{"id":28301,"date":"2017-06-14T00:26:44","date_gmt":"2017-06-13T16:26:44","guid":{"rendered":"https:\/\/case.ntu.edu.tw\/blog\/?p=28301"},"modified":"2018-07-29T09:47:26","modified_gmt":"2018-07-29T01:47:26","slug":"%e3%80%90%e5%a4%a7%e5%ae%87%e5%ae%99%e5%b0%8f%e6%95%85%e4%ba%8b%e3%80%9125-%e5%be%9e%e8%87%aa%e7%84%b6%e6%95%b8%e5%88%b0%e8%b6%85%e8%a4%87%e6%95%b8","status":"publish","type":"post","link":"https:\/\/case.ntu.edu.tw\/blog\/?p=28301","title":{"rendered":"\u3010\u5927\u5b87\u5b99\u5c0f\u6545\u4e8b\u301125 \u5f9e\u81ea\u7136\u6578\u5230\u8d85\u8907\u6578"},"content":{"rendered":"<p><strong>\u64b0\u6587\uff5c<\/strong>\u8449\u674e\u83ef<\/p>\n<p>\u300c\u4e0a\u5e1d(\u50c5\u50c5)\u5275\u9020\u6574\u6578\uff0c\u5176\u4ed6\u90fd\u662f\u4eba\u985e\u52aa\u529b\u7684\u6210\u679c\u3002\u300d\u9019\u662f\u5341\u4e5d\u4e16\u7d00\u5fb7\u570b\u6578\u5b78\u5bb6\u514b\u6c0f(L. Kronecker)\u7684\u540d\u8a00\u3002\u7136\u800c\uff0c\u5982\u679c\u6211\u5011\u5c07\u300c\u4e0a\u5e1d\u300d\u8996\u70ba\u300c\u5927\u81ea\u7136\u300d\u7684\u540c\u7fa9\u8a5e\uff0c\u5c31\u5fc5\u9808\u5c07\u300c\u6574\u6578\u300d\u6539\u70ba\u300c\u81ea\u7136\u6578\u300d\uff0c\u5538\u8d77\u4f86\u624d\u6703\u6bd4\u8f03\u901a\u9806\u3002<\/p>\n<p>\u4e8b\u5be6\u4e5f\u7684\u78ba\u5982\u6b64\uff0c\u56e0\u70ba\u300c\u96f6\u300d\u8207\u300c\u8ca0\u6574\u6578\u300d\u7121\u7591\u90fd\u662f\u4eba\u985e\u7684\u767c\u660e\u3002\u4f8b\u5982\u53e4\u4ee3\u4e2d\u570b\u4e26\u6c92\u6709\u96f6\u7684\u6982\u5ff5\uff0c\u6240\u4ee5\u300c\u96f6\u300d\u5176\u5be6\u662f\u5047\u501f\u5b57(\u539f\u610f\u662f\u96f6\u661f\u3001\u96f6\u788e)\u3002\u800c\u300c\u8ca0\u6574\u6578\u300d\u66f4\u4e0d\u80fd\u7b97\u662f\u81ea\u7136\u754c\u7684\u7522\u7269\uff0c\u8acb\u554f\u4f60\u6703\u4e0d\u6703\u8aaa\u300c\u6211\u5bb6\u9662\u5b50\u88e1\u6709\u8ca0\u4e09\u682a\u73ab\u7470\u300d\uff1f<\/p>\n<p>\u90a3\u9ebc\u5206\u6578\u5462\uff1f\u9019\u9ebc\u8aaa\u5427\uff0c\u5982\u679c\u4f60\u770b\u5230\u5207\u5f97\u6574\u6574\u9f4a\u9f4a\u7684\u534a\u96bb\u96de\uff0c\u6703\u4e0d\u6703\u4ee5\u70ba\u90a3\u662f\u5927\u81ea\u7136\u7684\u9b3c\u65a7\u795e\u5de5\uff1f\u63a8\u800c\u5ee3\u4e4b\uff0c\u53ea\u8981\u6263\u9664\u81ea\u7136\u6578\uff0c\u5176\u4ed6\u7684\u300c\u6709\u7406\u6578\u300d\u901a\u901a\u662f\u6578\u5b78\u5bb6\u7684\u5fc3\u8840\u7d50\u6676\u3002<\/p>\n<p>\u300c\u7121\u7406\u6578\u300d\u7576\u7136\u66f4\u662f\u4eba\u5de5\u7684\u7522\u7269\u3002\u8209\u4f8b\u800c\u8a00\uff0c\u53e4\u5e0c\u81d8\u7684\u5b78\u8005\u7d93\u904e\u591a\u5e74\u601d\u8fa8\uff0c\u624d\u7d42\u65bc\u78ba\u5b9a\u221a2\u662f\u7121\u7406\u6578\u3002\u64da\u8aaa\u7562\u6c0f\u5b78\u6d3e\u6709\u500b\u5b78\u751f\u751a\u81f3\u56e0\u6b64\u55aa\u547d\uff0c\u7576\u7136\u53ea\u662f\u8b20\u50b3\u7f77\u4e86\u3002<\/p>\n<p>\u7136\u5f8c\uff0c\u60f3\u5fc5\u4f60\u9084\u8a18\u5f97\u300c\u6709\u7406\u6578\u300d\u548c\u300c\u7121\u7406\u6578\u300d\u5408\u7a31\u70ba\u300c\u5be6\u6578\u300d(real