复数怎么计算 复数的概念与运算？
\u590d\u6570\u662f\u600e\u4e48\u8ba1\u7b97\u5f97\uff1f\u590d\u6570\u662f\u5f62\u5982 a \uff0b b i\u7684\u6570\u3002\u5f0f\u4e2da\uff0cb \u4e3a \u5b9e\u6570\uff0ci\u662f\u4e00\u4e2a\u6ee1\u8db3i^2 \uff1d\uff0d1\u7684\u6570\uff0c\u56e0\u4e3a\u4efb\u4f55\u5b9e\u6570\u7684\u5e73\u65b9\u4e0d\u7b49\u4e8e\uff0d1\uff0c\u6240\u4ee5i\u4e0d\u662f\u5b9e\u6570\uff0c\u800c\u662f\u5b9e\u6570\u4ee5\u5916\u7684\u65b0\u7684\u6570\u3002
\u5728\u590d\u6570a\uff0bbi\u4e2d\uff0ca\u79f0\u4e3a\u590d\u6570\u7684\u5b9e\u90e8\uff0cb\u79f0\u4e3a\u590d\u6570\u7684\u865a\u90e8\uff0ci\u79f0\u4e3a\u865a\u6570\u5355\u4f4d\u3002\u5f53\u865a\u90e8\u7b49\u4e8e\u96f6\u65f6\uff0c\u8fd9\u4e2a\u590d\u6570\u5c31\u662f\u5b9e\u6570\uff1b\u5f53\u865a\u90e8\u4e0d\u7b49\u4e8e\u96f6\u65f6\uff0c\u8fd9\u4e2a\u590d\u6570\u79f0\u4e3a\u865a\u6570\uff0c\u865a\u6570\u7684\u5b9e\u90e8\u5982\u679c\u7b49\u4e8e\u96f6\uff0c\u5219\u79f0\u4e3a\u7eaf\u865a\u6570\u3002\u7531\u4e0a\u53ef\u77e5\uff0c\u590d\u6570\u96c6\u5305\u542b\u4e86\u5b9e\u6570\u96c6\uff0c\u56e0\u800c\u662f\u5b9e\u6570\u96c6\u7684\u6269\u5f20\u3002\u590d\u6570\u6709\u591a\u79cd\u8868\u793a\u5f62\u5f0f\uff0c\u5e38\u7528\u5f62\u5f0f z \uff1d a \uff0b b i\u53eb\u505a\u4ee3\u6570\u5f0f\u3002\u6b64\u5916\u6709\u4e0b\u5217\u5f62\u5f0f\u3002
\u2460\u51e0\u4f55\u5f62\u5f0f\u3002\u590d\u6570 z \uff1d a \uff0b b i \u7528\u76f4\u89d2\u5750\u6807\u5e73\u9762\u4e0a\u70b9 Z \uff08 a \uff0c b \uff09\u8868\u793a\u3002\u8fd9\u79cd\u5f62\u5f0f\u4f7f\u590d\u6570\u7684\u95ee\u9898\u53ef\u4ee5\u501f\u52a9\u56fe\u5f62\u6765\u7814\u7a76\u3002\u4e5f\u53ef\u53cd\u8fc7\u6765\u7528\u590d\u6570\u7684\u7406\u8bba\u89e3\u51b3\u4e00\u4e9b\u51e0\u4f55\u95ee\u9898\u3002
\u2461\u5411\u91cf\u5f62\u5f0f\u3002\u590d\u6570 z \uff1d a \uff0b b i\u7528\u4e00\u4e2a\u4ee5\u539f\u70b9 O \u4e3a\u8d77\u70b9\uff0c\u70b9 Z \uff08 a \uff0c b \uff09\u4e3a\u7ec8\u70b9\u7684\u5411\u91cf O Z \u8868\u793a\u3002\u8fd9\u79cd\u5f62\u5f0f\u4f7f\u590d\u6570\u7684\u52a0\u3001\u51cf\u6cd5\u8fd0\u7b97\u5f97\u5230\u6070\u5f53\u7684\u51e0\u4f55\u89e3\u91ca\u3002
