find-all-a-b-R-such-that-1-a-bi-1-a-i-b-

Question Number 56183 by 121194 last updated on 11/Mar/19

find all a, b \in R such that

\frac{1}{a + b i} = \frac{1}{a} + \frac{i}{b}

Answered by kaivan.ahmadi last updated on 11/Mar/19

\frac{1}{a + b i} = \frac{b + a i}{a b} \Rightarrow a b + (a^{2} + b^{2}) i - a b = a b \Rightarrow

\frac{a b}{a^{2} + b^{2}} = i \Rightarrow \frac{a b}{a^{2} + b^{2}} \notin R

a \notin R \lor b \notin R