{"id":15860,"date":"2010-11-25T06:16:34","date_gmt":"2010-11-24T22:16:34","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=15860"},"modified":"2021-10-06T16:30:40","modified_gmt":"2021-10-06T08:30:40","slug":"%ef%bc%91%e7%9a%84%ef%bd%8e%e6%ac%a1%e6%96%b9%e6%a0%b9%ef%bc%88nth-root-of-1%ef%bc%89","status":"publish","type":"post","link":"http:\/\/localhost\/%ef%bc%91%e7%9a%84%ef%bd%8e%e6%ac%a1%e6%96%b9%e6%a0%b9%ef%bc%88nth-root-of-1%ef%bc%89\/","title":{"rendered":"\uff11\u7684\uff4e\u6b21\u65b9\u6839\uff08nth root of 1\uff09"},"content":{"rendered":"

\uff11\u7684\uff4e\u6b21\u65b9\u6839\uff08nth root of 1\uff09<\/strong><\/span>
\n\u570b\u7acb\u5c4f\u6771\u9ad8\u7d1a\u4e2d\u5b78\u6578\u5b78\u79d1\u694a\u74ca\u8339\u8001\u5e2b\/\u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u6d2a\u842c\u751f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n

\u5728\u4e00\u5143 $$n$$ \u6b21\u65b9\u7a0b\u5f0f\u4e2d\uff0c\u9996\u5148\u8a0e\u8ad6\u6700\u7c21\u55ae\u4e14\u91cd\u8981\u7684\u65b9\u7a0b\u5f0f $$x^n=1$$\uff0c\u5b83\u986f\u7136\u6709 $$1$$ \u9019\u4e00\u500b\u5be6\u6839\u3002\u6839\u64da\u4ee3\u6578\u57fa\u672c\u5b9a\u7406\u8207\u56e0\u5f0f\u5b9a\u7406\u5f97\u77e5\uff0c$$n$$ \u6b21\u591a\u9805\u5f0f\u65b9\u7a0b\u5f0f\u6070\u6709 $$n$$ \u500b\u8907\u6578\u6839\uff0c\u6240\u4ee5\u65b9\u7a0b\u5f0f $$x^n=1$$ \u5171\u6709 $$n$$ \u500b\u6839\u3002\u73fe\u5728\uff0c\u6211\u5011\u61c9\u7528\u68e3\u7f8e\u5f17\u5b9a\u7406\u6c42\u89e3\u51fa $$n$$ \u500b $$1$$ \u7684 $$n$$ \u6b21\u65b9\u6839\u3002<\/p>\n

\u5047\u8a2d\u8907\u6578 $$z$$ \u662f\u65b9\u7a0b\u5f0f $$x^n=1$$ \u7684\u6839\uff0c\u5373 $$z^n=1$$\u3002
\n\u4ee5\u6975\u5f0f\u8868\u793a\uff0c$$z=r(\\cos\\theta+i{\\sin}\\theta)$$\uff0c$$1=1(\\cos{0}+i{\\sin}0)$$\uff0c
\n\u4ee3\u5165 $$z^n=1$$ \u5f97 $$[r^n(\\cos\\theta+i{\\sin}\\theta)]=1(\\cos{0}+i{\\sin}0)$$\u3002
\n\u7531\u68e3\u7f8e\u5f17\u5b9a\u7406\u00a0$$[r^n(\\cos{n}\\theta+i{\\sin}n{\\theta})]=1(\\cos{0}+i{\\sin}0)$$
\n\u7531\u8907\u6578\u76f8\u7b49\u6027\u8cea\uff08$$r$$ \u662f\u5927\u65bc $$0$$ \u7684\u5be6\u6578\uff0c$$n\\theta$$ \u548c\u89d2\u5ea6 $$0$$ \u662f\u540c\u754c\u89d2\uff09<\/p>\n

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

\u96d6\u7136\u6574\u6578 $$k$$ \u6709\u7121\u9650\u591a\u500b\uff0c\u4e4d\u770b\u4e4b\u4e0b\u597d\u50cf\u4e5f\u6709\u7121\u9650\u591a\u500b $$\\theta$$\uff0c<\/p>\n

\u4f46 $$k=0,1,2,\\mbox{…},n-1$$ \u5f97\u51fa\u7684 $$n$$ \u500b\u8f3b\u89d2 $$\\theta$$\uff0c<\/p>\n

\u548c $$k=n,n+1,n+2,\\mbox{…},2n-1$$ \u5f97\u51fa\u7684 $$n$$ \u500b\u8f3b\u89d2 $$\\theta$$ \u662f\u540c\u754c\u89d2\uff0c<\/p>\n

\u800c\u4e14 $$\\sin\\theta$$\u3001$$\\cos\\theta$$ \u7684\u9031\u671f\u70ba $$2\\pi$$\uff0c<\/p>\n