\u2462\u4e09\u89d2\u5f62\u5f0f\u3002\u590d\u6570 z\uff1d a \uff0b b i\u5316\u4e3a\u4e09\u89d2\u5f62\u5f0f
z \uff1d\uff5c z \uff5c\uff08cos \u03b8 \uff0bisin \u03b8 \uff09 \u5f0f\u4e2d\uff5c z \uff5c= \uff0c\u53eb\u505a\u590d\u6570\u7684\u6a21\uff08\u6216\u7edd\u5bf9\u503c\uff09\uff1b \u03b8 \u662f\u4ee5 x \u8f74\u4e3a\u59cb\u8fb9\uff1b\u5411\u91cf O Z \u4e3a\u7ec8\u8fb9\u7684\u89d2\uff0c\u53eb\u505a\u590d\u6570\u7684\u8f90\u89d2\u3002\u8fd9\u79cd\u5f62\u5f0f\u4fbf\u4e8e\u4f5c\u590d\u6570\u7684\u4e58\u3001\u9664\u3001\u4e58\u65b9\u3001\u5f00\u65b9\u8fd0\u7b97\u3002
\u2463\u6307\u6570\u5f62\u5f0f\u3002\u5c06\u590d\u6570\u7684\u4e09\u89d2\u5f62\u5f0f z \uff1d\uff5c z \uff5c(cos \u03b8 \uff0bisin \u03b8 \uff09\u4e2d\u7684cos \u03b8 \uff0bisin \u03b8 \u6362\u4e3a e i q \uff0c\u590d\u6570\u5c31\u8868\u4e3a\u6307\u6570\u5f62\u5f0f
z \uff1d\uff5c z \uff5c e i q \uff0c \u590d\u6570\u7684\u4e58\u3001\u9664\u3001\u4e58\u65b9\u3001\u5f00\u65b9\u53ef\u4ee5\u6309\u7167\u5e42\u7684\u8fd0\u7b97\u6cd5\u5219\u8fdb\u884c\u3002
\u590d\u6570\u96c6\u4e0d\u540c\u4e8e\u5b9e\u6570\u96c6\u7684\u51e0\u4e2a\u7279\u70b9\u662f\uff1a\u5f00\u65b9\u8fd0\u7b97\u6c38\u8fdc\u53ef\u884c\uff1b\u4e00\u5143 n \u6b21\u590d\u7cfb\u6570\u65b9\u7a0b\u603b\u6709 n \u4e2a\u6839\uff08\u91cd\u6839\u6309\u91cd\u6570\u8ba1\uff09\uff1b\u590d\u6570\u4e0d\u80fd\u5efa\u7acb\u5927\u5c0f\u987a\u5e8f\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u6839\u636e\u5b9a\u4e49\uff0c\u82e5 (a\uff0cb\u2208R\uff09\uff0c\u5219 =a\uff0dbi\uff08a,b\u2208R\uff09\u3002\u5171\u8f6d\u590d\u6570\u6240\u5bf9\u5e94\u7684\u70b9\u5173\u4e8e\u5b9e\u8f74\u5bf9\u79f0\u3002\u4e24\u4e2a\u590d\u6570\uff1ax+yi\u4e0ex-yi\u79f0\u4e3a\u5171\u8f6d\u590d\u6570\uff0c\u5b83\u4eec\u7684\u5b9e\u90e8\u76f8\u7b49\uff0c\u865a\u90e8\u4e92\u4e3a\u76f8\u53cd\u6570\u3002
