{"id":9166,"date":"2025-04-13T14:18:34","date_gmt":"2025-04-13T05:18:34","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=9166"},"modified":"2025-04-24T17:48:35","modified_gmt":"2025-04-24T08:48:35","slug":"examples-of-arithmetic-of-complex-numbers","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/examples-of-arithmetic-of-complex-numbers\/","title":{"rendered":"\ubcf5\uc18c\uc218\uc758 \uc0ac\uce59\uacc4\uc0b0 \uc608\uc81c \ubaa8\uc74c"},"content":{"rendered":"<p><!--\n\n\n<div class=\"theorem\">\n\n<p>\n\uc138\uc885\uacfc\ud559\uc608\uc220\uc601\uc7ac\ud559\uad50 2022\ud559\ub144\ub3c4 1\ud559\uae30 \uc218\ud559 I 1\ucc28 \uc9c0\ud544\ud3c9\uac00\ub97c \uc900\ube44\ud560 \ub54c \uc774 \uae00\uc758 \u2018\ucc38\uace0\u2019\uc5d0 \ud574\ub2f9\ud558\ub294 \ubd80\ubd84\uc740 \uc77d\uc9c0 \uc54a\uc544\ub3c4 \ub429\ub2c8\ub2e4. (\ud765\ubbf8\ub97c \ub290\uaef4\uc11c \uc77d\ub294 \uac83\uc740 \ub9d0\ub9ac\uc9c0 \uc54a\uc2b5\ub2c8\ub2e4.)\n<\/p>\n\n<\/div>\n\n\n--><\/p>\n<div style=\"display: none; visibility: hidden;\">\n\\[<br \/>\n\\newcommand{\\complexI}{{\\mathbf{i}}}<br \/>\n\\]\n<\/div>\n<style type=\"text\/css\">\ndiv.proof p {\n\ttext-align: left;\n}\nspan.sasatemp {\n\tvisibility: hidden; display: none;\n}\n<\/style>\n<p>\ubcf5\uc18c\uc218\ub97c \ucc98\uc74c \uacf5\ubd80\ud560 \ub54c \ub3c4\uc6c0\uc774 \ub418\ub3c4\ub85d \ubcf5\uc18c\uc218\uc758 \uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \uc608\uc81c\uc640 \ud480\uc774\ub97c \ubaa8\uc558\uc2b5\ub2c8\ub2e4.<\/p>\n<p class=\"marginbottom2\">\ubaa8\ub4e0 \uc608\uc81c\uc640 \ud480\uc774\uc5d0\uc11c \\(\\complexI\\)\ub294 \ud5c8\uc218\ub2e8\uc704\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4.<!-- \\(\\mathbb{C}\\)\ub294 \ubcf5\uc18c\uc218 \uc804\uccb4\uc758 \uc9d1\ud569, \\(\\mathbb{R}\\)\uc740 \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569, \\(\\mathbb{Z}\\)\ub294 \uc815\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc744 \ub098\ud0c0\ub0c5\ub2c8\ub2e4. --><!-- \ub610\ud55c \ubcf5\uc18c\uc218 \\(z\\)\ub97c \\(z = a + b \\complexI\\)\uc640 \uac19\uc740 \uaf34\ub85c \ub098\ud0c0\ub0b4\uc5c8\uc744 \ub54c, \ubcc4\ub2e4\ub978 \uc5b8\uae09\uc774 \uc5c6\uc73c\uba74 \\(a\\)\uc640 \\(b\\)\ub294 \uc2e4\uc218\ub97c \ub098\ud0c0\ub0b4\ub294 \uac83\uc73c\ub85c \uc57d\uc18d\ud569\ub2c8\ub2e4. --><\/p>\n<p><span class=\"sasatemp\">\uac01 \ubb38\ud56d\uc5d0\uc11c \uad04\ud638 \uc548\uc5d0 \uc788\ub294 \ubc88\ud638\ub294 \uc138\uc885\uacfc\ud559\uc608\uc220\uc601\uc7ac\ud559\uad50 \uc218\ud559\u2160 \uad50\uacfc\uc11c\uc5d0 \uc2e4\ub9b0 \ubb38\ud56d \ubc88\ud638\uc785\ub2c8\ub2e4.<\/span><\/p>\n<p><!-- \n\n\n<div class=\"box\">\n\n<p><span class=\"theorem\">\uc608\uc81c 1.<\/span>\n\ub2e4\uc74c\uc744 \uacc4\uc0b0\ud558\uc5ec \\(a+b\\complexI\\) \uaf34\ub85c \ub098\ud0c0\ub0b4\uc2dc\uc624.<span class=\"sasatemp\"> (\uc720\uc81c 2.18)<\/span><\/p>\n\n\n\n\n<p>\n(1) \\((2+3\\complexI ) + (1+\\complexI )\\) <br \/>\n(2) \\((1+ \\sqrt{3} \\complexI )^2 - ( 1 - \\sqrt{3} \\complexI )^2 \\) <br \/>\n(3) \\(\\displaystyle \\left( \\frac{1-\\complexI }{1+ \\complexI} \\right)^2 \\) <br \/>\n(4) \\(\\displaystyle \\frac{1+\\complexI}{2-\\complexI}\\)\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\ud480\uc774<\/p>\n\n\n\n\n<p>(1) \\[\\begin{align}\n(2+3\\complexI) + (1+\\complexI) &= (2+1) + (3\\complexI+\\complexI)\\\\\n&= 3 + (3+1)\\complexI\\\\\n&= 3 + 4\\complexI\n\\end{align}\\]<\/p>\n\n\n\n\n\n<p>(2) \uba3c\uc800 \uac01 \ud56d\uc744 \uc804\uac1c\ud558\uc5ec \uacc4\uc0b0\ud55c\ub2e4.\n\\[\\begin{aligned}\n(1+ \\sqrt{3} \\complexI )^2 &= (1)^2 + 2(1)(\\sqrt{3} \\complexI) + (\\sqrt{3} \\complexI)^2\\\\[6pt]\n&= 1 + 2\\sqrt{3} \\complexI + 3 \\cdot (\\complexI)^2\\\\[6pt]\n&= 1 + 2\\sqrt{3} \\complexI + 3 \\cdot (-1)\\\\[6pt]\n&= 1 + 2\\sqrt{3} \\complexI - 3\\\\[6pt]\n&= -2 + 2\\sqrt{3} \\complexI \n\\end{aligned}\\]\n\ub2e4\uc74c\uc73c\ub85c \ub450 \ubc88\uc9f8 \ud56d\uc744 \uacc4\uc0b0\ud55c\ub2e4.