Question Number 83966 by Rio Michael last updated on 08/Mar/20
$$\mathrm{prove}\:\mathrm{that}\:\mathrm{for}\:\mathrm{any}\:\mathrm{complex}\:\mathrm{number}\:{z},\:\mathrm{if}\: \\ $$$$\:\mid{z}\mid\:<\:\mathrm{1},\:\mathrm{then}\:\mathrm{Re}\left({z}\:+\:\mathrm{1}\right)\:>\:\mathrm{0} \\ $$
Answered by mr W last updated on 08/Mar/20
$${let}\:{z}={a}+{bi} \\ $$$$\mid{z}\mid=\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }<\mathrm{1} \\ $$$$\Rightarrow{a}^{\mathrm{2}} +{b}^{\mathrm{2}} <\mathrm{1} \\ $$$$\Rightarrow{a}^{\mathrm{2}} <\mathrm{1}−{b}^{\mathrm{2}} \leqslant\mathrm{1} \\ $$$$\Rightarrow−\mathrm{1}<{a}<\mathrm{1} \\ $$$$\Rightarrow\mathrm{0}<{a}+\mathrm{1}<\mathrm{2} \\ $$$$ \\ $$$${Re}\left({z}+\mathrm{1}\right)={Re}\left({a}+\mathrm{1}+{bi}\right)={a}+\mathrm{1}>\mathrm{0} \\ $$
Commented by Rio Michael last updated on 08/Mar/20
$${thank}\:{you}\:{sir} \\ $$
Answered by TANMAY PANACEA last updated on 08/Mar/20
$${z}={rcos}\theta+{irsin}\theta \\ $$$$\mid{z}\mid={r}<\mathrm{1} \\ $$$${z}+\mathrm{1} \\ $$$$=\mathrm{1}+{rcos}\theta+{irsin}\theta \\ $$$$\mid{z}+\mathrm{1}\mid \\ $$$$=\sqrt{\left(\mathrm{1}+{rcos}\theta\right)^{\mathrm{2}} +\left({rsin}\theta\right)^{\mathrm{2}} }\: \\ $$$$=\sqrt{\mathrm{1}+\mathrm{2}{rcos}\theta+{r}^{\mathrm{2}} }\: \\ $$$$\sqrt{\mathrm{1}+\mathrm{2}{rcos}\theta+{r}^{\mathrm{2}} }\:>\sqrt{\mathrm{1}−\mathrm{2}{r}+{r}^{\mathrm{2}} }\:\:{when}\:{cos}\theta=−\mathrm{1} \\ $$$$\sqrt{\mathrm{1}+\mathrm{2}{rcos}\theta+{r}^{\mathrm{2}} }\:<\sqrt{\mathrm{1}+\mathrm{2}{r}+{r}^{\mathrm{2}} }\:\:{when}\:{cos}\theta=\mathrm{1} \\ $$$$\mathrm{1}+{r}>\sqrt{\mathrm{1}+\mathrm{2}{rcos}\theta+{r}^{\mathrm{2}} }\:\:>\mathrm{1}−{r}>\mathrm{0} \\ $$$$\boldsymbol{{note}}\:\mathrm{1}−\boldsymbol{{r}}>\mathrm{0} \\ $$$$ \\ $$
Commented by Rio Michael last updated on 08/Mar/20
$${thanks}\:{sir} \\ $$