Question Number 56183 by 121194 last updated on 11/Mar/19
$$\mathrm{find}\:\mathrm{all}\:{a},{b}\in\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{{a}+{bi}}=\frac{\mathrm{1}}{{a}}+\frac{{i}}{{b}} \\ $$
Answered by kaivan.ahmadi last updated on 11/Mar/19
$$\frac{\mathrm{1}}{{a}+{bi}}=\frac{{b}+{ai}}{{ab}}\Rightarrow{ab}+\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} \right){i}−{ab}={ab}\Rightarrow \\ $$$$\frac{{ab}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }={i}\Rightarrow\frac{{ab}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }\notin\mathbb{R} \\ $$$${a}\notin\mathbb{R}\:\vee\:{b}\notin\mathbb{R} \\ $$