\n\\[\\begin{aligned}\n(1 - \\sqrt{3} \\complexI)^2 &= (1)^2 + 2(1)(-\\sqrt{3} \\complexI) + (-\\sqrt{3} \\complexI)^2\\\\[6pt]\n&= 1 - 2\\sqrt{3} \\complexI + 3 \\cdot (\\complexI)^2\\\\[6pt]\n&= 1 - 2\\sqrt{3} \\complexI + 3 \\cdot (-1)\\\\[6pt]\n&= 1 - 2\\sqrt{3} \\complexI - 3\\\\[6pt]\n&= -2 - 2\\sqrt{3} \\complexI\n\\end{aligned}\\]\n\uc774\uc81c \ub450 \uacc4\uc0b0 \uacb0\uacfc\ub97c \ube7c\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\n(1+ \\sqrt{3} \\complexI )^2 - ( 1 - \\sqrt{3} \\complexI )^2 \n&= (-2 + 2\\sqrt{3} \\complexI) - (-2 - 2\\sqrt{3} \\complexI)\\\\[6pt]\n&= -2 + 2\\sqrt{3} \\complexI + 2 + 2\\sqrt{3} \\complexI\\\\[6pt]\n&= 0 + 4\\sqrt{3} \\complexI\\\\[6pt]\n&= 4\\sqrt{3} \\complexI\n\\end{aligned}\\]\n\n\ub530\ub77c\uc11c \\(a+b\\complexI\\) \uaf34\ub85c \ub098\ud0c0\ub0b4\uba74 \\(a=0\\), \\(b=4\\sqrt{3}\\)\uc774\ubbc0\ub85c \\(0+4\\sqrt{3}\\complexI\\) \ub610\ub294 \uac04\ub2e8\ud788 \\(4\\sqrt{3}\\complexI\\)\uc774\ub2e4.\n<\/p>\n\n\n\n<\/div>\n\n\n--><\/p>\n<p><!--\n\n\n<div class=\"box\">\n\n<p><span class=\"theorem\">\uc608\uc81c 2.<\/span>\n\ub2e4\uc74c\uc744 \uac04\ub2e8\ud788 \ud558\uc2dc\uc624. (\uc720\uc81c 2.20)<\/p>\n\n\n\n\n<p>\n(1) \\(\\complexI ^{4n} \\)<br \/>\n(2) \\(\\complexI ^{4n+3} \\)<br \/>\n(3) \\(\\displaystyle \\left( \\frac{1+\\complexI}{\\sqrt{2}}\\right)^{8n} + \\left( \\frac{1-\\complexI}{\\sqrt{2}}\\right)^{8n}\\) <br \/>\n(4) \\(\\displaystyle \\left( \\frac{1+\\complexI}{1-\\complexI}\\right)^{2020}\\)<br \/>\n(5) \\(\\displaystyle \\left( \\frac{1+\\sqrt{3} \\complexI }{2} \\right)^{100} + \\left( \\frac{1-\\sqrt{3} \\complexI }{2} \\right)^{98} \\)\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\ud480\uc774<\/p>\n\n\n\n\n<p>..\n\n\n<\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 1.<\/span><br \/>\n\\( a = 3+ \\sqrt{3} \\complexI ,\\) \\(b = 3-\\sqrt{3} \\complexI \\)\uc77c \ub54c,  \\(a^3 &#8211; a^2 b &#8211; ab^2 + b^3 \\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc720\uc81c 2.21)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(a^3 &#8211; a^2 b &#8211; ab^2 + b^3 \\)\uc758 \uac01 \ud56d\uc744 \uacc4\uc0b0\ud558\uc5ec \ub354\ud560 \uc218\ub3c4 \uc788\uc9c0\ub9cc, \uc6b0\uc120 \uc778\uc218\ubd84\ud574\ud558\uc5ec \uc815\ub9ac\ud55c \ub4a4 \uacc4\uc0b0\ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\na^3 &#8211; a^2 b &#8211; ab^2 + b^3<br \/>\n&#038;= a^2 (a-b) &#8211; b^2 (a-b) \\\\[6pt]<br \/>\n&#038;= (a^2 &#8211; b^2 )(a-b) \\\\[6pt]<br \/>\n&#038;= (a+b)(a-b)(a-b) \\\\[6pt]<br \/>\n&#038;= (a+b)(a-b)^2<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370<br \/>\n\\[a+b = 6 ,\\quad a-b = 2\\sqrt{3} \\complexI \\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[(a+b)(a-b)^2 = 6\\times (2\\sqrt{3} \\complexI)^2 = &#8211; 72\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 2.<\/span><br \/>\n\\(\\displaystyle x = \\frac{1+\\sqrt{3}\\complexI }{2}\\)\uc77c \ub54c, \\(x^4 &#8211; x^3 +3x &#8211; 2\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc720\uc81c 2.22)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\ubb38\uc81c\uc5d0\uc11c \uc8fc\uc5b4\uc9c4 \\(x\\)\uc758 \uac12\uc740 \ubc29\uc815\uc2dd \\(x^3 = -1\\)\uc758 \ud55c \uadfc\uc774\ub2e4. (\uc774 \uc0ac\uc2e4\uc744 \ubab0\ub790\ub2e4\uba74, \\(x\\)\uc758 \uac70\ub4ed\uc81c\uacf1\uc744 \uacc4\uc0b0\ud574 \ubcf4\uc544\uc57c \ud55c\ub2e4\u315c\u315c) \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\begin{aligned}<br \/>\nx^4 &#8211; x^3 + 3x &#8211; 2<br \/>\n&#038;= -x +1 +3x -2 \\\\[6pt]<br \/>\n&#038;= 2x -1 \\\\[6pt]<br \/>\n&#038;= 1 + \\sqrt{3}\\complexI -1 \\\\[6pt]<br \/>\n&#038;= \\sqrt{3} \\complexI<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<\/div>\n<p><!--\n\n\n<div class=\"box\">\n\n<p><span class=\"theorem\">\uc608\uc81c 3.<\/span>\n\uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \uc9d1\ud569 \\(A_n\\)\uc744\n\\[A_n = \\left\\{ z\\in\\mathbb{C} \\,\\vert\\, z^n = 1 \\right\\}\\]\n\uc774\ub77c\uace0 \uc815\uc758\ud558\uc790. \ub450 \uc870\uac74<br \/>\n\u3000(\uac00) \\(\\displaystyle \\frac{-1+\\sqrt{3}\\complexI}{2} \\in A_n ,\\)<br \/>\n\u3000(\ub098) \\(z\\in A_n\\)\uc774\uba74 \\(-z_n \\in A_n\\)\uc774\ub2e4<br \/>\n\ub97c \ubaa8\ub450 \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(n\\)\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc720\uc81c 2.23)<\/span>\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\ud480\uc774<\/p>\n\n\n\n\n<p>..\n\n\n<\/p>\n\n\n<\/div>\n\n\n--><\/p>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 3.