number)\u3002\u9019\u500b\u300c\u5be6\u300d\u975e\u5e38\u8010\u4eba\u5c0b\u5473\uff0c\u5b83\u8868\u660e\u4e86\u81ea\u5df1\u771f\u5be6\u5b58\u5728\u65bc\u5b87\u5b99\u4e2d\u3002\u6216\u8a31\u6211\u5011\u53ef\u4ee5\u9019\u9ebc\u60f3\uff0c\u53ea\u8981\u6709\u7a7a\u9593\uff0c\u5c31\u4e00\u5b9a\u6709\u8ddd\u96e2\uff1b\u5c07\u6240\u6709\u7684\u8ddd\u96e2\u653e\u5728\u4e00\u8d77\uff0c\u5c31\u662f\u4e00\u7d44\u5b8c\u6574\u7684\u5be6\u6578\u4e86\u3002<\/p>\n<p>\u63db\u6210\u6578\u5b78\u5bb6\u7684\u8aaa\u6cd5\uff0c\u5247\u662f\u300c\u76f4\u7dda\u300d\u548c\u300c\u5be6\u6578\u300d\u6709\u4e00\u5c0d\u4e00\u7684\u5c0d\u61c9\u95dc\u4fc2\u3002\u56e0\u6b64\uff0c\u5982\u679c\u4f60\u756b\u51fa\u4e00\u689d\u76f4\u7dda\uff0c\u5728\u4e0a\u9762\u6a19\u793a1, 2, 3...\u9019\u6a23\u7684\u523b\u5ea6\uff0c\u90a3\u9ebc\u9664\u4e86\u9019\u4e9b(\u81ea\u7136\u6578)\u523b\u5ea6\u4e4b\u5916\uff0c\u5176\u4ed6\u5404\u9ede\u6240\u4ee3\u8868\u7684\u6578\u90fd\u8ddf\u4e0a\u5e1d\u6c92\u6709\u95dc\u4fc2\uff0c\u8d77\u78bc\u662f\u6c92\u6709\u76f4\u63a5\u7684\u95dc\u4fc2\u3002<\/p>\n<p><span style=\"color: #008080;\"><strong>\u25cf\u865b\u5be6\u5171\u69cb<\/strong><\/span><\/p>\n<p>\u9023\u4e2d\u5b78\u751f\u90fd\u77e5\u9053\uff0c\u5be6\u6578\u4e4b\u5916\u9084\u6709\u865b\u6578(imaginary number)\u3002\u7576\u4f60\u5728\u89e3\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u7684\u6642\u5019\uff0c\u865b\u6578\u5df2\u7d93\u662f\u4e0d\u53ef\u6216\u7f3a\u7684\u5de5\u5177\u3002<\/p>\n<p>\u8010\u4eba\u5c0b\u5473\u7684\u662f\uff0c\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u7684\u6b77\u53f2\u8db3\u8db3\u6709\u5e7e\u5343\u5e74\uff0c\u865b\u6578\u537b\u662f\u5341\u516d\u4e16\u7d00\u624d\u51fa\u73fe\u7684\uff0c\u800c\u4e14\u7576\u6642\u9084\u4e0d\u53eb\u300c\u865b\u6578\u300d\u3002\u5728\u6b64\u4e4b\u524d\uff0c\u8af8\u5982x<sup>2<\/sup>+1=0\u9019\u6a23\u7684\u65b9\u7a0b\u5f0f\u90fd\u88ab\u8996\u70ba\u7121\u89e3\u3002<\/p>\n<p>\u300c\u5be6\u6578\u300d\u548c\u300c\u865b\u6578\u300d\u9019\u5169\u500b\u540d\u8a5e\u5247\u662f\u5341\u4e03\u4e16\u7d00\u7684\u7522\u7269\uff0c\u5169\u8005\u90fd\u662f\u7b1b\u5361\u5152\u7684\u767c\u660e\u3002\u5728\u5176\u7d93\u5178\u4e4b\u4f5c\u300a\u5e7e\u4f55\u5b78\u300b(1637)\u4e2d\uff0c\u7b1b\u5361\u5152\u983b\u983b\u4f7f\u7528\u300c\u865b\u6578\u300d\u4f86\u6307\u7a31\u8ca0\u6578\u7684\u5e73\u65b9\u6839\uff0c\u81f3\u65bc\u300c\u5be6\u6578\u300d\u5247\u662f\u4f34\u96a8\u300c\u865b\u6578\u300d\u800c\u751f\u7684\u540d\u8a5e\uff0c\u6b63\u5982\u300c\u6b50\u6c0f\u5e7e\u4f55\u300d\u4e4b\u65bc\u300c\u975e\u6b50\u5e7e\u4f55\u300d\u3002<\/p>\n<p>\u5f9e\u9019\u500b\u300c\u865b\u300d\u5b57\u5373\u53ef\u770b\u51fa\u7b1b\u5361\u5152\u81ea\u5df1\u76f8\u7576\u5fc3\u865b\u3002\u539f\u56e0\u5f88\u7c21\u55ae\uff0c\u9019\u7a2e\u865b\u7121\u7e39\u7df2\u3001\u6839\u672c\u4e0d\u5b58\u5728\u65bc\u81ea\u7136\u754c\u7684\u6771\u897f\uff0c\u5b83\u5728\u6578\u5b78\u3001\u7269\u7406\u4ee5\u53ca\u54f2\u5b78\u4e0a\u5230\u5e95\u8a72\u5982\u4f55\u5b9a\u4f4d\uff0c\u9023\u7b1b\u5361\u5152\u9019\u4f4d\u300c\u8fd1\u4ee3\u54f2\u5b78\u4e4b\u7236\u300d\u517c\u300c\u8fd1\u4ee3\u6578\u5b78\u4e4b\u7236\u300d\u4e5f\u8aaa\u4e0d\u6e05\u695a\u3002<\/p>\n<p>\u7b2c\u4e00\u500b\u52c9\u5f37\u5c07\u5b83\u8aaa\u6e05\u695a\u7684\u4eba\uff0c\u662f\u5927\u540d\u9f0e\u9f0e\u7684\u6578\u5b78\u738b\u5b50\u9ad8\u65af(Carl