{"id":54889,"date":"2014-08-10T05:00:43","date_gmt":"2014-08-09T21:00:43","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=54889"},"modified":"2021-10-06T16:18:04","modified_gmt":"2021-10-06T08:18:04","slug":"%e5%92%8c%e7%ae%97%e4%b8%ad%e7%9a%84%e8%a1%8c%e5%88%97%e5%bc%8f3%ef%bc%9a%e9%97%9c%e5%ad%9d%e5%92%8c%e7%9a%84%e3%80%8a%e8%a7%a3%e4%bc%8f%e9%a1%8c%e4%b9%8b%e6%b3%95%e3%80%8b%ef%bc%88%e4%b8%8b","status":"publish","type":"post","link":"http:\/\/localhost\/%e5%92%8c%e7%ae%97%e4%b8%ad%e7%9a%84%e8%a1%8c%e5%88%97%e5%bc%8f3%ef%bc%9a%e9%97%9c%e5%ad%9d%e5%92%8c%e7%9a%84%e3%80%8a%e8%a7%a3%e4%bc%8f%e9%a1%8c%e4%b9%8b%e6%b3%95%e3%80%8b%ef%bc%88%e4%b8%8b\/","title":{"rendered":"\u548c\u7b97\u4e2d\u7684\u884c\u5217\u5f0f(3)\uff1a\u95dc\u5b5d\u548c\u7684\u300a\u89e3\u4f0f\u984c\u4e4b\u6cd5\u300b\uff08\u4e0b\uff09\uff08Determinants in Wasan (3): Seki Takakazu\u2019s Kai Fukudai no Ho (Methods of Solving Secret Questions), Part 2\uff09"},"content":{"rendered":"

\u548c\u7b97\u4e2d\u7684\u884c\u5217\u5f0f(3)\uff1a\u95dc\u5b5d\u548c\u7684\u300a\u89e3\u4f0f\u984c\u4e4b\u6cd5\u300b\uff08\u4e0b\uff09\uff08Determinants in Wasan (3): Seki Takakazu\u2019s Kai Fukudai no Ho (Methods of Solving Secret Questions), Part 2\uff09
\n<\/strong><\/span>\u570b\u7acb\u81fa\u5357\u7b2c\u4e00\u9ad8\u7d1a\u4e2d\u5b78\u6797\u5009\u5104\u8001\u5e2b<\/strong><\/span><\/p>\n

\u9023\u7d50\uff1a\u548c\u7b97\u4e2d\u7684\u884c\u5217\u5f0f(2)\uff1a\u95dc\u5b5d\u548c\u7684\u300a\u89e3\u4f0f\u984c\u4e4b\u6cd5\u300b\uff08\u4e0a\uff09<\/a><\/p>\n

\u3008\u548c\u7b97\u4e2d\u7684\u884c\u5217\u5f0f(2)\uff1a\u95dc\u5b5d\u548c\u7684\u300a\u89e3\u4f0f\u984c\u4e4b\u6cd5\u300b\uff08\u4e0a\uff09\u3009\u4ecb\u7d39\u4e86\u95dc\u5b5d\u548c\u5982\u4f55\u5f9e\u89e3\u591a\u5143\u9ad8\u6b21\u65b9\u7a0b\u7d44\u4e2d\uff0c\u767c\u5c55\u51fa\u985e\u4f3c\u4eca\u65e5\u884c\u5217\u5f0f\u7684\u6982\u5ff5\u3002\u7136\u800c\uff0c\u5373\u4fbf\u662f\u4eca\u65e5\uff0c\u591a\u5143\u9ad8\u6b21\u65b9\u7a0b\u7d44\u6c42\u89e3\u4ecd\u662f\u4e00\u4ef6\u56f0\u96e3\u7684\u5de5\u4f5c\u3002\u6240\u4ee5\uff0c\u95dc\u5b5d\u548c\u80fd\u8655\u7406\u591a\u5143\u9ad8\u6b21\u65b9\u7a0b\u7d44\uff0c\u66f4\u986f\u5f97\u4ed6\u5728\u6578\u5b78\u4e0a\u7684\u9020\u8a63\u6df1\u539a\u3002\u4ee5\u4e0b\u900f\u904e\u5e7e\u500b\u7c21\u55ae\u7684\u5be6\u4f8b\uff0c\u8b93\u8b80\u8005\u66f4\u719f\u6089\u95dc\u5b5d\u548c\u7684\u65b9\u6cd5\uff0c\u4e5f\u6307\u51fa\u9019\u500b\u65b9\u6cd5\u4e5f\u6709\u7121\u80fd\u70ba\u529b\u7684\u6642\u5019\u3002<\/p>\n

