{"id":160,"date":"2023-07-15T22:44:08","date_gmt":"2023-07-15T22:44:08","guid":{"rendered":"https:\/\/mathority.org\/cn\/%e5%9b%9b%e5%85%83%e6%95%b0\/"},"modified":"2023-07-15T22:44:08","modified_gmt":"2023-07-15T22:44:08","slug":"%e5%9b%9b%e5%85%83%e6%95%b0","status":"publish","type":"post","link":"https:\/\/mathority.org\/cn\/%e5%9b%9b%e5%85%83%e6%95%b0\/","title":{"rendered":"\u4ec0\u4e48\u662f\u56db\u5143\u6570\uff1f"},"content":{"rendered":"

\u4ece\u8bcd\u6e90\u4e0a\u6765\u8bf4\uff0c\u56db\u5143\u6570\u6216\u56db\u5143\u6570<\/strong>\u6765\u81ea\u62c9\u4e01\u8bedquaterni<\/em> \u3002\u5728\u897f\u73ed\u7259\u8bed\u4e2d\uff0c\u8fd9\u4e2a\u8bcd\u7ffb\u8bd1\u4e3a\u201c\u56db\u70b9\u201d\u3002\u7136\u800c\uff0c\u5b83\u7684\u89e3\u91ca\u662f\u201c\u56db\u4e2a\u5143\u7d20\u7684\u6570\u91cf\u201d\u3002<\/p>\n

\u56db\u5143\u6570\u662f\u6700\u521d\u7531 William Rowan Hamilton \u521b\u5efa\u7684\u975e\u7f6e\u6362\u57df\u7684\u5143\u7d20\u3002\u56db\u5143\u6570\u88ab\u5b9a\u4e49\u4e3a\u6784\u6210\u8d85\u590d\u6570\u7684\u5b9e\u6570\u7684\u6269\u5c55\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5b83\u4eec\u4e0e\u590d\u6570<\/a>\u975e\u5e38\u76f8\u4f3c\u3002<\/p>\n

\u4e5f\u5c31\u662f\u8bf4\uff0c\u56db\u5143\u6570\u662f\u7531\u4e8e\u7c7b\u6bd4\u5f15\u8d77\u7684\u653e\u5927\u800c\u51fa\u73b0\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u590d\u6570\u662f\u901a\u8fc7\u865a\u6570\u5355\u4f4di<\/em>\u7684\u548c\u5bf9\u5b9e\u6570\u7684\u6269\u5c55\u800c\u4ea7\u751f\u7684\uff0c\u56e0\u6b64i \u7684<\/em>\u5e73\u65b9\u7b49\u4e8e -1\u3002\u5728\u7b2c\u4e00\u79cd\u60c5\u51b5\u4e0b\uff0c\u865a\u6570\u5355\u4f4d<\/strong>k<\/em> \u3001 i<\/em>\u548cj<\/em>\u88ab\u6dfb\u52a0\u5230\u5b9e\u6570\u4e2d\u3002<\/p>\n

\u56e0\u6b64\uff0c\u5bf9\u4e8e\u56db\u5143\u6570\uff0c\u6211\u4eec\u6709\uff1a i<\/em> 2<\/sup> = j<\/em> 2<\/sup> = k<\/em> 2<\/sup> = ijk<\/em> = -1\u3002\u8be5\u8868\u793a\u5bf9\u5e94\u4e8e\u51ef\u83b1\u8868<\/strong>\u4e2d\u7684\u6392\u5217\u3002\u5728\u8fd9\u4e00\u70b9\u4e0a\uff0c\u503c\u5f97\u4e00\u63d0\u7684\u662fi<\/em> \u3001 j<\/em> \u3001 k<\/em>\u548c 1 \u662f\u56db\u5143\u6570\u7684\u56db\u4e2a\u57fa\u672c\u652f\u67f1\u3002 <\/p>\n