<\/span><br \/>\n\\(z,\\) \\(w\\)\uac00 \ud5c8\uc218\uc77c \ub54c, \\(z+w\\)\uc640 \\(zw\\)\uac00 \ubaa8\ub450 \uc2e4\uc218\uc774\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(z = \\overline{w}\\)\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 8-(a))<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(z = a+b\\complexI ,\\) \\(w = c+d\\complexI \\)\ub77c\uace0 \ud558\uc790. \uc5ec\uae30\uc11c \\(a,\\) \\(b,\\) \\(c,\\) \\(d\\)\ub294 \ubaa8\ub450 \uc2e4\uc218\uc774\ub2e4. \uc774\ub54c \\(z+w\\)\uc640 \\(zw\\)\ub97c \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nz+w &#038;= (a+b\\complexI )+(c+d\\complexI ) = (a+c) + (b+d)\\complexI ,\\\\[6pt]<br \/>\nzw &#038;= (a+b\\complexI )(c+d\\complexI ) = (ac-bd) + (ad+bc)\\complexI<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>[\\(\\Rightarrow\\) \uc99d\uba85] \\(z+w\\)\uc640 \\(zw\\)\uac00 \ubaa8\ub450 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(z+w\\)\uc640 \\(zw\\)\ub97c \uacc4\uc0b0\ud55c \uacb0\uacfc\uc5d0\uc11c<br \/>\n\\[b+d = 0 , \\quad ad+bc = 0\\]<br \/>\n\uc774\ub2e4. \uc989 \\(b=-d\\)\uc774\uace0 \\(a=c\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\overline{w} = c-d\\complexI = a+b\\complexI = z\\]<br \/>\n\uc774\ub2e4.<\/p>\n<p>[\\(\\Leftarrow\\) \uc99d\uba85] \uc5ed\uc73c\ub85c, \\(z = \\overline{w}\\)\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[a+b\\complexI = z = \\overline{w} = c-d\\complexI\\]<br \/>\n\uc774\ub2e4. \uc2e4\uc218\ubd80\uc640 \ud5c8\uc218\ubd80\ub97c \uac01\uac01 \ube44\uad50\ud558\uba74<br \/>\n\\[b+d = 0 , \\quad ad+bc = 0\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(z+w\\)\uc640 \\(zw\\)\ub97c \uacc4\uc0b0\ud55c \uacb0\uacfc\uc5d0\uc11c \ubaa8\ub450 \ud5c8\uc218\ubd80\uac00 \\(0\\)\uc774 \ub41c\ub2e4. \uc989 \\(z+w\\)\uc640 \\(zw\\)\uac00 \ubaa8\ub450 \uc2e4\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 4.<\/span><br \/>\n\\(z\\)\uac00 \ubcf5\uc18c\uc218\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(z = \\overline{z}\\)\uac00 \uc131\ub9bd\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(z\\)\uac00 \uc2e4\uc218\uc784\uc744 \ubcf4\uc774\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 8-(b))<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(z= a+b\\complexI\\)\uc774\uace0 \\(a\\)\uc640 \\(b\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<p>[\\(\\Rightarrow\\) \uc99d\uba85] \\(z=\\overline{z}\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[a+b\\complexI = z = \\overline{z} = a-b\\complexI\\]<br \/>\n\uc774\ubbc0\ub85c \\(a=a\\) \uadf8\ub9ac\uace0 \\(b = -b\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(b=0\\)\uc774\uba70, \\(z=a+b\\complexI = a\\)\ub294 \uc2e4\uc218\uc774\ub2e4.<\/p>\n<p>[\\(\\Leftarrow\\) \uc99d\uba85] \\(z\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(z=a+b\\complexI\\)\uc5d0\uc11c \ud5c8\uc218\ubd80\uc778 \\(b\\)\uac00 \\(0\\)\uc774\ub2e4. \uc989 \\(z=a+b\\complexI = a\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[z = a+b\\complexI = a-b\\complexI = \\overline{z}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 5.<\/span><br \/>\n\\(z\\)\uac00 \ud5c8\uc218\uc774\uace0 \\(z+\\frac{1}{z}\\)\uc774 \uc2e4\uc218\uc77c \ub54c, \\(z\\overline{z}\\)\ub97c \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 8-(c))<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(z = a+b\\complexI\\)\ub77c\uace0 \ud558\uc790. \ubb38\uc81c\uc758 \uac00\uc815\uc5d0 \uc758\ud558\uc5ec \\(z\\)\uac00 \ud5c8\uc218\uc774\ubbc0\ub85c, \\(b\\ne 0\\)\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \\(z+\\frac{1}{z}\\)\uc744 \uacc4\uc0b0\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nz+\\frac{1}{z}<br \/>\n&#038;= (a+b\\complexI ) + \\frac{1}{a+b\\complexI} \\\\[6pt]<br \/>\n&#038;= (a+b\\complexI ) + \\frac{a-b\\complexI}{a^2 + b^2} \\\\[6pt]<br \/>\n&#038;= \\left( a + \\frac{a}{a^2 + b^2} \\right) + \\left( b &#8211; \\frac{b}{a^2 +b^2} \\right)\\complexI .<br \/>\n\\end{aligned}\\]<br \/>\n\uc774 \uac12\uc774 \uc2e4\uc218\uc774\ubbc0\ub85c, \ud5c8\uc218\ubd80\uac00 \\(0\\)\uc774\uc5b4\uc57c \ud55c\ub2e4. \uc989<br \/>\n\\[b &#8211; \\frac{b}{a^2 + b^2} = 0\\]<br \/>\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(b\\ne 0\\)\uc774\ubbc0\ub85c<br \/>\n\\[a^2 + b^2 = 1\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[z\\overline{z} = (a+b\\complexI )(a-b\\complexI ) = a^2 + b^2 = 1\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 6.