\u5728\u590d\u5e73\u9762\u4e0a\uff0c\u8868\u793a\u4e24\u4e2a\u5171\u8f6d\u590d\u6570\u7684\u70b9\u5173\u4e8eX\u8f74\u5bf9\u79f0\uff0c\u800c\u8fd9\u4e00\u70b9\u6b63\u662f"\u5171\u8f6d"\u4e00\u8bcd\u7684\u6765\u6e90----\u4e24\u5934\u725b\u5e73\u884c\u5730\u62c9\u4e00\u90e8\u7281\uff0c\u5b83\u4eec\u7684\u80a9\u8180\u4e0a\u8981\u5171\u67b6\u4e00\u4e2a\u6a2a\u6881\uff0c\u8fd9\u6a2a\u6881\u5c31\u53eb\u505a"\u8f6d"\u3002\u5982\u679c\u7528z\u8868\u793ax+yi\uff0c\u90a3\u4e48\u5728z\u5b57\u4e0a\u9762\u52a0\u4e2a"\u4e00"\u5c31\u8868\u793ax-yi\uff0c\u6216\u76f8\u53cd\u3002
1 \u52a0\u6cd5\u6cd5\u5219
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\u5373
2 \u4e58\u6cd5\u6cd5\u5219
\u590d\u6570\u7684\u4e58\u6cd5\u6cd5\u5219\uff1a\u628a\u4e24\u4e2a\u590d\u6570\u76f8\u4e58\uff0c\u7c7b\u4f3c\u4e24\u4e2a\u591a\u9879\u5f0f\u76f8\u4e58\uff0c\u7ed3\u679c\u4e2di2= -1\uff0c\u628a\u5b9e\u90e8\u4e0e\u865a\u90e8\u5206\u522b\u5408\u5e76\u3002\u4e24\u4e2a\u590d\u6570\u7684\u79ef\u4ecd\u7136\u662f\u4e00\u4e2a\u590d\u6570\u3002
\u5373
3 \u9664\u6cd5\u6cd5\u5219
\u590d\u6570\u9664\u6cd5\u5b9a\u4e49\uff1a\u6ee1\u8db3 \u7684\u590d\u6570 \u53eb\u590d\u6570a+bi\u9664\u4ee5\u590d\u6570c+di\u7684\u5546\u3002
\u8fd0\u7b97\u65b9\u6cd5\uff1a\u5c06\u5206\u5b50\u548c\u5206\u6bcd\u540c\u65f6\u4e58\u4ee5\u5206\u6bcd\u7684\u5171\u8f6d\u590d\u6570\uff0c\u518d\u7528\u4e58\u6cd5\u6cd5\u5219\u8fd0\u7b97\uff0c
\u5373
4 \u5f00\u65b9\u6cd5\u5219
\u82e5zn=r(cos\u03b8+isin\u03b8\uff09\uff0c\u5219 \uff08k=0\uff0c1\uff0c2\uff0c3\u2026n-1\uff09
\u6211\u4eec\u628a\u6570\u5b66\u5206\u6790\u4e2d\u57fa\u672c\u7684\u5b9e\u53d8\u521d\u7b49\u51fd\u6570\u63a8\u5e7f\u5230\u590d\u53d8\u521d\u7b49\u51fd\u6570\uff0c\u4f7f\u5f97\u5b9a\u4e49\u7684\u5404\u79cd\u590d\u53d8\u521d\u7b49\u51fd\u6570\uff0c\u5f53z\u53d8\u4e3a\u5b9e\u53d8\u6570x(y=0\uff09\u65f6\u4e0e\u76f8\u5e94\u7684\u5b9e\u53d8\u521d\u7b49\u51fd\u6570\u76f8\u540c\u3002
\u6ce8\u610f\u6839\u636e\u8fd9\u4e9b\u5b9a\u4e49\uff0c\u5728z\u4e3a\u4efb\u610f\u590d\u53d8\u6570\u65f6\uff0c