F. Gauss, 1777-1855)\u3002\u6216\u8a31\u4f60\u9084\u8a18\u5f97\uff0c\u6240\u8b02\u7684\u300c\u9ad8\u65af\u5e73\u9762\u300d\u5c31\u662f\u628a\u5e73\u9762\u76f4\u89d2\u5ea7\u6a19\u7684\u6a6b\u8ef8\u7576\u6210\u5be6\u6578\u8ef8\uff0c\u7e31\u8ef8\u7576\u6210\u865b\u6578\u8ef8\uff0c\u5176\u4e2d\u6bcf\u500b\u9ede\u525b\u597d\u5c0d\u61c9\u4e00\u500b\u8907\u6578(complex number)\uff0c\u6b63\u5982\u540c\u76f4\u7dda\u548c\u5be6\u6578\u6709\u4e00\u5c0d\u4e00\u7684\u5c0d\u61c9\u95dc\u4fc2\u3002(\u4e8b\u5be6\u4e0a\uff0c\u9ad8\u65af\u4e26\u975e\u7b2c\u4e00\u4f4d\u4f7f\u7528\u9ad8\u65af\u5e73\u9762\u7684\u6578\u5b78\u5bb6\u3002)<\/p>\n<p>\u9ad8\u65af\u767c\u8868\u76f8\u95dc\u8ad6\u6587\u662f1831\u5e74\u7684\u4e8b\uff0c\u5728\u6b64\u4e4b\u524d\uff0c\u4ed6\u4e5f\u66fe\u70ba\u300c\u865b\u6578\u7684\u54f2\u5b78\u610f\u7fa9\u300d\u7cfe\u7d50\u4e86\u4e09\u5341\u591a\u5e74\u3002\u6216\u8a31\u6b63\u56e0\u70ba\u5982\u6b64\uff0c\u4ed6\u5728\u8ad6\u6587\u4e2d\u7279\u5225\u5f37\u8abf\u300c\u865b\u6578\u300d\u662f\u500b\u5bb9\u6613\u9020\u6210\u8aa4\u5c0e\u7684\u540d\u8a5e\uff0c\u8a00\u4e0b\u4e4b\u610f\u5927\u5bb6\u6700\u597d\u656c\u800c\u9060\u4e4b\u3002\u5982\u679c\u6211\u5011\u7528\u5f8c\u898b\u4e4b\u660e\u4f86\u8a55\u65b7\uff0c\u9ad8\u65af\u6240\u767c\u660e\u7684\u300c\u8907\u6578\u300d\u5728\u8a9e\u610f\u4e0a\u7684\u78ba\u6bd4\u300c\u865b\u6578\u300d\u512a\u79c0\u8a31\u591a\u3002<\/p>\n<p><span style=\"color: #008080;\"><strong>\u25cf\u66f4\u4e0a\u4e00\u5c64\u6a13\uff1f<\/strong><\/span><\/p>\n<p>\u7576\u6578\u5b78\u5bb6\u6162\u6162\u63a5\u53d7\u4e86\u865b\u6578\u548c\u8907\u6578\u4e4b\u5f8c\uff0c\u81ea\u7136\u800c\u7136\u6703\u6709\u4eba\u60f3\u8981\u66f4\u4e0a\u4e00\u5c64\u6a13\uff0c\u9019\u662f\u6578\u5b78\u5bb6\u7684\u5929\u6027\u3002<\/p>\n<p>\u6bd4\u65b9\u8aaa\uff0c\u65e2\u7136\u5e73\u9762\u548c\u8907\u6578(a+bi)\u6709\u4e00\u5c0d\u4e00\u7684\u5c0d\u61c9\u95dc\u4fc2\uff0c\u90a3\u9ebc\u7a7a\u9593\u662f\u4e0d\u662f\u548c\u67d0\u7a2e\u300c\u8d85\u8907\u6578\u300d(a+bi+cj, \u5176\u4e2dj\u662fi\u7684\u63a8\u5ee3)\u5c0d\u61c9\u5462\uff1f\u9019\u500b\u554f\u984c\u6df1\u6df1\u5438\u5f15\u4e86\u4e00\u4f4d\u540d\u53eb\u54c8\u5bc6\u9813(William Hamilton, 