<\/span><br \/>\n\ubcf5\uc18c\uc218 \\(z=1+\\complexI \\)\uc5d0 \ub300\ud558\uc5ec<br \/>\n\\[\\frac{z}{\\overline{z}} + \\left( \\frac{z}{\\overline{z}} \\right)^2 + \\left( \\frac{z}{\\overline{z}} \\right)^3 + \\left( \\frac{z}{\\overline{z}} \\right)^4\\]<br \/>\n\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 9)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc6b0\uc120 \\(z\/\\overline{z}\\)\uc758 \uac12\uc744 \uad6c\ud558\uc790.<br \/>\n\\[\\frac{z}{\\overline{z}} = \\frac{1+\\complexI}{1-\\complexI} = \\frac{(1+\\complexI )^2}{2} = \\frac{ 1 + 2\\complexI -1}{2} = \\complexI .\\]<br \/>\n\uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \uac12\uc740<br \/>\n\\[\\complexI + \\complexI ^2 + \\complexI ^3 + \\complexI ^4 = \\complexI -1 -\\complexI +1 = 0\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 7.<\/span><br \/>\n\ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc815\uc218 \\(x\\)\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 10)<\/span><\/p>\n<p style=\"text-align: center\">\n\u201c\\(z = 3x+(2x-7)\\complexI \\)\uc77c \ub54c \\(z^2 + \\left( \\overline{z} \\right)^2\\)\uc774 \uc74c\uc218\uc774\ub2e4.\u201d\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc6b0\uc120 \\(z^2 + \\left( \\overline{z} \\right)^2\\)\uc774 \uc74c\uc218\uac00 \ub418\ub294 \uc870\uac74\uc744 \ucc3e\uc544\ubcf4\uc790. \\(z=a+b\\complexI\\)\uc774\uace0 \\(a,\\) \\(b\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790.<br \/>\n\\[\\begin{aligned}<br \/>\nz^2 + \\left( \\overline{z} \\right)^2<br \/>\n&#038;= (a+b\\complexI )^2 + ( a-b\\complexI )^2 \\\\[6pt]<br \/>\n&#038;= a^2 + 2ab\\complexI -b^2 + a^2 &#8211; 2ab\\complexI -b^2 \\\\[6pt]<br \/>\n&#038;= 2(a^2 -b^2 )<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c, \\(z^2 + \\left( \\overline{z} \\right)^2\\)\uc774 \uc74c\uc218\uac00 \ub418\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740<br \/>\n\\[a^2 < b^2\\]\n\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc5d0 \\(a = 3x ,\\) \\(b = 2x-7\\)\uc744 \ub300\uc785\ud558\uba74\n\\[ (3x)^2 < (2x-7)^2 \\]\n\uc774\ub2e4. \uc774 \ubd80\ub4f1\uc2dd\uc744 \ud480\uba74\n\\[\\begin{gathered}\n9x^2  < 4x^2 - 28x + 49,  \\\\[6pt]\n5x^2 + 28x -49  < 0\n\\end{gathered}\\]\n\uc774\ubbc0\ub85c\n\\[-7 < x < 1.4\\]\n\uc774\ub2e4. \uc774 \ubc94\uc704\uc5d0 \uc788\ub294 \uc815\uc218\ub294\n\\[-6 ,\\,\\, -5 ,\\,\\, -4 ,\\,\\, -3 ,\\,\\, -2 ,\\,\\, -1 ,\\,\\, 0 ,\\,\\, 1\\]\n\ub85c\uc11c, \uadf8 \uac1c\uc218\ub294 \\(8\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 8.<\/span><br \/>\n\ub450 \uc2e4\uc218 \\(a,\\) \\(b\\)\uc5d0 \ub300\ud558\uc5ec, \ubcf5\uc18c\uc218 \\(z = a+b\\complexI\\)\uac00 \\(z^2 + \\left(\\overline{z}\\right)^2 = 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(6a + 12 b^2 + 11\\)\uc758 \ucd5c\uc19f\uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 11)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(z^2 + \\left( \\overline{z} \\right)^2\\)\uc758 \uac12\uc744 \uacc4\uc0b0\ud558\uba74<br \/>\n\\[\\begin{aligned}<br \/>\nz^2 + \\left( \\overline{z} \\right)^2<br \/>\n&#038;= (a+b\\complexI )^2 + ( a-b\\complexI )^2 \\\\[6pt]<br \/>\n&#038;= a^2 + 2ab\\complexI -b^2 + a^2 &#8211; 2ab\\complexI -b^2 \\\\[6pt]<br \/>\n&#038;= 2(a^2 -b^2 )<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ub2e4. \uc774 \uac12\uc774 \\(0\\)\uc774 \ub418\ub824\uba74<br \/>\n\\[a^2 = b^2\\]<br \/>\n\uc774\uc5b4\uc57c \ud55c\ub2e4. \uc774\ub54c<br \/>\n\\[ 6a + 12b^2 +11 = 6a + 12a^2 +11 = 12a^2 + 6a + 11 \\]<br \/>\n\uc774\ub2e4. \uc774 \uac12\uc774 \ucd5c\uc18c\uac00 \ub418\ub294 \uac74<br \/>\n\\[a = &#8211; \\frac{1}{2} \\times \\frac{6}{12} = &#8211; \\frac{1}{4}\\]<br \/>\n\uc77c \ub54c\uc774\ub2e4. \uc774 \uac12\uc744 \ub300\uc785\ud558\uc5ec \uacc4\uc0b0\ud558\uba74<br \/>\n\\[12a^2 + 6a + 11 = 12 \\times \\left(-\\frac{1}{4}\\right)^2 + 6 \\times \\left(-\\frac{1}{4}\\right) + 11 = \\frac{41}{4} \\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 9.