\u4f8b1\uff1a\u89e3\u00a0$$\\left\\{ \\begin{array}{l} {(x – 1)^2} + {(y – 1)^2} = 1\\\\ {(x – 2)^2} + {(y – 2)^2} = 5\\\\ {(x – 3)^2} + {(y – 3)^2} = 13 \\end{array} \\right.$$\u3002<\/p>\n

\u3010\u95dc\u5b5d\u548c\u7684\u65b9\u6cd5\u3011\uff1a<\/p>\n

\u65b9\u7a0b\u7d44\u53ef\u6574\u7406\u6210\u00a0$$\\left\\{ \\begin{array}{l} ({y^2} – 2y + 1) – 2x + {x^2} = 0\\\\ ({y^2} – 4y + 3) – 4x + {x^2} = 0\\\\ ({y^2} – 6y + 5) – 6x + {x^2} = 0 \\end{array} \\right.$$\uff0c<\/p>\n

\u5229\u7528\u4fc2\u6578\u6240\u6210\u884c\u5217\u5f0f $$\\left| {\\begin{array}{*{20}{c}} {{y^2} – 2y + 1}&{ – 2}&1\\\\ {{y^2} – 4y + 3}&{ – 4}&1\\\\ {{y^2} – 6y + 5}&{ – 6}&1 \\end{array}} \\right| = 0$$\uff0c<\/p>\n

\u4f46\u5de6\u5f0f\u5c55\u958b\u5f8c\u5404\u9805\u5747\u6d88\u53bb\uff0c\u5f97\u5230 $$0=0$$ \u7684\u6046\u7b49\u5f0f\uff0c\u800c\u975e $$y$$\u00a0<\/i>\u7684\u65b9\u7a0b\u5f0f\uff0c\u56e0\u6b64\u7121\u5f9e\u6c42 $$y$$ \u4e4b\u503c\u3002<\/p>\n

\u3010\u5e7e\u4f55\u610f\u7fa9\u3011\uff1a<\/p>\n

\u4e09\u500b\u65b9\u7a0b\u5f0f\u7684\u4ee3\u8868\u7684\u5716\u5f62\u90fd\u662f\u5713\u5f62\uff0c\u5713\u5fc3\u5206\u5225\u662f $$(1,1)$$\u3001$$(2,2)$$\u3001$$(3,3)$$\uff0c<\/p>\n

\u534a\u5f91\u4f9d\u5e8f\u662f $$1$$\u3001$$\\sqrt 5$$\u3001$$\\sqrt{13}$$\uff0c\u5716\u5f62\u5982\u4e0b\uff1a<\/p>\n

\"54889_p1\"<\/a><\/p>\n

\u9019\u4e09\u500b\u5713\u5747\u4ea4\u65bc $$(1,0)$$\u3001$$(0,1)$$ \u5169\u9ede\uff0c\u56e0\u6b64 $$(x,y)=(1,0)$$\u3001$$(0,1)$$ \u662f\u65b9\u7a0b\u7d44\u7684\u5169\u7d44\u89e3\u3002<\/p>\n

\u4f8b2\uff1a\u89e3\u00a0$$\\left\\{ \\begin{array}{l} {(x – 1)^2} + {(y – 1)^2} = 1\\\\ {x^2} + {y^2} = 1\\\\ {(x + 1)^2} + {(y + 1)^2} = 1 \\end{array} \\right.$$\u3002<\/p>\n

\u3010\u95dc\u5b5d\u548c\u7684\u65b9\u6cd5\u3011\uff1a<\/p>\n

\u65b9\u7a0b\u7d44\u53ef\u6574\u7406\u6210\u00a0$$\\left\\{ \\begin{array}{l} ({y^2} – 2y + 1) – 2x + {x^2} = 0\\\\ ({y^2} – 1) + 0x + {x^2} = 0\\\\ ({y^2} + 2y + 1) + 2x + {x^2} = 0 \\end{array} \\right.$$\uff0c<\/p>\n

\u5229\u7528\u4fc2\u6578\u6240\u6210\u884c\u5217\u5f0f\u00a0$$\\left| {\\begin{array}{*{20}{c}} {{y^2} – 2y + 1}&{ – 2}&1\\\\ {{y^2} – 1}&0&1\\\\ {{y^2} + 2y + 1}&2&1 \\end{array}} \\right| = 0$$\uff0c<\/p>\n

