏‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐 = ) 𝒊𝒊 ‪ (𝟏𝟏 − 𝒊𝒊)(𝒊𝒊 − 𝟏𝟏)(𝟏𝟏 −‬‬ ‫𝟑𝟑‬ ‫=‬ ‫ ‬ ‫ ‬ ‫‪+‬‬ ‫ ‬ ‫ ‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 + 𝟒𝟒𝟒𝟒 −𝟑𝟑 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 − 𝟒𝟒𝟒𝟒 −𝟑𝟑 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 +‬‬ ‫) 𝟑𝟑𝒊𝒊 ‪𝑳𝑳𝑳𝑳𝑳𝑳 = (𝟏𝟏 − 𝒊𝒊)(𝒊𝒊𝟐𝟐 − 𝟏𝟏)(𝟏𝟏 −‬‬ ‫=‬ ‫‪+‬‬ ‫𝟐𝟐)𝟒𝟒( ‪(𝟑𝟑)𝟐𝟐 + (𝟒𝟒)𝟐𝟐 (𝟑𝟑)𝟐𝟐 +‬‬ ‫)]𝒊𝒊‪= (𝟏𝟏 − 𝒊𝒊)([−𝟏𝟏] − 𝟏𝟏)(𝟏𝟏 − [−‬‬ ‫𝟔𝟔‪−𝟑𝟑 − 𝟒𝟒𝟒𝟒 − 𝟑𝟑 + 𝟒𝟒𝟒𝟒 −‬‬ ‫=‬ ‫=‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 =‬ ‫)𝒊𝒊 ‪= (𝟏𝟏 − 𝒊𝒊)(−𝟐𝟐)(𝟏𝟏 +‬‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 = 𝟐𝟐𝟐𝟐 = 𝒊𝒊𝟒𝟒√ = 𝟒𝟒‪= −𝟐𝟐(𝟏𝟏 + 𝟏𝟏) = √−‬‬ ‫وزاري ‪ 2020‬دور اول اﺤ ﺎﺌﻲ ﻀﻊ ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ )اﻟجبر ﺔ( ﻨﺎﺘﺞ‪:‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫وزاري ‪ 2021‬دور ﺜﺎﻨﻲ اﺤ ﺎﺌﻲ اﺜبت ان‪:‬‬ ‫‪+‬‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫𝟒𝟒‬ ‫𝟐𝟐)𝒊𝒊 ‪(√𝟐𝟐 + 𝒊𝒊)𝟐𝟐 (√𝟐𝟐 −‬‬ ‫‪+‬‬ ‫=‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝟐𝟐𝟐𝟐 𝟐𝟐)𝒊𝒊 ‪(𝟑𝟑 + 𝒊𝒊)𝟐𝟐 (𝟑𝟑 −‬‬ ‫‪𝟐𝟐 +‬‬ ‫𝟐𝟐‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫ 𝒊𝒊 ‪ √𝟐𝟐 +‬‬ ‫ 𝒊𝒊 ‪ √𝟐𝟐 −‬‬ ‫= 𝑳𝑳𝑳𝑳𝑳𝑳‬ ‫‪+‬‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟑𝟑 + 𝒊𝒊)𝟐𝟐 (𝟑𝟑 −‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫=‬ ‫‪+‬‬ ‫=‬ ‫‪+‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟐𝟐 + 𝟐𝟐√𝟐𝟐𝒊𝒊 + 𝒊𝒊𝟐𝟐 𝟐𝟐 − 𝟐𝟐√𝟐𝟐𝒊𝒊 +‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟗𝟗 + 𝟔𝟔𝟔𝟔 + 𝒊𝒊𝟐𝟐 𝟗𝟗 − 𝟔𝟔𝟔𝟔 +‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊‬ ‫𝟏𝟏‬ ‫𝟏𝟏‬ ‫=‬ ‫‪+‬‬ ‫=‬ ‫‪+‬‬ ‫𝟏𝟏 ‪𝟐𝟐 + 𝟐𝟐√𝟐𝟐𝒊𝒊 − 𝟏𝟏 𝟐𝟐 − 𝟐𝟐√𝟐𝟐𝒊𝒊 −‬‬ ‫𝟏𝟏 ‪𝟗𝟗 + 𝟔𝟔𝟔𝟔 − 𝟏𝟏 𝟗𝟗 − 𝟔𝟔𝟔𝟔 −‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 −‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 +‬‬ ‫𝟏𝟏‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 −‬‬ ‫𝟏𝟏‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 +‬‬ ‫=‬ ‫ ‬ ‫‪ +‬‬ ‫ ‬ ‫ ‬ ‫=‬ ‫ ‬ ‫‪ +‬‬ ‫ ‬ ‫ ‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 + 𝟐𝟐√𝟐𝟐𝒊𝒊 𝟏𝟏 −‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 − 𝟐𝟐√𝟐𝟐𝒊𝒊 𝟏𝟏 +‬‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 + 𝟔𝟔𝟔𝟔 𝟖𝟖 −‬‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 − 𝟔𝟔𝟔𝟔 𝟖𝟖 +‬‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 −‬‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 +‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 −‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 +‬‬ ‫=‬ ‫‪+‬‬ ‫=‬ ‫‪+‬‬ ‫)𝟔𝟔( ‪(𝟖𝟖) +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫)𝟔𝟔( ‪(𝟖𝟖) +‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 − 𝟔𝟔𝟔𝟔 𝟖𝟖 +‬‬ ‫ 𝟐𝟐√𝟐𝟐 ‪(𝟏𝟏)𝟐𝟐 +‬‬ ‫ 𝟐𝟐√𝟐𝟐 ‪(𝟏𝟏)𝟐𝟐 +‬‬ ‫=‬ ‫‪+‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐 ‪𝟏𝟏 − 𝟐𝟐√𝟐𝟐𝒊𝒊 𝟏𝟏 +‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏‬ ‫=‬ ‫‪+‬‬ ‫𝟔𝟔𝟔𝟔 ‪𝟖𝟖 − 𝟔𝟔𝟔𝟔 + 𝟖𝟖 +‬‬ ‫𝟏𝟏𝟏𝟏‬ ‫𝟒𝟒‬ ‫𝟖𝟖 ‪𝟏𝟏 +‬‬ ‫𝟖𝟖 ‪𝟏𝟏 +‬‬ ‫=‬ ‫=‬ ‫=‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏‬ ‫𝟐𝟐𝟐𝟐 𝟏𝟏𝟏𝟏𝟏𝟏‬ ‫𝟏𝟏‬ ‫‪−‬‬ ‫𝒊𝒊𝟐𝟐√𝟐𝟐‬ ‫‪+‬‬ ‫𝟏𝟏‬ ‫𝟐𝟐 𝒊𝒊𝟐𝟐√𝟐𝟐 ‪+‬‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 =‬ ‫=‬ ‫=‬ ‫𝟗𝟗‬ ‫𝟗𝟗‬ ‫𝟐𝟐‬ ‫𝟎𝟎𝟎𝟎 ‪= +‬‬ ‫𝟗𝟗‬ ‫‪4‬‬ ‫‪set by: Sadiq Hussain‬‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫)𝟖𝟖 ‪−𝟔𝟔𝟔𝟔 + 𝟖𝟖 𝟔𝟔𝟔𝟔 + 𝟖𝟖 −𝟔𝟔𝟔𝟔 + 𝟖𝟖 − (𝟔𝟔𝟔𝟔 +‬‬ ‫𝟕𝟕)𝒊𝒊‪(𝟏𝟏+‬‬ ‫=‬ ‫‪−‬‬ ‫=‬ ‫وزاري ‪ 2021‬دور ﺜﺎﻨﻲ اﺤ ﺎﺌﻲ ﻀﻊ ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ‪:‬‬ ‫𝟏𝟏𝟏𝟏 ‪𝟗𝟗 + 𝟏𝟏𝟏𝟏 𝟗𝟗 +‬‬ ‫𝟏𝟏𝟏𝟏 ‪𝟗𝟗 +‬‬ ‫𝟖𝟖‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏‪−𝟔𝟔𝟔𝟔 + 𝟖𝟖 − 𝟔𝟔𝟔𝟔 − 𝟖𝟖 −‬‬ ‫أوﻻً ﺴنﻘوم ﺒت س ط سط اﻟمﻘدار اﻟمطﻠوب = 𝟕𝟕)𝒊𝒊 ‪(𝟏𝟏 +‬‬ ‫=‬ ‫=‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟑𝟑‬ ‫)𝒊𝒊 ‪(𝟏𝟏 + 𝒊𝒊)𝟕𝟕 = (𝟏𝟏 + 𝒊𝒊)𝟔𝟔 (𝟏𝟏 + 𝒊𝒊) = (𝟏𝟏 + 𝒊𝒊)𝟐𝟐 (𝟏𝟏 +‬‬ ‫𝟏𝟏𝟏𝟏‬ ‫‪= 𝟎𝟎 −‬‬ ‫𝒊𝒊‬ ‫𝟑𝟑‬ ‫𝟐𝟐𝟐𝟐‬ ‫)𝒊𝒊 ‪= 𝟏𝟏 + 𝟐𝟐𝟐𝟐 + 𝒊𝒊𝟐𝟐 (𝟏𝟏 +‬‬ ‫ﺃﺳﺌﻠﺔ ﺗﻄﺒﻴﻖ ﳋﻮﺍﺹ ﻣﺮﺍﻓﻖ ﺍﻟﻌﺪﺩ ﺍﳌﺮﻛﺐ‬ ‫)𝒊𝒊 ‪= [𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏]𝟑𝟑 (𝟏𝟏 +‬‬ ‫)𝒊𝒊 ‪= [𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏]𝟑𝟑 (𝟏𝟏 +‬‬ ‫ﻤثﺎل ‪ -8-‬اذا ﺎن 𝒊𝒊𝟐𝟐 ‪ 𝒄𝒄𝟏𝟏 = 𝟏𝟏 + 𝒊𝒊 , 𝒄𝒄𝟐𝟐 = 𝟑𝟑 −‬ﻓتحﻘق ﻤن‪:‬‬ ‫)𝒊𝒊 ‪= [𝟐𝟐𝟐𝟐]𝟑𝟑 (𝟏𝟏 + 𝒊𝒊) = 𝟖𝟖(−𝒊𝒊)(𝟏𝟏 +‬‬ ‫𝟖𝟖𝟖𝟖 ‪= −𝟖𝟖𝟖𝟖 − 𝟖𝟖𝒊𝒊𝟐𝟐 = 𝟖𝟖 −‬‬ ‫ ①‬ ‫ = 𝟐𝟐𝒄𝒄 ‪𝒄𝒄𝟏𝟏 ±‬‬ ‫𝒄𝒄 ‪𝒄𝒄𝟏𝟏 ±‬‬ ‫𝟐𝟐 ‬ ‫𝟕𝟕‬ ‫)𝒊𝒊 ‪(𝟏𝟏 +‬‬ ‫𝟖𝟖 𝟖𝟖 𝟖𝟖𝟖𝟖 ‪𝟖𝟖 −‬‬ ‫ ②‬ ‫𝒄𝒄 = 𝟐𝟐𝒄𝒄 × 𝟏𝟏𝒄𝒄‬ ‫ × 𝟏𝟏 ‬ ‫𝟐𝟐𝒄𝒄‬ ‫⟹‬ ‫=‬ ‫𝒊𝒊 ‪= − 𝒊𝒊 = 𝟏𝟏 −‬‬ ‫𝟖𝟖‬ ‫𝟖𝟖‬ ‫𝟖𝟖 𝟖𝟖‬ ‫ = 𝑳𝑳𝑳𝑳𝑳𝑳 ①‬ ‫𝒊𝒊 ‪𝒄𝒄𝟏𝟏 + 𝒄𝒄𝟐𝟐 = (𝟏𝟏 + 𝒊𝒊) + (𝟐𝟐 − 𝟐𝟐𝟐𝟐) = 𝟑𝟑 −‬‬ ‫= 𝒃𝒃𝒃𝒃 ‪ 𝒂𝒂 +‬اﺜبت ان‬ ‫𝒊𝒊‪𝟐𝟐+‬‬ ‫وزاري ‪ 2022‬دور اول اذا ﺎن‬ ‫𝒊𝒊 ‪= 𝟑𝟑 +‬‬ ‫𝟑𝟑‬ ‫𝒊𝒊‪𝟏𝟏−‬‬ ‫𝟑𝟑‬ ‫𝟕𝟕 = 𝒃𝒃 ‪𝟐𝟐 𝒂𝒂 +‬‬ ‫𝒄𝒄 ‪ 𝟏𝟏 +‬‬ ‫𝒄𝒄 = 𝑹𝑹𝑹𝑹𝑹𝑹‬ ‫)𝟐𝟐𝟐𝟐 ‪ 𝟐𝟐 = (𝟏𝟏 + 𝒊𝒊) + (𝟐𝟐 −‬‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 = 𝒊𝒊 ‪= 𝟏𝟏 − 𝒊𝒊 + 𝟐𝟐 + 𝟐𝟐𝟐𝟐 = 𝟑𝟑 +‬‬ ‫أوﻻً ﻨحﻞ اﻟمﻌﺎدﻟﺔ اﻟمﻌطﺎة‬ ‫𝒊𝒊 ‪𝟐𝟐 +‬‬ ‫𝒄𝒄 = 𝑳𝑳𝑳𝑳𝑳𝑳‬ ‫)𝟐𝟐𝟐𝟐 ‪𝟏𝟏 − 𝒄𝒄𝟐𝟐 = (𝟏𝟏 + 𝒊𝒊) − (𝟐𝟐 −‬‬ ‫ ‬ ‫= 𝒃𝒃𝒃𝒃 ‪𝒂𝒂 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫𝟑𝟑𝟑𝟑 ‪= 𝟏𝟏 + 𝒊𝒊 − 𝟐𝟐 + 𝟐𝟐𝟐𝟐 = −𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪(𝒂𝒂 + 𝒃𝒃𝒃𝒃)(𝟏𝟏 − 𝒊𝒊) = 𝟐𝟐 +‬‬ ‫𝟑𝟑𝟑𝟑 ‪= −𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝒂𝒂 − 𝒂𝒂𝒂𝒂 + 𝒃𝒃𝒃𝒃 + 𝒃𝒃 = 𝟐𝟐 +‬‬ ‫𝒄𝒄 ‪ 𝟏𝟏 −‬‬ ‫𝒄𝒄 = 𝑹𝑹𝑹𝑹𝑹𝑹‬ ‫)𝟐𝟐𝟐𝟐 ‪ 𝟐𝟐 = (𝟏𝟏 + 𝒊𝒊) − (𝟐𝟐 −‬‬ ‫𝒊𝒊 ‪(𝒂𝒂 + 𝒃𝒃) + (−𝒂𝒂 + 𝒃𝒃)𝒊𝒊 = 𝟐𝟐 +‬‬ ‫)𝟐𝟐𝟐𝟐 ‪= (𝟏𝟏 − 𝒊𝒊) − (𝟐𝟐 +‬‬ ‫ ‬ ‫𝟐𝟐 = 𝒃𝒃 ‪𝒂𝒂 +‬‬ ‫𝟏𝟏 = 𝒃𝒃 ‪−𝒂𝒂 +‬‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 = 𝟑𝟑𝟑𝟑 ‪= 𝟏𝟏 − 𝒊𝒊 − 𝟐𝟐 − 𝟐𝟐𝟐𝟐 = −𝟏𝟏 −‬‬ ‫ ‬ ‫𝒃𝒃 ‪𝒂𝒂 = 𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝒂𝒂 = 𝒃𝒃 −‬‬ ‫ = 𝑳𝑳𝑳𝑳𝑳𝑳 ②‬ ‫)𝟐𝟐𝟐𝟐 ‪𝒄𝒄𝟏𝟏 × 𝒄𝒄𝟐𝟐 = (𝟏𝟏 + 𝒊𝒊)(𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝟐𝟐 − 𝒃𝒃 = 𝒃𝒃 −‬‬ ‫𝟒𝟒 = 𝟐𝟐 ‪= 𝟐𝟐 − 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐 − 𝟐𝟐𝒊𝒊𝟐𝟐 = 𝟐𝟐 +‬‬ ‫𝟐𝟐 ‪−𝒃𝒃 − 𝒃𝒃 = −𝟏𝟏 −‬‬ ‫𝟒𝟒 =‬ ‫𝟑𝟑‪−𝟐𝟐𝟐𝟐 = −‬‬ ‫𝟑𝟑‬ ‫𝒄𝒄 × 𝟏𝟏 ‬ ‫𝒄𝒄 = 𝑹𝑹𝑹𝑹𝑹𝑹‬ ‫)𝟐𝟐𝟐𝟐 ‪ 𝟐𝟐 = (𝟏𝟏 + 𝒊𝒊) × (𝟐𝟐 −‬‬ ‫= 𝒃𝒃‬ ‫𝟐𝟐‬ ‫𝟐𝟐 ‪= (𝟏𝟏 − 𝒊𝒊)(𝟐𝟐 + 𝟐𝟐𝟐𝟐) = 𝟐𝟐 + 𝟐𝟐𝟐𝟐 − 𝟐𝟐𝟐𝟐 − 𝟐𝟐𝒊𝒊𝟐𝟐 = 𝟐𝟐 +‬‬ ‫𝟏𝟏 ‪∵ 𝒂𝒂 = 𝒃𝒃 −‬‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 = 𝟒𝟒 =‬ ‫𝟑𝟑‬ ‫𝟐𝟐‬ ‫𝟏𝟏 𝟐𝟐 𝟑𝟑‬ ‫= ‪𝒂𝒂 = − 𝟏𝟏 = −‬‬ ‫ﻤثﺎل ‪ -11-‬اذا ﺎن 𝒊𝒊𝟐𝟐 ‪ 𝒄𝒄𝟐𝟐 = 𝟏𝟏 + 𝒊𝒊 , 𝒄𝒄𝟏𝟏 = 𝟑𝟑 −‬ﻓتحﻘق ﻤن‪:‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫𝟐𝟐 𝟐𝟐 𝟐𝟐‬ ‫ ‬ ‫𝟏𝟏𝒄𝒄‬ ‫ ‬ ‫𝟏𝟏𝒄𝒄‬ ‫𝒂𝒂 𝟐𝟐‬ ‫𝟑𝟑‬ ‫اﻵن ﻨثبت إن 𝟕𝟕 = 𝟑𝟑𝒃𝒃 ‪+‬‬ ‫= ‬ ‫𝟐𝟐𝒄𝒄‬ ‫ ‬ ‫𝟐𝟐𝒄𝒄‬ ‫𝟑𝟑‬ ‫𝟑𝟑‬ ‫𝟑𝟑 𝟏𝟏‬ ‫𝟑𝟑 𝟑𝟑‬ ‫ ‪𝑳𝑳𝑳𝑳𝑳𝑳 = 𝟐𝟐 𝒂𝒂 + 𝒃𝒃 = 𝟐𝟐 +‬‬ ‫ ‬ ‫𝟏𝟏𝒄𝒄‬ ‫𝟐𝟐𝟐𝟐 ‪𝟑𝟑 −‬‬ ‫𝒊𝒊 ‪𝟑𝟑 − 𝟐𝟐𝟐𝟐 𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫ = = 𝑳𝑳𝑳𝑳𝑳𝑳‬ ‫ = ‬ ‫ ‬ ‫ ‬ ‫𝟏𝟏‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟕𝟕‬ ‫𝟐𝟐𝒄𝒄‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫ 𝟐𝟐 = 𝟐𝟐 = ‪= 𝟐𝟐 +‬‬ ‫𝟐𝟐𝒊𝒊𝟐𝟐 ‪𝟑𝟑 − 𝟑𝟑𝟑𝟑 − 𝟐𝟐𝟐𝟐 +‬‬ ‫𝟖𝟖 𝟖𝟖‬ ‫𝟖𝟖‬ ‫𝟐𝟐‬ ‫ =‬ ‫ ‬ ‫𝑹𝑹𝑹𝑹𝑹𝑹 = 𝟕𝟕 =‬ ‫𝟐𝟐)𝟏𝟏( ‪(𝟏𝟏)𝟐𝟐 +‬‬ ‫وزاري ‪ 2022‬دور ﺜﺎﻨﻲ اﺤ ﺎﺌﻲ ﻀﻊ ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ‪:‬‬ ‫𝟐𝟐 ‪𝟑𝟑 − 𝟓𝟓𝟓𝟓 −‬‬ ‫𝟓𝟓𝟓𝟓 ‪𝟏𝟏 −‬‬ ‫ =‬ ‫ = ‬ ‫ ‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟏𝟏 +‬‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟏𝟏 −‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫𝟐𝟐‬ ‫‪−‬‬ ‫𝟐𝟐)𝟐𝟐𝟐𝟐 ‪(𝟏𝟏 + 𝟐𝟐𝟐𝟐)𝟐𝟐 (𝟏𝟏 −‬‬ ‫𝟓𝟓 𝟏𝟏‬ ‫𝟓𝟓 𝟏𝟏‬ ‫𝟐𝟐‬ ‫𝒊𝒊 ‪= − 𝒊𝒊 = +‬‬ ‫)𝒊𝒊 ‪(𝟏𝟏 +‬‬ ‫𝟐𝟐)𝒊𝒊 ‪(𝟏𝟏 −‬‬ ‫𝟐𝟐 𝟐𝟐‬ ‫𝟐𝟐 𝟐𝟐‬ ‫‪−‬‬ ‫𝟐𝟐𝟐𝟐 ‪ 𝟏𝟏 𝟑𝟑 −‬‬ ‫𝒄𝒄‬ ‫𝒊𝒊 ‪𝟑𝟑 + 𝟐𝟐𝟐𝟐 𝟏𝟏 +‬‬ ‫𝟐𝟐)𝟐𝟐𝟐𝟐 ‪(𝟏𝟏 + 𝟐𝟐𝟐𝟐)𝟐𝟐 (𝟏𝟏 −‬‬ ‫= 𝑹𝑹𝑹𝑹𝑹𝑹‬ ‫=‬ ‫ =‬ ‫ ‬ ‫ ‬ ‫𝟐𝟐𝒊𝒊 ‪𝟏𝟏 + 𝟐𝟐𝟐𝟐 +‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟏𝟏 − 𝟐𝟐𝟐𝟐 +‬‬ ‫ ‬ ‫𝟐𝟐𝒄𝒄‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 − 𝒊𝒊 𝟏𝟏 +‬‬ ‫=‬ ‫‪−‬‬ ‫𝟐𝟐 ‪𝟑𝟑 + 𝟑𝟑𝟑𝟑 + 𝟐𝟐𝟐𝟐 + 𝟐𝟐𝒊𝒊𝟐𝟐 𝟑𝟑 + 𝟓𝟓𝟓𝟓 −‬‬ ‫𝟐𝟐𝒊𝒊𝟒𝟒 ‪𝟏𝟏 + 𝟒𝟒𝟒𝟒 + 𝟒𝟒𝒊𝒊𝟐𝟐 𝟏𝟏 − 𝟒𝟒𝟒𝟒 +‬‬ ‫=‬ ‫=‬ ‫𝟏𝟏 ‪𝟏𝟏 + 𝟐𝟐𝟐𝟐 − 𝟏𝟏 𝟏𝟏 − 𝟐𝟐𝟐𝟐 −‬‬ ‫𝟐𝟐)𝟏𝟏( ‪(𝟏𝟏)𝟐𝟐 +‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫=‬ ‫‪−‬‬ ‫𝟒𝟒 ‪𝟏𝟏 + 𝟒𝟒𝟒𝟒 − 𝟒𝟒 𝟏𝟏 − 𝟒𝟒𝟒𝟒 −‬‬ ‫𝟓𝟓 𝟏𝟏 𝟓𝟓𝟓𝟓 ‪𝟏𝟏 +‬‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟐𝟐𝟐𝟐‪−‬‬ ‫=‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 = 𝒊𝒊 ‪= +‬‬ ‫=‬ ‫‪−‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐 𝟐𝟐‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 + 𝟒𝟒𝟒𝟒 −𝟑𝟑 −‬‬ ‫وزاري ‪ 2018‬دور ﺜﺎﻨﻲ اﺤ ﺎﺌﻲ اذا ﻋﻠمت ان 𝒊𝒊 ‪ 𝒙𝒙 = 𝟖𝟖 −‬و ﺎن‬ ‫𝟐𝟐𝟐𝟐‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟐𝟐𝟐𝟐 −𝟑𝟑 +‬‬ ‫=‬ ‫ ‬ ‫‪ −‬‬ ‫ ‬ ‫ ‬ ‫𝒊𝒊 ‪ 𝒚𝒚 = 𝟐𝟐 +‬ﺘحﻘق ﻤن ان 𝒚𝒚 ∙ 𝒙𝒙 = 𝒙𝒙𝒙𝒙‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 + 𝟒𝟒𝟒𝟒 −𝟑𝟑 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪−𝟑𝟑 − 𝟒𝟒𝟒𝟒 −𝟑𝟑 +‬‬ ‫𝟐𝟐𝒊𝒊𝟖𝟖 ‪−𝟔𝟔𝒊𝒊 −‬‬ ‫𝟐𝟐𝒊𝒊𝟖𝟖 ‪𝟔𝟔𝟔𝟔 −‬‬ ‫=‬ ‫‪−‬‬ ‫) 𝟐𝟐𝟒𝟒( ‪(−𝟑𝟑)𝟐𝟐 + (𝟒𝟒𝟐𝟐 ) (−𝟑𝟑)𝟐𝟐 +‬‬ ‫‪ 5‬ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍ ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫𝟐𝟐‬ ‫)𝒊𝒊 ‪𝑳𝑳𝑳𝑳𝑳𝑳 = (𝒙𝒙𝒙𝒙) = (𝟖𝟖 − 𝒊𝒊)(𝟐𝟐 +‬‬ ‫= 𝒙𝒙‬ ‫𝟏𝟏‪= −‬‬ ‫𝟐𝟐‪−‬‬ ‫) 𝟐𝟐𝒊𝒊 ‪= (𝟏𝟏𝟏𝟏 + 𝟖𝟖𝟖𝟖 − 𝟐𝟐𝟐𝟐 −‬‬ ‫ﺘمر ن ‪ -2-‬ﺠد ق مﺔ ﻞ ﻤن 𝒙𝒙 ‪ 𝒚𝒚,‬اﻟحق ﻘیتین اﻟﻠتین ﺘحﻘﻘﺎن‬ ‫)𝟔𝟔𝟔𝟔 ‪= (𝟏𝟏𝟏𝟏 + 𝟔𝟔𝟔𝟔 + 𝟏𝟏) = (𝟏𝟏𝟏𝟏 +‬‬ ‫اﻟمﻌﺎدﻻت اﻻﺘ ﺔ‪:‬‬ ‫𝟔𝟔𝟔𝟔 ‪= 𝟏𝟏𝟏𝟏 −‬‬ ‫)𝟐𝟐𝟐𝟐 ‪𝒂𝒂) 𝒚𝒚 + 𝟓𝟓𝟓𝟓 = (𝟐𝟐𝟐𝟐 + 𝒊𝒊)(𝒙𝒙 +‬‬ ‫)𝒊𝒊 ‪𝑹𝑹𝑹𝑹𝑹𝑹 = 𝒙𝒙 ⋅ 𝒚𝒚 = (𝟖𝟖 − 𝒊𝒊) ⋅ (𝟐𝟐 + 𝒊𝒊) = (𝟖𝟖 + 𝒊𝒊)(𝟐𝟐 −‬‬ ‫𝟐𝟐‬ ‫𝒊𝒊𝟐𝟐 ‪𝒚𝒚 + 𝟓𝟓𝟓𝟓 = 𝟐𝟐𝒙𝒙 + 𝟒𝟒𝟒𝟒𝟒𝟒 + 𝒙𝒙𝒙𝒙 +‬‬ ‫𝟐𝟐‬ ‫𝟏𝟏 ‪= 𝟏𝟏𝟏𝟏 − 𝟖𝟖𝟖𝟖 + 𝟐𝟐𝟐𝟐 − 𝒊𝒊𝟐𝟐 = 𝟏𝟏𝟏𝟏 − 𝟔𝟔𝒊𝒊 +‬‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 = 𝟔𝟔𝟔𝟔 ‪= 𝟏𝟏𝟏𝟏 −‬‬ ‫𝟓𝟓𝟓𝟓𝟓𝟓 ‪𝒚𝒚 + 𝟓𝟓𝟓𝟓 = 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟐𝟐 +‬‬ ‫ ‬ ‫وزاري ‪ 2019‬دور ﺜﺎﻟث اﺤ ﺎﺌﻲ اذا ﺎن 𝟐𝟐)𝟐𝟐𝟐𝟐 ‪𝒚𝒚 = ، 𝒙𝒙 = (𝟑𝟑 −‬‬ ‫𝟐𝟐 ‪𝒚𝒚 = 𝟐𝟐𝒙𝒙𝟐𝟐 −‬‬ ‫𝟓𝟓𝟓𝟓 = 𝟓𝟓‬ ‫𝒊𝒊‪𝟑𝟑−‬‬ ‫ ‬ ‫𝟏𝟏 = 𝒙𝒙‬ ‫‪ ،‬ﺠد 𝒚𝒚 ‪ 𝒙𝒙,‬ﺎﻟص ﻐﺔ اﻟﻌﺎد ﺔ‪ ،‬ﺜم اﺜبت ان‪:‬‬ ‫𝒊𝒊‪𝟏𝟏+‬‬ ‫𝟎𝟎 = 𝟐𝟐 ‪∵ 𝒚𝒚 = 𝟐𝟐𝒙𝒙𝟐𝟐 − 𝟐𝟐 = 𝟐𝟐(𝟏𝟏)𝟐𝟐 − 𝟐𝟐 = 𝟐𝟐 −‬‬ ‫𝒚𝒚 ‪𝒙𝒙 + 𝒚𝒚 = 𝒙𝒙 +‬‬ ‫𝟐𝟐‬ ‫𝟏𝟏 ‪𝒃𝒃) 𝟖𝟖𝟖𝟖 = (𝒙𝒙 + 𝟐𝟐𝟐𝟐)(𝒚𝒚 + 𝟐𝟐𝟐𝟐) +‬‬ ‫𝒙𝒙‬ ‫=‬ ‫(‬ ‫𝟑𝟑‬ ‫‪−‬‬ ‫𝟐𝟐𝟐𝟐‬ ‫)‬ ‫=‬ ‫𝟗𝟗‬ ‫𝟒𝟒 ‪− 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟒𝟒𝒊𝒊𝟐𝟐 = 𝟗𝟗 − 𝟏𝟏𝟏𝟏𝟏𝟏 −‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟓𝟓 −‬‬ ‫𝟏𝟏 ‪𝟖𝟖𝟖𝟖 = (𝒙𝒙 + 𝟐𝟐𝟐𝟐)(𝒚𝒚 + 𝟐𝟐𝟐𝟐) +‬‬ ‫𝟑𝟑‬ ‫‪−‬‬ ‫𝒊𝒊‬ ‫𝟑𝟑‬ ‫‪−‬‬ ‫𝒊𝒊 ‪𝒊𝒊 𝟏𝟏 −‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟑𝟑 − 𝟑𝟑𝟑𝟑 − 𝒊𝒊 +‬‬ ‫𝟏𝟏 ‪𝟖𝟖𝟖𝟖 = 𝒙𝒙𝒙𝒙 + 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟒𝟒𝒊𝒊𝟐𝟐 +‬‬ ‫= 𝒚𝒚‬ ‫=‬ ‫ ‬ ‫= ‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫𝟏𝟏 ‪𝟖𝟖𝟖𝟖 = 𝒙𝒙𝒙𝒙 + 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝟐𝟐𝟐𝟐𝟐𝟐 − 𝟒𝟒 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝟑𝟑 − 𝟒𝟒𝟒𝟒 − 𝟏𝟏 𝟐𝟐 −‬‬ ‫𝒊𝒊)𝟐𝟐𝟐𝟐 ‪𝟎𝟎 + 𝟖𝟖𝟖𝟖 = (𝒙𝒙𝒙𝒙 − 𝟑𝟑) + (𝟐𝟐𝟐𝟐 +‬‬ ‫=‬ ‫=‬ ‫𝟐𝟐𝟐𝟐 ‪= 𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫ ‬ ‫𝟎𝟎 = 𝟑𝟑 ‪𝒙𝒙𝒙𝒙 −‬‬ ‫𝟖𝟖 = 𝟐𝟐𝟐𝟐 ‪𝟐𝟐𝟐𝟐 +‬‬ ‫)𝟏𝟏𝟏𝟏𝟏𝟏 ‪𝑳𝑳𝑳𝑳𝑳𝑳 = 𝒙𝒙 + 𝒚𝒚 = (𝟓𝟓 − 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟏𝟏 − 𝟐𝟐𝟐𝟐) = (𝟔𝟔 −‬‬ ‫𝟑𝟑 = 𝒙𝒙𝒙𝒙‬ ‫𝟒𝟒 = 𝒚𝒚 ‪𝒙𝒙 +‬‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟔𝟔 +‬‬ ‫𝟑𝟑‬ ‫)𝟐𝟐𝟐𝟐 ‪𝑹𝑹𝑹𝑹𝑹𝑹 = 𝒙𝒙 + 𝒚𝒚 = (𝟓𝟓 − 𝟏𝟏𝟏𝟏𝟏𝟏) + (𝟏𝟏 −‬‬ ‫= 𝒙𝒙‬ ‫𝒚𝒚 ‪𝒙𝒙 = 𝟒𝟒 −‬‬ ‫𝒚𝒚‬ ‫ ‬ ‫𝟏𝟏𝟏𝟏𝟏𝟏 ‪= 𝟓𝟓 + 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟏𝟏 + 𝟐𝟐𝟐𝟐 = 𝟔𝟔 +‬‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 =‬ ‫𝟑𝟑‬ ‫𝒚𝒚 ‪∴ = 𝟒𝟒 −‬‬ ‫𝒚𝒚‬ ‫وزاري ‪ 2020‬دور ﺜﺎﻨﻲ ﺘكمیﻠﻲ اﺤ ﺎﺌﻲ اذا ﺎن )𝒊𝒊 ‪𝒙𝒙 = (𝟑𝟑 +‬‬ ‫𝟑𝟑 = 𝟐𝟐𝒚𝒚 ‪𝟒𝟒𝟒𝟒 −‬‬ ‫‪ 𝒚𝒚 = (𝟏𝟏 − 𝒊𝒊) ،‬ﻓتحﻘق ﻤن ان‪𝒙𝒙 ∙ 𝒚𝒚 = 𝒙𝒙 ∙ 𝒚𝒚 :‬‬ ‫𝟎𝟎 = 𝟑𝟑 ‪𝒚𝒚𝟐𝟐 − 𝟒𝟒𝟒𝟒 +‬‬ ‫) 𝟐𝟐𝒊𝒊 ‪𝑳𝑳𝑳𝑳𝑳𝑳 = (𝒙𝒙 ∙ 𝒚𝒚) = (𝟏𝟏 − 𝒊𝒊)(𝟑𝟑 + 𝒊𝒊) = (𝟑𝟑 + 𝒊𝒊 − 𝟑𝟑𝟑𝟑 −‬‬ ‫𝟎𝟎 = )𝟏𝟏 ‪(𝒚𝒚 − 𝟑𝟑)(𝒚𝒚 −‬‬ ‫𝟐𝟐𝟐𝟐 ‪= (𝟑𝟑 − 𝟐𝟐𝟐𝟐 + 𝟏𝟏) = (𝟒𝟒 − 𝟐𝟐𝟐𝟐) = 𝟒𝟒 +‬‬ ‫ ‬ ‫𝟎𝟎 = 𝟑𝟑 ‪ 𝒚𝒚 −‬أﻤﺎ‬ ‫𝟎𝟎 = 𝟏𝟏 ‪ 𝒚𝒚 −‬أو‬ ‫)𝒊𝒊 ‪𝑹𝑹𝑹𝑹𝑹𝑹 = 𝒙𝒙 ⋅ 𝒚𝒚 = (𝟏𝟏 − 𝒊𝒊) ⋅ (𝟑𝟑 + 𝒊𝒊) = (𝟏𝟏 + 𝒊𝒊)(𝟑𝟑 −‬‬ ‫𝟑𝟑 = 𝒚𝒚‬ ‫𝟏𝟏 = 𝒚𝒚‬ ‫𝟏𝟏 ‪= 𝟑𝟑 − 𝒊𝒊 + 𝟑𝟑𝟑𝟑 − 𝒊𝒊𝟐𝟐 = 𝟑𝟑 + 𝟐𝟐𝟐𝟐 +‬‬ ‫𝒚𝒚 ‪∵ 𝒙𝒙 = 𝟒𝟒 −‬‬ ‫𝒚𝒚 ‪∵ 𝒙𝒙 = 𝟒𝟒 −‬‬ ‫𝑳𝑳𝑳𝑳𝑳𝑳 = 𝟐𝟐𝟐𝟐 ‪= 𝟒𝟒 +‬‬ ‫𝟏𝟏 = )𝟑𝟑( ‪𝒙𝒙 = 𝟒𝟒 −‬‬ ‫𝟑𝟑 = )𝟏𝟏( ‪𝒙𝒙 = 𝟒𝟒 −‬‬ ‫ﺃﺳﺌﻠﺔ "ﺟﺪ ﻗﻴﻢ 𝒚𝒚 ‪"𝒙𝒙,‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫ )𝒄𝒄‬ ‫𝟐𝟐)𝟐𝟐𝟐𝟐 ‪ + (𝒙𝒙 + 𝒚𝒚𝒚𝒚) = (𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫ﻤثﺎل ‪ -3-‬ﺠد ق مﺔ ﻞ ﻤن 𝒚𝒚 ‪ 𝒙𝒙 ,‬اﻟحق ﻘیتین اﻟﻠتین ﺘحﻘﻘﺎن اﻟمﻌﺎدﻟﺔ‬ ‫𝒊𝒊 ‪𝟏𝟏 − 𝒊𝒊 𝟏𝟏 −‬‬ ‫ ‬ ‫𝟐𝟐𝒊𝒊𝟒𝟒 ‪ + (𝒙𝒙 + 𝒚𝒚𝒚𝒚) = 𝟏𝟏 + 𝟒𝟒𝟒𝟒 +‬‬ ‫ﻓﻲ ﻞ ﻤمﺎ ﺄﺘﻲ‪:‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝟐𝟐𝒊𝒊 ‪𝟏𝟏 − 𝟐𝟐𝟐𝟐 +‬‬ ‫𝒊𝒊)𝟏𝟏 ‪① 𝟐𝟐𝒙𝒙 − 𝟏𝟏 + 𝟐𝟐𝟐𝟐 = 𝟏𝟏 + (𝒚𝒚 +‬‬ ‫𝟒𝟒 ‪+ 𝒙𝒙 + 𝒚𝒚𝒚𝒚 = 𝟏𝟏 + 𝟒𝟒𝟒𝟒 −‬‬ ‫ ‬ ‫𝟏𝟏 = 𝟏𝟏 ‪𝟐𝟐𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝟐𝟐 = 𝒚𝒚 +‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫𝟏𝟏 ‪𝟏𝟏 − 𝟐𝟐𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝟐𝟐𝟐𝟐 = 𝟏𝟏 +‬‬ ‫𝒚𝒚 = 𝟏𝟏 ‪𝟐𝟐 −‬‬ ‫𝟒𝟒𝟒𝟒 ‪+ 𝒙𝒙 + 𝒚𝒚𝒚𝒚 = −𝟑𝟑 +‬‬ ‫𝟐𝟐 = 𝟐𝟐𝟐𝟐‬ ‫𝟏𝟏 = 𝒚𝒚‬ ‫𝟐𝟐‬ ‫𝒊𝒊𝟐𝟐‪−‬‬ ‫𝟏𝟏 = 𝒙𝒙‬ ‫𝟒𝟒𝟒𝟒 ‪+ 𝒙𝒙 + 𝒚𝒚𝒚𝒚 = −𝟑𝟑 +‬‬ ‫𝟖𝟖𝟖𝟖𝟖𝟖 ‪② 𝟑𝟑𝟑𝟑 + 𝟒𝟒𝟒𝟒 = 𝟐𝟐 +‬‬ ‫𝟐𝟐‬ ‫ ‬ ‫𝟐𝟐 = 𝟑𝟑𝟑𝟑‬ ‫𝟖𝟖𝟖𝟖 = 𝟒𝟒‬ ‫𝟒𝟒𝟒𝟒 ‪−𝒊𝒊 + 𝒙𝒙 + 𝒚𝒚𝒚𝒚 = −𝟑𝟑 +‬‬ ‫𝟒𝟒𝟒𝟒 ‪𝒙𝒙 + (𝒚𝒚 − 𝟏𝟏)𝒊𝒊 = −𝟑𝟑 +‬‬ ‫𝟐𝟐‬ ‫𝟏𝟏 𝟒𝟒‬ ‫= 𝒙𝒙‬ ‫= = 𝒚𝒚‬ ‫ ‬ ‫𝟑𝟑‪𝒙𝒙 = −‬‬ ‫𝟒𝟒 = 𝟏𝟏 ‪𝒚𝒚 −‬‬ ‫𝟑𝟑‬ ‫𝟐𝟐 𝟖𝟖‬ ‫𝟑𝟑𝟑𝟑 ‪③ (𝟐𝟐𝟐𝟐 + 𝟏𝟏) − (𝟐𝟐𝟐𝟐 − 𝟏𝟏)𝒊𝒊 = −𝟖𝟖 +‬‬ ‫𝟓𝟓 = 𝟏𝟏 ‪𝒚𝒚 = 𝟒𝟒 +‬‬ ‫ ‬ ‫𝟖𝟖‪𝟐𝟐𝟐𝟐 + 𝟏𝟏 = −‬‬ ‫𝟑𝟑 = )𝟏𝟏 ‪−(𝟐𝟐𝟐𝟐 −‬‬ ‫𝟏𝟏 ‪𝟐𝟐𝟐𝟐 = −𝟖𝟖 −‬‬ ‫𝟑𝟑 = 𝟏𝟏 ‪−𝟐𝟐𝟐𝟐 +‬‬ ‫𝟗𝟗‪𝟐𝟐𝟐𝟐 = −‬‬ ‫𝟏𝟏 ‪−𝟐𝟐𝟐𝟐 = 𝟑𝟑 −‬‬ ‫𝟗𝟗‪−‬‬ ‫= 𝒚𝒚‬ ‫𝟐𝟐 = 𝟐𝟐𝟐𝟐‪−‬‬ ‫𝟐𝟐‬ ‫‪6‬‬ ‫‪set by: Sadiq Hussain‬‬  ‫ﺇﻋﺪﺍﺩ ﺍﻻﺳﺘﺎﺫ‪ :‬ﺻﺎﺩﻕ ﺣﺴﲔ‬ ‫ﺍﻟﻔﺼﻞ ﺍﻻﻭﻝ‪ :‬ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ | ﳎﻤﻮﻋﺔ ﺍﻻﻋﺪﺍﺩ ﺍﳌﺮﻛﺒﺔ ﻭﺍﻟﻌﻤﻠﻴﺎﺕ ﻋﻠﻴﻬﺎ‬ ‫𝟏𝟏𝟏𝟏‪𝒚𝒚 = −‬‬ ‫𝒊𝒊 ‪𝟐𝟐 −‬‬ ‫𝒊𝒊 ‪𝟑𝟑 −‬‬ ‫𝟏𝟏‬ ‫)𝒅𝒅‬ ‫‪𝒙𝒙 +‬‬ ‫= 𝒚𝒚‬ ‫وزاري ‪ 2017‬دور اول اﺤ ﺎﺌﻲ ‪ -‬ﺨﺎرج ﺠد ق مﺔ 𝑹𝑹 ∈ 𝒚𝒚 ‪ 𝒙𝒙,‬اذا‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟐𝟐 +‬‬ ‫𝒊𝒊‬ ‫𝒊𝒊‪𝟏𝟏−‬‬ ‫𝒊𝒊 ‪𝟐𝟐 − 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝟑𝟑 − 𝒊𝒊 𝟐𝟐 −‬‬ ‫𝒊𝒊‪𝟏𝟏 −‬‬ ‫ﻋﻠمت ان‪𝒙𝒙 + (𝟏𝟏 + 𝟑𝟑𝟑𝟑)𝟐𝟐 𝒚𝒚 = (𝟏𝟏 − 𝒊𝒊)(𝟏𝟏 + 𝟑𝟑𝟑𝟑) :‬‬ ‫ ‬ ‫‪ 𝒙𝒙 +‬‬ ‫ ‬ ‫ = 𝒚𝒚 ‬ ‫𝒊𝒊‪𝟏𝟏+‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝒊𝒊 ‪𝟐𝟐 + 𝒊𝒊 𝟐𝟐 −‬‬ ‫𝒊𝒊‪𝒊𝒊 −‬‬ ‫𝒊𝒊 ‪𝟏𝟏 −‬‬ ‫𝟐𝟐‬ ‫𝟐𝟐‬ ‫)𝟑𝟑𝟑𝟑 ‪𝒙𝒙 + (𝟏𝟏 + 𝟑𝟑𝟑𝟑)𝟐𝟐 𝒚𝒚 = (𝟏𝟏 − 𝒊𝒊)(𝟏𝟏 +‬‬ ‫𝒊𝒊 ‪𝟐𝟐 − 𝟐𝟐𝟐𝟐 − 𝒊𝒊 +‬‬ ‫𝒊𝒊 ‪𝟔𝟔 − 𝟑𝟑𝟑𝟑 − 𝟐𝟐𝟐𝟐 +‬‬ ‫𝒊𝒊‪−‬‬ ‫𝒊𝒊 ‪𝟏𝟏 +‬‬ ‫‪𝒙𝒙 +‬‬ ‫𝟐𝟐 = 𝒚𝒚‬ ‫𝒊𝒊 ‪𝟏𝟏 − 𝒊𝒊 𝟏𝟏 −‬‬ ‫𝟏𝟏 ‪𝟏𝟏 +‬‬ ‫𝟏𝟏 ‪𝟒𝟒 +‬‬ ‫𝒊𝒊‪−‬‬ ‫ ‬ ‫𝒚𝒚 𝟐𝟐𝒊𝒊𝟗𝟗 ‪ 𝒙𝒙 + 𝟏𝟏 + 𝟔𝟔𝟔𝟔 +‬‬ ‫𝟏𝟏 ‪𝟐𝟐 − 𝟑𝟑𝟑𝟑 −‬‬ ‫𝟏𝟏 ‪𝟔𝟔 − 𝟓𝟓𝟓𝟓 −‬‬ ‫𝒊𝒊‪−‬‬ ‫𝒊𝒊 ‪𝟏𝟏 + 𝒊𝒊 𝟏𝟏 −‬‬ ‫‪𝒙𝒙 +‬‬ ‫= 𝒚𝒚‬ ‫𝟐𝟐𝒊𝒊𝟑𝟑 ‪= 𝟏𝟏 + 𝟑𝟑𝟑𝟑 − 𝒊𝒊 −‬‬ ‫𝟐𝟐‬