\n\n\n\n\n\n\n\n
\u00d7<\/td>\n 1<\/td>\n\u54df<\/em><\/td>\nj<\/em><\/td>\n\u4ec0\u4e48<\/em><\/td>\n<\/tr>\n
1<\/td>\n 1<\/td>\n\u54df<\/em><\/td>\nj<\/em><\/td>\n\u4ec0\u4e48<\/em><\/td>\n<\/tr>\n
\u54df<\/em><\/td>\n\u54df<\/em><\/td>\n-1<\/td>\n\u4ec0\u4e48<\/em><\/td>\n-j<\/em><\/td>\n<\/tr>\n
j<\/em><\/td>\n j<\/em><\/td>\n -k<\/em><\/td>\n -1<\/td>\n\u54df<\/em><\/td>\n<\/tr>\n
\u4ec0\u4e48<\/em><\/td>\n\u4ec0\u4e48<\/em><\/td>\nj<\/em><\/td>\n -\u54df<\/em><\/td>\n-1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>
\u51ef\u83b1\u7684\u684c\u5b50<\/figcaption><\/figure>\n

\u5a01\u5ec9\u00b7\u6c49\u5bc6\u5c14\u987f (William Hamilton) \u4e8e 1843 \u5e74\u53d1\u660e\u4e86\u56db\u5143\u6570\uff0c\u8fd9\u79cd\u65b9\u6cd5\u4f7f\u4ed6\u80fd\u591f\u5bf9\u5411\u91cf\u8fdb\u884c\u4e58\u9664\u6cd5\u3001\u65cb\u8f6c\u5411\u91cf\u548c\u62c9\u4f38\u5411\u91cf\u3002<\/p>\n

\u56db\u5143\u6570\u662f\u5982\u4f55\u5236\u4f5c\u7684\uff1f<\/span><\/h2>\n

\u56db\u5143\u6570\u5f62\u6210\u4e86\u4e00\u4e2a\u7f8e\u4e3d\u7684\u4ee3\u6570\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5bf9\u8c61\u90fd\u5305\u542b 4 \u4e2a\u53d8\u91cf<\/strong>\u3002\u4e8b\u5b9e\u4e0a\uff0c\u5b83\u4eec\u6709\u65f6\u88ab\u79f0\u4e3a\u6b27\u62c9\u53c2\u6570\uff0c\u4e0d\u5e94\u4e0e\u6b27\u62c9\u89d2\u6df7\u6dc6\u3002\u8fd9\u4e9b\u5bf9\u8c61\u53ef\u4ee5\u4f5c\u4e3a\u5355\u4e2a\u5355\u5143\u4ee5\u7c7b\u4f3c\u4e8e\u5e38\u89c4\u6570\u5b57\u4ee3\u6570\u7684\u65b9\u5f0f\u8fdb\u884c\u52a0\u6cd5\u548c\u4e58\u6cd5\u3002<\/p>\n

\u7136\u800c\uff0c\u8fd8\u662f\u6709\u533a\u522b\u7684\u3002\u7528\u6570\u5b66\u672f\u8bed\u6765\u8bf4\uff0c\u56db\u5143\u6570\u4e58\u6cd5\u662f\u4e0d\u53ef\u4ea4\u6362\u7684\u3002<\/p>\n

\u56db\u5143\u6570\u6709 4 \u4e2a\u7ef4\u5ea6\u3002\u6bcf\u4e2a\u56db\u5143\u6570\u7531 4 \u4e2a\u6807\u91cf<\/strong>\u3001\u4e00\u4e2a\u5b9e\u7ef4\u548c 3 \u4e2a\u865a\u7ef4\u7ec4\u6210\u3002\u8fd9\u4e9b\u865a\u6570\u7ef4\u5ea6\u4e2d\u7684\u6bcf\u4e00\u4e2a\u90fd\u6709\u4e00\u4e2a\u5355\u4f4d\u503c -1 \u7684\u5e73\u65b9\u6839\u3002\u7136\u800c\uff0c\u5b83\u4eec\u662f\u4e0d\u540c\u7684 -1 \u5e73\u65b9\u6839\uff0c\u5f7c\u6b64\u5782\u76f4\uff0c\u79f0\u4e3ai<\/em> \u3001 j<\/em>\u548ck<\/em> \u3002\u56e0\u6b64\uff0c\u56db\u5143\u6570\u53ef\u4ee5\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n

