证明：（a+bi）/（c+di）=（a*a+b*b）^(1/2)/(c*c+d*d)^(1/2)应该是：“（a+bi）/（c+di）”的模=（a*a+b*b）^(1/2)/(c*c+d*d)^(1/2)

Question

证明：（a+bi）/（c+di）=（a*a+b*b）^(1/2)/(c*c+d*d)^(1/2)
应该是：“（a+bi）/（c+di）”的模=（a*a+b*b）^(1/2)/(c*c+d*d)^(1/2)

Answer 1

(a+bi)/(c+di)
=[(a+bi)(c+di)]/[(c+di)(c-di)]
=[(ac-bd)+(bc+ad)i]/(c²+d²)
|(a+bi)/(c+di)|=√[(ac-bd)/(c²+d²)]²+[(bc+ad)/(c²+d²)]²
=√[(a²c²+b²d²+b²c²+a²d²)/(c²+d²)²]
=√[(a²+b²)(c²+d²)/(c²+d²)²]
=√[(a²+b²)/(c²+d²)]