\u6545\u53d6 $$k=0,1,2,\\mbox{…},n-1$$\uff0c\u5373\u8907\u6578 $$z$$ \u7684\u4e3b\u8f3b\u89d2 $$\\theta$$ \u70ba $$0,\\frac{2\\pi}{n},\\frac{4\\pi}{n},\\mbox{…}\\mbox{…},\\frac{2(n-1)\\pi}{n}$$\u3002<\/p>\n

\u56e0\u6b64\uff0c$$1$$ \u7684 $$n$$ \u500b $$n$$ \u6b21\u65b9\u6839\u70ba<\/p>\n

$$\\displaystyle z_k=\\cos\\frac{2k\\pi}{n}+i\\sin\\frac{2k\\pi}{n},~k=0,1,2,……,n-1$$<\/p>\n

\u7531\u65bc $$|z_k|=1$$\uff0c\u6240\u4ee5\u5728\u8907\u6578\u5e73\u9762\u4e0a\uff0c$$n$$ \u500b\u6839\u6240\u4ee3\u8868\u7684 $$n$$ \u500b\u9ede\u90fd\u5728\u55ae\u4f4d\u5713\u4e0a\uff0c\u4e26\u4e14\u5e73\u5747\u5206\u5e03\u5728\u55ae\u4f4d\u5713\u7684\u5167\u63a5\u6b63 $$n$$ \u908a\u5f62\u7684\u9802\u9ede\u4e0a\uff0c\u89d2\u5ea6\u9593\u9694 $$\\frac{2\\pi}{n}$$\uff0c\u5982\u4e0b\u5716\u3002<\/p>\n

\"15860_p1\"<\/a><\/p>\n

\u5e95\u4e0b\uff0c\u5206\u5225\u4ee5 $$1$$ \u7684 $$3$$ \u6b21\u65b9\u6839\u3001$$4$$ \u6b21\u65b9\u6839\u3001$$5$$ \u6b21\u65b9\u6839\u8207\u4f8b\u8aaa\u660e\u3002<\/p>\n

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

\u5982\u6b64\u6f02\u4eae\u7684 $$n$$ \u500b\u6839\u53ca\u5176\u5716\u5f62\u61c9\u7528\u68e3\u7f8e\u5f17\u5b9a\u7406\u4fbf\u53ef\u89e3\u51fa\uff0c\u4f46\u662f\uff0c\u8a31\u591a\u5b78\u751f\u5c0d\u65bc $$1$$ \u6709 $$n$$ \u500b\u76f8\u7570\u7684 $$n$$ \u6b21\u65b9\u6839\uff0c\u7e3d\u662f\u89ba\u5f97\u56f0\u60d1\u548c\u9a5a\u8a1d\uff0c\u800c\u4e14\u770b\u5230\u6839\u7adf\u7136\u662f\u8907\u6578\u66f4\u611f\u5230\u77a0\u76ee\u7d50\u820c\u3002\u5176\u5be6\uff0c\u5728\u68e3\u7f8e\u5f17\u6240\u8655\u7684\u6642\u4ee3\uff0c\u5c08\u696d\u7684\u6578\u5b78\u5bb6\u5c0d\u8907\u6578\u7121\u7406\u6839\u6216\u662f\u628a\u8907\u6578\u7d0d\u5165\u4e09\u89d2\u51fd\u6578\u5f62\u5f0f\u4e5f\u6703\u5927\u611f\u8a1d\u7570\u3002<\/p>\n<\/div>","protected":false},"excerpt":{"rendered":"

\u7531\u300c\u4ee3\u6578\u57fa\u672c\u5b9a\u7406\u300d\u8207\u300c\u56e0\u5f0f\u5b9a\u7406\u300d\u5f97\u77e5\uff0c\u8907\u4fc2\u6578n\u6b21\u591a\u9805\u5f0f\u65b9\u7a0b\u5f0f\u6070\u6709n\u500b\u8907\u6578\u6839\u3002\u61c9\u7528\u300c\u68e3\u7f8e\u5f17\u5b9a\u7406\u300d\u4fbf\u53ef\u89e3\u51fa\u9019n\u500b\u4ee4\u4eba\u9a5a\u8c54\u7684\u8907\u6578\u6839\u3002<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[111,216],"tags":[453,444,431],"class_list":["post-15860","post","type-post","status-publish","format-standard","hentry","category-mathematics00","category-math02","tag-1-n-","tag-444","tag-431","loop-entry","cat-111","cat-216","no-thumbnail"],"views":12317,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15860","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=15860"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15860\/revisions"}],"predecessor-version":[{"id":89300,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/15860\/revisions\/89300"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=15860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=15860"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=15860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}