x = (a, b, c, d) \u5199\u4f5c x = a + bi + cj + dk<\/p>\n

\u56e0\u6b64\uff0ca\u3001b\u3001c\u548cd\u8868\u793a\u7531\u6bcf\u4e2a\u56db\u5143\u6570\u660e\u786e\u5b9a\u4e49\u7684\u5b9e\u6570\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b57 1\u3001 i<\/em> \u3001 j<\/em>\u548ck<\/em>\u662f\u57fa\u672c\u6570\u5b57\u3002\u5982\u679c\u6211\u4eec\u60f3\u7528\u96c6\u5408\u6765\u8868\u793a\u56db\u5143\u6570<\/strong>\uff0c\u53ef\u4ee5\u8fd9\u6837\u505a\uff1a \u5047\u8bbeIR 4<\/sup>\u8868\u793a\u96c6\u5408\uff0c\u5219\u8868\u8fbe\u5f0f\u4e3a\uff1a IR4= {a + bi + cj + dk: a, b, c, d \u2208 IR}<\/p>\n

\u8fd9\u4e2a\u96c6\u5408\u4e0e\u771f\u5b9e\u7684\u56db\u7ef4\u7a7a\u95f4\u662f\u4e00\u81f4\u7684\u3002\u6b63\u5982\u4e00\u7ec4\u5b9e\u6570\u5bf9\u5e94\u4e8e\u4e00\u7ef4\u7a7a\u95f4\uff0c\u4e00\u7ec4\u590d\u6570\u5bf9\u5e94\u4e8e\u4e8c\u7ef4\u7a7a\u95f4\u3002<\/p>\n

\u56db\u5143\u6570\u7684\u4ee3\u6570\u7ed3\u6784\u662f\u4ec0\u4e48\uff1f<\/span><\/h2>\n

\u56db\u5143\u6570\u8bf4\u660e\u4e86\u4e0d\u89c4\u5219\u4f53<\/strong>\u3002\u8fd9\u610f\u5473\u7740\u5b83\u662f\u4e00\u4e2a\u7c7b\u4f3c\u4e8e\u57df\u7684\u4ee3\u6570\u7ed3\u6784\u3002\u7136\u800c\uff0c\u5b83\u5728\u4e58\u6cd5\u4e2d\u4e0d\u53ef\u4ea4\u6362\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5b83\u6ee1\u8db3\u4e86\u7269\u4f53\u7684\u6240\u6709\u54c1\u8d28\uff0c\u4f46\u5176\u7ed3\u679c\u4e0d\u53ef\u4ea4\u6362\u3002<\/p>\n

\u56db\u5143\u6570\u4e58\u6cd5\u662f\u7ed3\u5408\u5f8b\u3002\u6b64\u5916\uff0c\u6bcf\u4e2a\u975e\u96f6\u56db\u5143\u6570\u90fd\u6709\u4e00\u4e2a\u552f\u4e00\u7684\u9006<\/strong>\u3002\u4e0e\u590d\u6570\u76f8\u6bd4\uff0c\u56db\u5143\u6570\u4e0d\u6784\u6210\u5173\u8054\u4ee3\u6570\u3002<\/p>\n

\u6700\u540e\uff0c\u4e0e\u590d\u6570\u548c\u5b9e\u6570\u8868\u793a\u5355\u4f4d\u6216\u53cc\u7a7a\u95f4\u7684\u6b27\u51e0\u91cc\u5fb7\u5411\u91cf\u7ef4\u5ea6\u76f8\u540c\uff0c\u56db\u5143\u6570\u521b\u5efa\u4e86\u56db\u7ef4\u6b27\u51e0\u91cc\u5fb7\u5411\u91cf\u533a\u57df\u3002<\/p>\n