1805-1865)\u7684\u611b\u723e\u862d\u5b78\u8005\u3002<\/p>\n<p>\u54c8\u5bc6\u9813\u82e6\u601d\u591a\u5e74\uff0c\u59cb\u7d42\u7121\u6cd5\u627e\u5230\u5713\u6eff\u7684\u7b54\u6848\u3002\u76f4\u52301843\u5e74\u5e95\uff0c\u4ed6\u8d70\u904e\u4e00\u5ea7\u77f3\u6a4b\uff0c\u5ffd\u7136\u9748\u611f\u5927\u767c\uff0c\u7acb\u523b\u628a\u9813\u609f\u7684\u516c\u5f0f\u523b\u5728\u6a4b\u4e0a(\u6240\u4ee5\u4ed6\u53ef\u7b97\u662f\u5857\u9d09\u85dd\u8853\u7684\u59cb\u7956)\u3002<\/p>\n<p>\u54c8\u5bc6\u9813\u7684\u5857\u9d09\u65e9\u5df2\u78e8\u640d\u6b86\u76e1\uff0c\u4f46\u5982\u679c\u4f60\u73fe\u5728\u53bb\u90fd\u67cf\u6797\u89c0\u5149\uff0c\u6703\u770b\u5230\u90a3\u5ea7\u6a4b\u4e0a\u6709\u4e00\u584a\u77f3\u677f\uff0c\u4e0a\u9762\u523b\u8457\u9019\u9ebc\u5e7e\u53e5\u8a71\uff1a<\/p>\n<p><em>\u3000\u30001843\u5e7410\u670816\u65e5\u9019\u5929<\/em><\/p>\n<p><em>\u3000\u3000\u7576\u4ed6\u8def\u904e\u6b64\u5730<\/em><\/p>\n<p><em>\u3000\u3000\u54c8\u5bc6\u9813\u7235\u58eb<\/em><\/p>\n<p><em>\u3000\u3000\u9748\u5149\u4e00\u9583<\/em><\/p>\n<p><em>\u3000\u3000\u767c\u89ba\u56db\u5143\u6578\u7684\u4e58\u6cd5<\/em><\/p>\n<p><em>\u3000\u3000\u670d\u5f9e\u5982\u4e0b\u7684\u57fa\u672c\u516c\u5f0f<\/em><\/p>\n<p><em>\u3000\u3000i<sup>2<\/sup>=j<sup>2<\/sup>=k<sup>2<\/sup>=ijk=-1<\/em><\/p>\n<p><em>\u3000\u3000\u9806\u624b\u5c07\u5b83\u523b\u5728\u9019\u5ea7\u6a4b\u4e0a<\/em><\/p>\n<p>(\u90fd\u67cf\u6797\u5e02\u653f\u5e9c\u5fd8\u4e86\u52a0\u4e0a\u4e00\u53e5\uff1a<em>\u7279\u6b8a\u6848\u4f8b\uff0c\u8acb\u52ff\u6a21\u4eff\uff0c\u5426\u5247\u2026\u2026<\/em>)<\/p>\n<figure id=\"attachment_28302\" aria-describedby=\"caption-attachment-28302\" style=\"width: 700px\" class=\"wp-caption aligncenter\"><img fetchpriority=\"high\" decoding=\"async\" class=\"wp-image-28302\" src=\"http:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2017\/06\/25-1.jpg\" alt=\"\" width=\"700\" height=\"270\" \/><figcaption id=\"caption-attachment-28302\" class=\"wp-caption-text\">\u5716\u50cf\u8207\u6587\u5b57\u4f86\u6e90\uff1a\u7dad\u57fa\u767e\u79d1<\/figcaption><\/figure>\n<p>\u63a5\u4e0b\u4f86\uff0c\u7576\u7136\u8981\u89e3\u91cb\u4e00\u4e0b\u9019\u500b\u516c\u5f0f\u80cc\u5f8c\u7684\u610f\u7fa9\u3002<\/p>\n<p>\u5728\u6b64\u4e4b\u524d\uff0c\u54c8\u5bc6\u9813\u4e00\u76f4\u60f3\u8981\u5efa\u69cba+bi+cj\u9019\u6a23\u7684\u8d85\u8907\u6578\uff0c\u537b\u59cb\u7d42\u7121\u6cd5\u767c\u660e\u5408\u7406\u7684\u4e58\u6cd5\u898f\u5247\u3002\u6bd4\u65b9\u8aaa\uff0c\u65e2\u7136j\u662fi\u7684\u63a8\u5ee3\uff0ci<em><sup>2<\/sup><\/em>=j<em><sup>2<\/sup><\/em>=-1\u662f\u7406\u6240\u7576\u7136\u7684\u4e8b\uff0c\u53ef\u662fij\u8a72\u7b49\u65bc\u4ec0\u9ebc\u5462\uff1f\u76f4\u5230\u9019\u4e00\u5929\uff0c\u4ed6\u624d\u9a5a\u89ba\u300c\u66f4\u4e0a\u4e00\u5c64\u6a13\u300d\u6ce8\u5b9a\u662f\u6b7b\u8def\uff0c\u5fc5\u9808\u4e00\u53e3\u6c23\u66f4\u4e0a\u5169\u5c64\u6a13\u624d\u884c\u3002\u63db\u53e5\u8a71\u8aaa\uff0c\u5fc5\u9808\u5c07a+bi\u64f4\u5145\u6210a+bi+cj+dk(j, k\u90fd\u662fi\u7684\u63a8\u5ee3)\uff0c\u9019\u6a23\u7684\u6578\u7cfb\u7d50\u69cb\u624d\u6703\u64c1\u6709\u5408\u7406\u7684\u4e58\u6cd5\u3002<\/p>\n<p>\u54c8\u5bc6\u9813\u5c07\u9019\u500b\u300c\u8d85\u8907\u6578\u300d\u547d\u540d\u70ba\u300c\u56db\u5143\u6578\u300d(quaternion)\uff0c\u9