复数的除法在线一键计算器

复数的除法运算

复数除法是复数四则运算中最复杂的一种，掌握代数形式和极坐标形式两种算法，可轻松应对各类计算。

一、代数形式的复数除法

设 $z_1 = a + bi$，$z_2 = c + di$（$z_2 \neq 0$），利用共轭复数分母实数化：文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html

$\dfrac{z_1}{z_2} = \dfrac{a+bi}{c+di} = \dfrac{(a+bi)(c-di)}{(c+di)(c-di)} = \dfrac{(ac+bd)+(bc-ad)i}{c^2+d^2}$文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html

二、极坐标形式的复数除法

设 $z_1 = r_1(\cos\theta_1 + i\sin\theta_1) = r_1 e^{i\theta_1}$，$z_2 = r_2 e^{i\theta_2}$：文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html

$\dfrac{z_1}{z_2} = \dfrac{r_1}{r_2}\left[\cos(\theta_1-\theta_2) + i\sin(\theta_1-\theta_2)\right] = \dfrac{r_1}{r_2}e^{i(\theta_1-\theta_2)}$文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html

口诀：商的模等于模的商，商的辐角等于辐角的差文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html

三、计算示例

$\dfrac{3+2i}{1-i} = \dfrac{(3+2i)(1+i)}{(1-i)(1+i)} = \dfrac{3+3i+2i+2i^2}{1+1} = \dfrac{3+5i-2}{2} = \dfrac{1+5i}{2} = 0.5 + 2.5i$文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html 文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html文章源自公式库网-https://www.gongshiku.com/html/201911/fushudechufazaixianyijianjisuanqi.html