\u56db\u5143\u6570\u5728\u77e9\u9635\u4e2d\u5982\u4f55\u8868\u793a\uff1f<\/span><\/h2>\n

\u77e9\u9635\u8868\u793a<\/strong>\u4e5f\u662f\u56db\u5143\u6570\u7684\u7279\u5f81\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u5e94\u7528\u6570\u5b66\u77e9\u9635\u6765\u8868\u8fbe\u5b83\u3002\u4f8b\u5982\uff0c\u5982\u679c\u6211\u4eec\u6709\u56db\u5143\u6570 p = a + bi + cj + dk\uff0c\u5219\u53ef\u4ee5\u5c06\u5176\u8868\u793a\u4e3a\u590d\u6570 2 x 2 \u77e9\u9635\uff0c\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n

\u5728\u56db\u5143\u6570\u4e2d\u4f7f\u7528\u77e9\u9635\u8868\u793a\u7684\u53e6\u4e00\u79cd\u65b9\u6cd5\u662f\u4f7f\u7528\u5b9e 4 x 4 \u77e9\u9635<\/strong>\u3002\u6b64\u5916\uff0c\u901a\u8fc7\u4f7f\u7528\u77e9\u9635\u6765\u8868\u793a\u56db\u5143\u6570\uff0c\u53ef\u4ee5\u5c06\u5b83\u4eec\u8868\u793a\u4e3a\u4e24\u4e2a\u5411\u91cf\u7684\u5185\u79ef\u3002\u56e0\u6b64\uff0c\u4e00\u4e2a\u5206\u91cf\u4e3a\uff1a = (a1, a2, a3, a4) \uff0c\u53e6\u4e00\u4e2a\u5206\u91cf\u4e3a {1, i, j, k<\/em> }\u3002<\/p>\n

\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u751f\u6210\u5b9e\u90e8\u7684\u5143\u7d20a 1<\/sub>\u88ab\u5355\u72ec\u5199\u5165\u3002\u6b64\u5916\uff0c\u5bf9\u4e8e\u6807\u91cf\u79ef\uff0c\u4ec5\u8003\u8651\u4e09\u4e2a\u57fa\u6570<\/strong>i\u3001j\u3001k<\/em> \uff1a<\/p>\n

x = (a1, a) = (a1, a2, a3, a4)<\/p>\n

\u56db\u5143\u6570\u53ef\u4ee5\u8fdb\u884c\u54ea\u4e9b\u57fa\u672c\u8fd0\u7b97\uff1f<\/span><\/h2>\n

\u4e3a\u4e86\u6dfb\u52a0\u5e76\u83b7\u5f97\u4e00\u4e2a\u56db\u5143\u6570\u4e0e\u53e6\u4e00\u4e2a\u56db\u5143\u6570\u4e4b\u95f4\u7684\u4e58\u79ef\uff0c\u9700\u8981\u5e94\u7528\u590d\u6570\u7b97\u672f\u3002\u5176\u5de5\u4f5c\u539f\u7406\u4e0e\u4e4b\u524d IR 4<\/sup>\u7ec4<\/strong>\u7684\u60c5\u51b5\u76f8\u540c\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u8be5\u96c6\u5408\u52a0\u4e0a\u5176\u4f59\u64cd\u4f5c\u8865\u507f\u4e86\u7269\u4f53\u7684\u6240\u6709\u54c1\u8d28\u3002\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u552f\u4e00\u76f8\u5173\u7684\u662f\u4ea7\u54c1\u4e0d\u901a\u52e4\u3002<\/p>\n

\u5982\u6709\u8865\u5145\uff0c\u5219\u9010\u9879\u8fdb\u884c\u3002\u65e0\u8bba\u5982\u4f55\uff0c\u5b83\u7684\u5de5\u4f5c\u539f\u7406\u4e0e\u590d\u6570\u76f8\u540c\u3002\u4e5f\u5c31\u662f\u8bf4\uff1a<\/p>\n