019\u500b\u540d\u8a5e\u5f88\u5bb9\u6613\u9867\u540d\u601d\u7fa9\uff0c\u7528\u4e0d\u8457\u591a\u505a\u89e3\u91cb\u3002<\/p>\n<p>\u6216\u8a31\u4f60\u6703\u64d4\u5fc3\u56db\u5143\u6578\u7121\u6cd5\u548c\u4e09\u7dad\u7a7a\u9593\u4e00\u4e00\u5c0d\u61c9\uff0c\u6240\u4ee5\u6c92\u6709\u5e7e\u4f55\u610f\u7fa9\u3002\u5176\u5be6\uff0c\u60f3\u8981\u5f97\u5230\u9019\u6a23\u7684\u5c0d\u61c9\u95dc\u4fc2\uff0c\u53ea\u8981\u628a\u56db\u5143\u6578\u7684\u300c\u5be6\u90e8\u300d\u5b9a\u70ba\u96f6\u5c31\u884c\u4e86\u3002\u63db\u8a00\u4e4b\uff0c\u8981\u5c0d\u61c9\u4e09\u7dad\u5ea7\u6a19\uff0cbi+cj+dk\u662f\u518d\u81ea\u7136\u4e0d\u904e\u7684\u9078\u64c7\u3002<\/p>\n<p>\u56db\u5143\u6578\u6709\u4e00\u500b\u7279\u6b8a\u6027\u8cea(\u53ef\u5f9e\u4e0a\u9762\u7684\u516c\u5f0f\u5c0e\u51fa\u4f86)\uff0c\u90a3\u5c31\u662fi, j, k\u4e09\u8005\u7684\u4e58\u6cd5\u5fc5\u9808\u6eff\u8db3\u300c\u53cd\u4ea4\u63db\u300d\u95dc\u4fc2\uff0c\u4f8b\u5982ij=-ji=k. \u5982\u679c\u6211\u5011\u5c07i, j, k\u8996\u70ba\u77e9\u9663\uff0c\u9019\u7a2e\u53cd\u4ea4\u63db\u95dc\u4fc2\u5176\u5be6\u4e0d\u96e3\u7406\u89e3\u3002\u4f46\u662f\u7576\u6642(1843\u5e74)\u77e9\u9663\u7406\u8ad6\u5c1a\u672a\u6b63\u5f0f\u8a95\u751f\uff0c\u7531\u6b64\u53ef\u898b\u54c8\u5bc6\u9813\u662f\u591a\u9ebc\u85dd\u9ad8\u81bd\u5927\u3002<\/p>\n<p><span style=\"color: #008080;\"><strong>\u25cf\u518d\u4e0a\u5e7e\u5c64\u6a13\uff1f<\/strong><\/span><\/p>\n<p>\u770b\u5230\u9019\u88e1\uff0c\u4f60\u6703\u4e0d\u6703\u958b\u59cb\u52d5\u8166\u7b4b\uff0c\u60f3\u628a\u56db\u5143\u6578\u518d\u63a8\u5ee3\u6210\u66f4\u8d85\u7d1a\u7684\u8907\u6578\uff1f<\/p>\n<p>\u6839\u64da\u6b77\u53f2\u6559\u8a13\uff0c\u4e0b\u4e00\u500b\u8d85\u8907\u6578\u61c9\u8a72\u4e0d\u662f\u4e94\u5143\u6578\uff0c\u4e5f\u4e0d\u592a\u53ef\u80fd\u662f\u516d\u5143\u6578\u3002\u53ef\u662f\u5230\u5e95\u662f\u5e7e\u5143\u6578\u5462\uff1f\u5176\u5be6\uff0c\u524d\u9762\u5e7e\u500b\u6578\u7cfb\u7684\u300c\u7dad\u5ea6\u300d(\u57fa\u5e95\u7684\u6578\u76ee)\u63d0\u4f9b\u4e86\u5f88\u660e\u986f\u7684\u7dda\u7d22\u3002<\/p>\n<p>\u5f9e\u5be6\u6578\u5230\u8907\u6578\uff0c\u518d\u5f9e\u8907\u6578\u5230\u56db\u5143\u6578\uff0c\u7dad\u5ea6\u7684\u8b8a\u5316\u662f1\u21922\u21924, \u56e0\u6b64\u6211\u5011\u53ef\u4ee5\u5927\u81bd\u9810\u8a00\uff0c\u4e0b\u4e00\u500b\u8d85\u8907\u6578\u61c9\u8a72\u662f\u300c\u516b\u5143\u6578\u300d(octonion)\u3002<\/p>\n<p>\u4e8b\u5be6\u4e0a\uff0c\u5728\u54c8\u5bc6\u9813\u767c\u660e\u56db\u5143\u6578\u4e4b\u5f8c\u77ed\u77ed\u5169\u500b\u6708\uff0c\u4ed6\u7684\u670b\u53cb(John Graves)\u5c31\u767c\u660e\u4e86\u516b\u5143\u6578(\u4e3b\u8981\u662f\u627e\u5230\u6b63\u78ba\u7684\u4e58\u6cd5\u898f\u5247)\uff0c\u4f46\u7531\u65bc\u8ad6\u6587\u767c\u8868\u8f03\u9072\uff0c\u6240\u4ee5\u6839\u64da\u6b63\u5f0f\u7684\u6578\u5b78\u53f2\uff0c\u516b\u5143\u6578\u7684\u767c\u660e\u8005\u53e6\u6709\u5176\u4eba(Arthur