<\/span><br \/>\n\ud5c8\uc218 \\(z = (\\complexI -2 )x^2 &#8211; 3x\\complexI &#8211; 4\\complexI + 32\\)\uac00 \\(z + \\overline{z} = 0\\)\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \uc2e4\uc218 \\(x\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc5f0\uc2b5\ubb38\uc81c 12)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(z + \\overline{z} =0\\)\uc774 \ub418\uae30 \uc704\ud55c \uc870\uac74\uc744 \uad6c\ud558\uc790. \\(z=a+b\\complexI\\)\ub77c\uace0 \ud558\uba74<br \/>\n\\[z+\\overline{z} = (a+b\\complexI ) + (a- b\\complexI ) = 2a\\]<br \/>\n\uc774\ubbc0\ub85c, \uad6c\ud558\ub294 \uc870\uac74\uc740<br \/>\n\\[a=0 \\tag{*}\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8<br \/>\n\\[z = (-2x^2 +32) + (x^2 -3x -4 ) \\complexI \\tag{**}\\]<br \/>\n\uc774\ubbc0\ub85c, (*)\uc5d0 \\(a = -2x^2 +32\\)\ub97c \ub300\uc785\ud558\uba74<br \/>\n\\[-2x^2 +32 = 0\\]<br \/>\n\uc774\ub2e4. \uc774 \ubc29\uc815\uc2dd\uc744 \ud480\uba74<br \/>\n\\[x=\\pm 4\\]<br \/>\n\uc774\ub2e4. \ud55c\ud3b8 \\(z\\)\uac00 \ud5c8\uc218\uc774\ubbc0\ub85c, (**)\uc5d0\uc11c \ud5c8\uc218\ubd80\uac00 \\(0\\)\uc774 \uc544\ub2c8\ub2e4. \uc989<br \/>\n\\[x^2 -3x -4 \\ne 0\\]<br \/>\n\uc774\ub2e4. \\(x=4\\)\uc77c \ub54c \\(x^2 -3x -4\\)\uc758 \uac12\uc774 \\(0\\)\uc774\uace0, \\(x=-4\\)\uc77c \ub54c<br \/>\n\\[x^2 -3x -4 = 16 + 12 -4 \\ne 0\\]<br \/>\n\uc774\ubbc0\ub85c, \uad6c\ud558\ub294 \uac12\uc740<br \/>\n\\[x = -4\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 10.<\/span><br \/>\n\ub124 \uac1c\uc758 \uc6d0\uc18c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc9d1\ud569<br \/>\n\\[A = \\left\\{ a , \\,\\, a^2 ,\\,\\, a^3 ,\\,\\, a^4 \\right\\}\\]<br \/>\n\uc774 \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\ub2e4. \uc774\ub54c \\(a^{10}\\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc885\ud569\ubb38\uc81c 8)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\uc6b0\uc120 \\(a=0\\) \ub610\ub294 \\(a=1\\) \ub610\ub294 \\(a=-1\\)\uc774\uba74 \\(A\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \\(2\\) \uc774\ud558\uac00 \ub418\ubbc0\ub85c, \\(a\\)\uc758 \uac12\uc740 \\(0,\\) \\(1\\) \\(-1\\) \uc911 \uc5b4\ub290 \uac83\ub3c4 \ub420 \uc218 \uc5c6\ub2e4.<\/p>\n<p>\uc774\uc81c \\(A\\)\uc758 \uc6d0\uc18c \uac01\uac01\uc5d0 \\(a\\)\ub97c \uacf1\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ub124 \uac1c\uc758 \uc218\ub97c \ub9cc\ub4e4\uc790.<br \/>\n\\[a^2 ,\\quad a^3 ,\\quad a^4 ,\\quad a^5 . \\tag{*}\\]<br \/>\n\uc704 \ub124 \uac1c\uc758 \uc218\ub294 \ubaa8\ub450 \uc11c\ub85c \ub2ec\ub77c\uc57c \ud55c\ub2e4. \uc65c\ub0d0\ud558\uba74, \ub9cc\uc57d \ub124 \uac1c\uc758 \uc218 \uc911 \uac19\uc740 \uac83\uc774 \uc874\uc7ac\ud55c\ub2e4\uba74, \uadf8 \ub450 \uac1c\uc758 \uac12\uc744 \\(a\\)\ub85c \ub098\ub204\uc5b4\ub3c4 \uac19\uc544\uc57c \ud558\ub294\ub370, \uadf8\ub7ec\uba74 \\(A\\)\uc758 \uc6d0\uc18c\uac00 \ubaa8\ub450 \ub2e4\ub974\ub2e4\ub294 \uac00\uc815\uc5d0 \ubaa8\uc21c\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. (\uc608\ub97c \ub4e4\uc5b4 \\(a^2 = a^5\\)\ub77c\uba74, \uc591\ubcc0\uc744 \\(a\\)\ub85c \ub098\ub204\uc5b4 \\(a = a^4\\)\uc774 \ub418\ubbc0\ub85c \ubaa8\uc21c\uc774\ub2e4.)<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c (*)\uc758 \uc6d0\uc18c\ub294 \ubaa8\ub450 \ub2e4\ub974\ub2e4. \uadf8\ub7f0\ub370 \\(A\\)\uac00 \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\uc73c\ubbc0\ub85c, (*)\uc758 \uc6d0\uc18c\ub294 \ubaa8\ub450 \\(A\\)\uc5d0 \uc18d\ud55c\ub2e4. \ud2b9\ud788 \\(a^2 ,\\) \\(a^3 ,\\) \\(a^4\\)\uc774 \uc774\ubbf8 \\(A\\)\uc5d0 \uc788\uc73c\ubbc0\ub85c, \\(a^5\\)\uc758 \uac12\uc774 \\(a\\)\uc640 \uac19\uc744 \uc218\ubc16\uc5d0 \uc5c6\ub2e4. \uc989 \ub2e4\uc74c\uacfc \uac19\uc740 \ubc29\uc815\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[a^5 = a\\]<br \/>\n\uc774 \ubc29\uc815\uc2dd\uc744 \ud480\uba74<br \/>\n\\[\\begin{aligned}<br \/>\na^5 -a &#038;=0 , \\\\[6pt]<br \/>\na(a^4 -1 ) &#038;= 0 , \\\\[6pt]<br \/>\na( a^2 +1) (a+1)(a-1) &#038;= 0<br \/>\n\\end{aligned}\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[a = 1 \\quad\\text{or}\\quad a=-1 \\quad\\text{or}\\quad a=\\complexI \\quad\\text{or}\\quad a=-\\complexI \\]<br \/>\n\uc774\ub2e4. \uadf8\ub7f0\ub370 \\(1\\)\uacfc \\(-1\\)\uc740 \\(a\\)\uc758 \uac12\uc774 \ub420 \uc218 \uc5c6\uc73c\ubbc0\ub85c, \\(a\\)\uc758 \uac12\uc774 \ub420 \uc218 \uc788\ub294 \ud6c4\ubcf4\ub294<br \/>\n\\[\\complexI ,\\quad -\\complexI\\]<br \/>\n\ubfd0\uc774\ub2e4. \uc774 \ub450 \uac1c\uc758 \uac12 \ubaa8\ub450 \\(A\\)\uc758 \uc6d0\uc18c\uc758 \uac1c\uc218\uac00 \\(4\\)\ub77c\ub294 \uac00\uc815\uacfc \\(A\\)\uac00 \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\ub2e4\ub294 \uac00\uc815\uc5d0 \ubd80\ud569\ud55c\ub2e4.