\u2460\uff0e\u54ea\u4e9b\u76f8\u5e94\u7684\u5b9e\u53d8\u521d\u7b49\u51fd\u6570\u7684\u6027\u8d28\u88ab\u4fdd\u7559\u4e0b\u6765
\u2461\uff0e\u54ea\u4e9b\u76f8\u5e94\u7684\u5b9e\u53d8\u521d\u7b49\u51fd\u6570\u7684\u6027\u8d28\u4e0d\u518d\u6210\u7acb
\u2462\uff0e\u51fa\u73b0\u4e86\u54ea\u4e9b\u76f8\u5e94\u7684\u5b9e\u53d8\u521d\u7b49\u51fd\u6570\u6240\u6ca1\u6709\u7684\u65b0\u7684\u6027\u8d28\u3002
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\u52a0\u6cd5\u6cd5\u5219
\u590d\u6570\u7684\u52a0\u6cd5\u6309\u7167\u4ee5\u4e0b\u89c4\u5b9a\u7684\u6cd5\u5219\u8fdb\u884c\uff1a\u8bbez1=a+bi\uff0cz2=c+di\u662f\u4efb\u610f\u4e24\u4e2a\u590d\u6570\uff0c
\u5219\u5b83\u4eec\u7684\u548c\u662f (a+bi)+(c+di)=(a+c)+(b+d)i\u3002
\u4e24\u4e2a\u590d\u6570\u7684\u548c\u4f9d\u7136\u662f\u590d\u6570\uff0c\u5b83\u7684\u5b9e\u90e8\u662f\u539f\u6765\u4e24\u4e2a\u590d\u6570\u5b9e\u90e8\u7684\u548c\uff0c\u5b83\u7684\u865a\u90e8\u662f\u539f\u6765\u4e24\u4e2a\u865a\u90e8\u7684\u548c\u3002
\u590d\u6570\u7684\u52a0\u6cd5\u6ee1\u8db3\u4ea4\u6362\u5f8b\u548c\u7ed3\u5408\u5f8b\uff0c
\u5373\u5bf9\u4efb\u610f\u590d\u6570z1\uff0cz2\uff0cz3\uff0c\u6709\uff1a z1+z2=z2+z1\uff1b(z1+z2)+z3=z1+(z2+z3)\u3002
\u51cf\u6cd5\u6cd5\u5219
\u590d\u6570\u7684\u51cf\u6cd5\u6309\u7167\u4ee5\u4e0b\u89c4\u5b9a\u7684\u6cd5\u5219\u8fdb\u884c\uff1a\u8bbez1=a+bi\uff0cz2=c+di\u662f\u4efb\u610f\u4e24\u4e2a\u590d\u6570\uff0c
\u5219\u5b83\u4eec\u7684\u5dee\u662f (a+bi)-(c+di)=(a-c)+(b-d)i\u3002
\u4e24\u4e2a\u590d\u6570\u7684\u5dee\u4f9d\u7136\u662f\u590d\u6570\uff0c\u5b83\u7684\u5b9e\u90e8\u662f\u539f\u6765\u4e24\u4e2a\u590d\u6570\u5b9e\u90e8\u7684\u5dee\uff0c\u5b83\u7684\u865a\u90e8\u662f\u539f\u6765\u4e24\u4e2a\u865a\u90e8\u7684\u5dee\u3002
\u53c2\u8003\u8d44\u6599\uff1a\u767e\u5ea6\u767e\u79d1-\u590d\u6570
绛旓細鍒欏畠浠殑鍜屾槸 (a+bi)+(c+di)=(a+c)+(b+d)i銆涓や釜澶嶆暟鐨勫拰渚濈劧鏄鏁锛屽畠鐨勫疄閮ㄦ槸鍘熸潵涓や釜澶嶆暟瀹為儴鐨勫拰锛屽畠鐨勮櫄閮ㄦ槸鍘熸潵涓や釜铏氶儴鐨勫拰銆2銆佸噺娉曟硶鍒 澶嶆暟鐨勫噺娉曟寜鐓т互涓嬭瀹氱殑娉曞垯杩涜锛氳z1=a+bi锛寊2=c+di鏄换鎰忎袱涓鏁帮紝鍒欏畠浠殑宸槸 (a+bi)-(c+di)=(a-c)+(b-d)i銆備袱涓...