(a1 + b1i + c1j + d1k) + (a2 + b2i + c2j + d2k) = (a1 + a2) + (b1 + b2)i + (c1 + c2)j + (d1 + d2)k\u3002<\/p>\n

\u5bf9\u4e8e\u4ea7\u54c1\u6765\u8bf4\uff0c\u5b83\u662f\u4ece\u4e00\u4e2a\u7ec4\u4ef6\u5e94\u7528\u5230\u53e6\u4e00\u4e2a\u7ec4\u4ef6<\/strong>\u3002\u6839\u636e\u8fd9\u4e2a\uff0c\u5b83\u770b\u8d77\u6765\u50cf\u8fd9\u6837\uff1a<\/p>\n

ab = (a1b1 \u2013 a2b2 \u2013 a3b3 \u2013 a4b4) + (a1b2 + a2b1 + a3b4 \u2013 a4b3)i + (a1b3 \u2013 a2b4 + a3b1 \u2013 a4b2)j + (a1b4 + a2b3 \u2013 a3b2 + a4b3)k<\/p>\n

\u6b63\u5982\u6211\u4eec\u4e4b\u524d\u5df2\u7ecf\u6307\u51fa\u7684\uff0c\u56db\u5143\u6570\u7684\u4e58\u79ef\u6c38\u8fdc\u4e0d\u53ef\u4ea4\u6362\u3002\u76f8\u53cd\uff0c\u5b83\u603b\u662f\u5173\u8054\u7684<\/strong>\u3002\u524d\u9762\u8be6\u7ec6\u9610\u8ff0\u7684\u64cd\u4f5c\u53ef\u4ee5\u901a\u8fc7\u66ff\u6362\u8868\u793a\u6765\u6267\u884c\u3002<\/p>\n

\u56db\u5143\u6570\u6709\u54ea\u4e9b\u5e94\u7528\uff1f<\/span><\/h2>\n

\u56db\u5143\u6570\u8fdc\u8fdc\u8d85\u51fa\u4e86\u6570\u5b66\u7814\u7a76\u7684\u8303\u56f4\u3002\u76ee\u524d\u5b83\u4eec\u6709\u5404\u79cd\u5e94\u7528\u3002\u9996\u5148\uff0c\u5b83\u4eec\u7528\u4e8e\u9a8c\u8bc1\u6570\u8bba<\/strong>\u4e2d\u7684\u7b54\u6848\u3002\u62c9\u683c\u6717\u65e5\u5b9a\u7406\u5c31\u662f\u4e00\u4e2a\u4f8b\u5b50\uff0c\u8be5\u5b9a\u7406\u6307\u51fa\u4efb\u4f55\u81ea\u7136\u6570\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a 4 \u4e2a\u5b8c\u5168\u5e73\u65b9\u548c\u3002<\/p>\n

\u53e6\u4e00\u65b9\u9762\uff0c\u5b83\u5728\u7269\u7406\u5b66\u9886\u57df\u4e5f\u6709\u5e94\u7528\u3002\u56db\u5143\u6570\u5bf9\u4e8e\u91cf\u5b50\u529b\u5b66\u3001\u7535\u78c1\u5b66\u7b49\u975e\u5e38\u6709\u7528\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4ece\u8bcd\u6e90\u4e0a\u6765\u8bf4\uff0c\u56db\u5143\u6570\u6216\u56db\u5143\u6570\u6765\u81ea\u62c9\u4e01\u8bedquaterni \u3002\u5728\u897f\u73ed\u7259\u8bed\u4e2d\uff0c\u8fd9\u4e2a\u8bcd\u7ffb\u8bd1\u4e3a\u201c\u56db\u70b9\u201d\u3002\u7136\u800c\uff0c\u5b83\u7684\u89e3\u91ca\u662f …<\/p>\n

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