<\/p>\n<p>\\(a=\\complexI\\)\uc77c \ub54c \\(a^{10} = -1\\)\uc774\uace0, \\(a=-\\complexI\\)\uc77c \ub54c\uc5d0\ub3c4 \\(a^{10} = -1\\)\uc774\ubbc0\ub85c, \uad6c\ud558\ub294 \uac12\uc740 \\(-1\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 11.<\/span><br \/>\n\ub450 \uc870\uac74 \\(z\\overline{z} = 5\\)\uc640 \\(z^2 + \\left( \\overline{z} \\right)^2 = 6\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \ubcf5\uc18c\uc218 \\(z\\)\uc758 \ud569\uc744 \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc885\ud569\ubb38\uc81c 9)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\\(z=a+b\\complexI \\)\ub77c\uace0 \ud558\uace0,  \\(a\\)\uc640 \\(b\\)\ub97c \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \ubb38\uc81c\uc5d0\uc11c \uc81c\uc2dc\ud55c \ub450 \uc870\uac74\uc744 \\(a\\)\uc640 \\(b\\)\uc5d0 \ub300\ud55c \uc870\uac74\uc73c\ub85c \ubc14\uafb8\uc5b4 \ubcf4\uc790. \uc6b0\uc120 \uccab \ubc88\uc9f8 \uc870\uac74\uc740<br \/>\n\\[z\\overline{z} = a^2 + b^2 = 5 \\tag{*}\\]<br \/>\n\uc774\ub2e4. \ub2e4\uc74c\uc73c\ub85c \ub450 \ubc88\uc9f8 \uc870\uac74\uc740<br \/>\n\\[ z^2 + \\left( \\overline{z} \\right)^2 = a^2 + 2ab\\complexI &#8211; b^2 + a^2 &#8211; 2ab\\complexI -b^2 = 2(a^2 &#8211; b^2 ) =6\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[ a^2 &#8211; b^2 =  3\\tag{**}\\]<br \/>\n\uc774\ub2e4. \ub450 \ub4f1\uc2dd (*)\uacfc (**)\uc744 \uc5f0\ub9bd\ud558\uc5ec \ud480\uba74<br \/>\n\\[a^2 = 4 ,\\quad b^2 = 1\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c, \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubcf5\uc18c\uc218 \\(z\\)\ub97c \ub098\uc5f4\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{aligned}<br \/>\nz &#038;= +2 +  \\complexI , \\\\[6pt]<br \/>\nz &#038;= +2 &#8211;  \\complexI , \\\\[6pt]<br \/>\nz &#038;= -2 +  \\complexI , \\\\[6pt]<br \/>\nz &#038;= -2 &#8211;  \\complexI .<br \/>\n\\end{aligned}\\]<br \/>\n\uc774 \uac12\uc744 \ubaa8\ub450 \ub354\ud558\uba74 \\(0\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"box\">\n<p><span class=\"theorem\">\uc608\uc81c 12.<\/span><br \/>\n\uc11c\ub85c \ub2e4\ub978 \uc138 \ubcf5\uc18c\uc218\uac00 \uc788\uace0, \uc774\ub4e4\uc758 \ud569\uc740 \\(0\\)\uc774\ub2e4. \uc774\ub4e4 \uc911 \\(2\\)\uac1c\uc529 \ubf51\uc544\uc11c \uacf1\uc744 \ub9cc\ub4e4\uc5b4 \uc774\ub8e8\uc5b4\uc9c4 \uc138 \ubcf5\uc18c\uc218\uc758 \uc9d1\ud569\uc740 \ucc98\uc74c \uc138 \ubcf5\uc18c\uc218\uc758 \uc9d1\ud569\uacfc \uac19\ub2e4. \uc774\ub7ec\ud55c \uc138 \ubcf5\uc18c\uc218\ub97c \uad6c\ud558\uc2dc\uc624. <span class=\"sasatemp\">(\uc885\ud569\ubb38\uc81c 10)<\/span>\n<\/p>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\ud480\uc774<\/p>\n<p>\ub9cc\uc57d \uc138 \ubcf5\uc18c\uc218<br \/>\n\\[x,\\quad y,\\quad z \\tag{*}\\]<br \/>\n\uc911\uc5d0 \\(0\\)\uc774 \uc874\uc7ac\ud55c\ub2e4\uba74<br \/>\n\\[xy ,\\quad yz ,\\quad xz \\tag{**}\\]<br \/>\n\uc911\uc5d0 \\(0\\)\uc774 \ub450 \uac1c\uac00 \uc874\uc7ac\ud558\uac8c \ub41c\ub2e4. \uc774\uac83\uc740 (*)\uacfc (**)\uc774 \ud558\ub098\uc529 \ub300\uc751\ub418\uc5b4\uc57c \ud55c\ub2e4\ub294 \ubb38\uc81c\uc758 \uc870\uac74\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x,\\) \\(y,\\) \\(z\\) \uc911\uc5d0 \\(0\\)\uc774 \uc874\uc7ac\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\n<p>\ub2e4\uc74c\uc73c\ub85c (*)\uc758 \uc138 \ubcf5\uc18c\uc218\ub97c (**)\uc758 \uc138 \ubcf5\uc18c\uc218\uc5d0 \ub300\uc751\uc2dc\ud0a4\ub294 \ubc29\ubc95\uc744 \uc0dd\uac01\ud558\uc790. \ub9cc\uc57d (*)\uc758 \uc138 \ubcf5\uc18c\uc218\uac00 \uc790\uc2e0\uc744 \uc778\uc218\ub85c \uac16\uc9c0 \uc54a\ub294 \ubcf5\uc18c\uc218\uc5d0\ub9cc \ub300\uc751\ub41c\ub2e4\uba74(\uc5c4\ubc00\ud558\uac8c \ud45c\ud604\ud558\uba74, \u201c\uc790\uc2e0\uc744 \ub098\ud0c0\ub0b4\ub294 \ubb38\uc790\uac00 \uc544\ub2cc \ubb38\uc790\uc758 \uacf1\uc73c\ub85c\ub9cc \ud45c\ud604\ub418\ub294 \ubcf5\uc18c\uc218\uc5d0 \ub300\uc751\ub41c\ub2e4\uba74\u201d\uc774\ub77c\uace0 \ud574\uc57c \ud55c\ub2e4), \uc989<br \/>\n\\[x = yz ,\\quad y = xz ,\\quad z = xy \\tag{***}\\]<br \/>\n\ub77c\uba74,<br \/>\n\\[xyz = x^2 = y^2 = z^2\\]<br \/>\n\uc774 \uc131\ub9bd\ud558\ubbc0\ub85c, \\(x,\\) \\(y,\\) \\(z\\) \uc911 \ub450 \uac1c\ub294 \uc11c\ub85c \uac19\uc544\uc57c \ud55c\ub2e4. \uc774\uac83\uc740 \\(x,\\) \\(y,\\) \\(z\\)\uac00 \uc11c\ub85c \ub2e4\ub978 \uc138 \ubcf5\uc18c\uc218\ub77c\ub294 \ubb38\uc81c\uc758 \uc870\uac74\uc5d0 \ubaa8\uc21c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c (***)\uacfc \uac19\uc774 \ub300\uc751\ub418\uc9c0\ub294 \uc54a\ub294\ub2e4. \uc989<br \/>\n\\[x = xy \\quad\\text{or}\\quad x=xz \\quad\\text{or}\\quad y=yx \\quad\\text{or}\\quad y=yz \\quad\\text{or}\\quad z=xz \\quad\\text{or}\\quad z = yz\\]<br \/>\n\uc640 \uac19\uc774, \\(x,\\) \\(y,\\) \\(z\\) \uc911 \uc801\uc5b4\ub3c4 \ud558\ub098\ub294 (**)\uc758 \uc138 \uc218 \uc911 \uc790\uc2e0\uc744 \uc778\uc218\ub85c \uac16\ub294 \uc2dd\uc774 \ub098\ud0c0\ub0b4\ub294 \uc218\uc5d0 \ub300\uc751\ub418\uc5b4\uc57c \ud55c\ub2e4. \uc5ec\uc12f \uac00\uc9c0 \uacbd\uc6b0 \uc911 \uc5b4\ub290 \uacbd\uc6b0\ub4e0, \\(x,\\) \\(y,\\) \\(z\\) \uc911 \ud558\ub098\uac00 \\(1\\)\uc774 \ub41c\ub2e4\ub294 \uc0ac\uc2e4\uc744 \uc54c \uc218 \uc788\ub2e4. (\uc608\ub97c \ub4e4\uc5b4 \\(y = yz\\)\ub77c\uba74, \\(z=1\\)\uc774 \ub41c\ub2e4.)<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \uc138 \ubcf5\uc18c\uc218 \uc911 \ud558\ub098\ub97c \\(1\\)\uc774\ub77c\uace0 \ud558\uace0, \ub2e4\ub978 \ub450 \ubcf5\uc18c\uc218\ub97c \\(x,\\) \\(y\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubb38\uc81c\uc758 \uc870\uac74\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[x+y+1 = 0\\]<br \/>\n\uc774\uace0<br \/>\n\\[xy=1\\]<br \/>\n\uc774 \uc131\ub9bd\ud574\uc57c \ud55c\ub2e4. \ub450 \uc2dd\uc744 \uc5f0\ub9bd\ud558\uc5ec \ud480\uba74, \\(x\\)\uc640 \\(y\\)\uc758 \uac12\uc740 \ub450 \ubcf5\uc18c\uc218<br \/>\n\\[\\frac{-1 + \\sqrt{3} \\complexI}{2} ,\\quad \\frac{-1 &#8211; \\sqrt{3} \\complexI}{2}\\]<br \/>\n\uc5d0 \ud558\ub098\uc529 \ub300\uc751\ub41c\ub2e4. \ubb3c\ub860, \\(x\\)\uc640 \\(y\\)\uc758 \uac12\uc740 \uc11c\ub85c \ubc14\ub014 \uc218 \uc788\ub2e4.<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c, \uc138 \ubcf5\uc18c\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[1 ,\\quad \\frac{-1 + \\sqrt{3} \\complexI}{2} ,\\quad \\frac{-1 &#8211; \\sqrt{3} \\complexI}{2} .\\]\n<\/p>\n<\/div>\n<p><!--\n\n\n<div class=\"example\">\n\n\n<p><span class=\"theorem\">\ucc38\uace0.<\/span>\n\uc0ac\uc2e4 \uc704 \ubb38\uc81c\uc758 \ud480\uc774\ub294 \ub9e4\uc6b0 \uac04\ub2e8\ud558\uac8c \ud560 \uc218 \uc788\ub2e4. \uc11c\ub85c \ub2e4\ub978 \\(n\\)\uac1c\uc758 \ubcf5\uc18c\uc218\ub97c \uc6d0\uc18c\ub85c \uac00\uc9c0\uace0 \uc788\uace0, \uc6d0\uc18c\uc758 \ud569\uc774 \\(0\\)\uc774\uba74\uc11c, \uacf1\uc148\uc5d0 \ub300\ud558\uc5ec \ub2eb\ud600 \uc788\ub294 \uc9d1\ud569\uc740 \ud56d\uc0c1 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\nU_n = \\left\\{ \\cos \\left( \\frac{2 \\pi j}{n} \\right) + \\complexI \\sin \\left( \\frac{2 \\pi j}{n} \\right) \\,\\Big\\vert\\,  j = 0 ,\\, 1,\\, 2,\\, \\cdots ,\\, n-1 \\right\\}\n\\]\n\uc774 \uc9d1\ud569\uc758 \uc6d0\uc18c\ub294 \ubcf5\uc18c\ud3c9\uba74\uc5d0\uc11c \uc911\uc2ec\uc774 \uc6d0\uc810\uc778 \ub2e8\uc704\uc6d0\uc744 \uadf8\ub9b0 \ub4a4, \uc2e4\uc218 \\(1\\)\uc5d0 \ud574\ub2f9\ud558\ub294 \uacf3\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec \uc6d0\uc744 \\(n\\)\ub4f1\ubd84\ud558\ub3c4\ub85d \uc810\uc744 \ucc0d\uc5c8\uc744 \ub54c, \uac01 \uc810\uc774 \ub098\ud0c0\ub0b4\ub294 \ubcf5\uc18c\uc218\uc774\ub2e4. \uc774 \uc9d1\ud569\uc758 \uc6d0\uc18c\ub97c \u201c\ub2e8\uc704\uc6d0\uc758 \\(n\\)\uc81c\uacf1\uadfc\u201d(n-th root of unity)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n\n\n\n\n<p>\ub300\uc218\uc801\uc73c\ub85c \ubcf4\uc558\uc744 \ub54c \uacf1\uc148 \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(U_n\\)\uc740 \ubc95 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ud569\ub3d9\uc778 \ub367\uc148 \uc5f0\uc0b0\uc774 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(\\mathbb{Z} _n\\)\uacfc \ub3d9\uc77c\ud55c \uad6c\uc870\ub97c \uac00\uc9c4\ub2e4.<\/p>\n\n\n\n\n<p>\\(n=3\\)\uc77c \ub54c \\(U_n\\)\uc758 \uc6d0\uc18c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n\\[\\begin{aligned}\n\\cos \\frac{0}{3} + \\complexI \\sin \\frac{0}{3} &= 1 ,\\\\[3pt]\n\\cos \\frac{2\\pi}{3} + \\complexI \\sin \\frac{2\\pi}{3} &= \\frac{-1 + \\sqrt{3} \\complexI}{2} ,\\\\[3pt]\n\\cos \\frac{4\\pi}{3} + \\complexI \\sin \\frac{4\\pi}{3} &= \\frac{-1 - \\sqrt{3} \\complexI}{2} .