Cayley)\u3002<\/p>\n<p>\u81f3\u65bc\u516b\u5143\u6578\u6709\u4e9b\u4ec0\u9ebc\u7279\u6b8a\u898f\u5247\uff0c\u6211\u5011\u53ef\u4ee5\u7528\u6b78\u7d0d\u6cd5\u4f86\u601d\u8003\u9019\u500b\u554f\u984c\u3002<\/p>\n<p>\u5728\u5be6\u6578\u63a8\u5ee3\u5230\u8907\u6578\u7684\u904e\u7a0b\u4e2d\uff0c\u300c\u6b21\u5e8f\u300d\u7684\u6982\u5ff5\u88ab\u72a7\u7272\u4e86\u3002\u56e0\u70ba\u5be6\u6578\u5177\u6709\u660e\u78ba\u7684\u5927\u5c0f\u6b21\u5e8f(\u4f8b\u59822&gt;1&gt;-1)\uff0c\u800c\u8907\u6578\u537b\u6c92\u6709(\u4f60\u4e0d\u80fd\u8aaa2+i&gt;1+i, \u4e5f\u4e0d\u80fd\u8aaa2+i&lt;1+i)\u3002<\/p>\n<p>\u8907\u6578\u63a8\u5ee3\u5230\u56db\u5143\u6578\u7684\u904e\u7a0b\u4e2d\u53c8\u72a7\u7272\u4e86\u4ec0\u9ebc\u5462\uff1f\u5176\u5be6\u525b\u624d\u5df2\u7d93\u63d0\u5230\uff0c\u72a7\u7272\u4e86\u300c\u4e58\u6cd5\u7684\u4ea4\u63db\u5f8b\u300d\u3002<\/p>\n<p>\u90a3\u9ebc\uff0c\u7576\u6211\u5011\u5c07\u56db\u5143\u6578\u63a8\u5ee3\u5230\u516b\u5143\u6578\uff0c\u9084\u80fd\u518d\u72a7\u7272\u4ec0\u9ebc\u5462\uff1f\u90a3\u5c31\u662f\u6216\u8a31\u6bd4\u300c\u4ea4\u63db\u5f8b\u300d\u66f4\u57fa\u672c\u7684\u300c\u7d50\u5408\u5f8b\u300d\u3002<\/p>\n<p>\u6240\u4ee5\u8aaa\uff0c\u516b\u5143\u6578\u6700\u5927\u7684\u7279\u8272\u5c31\u662f\u9023\u4e58\u6cd5\u7684\u7d50\u5408\u5f8b\u90fd\u4e0d\u898b\u4e86\u3002\u5982\u679c\u4f60\u60f3\u8b93\u4e09\u500b\u516b\u5143\u6578x, y, z\u4f9d\u7167\u9019\u500b\u9806\u5e8f\u76f8\u4e58\uff0c\u5beb\u6210xyz\u6839\u672c\u6c92\u6709\u610f\u7fa9\uff0c\u5fc5\u9808\u5beb\u6210(xy)z\u6216\u662fx(yz)\uff0c\u524d\u8005\u4ee3\u8868\u5148\u9032\u884cxy\u7684\u4e58\u6cd5\uff0c\u5f8c\u8005\u5247\u4ee5\u6b64\u985e\u63a8\u3002<\/p>\n<p>\u524d\u9762\u63d0\u5230\uff0c\u53ea\u8981\u4f60\u719f\u6089\u77e9\u9663\uff0c\u5c31\u4e0d\u81f3\u65bc\u5c0d\u300c\u975e\u4ea4\u63db\u6027\u300d\u592a\u904e\u964c\u751f\u3002\u63db\u6210\u6578\u5b78\u7684\u8aaa\u6cd5\uff0c\u5c31\u662f\u56db\u5143\u6578\u7684i, j, k\u90fd\u53ef\u4ee5\u7528\u77e9\u9663\u4f86\u8868\u73fe\uff0c\u6240\u4ee5a+bi+cj+dk\u5176\u5be6\u5c31\u662f\u56db\u500b\u77e9\u9663\u76f8\u52a0\uff0c\u7d50\u679c\u7576\u7136\u9084\u662f\u4e00\u500b\u77e9\u9663\uff0c\u4e26\u6c92\u6709\u4ec0\u9ebc\u795e\u79d8\u7684\u3002<\/p>\n<p>\u4f46\u7531\u65bc\u77e9\u9663\u4e58\u6cd5\u670d\u5f9e\u7d50\u5408\u5f8b\uff0c\u800c\u516b\u5143\u6578\u4e26\u4e0d\u670d\u5f9e\uff0c\u9019\u610f\u5473\u8457\u4f60\u6839\u672c\u4e0d\u80fd\u7528\u77e9\u9663\u4f86\u8868\u73fe\u516b\u5143\u6578\uff0c\u56e0\u6b64\u5c31\u9023\u6578\u5b78\u5bb6\u90fd\u5c0d\u5b83\u6709\u9ede\u982d\u75db\u3002<\/p>\n<p><span style=\"color: #008080;\"><strong>\u25cf\u5230\u4e86\u76e1\u982d\uff1f<\/strong><\/span><\/p>\n<p>\u5982\u679c\u4f60\u60f3\u518d\u63a5\u518d\u53b2\uff0c\u5c07\u516b\u5143\u6578\u64f4\u5145\u70ba\u5341\u516d\u5143\u6578\uff0c\u90a3\u9ebc\u6839\u64da\u4e0a\u8ff0\u7684\u300c\u72a7\u7272\u898f\u5f8b\u300d\uff0c\u4f60\u5f97\u5148\u627e\u51fa\u9084\u6709\u4ec0\u9ebc\u80fd\u72a7\u7272\u7684\u3002\u53ef\u662f\u4e0d\u8ad6\u4f60\u591a\u9ebc\u52aa\u529b\uff0c\u6050\u6015\u90fd\u662f\u5f92\u52de\u7121\u529f\u3002\u9019\u5c31\u4ee3\u8868\u516b\u5143\u6578\u7684\u72a7\u7272\u5df2\u7d93\u5230\u4e86\u76e1\u982d\uff0c\u6240\u4ee5\u5b83\u5c31\u662f\u6700\u9ad8\u7dad\u7684\u8d85\u8907\u6578\u3002<\/p>\n<p>\u9019\u662f\u500b\u5f88\u91cd\u8981\u7684\u6578\u5b78\u5b9a\u7406\uff0c\u5b83\u6709\u500b\u53ef\u611b\u7684\u66b1\u7a31\u300c1,2,4,8\u5b9a\u7406\u300d\u3002<\/p>\n<p>\u7136\u800c\uff0c\u9019\u662f\u5426\u4ee3\u8868\u8d85\u8907\u6578\u7684\u6f14\u5316\u771f\u7684\u5230\u4e86\u76e1\u982d\uff1f\u7b54\u6848\u5247\u662f\u898b\u4ec1\u898b\u667a\u3002\u56e0\u70ba\u4e0a\u8ff0\u7684\u56db\u5143\u6578\u8207\u516b\u5143\u6578\u53ea\u80fd\u7b97\u662f\u8d85\u8907\u6578\u7684\u4e00\u7a2e\u300c\u6b63\u7d71\u300d\u985e\u578b\uff0c\u6211\u5011\u82e5\u5c0d\u8840\u7d71\u7684\u7d14\u6b63\u7a0d\u5fae\u653e\u9b06\u4e00\u9ede\uff0c\u5c31\u80fd\u53e6\u95e2\u8e4a\u5f91\uff0c\u767c\u660e\u51fa\u66f4\u591a\u7684\u8d85\u8907\u6578\uff0c\u4e0d\u904e\u90a3\u7576\u7136\u662f\u53e6\u4e00\u500b\u6545\u4e8b\u4e86\u3002<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-28303\" src=\"http:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2017\/06\/25-2.jpg\" alt=\"\" width=\"533\" height=\"400\" \/><\/p>\n<p>\u8a3b\u4e00\uff1a\u4f55\u8b02\u6b63\u7d71\uff1f<\/p>\n<p>\u8907\u6578z=a+bi\u7684\u7d55\u5c0d\u503c\u5e73\u65b9=|z|<sup>2<\/sup>=zz*=(a+bi)(a-bi)=a<sup>2<\/sup>+b<sup>2<\/sup>.<\/p>\n<p>\u82e5z, w\u7686\u70ba\u8907\u6578\uff0c\u90a3\u9ebc|z|<sup>2<\/sup>|w|<sup>2<\/sup>=zz*ww*=zwz*w*=(zw)(zw)*=|zw|<sup>2<\/sup>.<\/p>\n<p>\u56e0\u6b64\u6211\u5011\u5f97\u5230|z||w|=|zw|\u9019\u500b\u95dc\u4fc2\u3002<\/p>\n<p>\u5c07z, w\u63a8\u5ee3\u6210\u56db\u5143\u6578\uff0c\u96d6\u7136\u4e58\u6cd5\u6c92\u6709\u4ea4\u63db\u5f8b\uff0c\u6211\u5011\u80fd\u5920\u8b49\u660e|z||w|=|zw|\u4ecd\u820a\u6210\u7acb\u3002<\/p>\n<p>\u5c07z, w\u63a8\u5ee3\u6210\u516b\u5143\u6578\uff0c\u96d6\u7136\u4e58\u6cd5\u6c92\u6709\u4ea4\u63db\u5f8b\u548c\u7d50\u5408\u5f8b\uff0c|z||w|=|zw|\u540c\u6a23\u6210\u7acb\u3002<\/p>\n<p>\u6240\u4ee5\u670d\u5f9e|z||w|=|zw|\u7684\u6578\u7cfb\u5c31\u662f\u6211\u5011\u6240\u8b02\u7684\u6b63\u7d71\u6578\u7cfb(\u6578\u5b78\u5bb6\u7576\u7136\u53e6\u6709\u540d\u7a31)\u3002<\/p>\n<p>\u8a3b\u4e8c\uff1a\u82e5\u7528\u5be6\u4fc2\u6578\u4f86\u5c55\u958b\uff0c\u90a3\u9ebc\u5c0d\u65bc\u8907\u6578\u800c\u8a00\uff0c|z|<sup>2<\/sup>|w|<sup>2<\/sup>=|zw|<sup>2<\/sup>\u9019\u500b\u95dc\u4fc2\u5c0d\u61c9\u65bcFibonacci\u6046\u7b49\u5f0f(\u80fd\u5beb\u6210\u5e73\u65b9\u548c\u7684\u5169\u500b\u6574\u6578\u76f8\u4e58\uff0c\u7d50\u679c\u4ecd\u80fd\u5beb\u6210\u5e73\u65b9\u548c)\uff1a(a<sup>2<\/sup>+b<sup>2<\/sup>)(c<sup>2<\/sup>+d<sup>2<\/sup>)=(ac+bd)<sup>2<\/sup>+(ad-bc)<sup>2<\/sup>=(ac-bd)<sup>2<\/sup>+(ad+bc)<sup>2<\/sup>.