\u5f97\u5230 $$8 = 0 \\to\\leftarrow $$\uff0c\u6545\u6b64\u984c\u7121\u89e3\u3002<\/p>\n

\u3010\u5e7e\u4f55\u610f\u7fa9\u3011\uff1a<\/p>\n

\u4e09\u500b\u65b9\u7a0b\u5f0f\u7684\u4ee3\u8868\u7684\u5716\u5f62\u90fd\u662f\u5713\u5f62\uff0c\u5713\u5fc3\u5206\u5225\u662f $$(1,1)$$\u3001$$(0,0)$$\u3001$$(-1,-1)$$\uff0c<\/p>\n

\u534a\u5f91\u90fd\u662f1\uff0c\u4e09\u500b\u5713\u4e26\u6c92\u6709\u5171\u540c\u7684\u4ea4\u9ede\uff0c\u6545\u65b9\u7a0b\u7d44\u7121\u89e3\u3002\u5716\u5f62\u5982\u4e0b\uff1a<\/p>\n

\"54889_p2\"<\/a><\/p>\n

\u4f8b3\uff1a\u89e3 $$\\left\\{ \\begin{array}{l} {x^2} + {y^2} = 1\\\\ 4{x^2} + {y^2} – 4 = 0\\\\ {y^2} – 4x + 4 = 0 \\end{array} \\right.$$\u3002<\/p>\n

\u3010\u95dc\u5b5d\u548c\u7684\u65b9\u6cd5\u3011\uff1a<\/p>\n

\u65b9\u7a0b\u7d44\u53ef\u6574\u7406\u6210 $$\\left\\{ \\begin{array}{l} ({y^2} – 1) + 0x + {x^2} = 0\\\\ ({y^2} – 4) + 0x + 4{x^2} = 0\\\\ ({y^2} + 4) – 4x + 0{x^2} = 0 \\end{array} \\right.$$\uff0c<\/p>\n

\u5229\u7528\u4fc2\u6578\u6240\u6210\u884c\u5217\u5f0f\u00a0$$\\left| {\\begin{array}{*{20}{c}} {{y^2} – 1}&0&1\\\\ {{y^2} – 4}&0&4\\\\ {{y^2} + 4}&{ – 4}&0 \\end{array}} \\right| = 0 \\Rightarrow 12{y^2} = 0\\Rightarrow y = 0\\Rightarrow x = 1$$\u3002<\/p>\n

\u3010\u5e7e\u4f55\u610f\u7fa9\u3011\uff1a<\/p>\n

\u4e09\u500b\u65b9\u7a0b\u5f0f\u7684\u4ee3\u8868\u7684\u5716\u5f62\u5206\u5225\u662f\u5713\u5f62\u3001\u6a62\u5713\u8207\u62cb\u7269\u7dda\uff0c\u53ea\u76f8\u4ea4\u65bc $$(1,0)$$\uff0c<\/p>\n

\u6545 $$(1,0)=(x,y)$$\u00a0\u662f\u65b9\u7a0b\u7d44\u7684\u552f\u4e00\u89e3\u3002\u5716\u5f62\u5982\u4e0b\uff1a<\/p>\n

\"54889_p3\"<\/a><\/p>\n

\u4f8b4\uff1a\u89e3\u00a0$$\\left\\{ \\begin{array}{l} {x^2} + {y^2} = 1\\\\ 4{x^2} + {y^2} – 4 = 0\\\\ {y^2} – 4x + 8 = 0 \\end{array} \\right.$$\u3002<\/p>\n

\u3010\u95dc\u5b5d\u548c\u7684\u65b9\u6cd5\u3011\uff1a<\/p>\n

\u65b9\u7a0b\u7d44\u53ef\u6574\u7406\u6210\u00a0$$\\left\\{ \\begin{array}{l} ({y^2} – 1) + 0x + {x^2} = 0\\\\ ({y^2} – 4) + 0x + 4{x^2} = 0\\\\ ({y^2} + 8) – 4x + 0{x^2} = 0 \\end{array} \\right.$$\uff0c<\/p>\n

\u5229\u7528\u4fc2\u6578\u6240\u6210\u884c\u5217\u5f0f\u00a0$$\\left| {\\begin{array}{*{20}{c}} {{y^2} – 1}&0&1\\\\ {{y^2} – 4}&0&4\\\\ {{y^2} + 8}&{ – 4}&0 \\end{array}} \\right| = 0\\Rightarrow 12{y^2} = 0\\Rightarrow y = 0\\Rightarrow x = 1\\to \\leftarrow $$<\/p>\n