\n\\end{aligned}\\]\n\uc774 \uac12\uc740 \uc55e\uc5d0\uc11c \uad6c\ud55c \uc138 \uac1c\uc758 \uac12\uacfc \uc77c\uce58\ud55c\ub2e4. \uc138 \uc6d0\uc18c\ub97c \uc21c\uc11c\ub300\ub85c \\(0,\\) \\(1,\\) \\(2\\)\uc5d0 \ub300\uc751\uc2dc\ud0a4\uba74\n\\[\\begin{aligned}\n0+0 &\\equiv 0 \\quad (\\text{mod }3), \\\\[6pt]\n0+1 &\\equiv 1 \\quad (\\text{mod }3), \\\\[6pt]\n0+2 &\\equiv 2 \\quad (\\text{mod }3), \\\\[6pt]\n1+0 &\\equiv 1 \\quad (\\text{mod }3), \\\\[6pt]\n1+1 &\\equiv 2 \\quad (\\text{mod }3), \\\\[6pt]\n1+2 &\\equiv 0 \\quad (\\text{mod }3), \\\\[6pt]\n2+0 &\\equiv 2 \\quad (\\text{mod }3), \\\\[6pt]\n2+1 &\\equiv 0 \\quad (\\text{mod }3), \\\\[6pt]\n1+2 &\\equiv 1 \\quad (\\text{mod }3)\n\\end{aligned}\\]\n\uc774\ubbc0\ub85c, \uacf1\uc148\uc774 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(U_3\\)\uc740 \ubc95 \\(3\\)\uc5d0 \ub300\ud558\uc5ec \ud569\ub3d9\uc778 \ub367\uc148\uc774 \uc8fc\uc5b4\uc9c4 \uc9d1\ud569 \\(\\mathbb{Z}_3 = \\left\\{ 0 ,\\, 1 ,\\, 2 \\right\\}\\)\uc758 \uad6c\uc870\uc640 \uc77c\uce58\ud55c\ub2e4.\n<\/div>\n\n\n--><\/p>\n<p><!-- ############################\n\n\\[\\begin{aligned}\n.. \\quad\n&\\Longleftrightarrow \\quad .. \\\\[6pt]\n\n\\end{aligned}\\]\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"box\">\n\n<p><span class=\"theorem\">\uc608\uc81c n.<\/span>\n\ub0b4\uc6a9\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\ud480\uc774<\/p>\n\n\n\n\n<p>..\n\n\n<\/p>\n\n\n<\/div>\n\n\n\n--><\/p>\n<p><!--\n\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"theorem\">\ucc38\uace0.<\/span>\n\uc704 \uc99d\uba85\uc740 \uc6d0\ub798 \ub2e4\uc74c\uacfc \uac19\uc774 \ud574\uc57c \ub354 \uc5c4\ubc00\ud558\ub2e4.\n\\[\\begin{aligned}\nA=B \\quad\n&\\Longleftrightarrow \\quad (\\forall x ( x\\in A \\,\\leftrightarrow\\, x\\in B)) \\\\[6pt]\n&\\Longleftrightarrow \\quad (\\forall x ( ( x\\in A \\,\\rightarrow \\, x\\in B) \\,\\wedge\\, (x\\in B \\,\\rightarrow \\,x\\in A))) \\\\[6pt]\n&\\Longleftrightarrow \\quad ((\\forall x  ( x\\in A \\,\\rightarrow \\, x\\in B)) \\,\\wedge\\, (\\forall x (x\\in B \\,\\rightarrow \\,x\\in A))) \\\\[6pt]\n&\\Longleftrightarrow \\quad (A\\subset B \\,\\wedge\\, B\\subset A ).\n\\end{aligned}\\]\n\uc774 \uc99d\uba85 \uacfc\uc815\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \ubc95\uce59\uc774 \uc0ac\uc6a9\ub418\uc5c8\ub2e4.\n\\[[\\forall x (p(x) \\wedge q(x))] \\quad\\Longleftrightarrow\\quad   [(\\forall x (p(x))) \\wedge (\\forall x (q(x)))] \\]\n<\/p>\n\n\n<\/div>\n\n\n\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\\( \\newcommand{\\complexI}{{\\mathbf{i}}} \\) \ubcf5\uc18c\uc218\ub97c \ucc98\uc74c \uacf5\ubd80\ud560 \ub54c \ub3c4\uc6c0\uc774 \ub418\ub3c4\ub85d \ubcf5\uc18c\uc218\uc758 \uc0ac\uce59\uacc4\uc0b0\uacfc \uad00\ub828\ub41c \uc608\uc81c\uc640 \ud480\uc774\ub97c \ubaa8\uc558\uc2b5\ub2c8\ub2e4. \ubaa8\ub4e0 \uc608\uc81c\uc640 \ud480\uc774\uc5d0\uc11c \\(\\complexI\\)\ub294 \ud5c8\uc218\ub2e8\uc704\ub97c \ub098\ud0c0\ub0c5\ub2c8\ub2e4. \uac01 \ubb38\ud56d\uc5d0\uc11c \uad04\ud638 \uc548\uc5d0 \uc788\ub294 \ubc88\ud638\ub294 \uc138\uc885\uacfc\ud559\uc608\uc220\uc601\uc7ac\ud559\uad50 \uc218\ud559\u2160 \uad50\uacfc\uc11c\uc5d0 \uc2e4\ub9b0 \ubb38\ud56d \ubc88\ud638\uc785\ub2c8\ub2e4. \uc608\uc81c 1. \\( a = 3+ \\sqrt{3} \\complexI ,\\) \\(b = 3-\\sqrt{3} \\complexI \\)\uc77c \ub54c, \\(a^3 &#8211; a^2 b &#8211; ab^2 + b^3 \\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. (\uc720\uc81c 2.21) \ud480\uc774 \\(a^3 &#8211;&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"no","_lmt_disable":"","footnotes":""},"categories":[54,52],"tags":[585,584,586],"class_list":["post-9166","post","type-post","status-publish","format-standard","hentry","category-basic-mathematics","category-complex-analysis","tag-complex-number","tag-584","tag-586"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/9166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=9166"}],"version-history":[{"count":56,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/9166\/revisions"}],"predecessor-version":[{"id":9228,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/9166\/revisions\/9228"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=9166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=9166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=9166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}