<\/p>\n<p>\u63a8\u800c\u5ee3\u4e4b\uff0c\u56db\u5143\u6578\u6240\u5c0d\u61c9\u7684\u95dc\u4fc2\u5f0f\u70ba<a href=\"https:\/\/en.wikipedia.org\/wiki\/Euler%27s_four-square_identity\" target=\"_blank\" rel=\"noopener noreferrer\">Euler\u56db\u5e73\u65b9\u548c\u6046\u7b49\u5f0f<\/a>\uff0c\u800c\u516b\u5143\u6578\u5c0d\u61c9\u7684\u95dc\u4fc2\u5f0f\u5247\u662f<a href=\"https:\/\/en.wikipedia.org\/wiki\/Degen%27s_eight-square_identity\" target=\"_blank\" rel=\"noopener noreferrer\">Degen\u516b\u5e73\u65b9\u548c\u6046\u7b49\u5f0f<\/a>\u3002\u6839\u64da\u300c1,2,4,8\u5b9a\u7406\u300d\uff0c\u9019\u985e\u7684\u95dc\u4fc2\u5f0f\u5230\u6b64\u70ba\u6b62\uff0c\u5343\u842c\u5225\u4e0d\u4fe1\u90aa\uff0c\u767d\u767d\u6d6a\u8cbb\u6642\u9593\uff01<\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/line.me\/R\/ti\/p\/%40fcx6182w\" target=\"_blank\" rel=\"noopener noreferrer\"><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/scdn.line-apps.com\/n\/line_add_friends\/btn\/zh-Hant.png\" alt=\"\u52a0\u5165\u597d\u53cb\" width=\"212\" height=\"36\" border=\"0\"><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5982\u679c\u4f60\u60f3\u518d\u63a5\u518d\u53b2\uff0c\u5c07\u516b\u5143\u6578\u64f4\u5145\u70ba\u5341\u516d\u5143\u6578\uff0c\u90a3\u9ebc\u6839\u64da\u4e0a\u8ff0\u7684\u300c\u72a7\u7272\u898f\u5f8b\u300d\uff0c\u4f60\u5f97\u5148\u627e\u51fa\u9084\u6709\u4ec0\u9ebc\u80fd\u72a7\u7272\u7684\u3002\u53ef\u662f\u4e0d\u8ad6\u4f60\u591a\u9ebc\u52aa\u529b\uff0c\u6050\u6015\u90fd\u662f\u5f92\u52de\u7121\u529f\u3002\u9019\u5c31\u4ee3\u8868\u516b\u5143\u6578\u7684\u72a7\u7272\u5df2\u7d93\u5230\u4e86\u76e1\u982d\uff0c\u6240\u4ee5\u5b83\u5c31\u662f\u6700\u9ad8\u7dad\u7684\u8d85\u8907\u6578\u3002\u9019\u662f\u500b\u5f88\u91cd\u8981\u7684\u6578\u5b78\u5b9a\u7406\uff0c\u5b83\u6709\u500b\u53ef\u611b\u7684\u66b1\u7a31\u300c1,2,4,8\u5b9a\u7406\u300d\u3002<\/p>\n","protected":false},"author":20,"featured_media":28304,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3112,3293,2977],"tags":[3440,3439,1766,3437,3438],"aioseo_notices":[],"jetpack_featured_media_url":"https:\/\/case.ntu.edu.tw\/blog\/wp-content\/uploads\/2017\/06\/25-0.jpg","_links":{"self":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts\/28301"}],"collection":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/users\/20"}],"replies":[{"embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=28301"}],"version-history":[{"count":2,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts\/28301\/revisions"}],"predecessor-version":[{"id":31778,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/posts\/28301\/revisions\/31778"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=\/wp\/v2\/media\/28304"}],"wp:attachment":[{"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=28301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=28301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/case.ntu.edu.tw\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=28301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}