\u56e0\u70ba $$x=1,y=0$$ \u4ee3\u5165\u7b2c\u4e09\u500b\u65b9\u7a0b\u5f0f $$y^2-4x+8=0$$ \u4e0d\u5408\u3002<\/p>\n

\u3010\u5e7e\u4f55\u610f\u7fa9\u3011\uff1a<\/p>\n

\u4e09\u500b\u65b9\u7a0b\u5f0f\u7684\u4ee3\u8868\u7684\u5716\u5f62\u5206\u5225\u662f\u5713\u5f62\u3001\u6a62\u5713\u8207\u62cb\u7269\u7dda\uff0c\u4f46\u6c92\u6709\u5171\u540c\u7684\u4ea4\u9ede\uff0c\u6545\u65b9\u7a0b\u7d44\u7121\u89e3\u3002<\/p>\n

\u5716\u5f62\u5982\u4e0b\uff1a<\/p>\n

\"54889_p4\"<\/a><\/p>\n

\u7531\u4e0a\u8ff04\u500b\u4f8b\u5b50\u53ef\u770b\u51fa\uff0c\u5229\u7528\u4fc2\u6578\u884c\u5217\u5f0f\u7b49\u65bc $$0$$ \u4e0d\u4e00\u5b9a\u80fd\u6c42\u5f97\u7684 $$y$$\u00a0<\/i>\u7684\u65b9\u7a0b\u5f0f\uff0c<\/p>\n

\u5c31\u7b97\u53ef\u4ee5\uff0c\u89e3\u51fa\u7684 $$y,x$$\u00a0<\/i>\u4ecd\u8981\u4ee3\u56de\u539f\u65b9\u7a0b\u7d44\u6aa2\u9a57\uff0c\u4e0d\u4e00\u5b9a\u90fd\u6703\u7b26\u5408\u3002<\/p>\n

\u90a3\u70ba\u4ec0\u9ebc\u6703\u6709\u4e0d\u5408\u7684\u60c5\u5f62\u51fa\u73fe\u5462\uff1f<\/p>\n

\u6211\u5011\u53ef\u4ee5\u63db\u500b\u65b9\u5f0f\u4f86\u770b\u95dc\u5b5d\u548c\u7684\u65b9\u6cd5\uff0c\u5c31\u662f\u5c07\u65b9\u7a0b\u7d44\u6574\u7406\u6210 \"54889_eq1\"<\/a>\u00a0\u5f8c\uff0c<\/p>\n

\u82e5\u5c07 $$x^2$$\u00a0<\/i>\u770b\u6210\u8b8a\u6578 $$y’$$\uff0c\u90a3\u9ebc\u7531\u4e09\u500b\u4e8c\u5143\u4e00\u6b21\u65b9\u7a0b\u7d44\u8981\u6709\u5171\u540c\u7684\u89e3\uff0c\u5f97\u5230\u4fc2\u6578\u884c\u5217\u5f0f\"54862_eq5\"<\/a><\/p>\n

\u8aaa\u5de7\u4e0d\u5de7\uff0c\u9019\u6070\u597d\u5c31\u662f\u840a\u5e03\u5c3c\u8332\u767c\u5c55\u51fa\u884c\u5217\u5f0f\u6982\u5ff5\u7684\u65b9\u5f0f\uff0c\u8acb\u53c3\u898b\u672c\u7db2\u7ad9\u3008\u884c\u5217\u5f0f\u7684\u6feb\u89f4\uff1a\u840a\u5e03\u5c3c\u8332 (1)\u3001(2)\u3009\u3002\u6771\u3001\u897f\u65b9\u884c\u5217\u5f0f\u7684\u958b\u5275\u8005\uff0c\u7adf\u7136\u53ef\u4ee5\u5728\u9019\u88e1\u300c\u5408\u9cf4\u300d\uff01<\/p>\n

\u4f46\u554f\u984c\u4f86\u4e86\uff0c$$y’=x^2$$\u00a0<\/i>\u4e26\u4e0d\u662f\u8207 $$x$$\u00a0<\/i>\u7368\u7acb\u7684\u8b8a\u6578\uff0c\u56e0\u6b64\uff0c\u5229\u7528\u00a0\"54862_eq5\"<\/a>\u00a0\u6c42\u5f97\u7684\u89e3\uff0c<\/p>\n