绛旓細瀵逛簬涓や釜澶嶆暟 a+bi 鍜 c+di锛屽畠浠殑鍟嗗彲浠ラ氳繃浠ヤ笅鍏紡璁＄畻锛(a+bi) / (c+di) = [(a+bi)*(c-di)] / [(c+di)*(c-di)] = [(ac+bd) + (bc-ad)i] / (c^2 + d^2)銆備緥濡傦紝(2+3i) / (4+5i) = [(2*4 + 3*5) + (3*4 - 2*5)i] / (4^2 + 5^2) =...
绛旓細澶嶆暟鐨勫姞娉曟硶鍒欙細璁緕1=a+bi锛寊2=c+di鏄换鎰忎袱涓鏁銆備袱鑰呭拰鐨勫疄閮ㄦ槸鍘熸潵涓や釜澶嶆暟瀹為儴鐨勫拰锛屽畠鐨勮櫄閮ㄦ槸鍘熸潵涓や釜铏氶儴鐨勫拰銆備袱涓鏁扮殑鍜屼緷鐒舵槸澶嶆暟銆傚嵆 2.涔樻硶娉曞垯 澶嶆暟鐨勪箻娉曟硶鍒欙細鎶婁袱涓鏁扮浉涔橈紝绫讳技涓や釜澶氶」寮忕浉涔橈紝缁撴灉涓璱2= -1锛屾妸瀹為儴涓庤櫄閮ㄥ垎鍒悎骞躲備袱涓鏁扮殑绉粛鐒舵槸涓涓鏁般傚嵆...
绛旓細澶嶆暟鐨勫叕寮忔槸z=a+bi锛岃繍绠楁硶鍒欐湁鍔犲噺娉曞拰涔橀櫎娉曪紝鍖呮嫭瀵规暟娉曞垯鍜屾寚鏁版硶鍒欍傚鏁拌繍绠楁硶鍒欐湁锛鍔犲噺娉曘佷箻闄ゆ硶銆備袱涓鏁扮殑鍜屼緷鐒舵槸澶嶆暟锛屽畠鐨勫疄閮ㄦ槸鍘熸潵涓や釜澶嶆暟瀹為儴鐨勫拰锛屽畠鐨勮櫄閮ㄦ槸鍘熸潵涓や釜铏氶儴鐨勫拰銆傚鏁扮殑鍔犳硶婊¤冻浜ゆ崲寰嬪拰缁撳悎寰嬨傛澶栵紝澶嶆暟浣滀负骞傚拰瀵规暟鐨勫簳鏁般佹寚鏁般佺湡鏁版椂锛屽叾杩愮畻瑙勫垯鍙敱娆ф媺鍏...
绛旓細7銆佸鏁版病鏈夌偣涔橈紱8銆佸鏁版病鏈夊弶涔橈紱銆傝礋鏁扮殑璁＄畻鏂规硶 璐熸暟鐨勮绠楁柟娉曪細 1.璐熸暟鐩稿姞锛屾妸璐熷彿甯﹀叆锛屾暟瀛楃浉鍔犲嵆鍙銆備緥濡傦細锛-3锛+(-2)=-5 2.璐熸暟鐩稿噺锛岃鍑忔暟澶х殑锛岃礋鍙风户缁甫鍏ワ紝鍑忔暟澶х殑锛岃礋鍙锋敼鎴愭鍙凤紝鏁板瓧鐢ㄥぇ鐨勫噺鎺夊皬鐨勫氨鍙互浜嗐備緥濡傦細锛-9锛-(-3)=-6;(-3)-(-9)=+6 3.璐熸暟...
