Question Number 127261 by mohammad17 last updated on 28/Dec/20
$${for}\:{any}\:{complex}\:{number}\:{if}\:{im}\left({z}\right)>\mathrm{0} \\ $$ $${then}\:{im}\left(\frac{\mathrm{1}}{{z}}\right)>\mathrm{0}\:\:\:\:{prove}\:{this}\:? \\ $$
Commented bymr W last updated on 28/Dec/20
$${wrong}! \\ $$ $${if}\:{im}\left({z}\right)>\mathrm{0}\:{then}\:{im}\left(\frac{\mathrm{1}}{{z}}\right)<\mathrm{0}. \\ $$
Answered by mr W last updated on 28/Dec/20
$${z}={a}+{bi} \\ $$ $${im}\left({z}\right)={b}>\mathrm{0} \\ $$ $$\frac{\mathrm{1}}{{z}}=\frac{\mathrm{1}}{{a}+{bi}}=\frac{{a}−{bi}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} } \\ $$ $${im}\left(\frac{\mathrm{1}}{{z}}\right)=−\frac{{b}}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }<\mathrm{0} \\ $$
Commented bymohammad17 last updated on 28/Dec/20
$${thank}\:{you}\:{sir} \\ $$