\u4ecd\u5fc5\u9808\u4ee3\u56de\u539f\u65b9\u7a0b\u7d44\u6aa2\u9a57\u662f\u5426\u6709\u4e0d\u5408\u7684\u60c5\u5f62\u51fa\u73fe\u3002<\/p>\n

\u7c21\u8a00\u4e4b\uff0c\u591a\u5143\u9ad8\u6b21\u65b9\u7a0b\u7d44\u7684\u984c\u76ee\uff0c\u6703\u96a8\u8457\u689d\u4ef6\u7684\u591a\u5be1\uff08\u65b9\u7a0b\u5f0f\u7684\u6578\u76ee\uff09\u800c\u6709\u4e0d\u540c\u7684\u89e3\uff0c\u751a\u81f3\u89e3\u6cd5\u9084\u6703\u76f8\u7576\u7e41\u96dc\u3002\u6bd4\u5982\u8aaa\uff0c\u7576\u672a\u77e5\u6578\u500b\u6578\u4e0d\u53ea\u5169\u500b\u7684\u6642\u5019\uff0c\u5c31\u8981\u53cd\u8986\u6d88\u53bb\u672a\u77e5\u6578\u500b\u6578\u624d\u884c\u3002<\/p>\n

\u4f8b\u5982\u95dc\u5b5d\u548c\u8207\u5169\u500b\u5b78\u751f\u5efa\u90e8\u8ce2\u660e\u3001\u5efa\u90e8\u8ce2\u5f18\u6240\u5beb\u7684\u300a\u5927\u6210\u7b97\u7d93\u300b\u4e2d\uff0c\u5c31\u6709\u9019\u9ebc\u4e00\u500b\u76f8\u7576\u65bc\u56db\u5143\u56db\u6b21\u65b9\u7a0b\u7d44 $$\\left\\{ \\begin{array}{l} {x_1}^4 + {x_2}^3 + {x_3} + {x_4} = {A_{1}}\\\\ {x_1}^3 + {x_2}^2 + {x_3}^3 + {x_4}^4 = {A_{2}}\\\\ {x_1}^2 + {x_2} + {x_3}^4 + {x_4}^3 = {A_{3}}\\\\ {x_1} + {x_2}^4 + {x_3}^2 + {x_4}^2 = {A_{4}} \\end{array} \\right.$$\u00a0\u7684\u554f\u984c\uff0c\u8f49\u8f09\u65bc\u4e0b\uff0c\u8b93\u6709\u8208\u8da3\u7684\u8b80\u8005\u6311\u6230\u770b\u770b\uff01<\/p>\n

\u5047\u5982\u6709\u7532\u4e59\u4e19\u4e01\u5e73\u65b9\u5404\u4e00\u3002\u7532\u4e91\uff1a\u7532\u65b9\u4e09\u4e58\u51aa[\u8a3b\uff1a\u56db\u6b21\u65b9]\u3001\u4e59\u65b9\u518d\u4e58\u51aa[\u8a3b\uff1a\u4e09\u6b21\u65b9]\u3001\u4e19\u65b9\u3001\u4e01\u65b9\u76f8\u4f75\u5171\u82e5\u5e72\uff1b\u4e59\u4e91\uff1a\u7532\u65b9\u518d\u4e58\u51aa\u3001\u4e59\u65b9\u51aa[\u8a3b\uff1a\u4e8c\u6b21\u65b9]\u3001\u4e19\u65b9\u518d\u4e58\u51aa\u3001\u4e01\u65b9\u4e09\u4e58\u51aa\u76f8\u4f75\u5171\u82e5\u5e72\uff1b\u4e19\u4e91\uff1a\u7532\u65b9\u51aa\u3001\u4e59\u65b9\u3001\u4e19\u65b9\u4e09\u4e58\u51aa\u3001\u4e01\u65b9\u518d\u4e58\u51aa\u76f8\u4f75\u5171\u82e5\u5e72\uff1b\u4e01\u4e91\uff1a\u7532\u65b9\u3001\u4e59\u65b9\u4e09\u4e58\u51aa\u3001\u4e19\u65b9\u51aa\u3001\u4e01\u65b9\u51aa\u76f8\u4f75\u5171\u82e5\u5e72\u3002\u554f\u7532\u65b9\uff1f<\/strong><\/p>\n

\u9023\u7d50\uff1a\u548c\u7b97\u4e2d\u7684\u884c\u5217\u5f0f(4)\uff1a\u964d\u968e\u5c55\u958b\u6cd5<\/a><\/p>\n

\u53c3\u8003\u8cc7\u6599\uff1a<\/p>\n