绛旓細^2=(3^2-2^2)+2*3*2i=8-4i銆8.瀵规暟杩愮畻锛氫竴涓澶嶆暟鐨勮嚜鐒跺鏁版槸鍏跺疄閮ㄥ拰铏氶儴鍒嗗埆杩涜鑷劧瀵规暟杩愮畻鐨勭粨鏋溿備緥濡傦紝log(3+2i)=log(3)+log(2i)=log(3)+log(-1/2)*i銆備互涓婂氨鏄鏁扮殑涓昏杩愮畻瑙勫垯锛岄渶瑕佹敞鎰忕殑鏄紝鐢变簬澶嶆暟鐨勫鏉傛э紝杩欎簺杩愮畻閫氬父闇瑕侀氳繃娆ф媺鍏紡绛夊伐鍏疯繘琛岀畝鍖栧拰璁＄畻銆
绛旓細澶嶆暟鐨勫姞娉曟弧瓒充氦鎹㈠緥鍜岀粨鍚堝緥锛屽嵆瀵逛换鎰忓鏁皕1锛寊2锛寊3锛屾湁锛 z1+z2=z2+z1锛(z1+z2)+z3=z1+(z2+z3)銆傚噺娉曟硶鍒 澶嶆暟鐨勫噺娉曟寜鐓т互涓嬭瀹氱殑娉曞垯杩涜锛璁緕1=a+bi锛寊2=c+di鏄换鎰忎袱涓鏁锛屽垯瀹冧滑鐨勫樊鏄 (a+bi)-(c+di)=(a-c)+(b-d)i銆備袱涓鏁扮殑宸緷鐒舵槸澶嶆暟锛屽畠鐨勫疄閮...
绛旓細闄ゆ硶锛氭妸闄ゆ硶鍐欐垚鍒嗘暟鐨勫舰寮忥紝鍐嶅皢鍒嗘瘝瀹炴暟鍖栵紙灏辨槸涔樺叾鍏辫江澶嶆暟锛(a+bi)/(c+di)=(a+bi)*(c-di)/[(c+di)(c-di)]=[ac+bd-(ad-bc)i]/(c^2+d^2)鍦ㄥ疄鏁板煙涓婂畾涔変簩鍏冩湁搴忓z=(a,b)骞惰瀹氭湁搴忓涔嬮棿鏈夎繍绠椻+鈥濄佲溍椻濓紙璁皕1=(a, b)锛寊2=(c, d)锛夛細z1 + z2=(a...
绛旓細澶嶆暟璁＄畻鍏紡濡備笅锛1銆佸姞娉曡繍绠:璁緕1=a+bi锛寊2=c+di鏄换鎰忎袱涓鏁锛屽畠鐨勫疄閮ㄦ槸鍘熸潵涓や釜澶嶆暟瀹為儴鐨勫拰锛屽畠鐨勮櫄閮ㄦ槸鍘熸潵涓や釜铏氶儴鐨勫拰锛屽嵆(a+bi)+(c+di)=(a+c)+(b+d)i銆2銆佷箻娉曡繍绠:璁緕1=abi锛寊2=c+di鏄换鎰忎袱涓鏁帮紝鍒(a+bi)(c+di)=(ac-bd)+(bc+ad)i銆3銆侀櫎娉曡繍绠:澶嶆暟...
绛旓細澶嶆暟鐨勮绠楁柟娉曞涓嬶細1銆佸姞娉曟硶鍒欙細璁緕1=a+bi锛寊2=c+di鏄换鎰忎袱涓鏁銆傝繍绠楁柟娉曪細灏嗗垎瀛愬拰鍒嗘瘝鍚屾椂涔樹互鍒嗘瘝鐨勫叡杞鏁帮紝鍐嶇敤涔樻硶娉曞垯杩愮畻銆2銆佷箻娉曟硶鍒欙細澶嶆暟鐨勪箻娉曟硶鍒欙細璁緕1=a+bi锛寊2=c+di鏄换鎰忎袱涓鏁般傝繍绠楁柟娉曪細涓や釜澶嶆暟鐩镐箻锛屾妸瀹為儴鐩镐箻锛岃櫄閮ㄧ浉涔橈紝鐒跺悗寮鏂广
