Find the square root of the following complex numbers:
Question:

Find the square root of the following complex numbers:

(i) $-5+12 i$

 

(ii) $-7-24 i$

(iii) $1-\mathrm{i}$

(iv) $-8-6 i$

 

(v) $8-15 i$

(vi) $-11-60 \sqrt{-1}$

 

(vii) $1+4 \sqrt{-3}$

(viii) $4 i$

(ix) -i

Solution:

$\sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right] \quad$, if $\operatorname{Im}(z)>0$

$\sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$, if $\operatorname{Im}(z)<0$

(i) $z=-5+12 i, \operatorname{Re}(z)=-5$ and $|z|=\sqrt{25+144}=13$

Here, $\operatorname{Im}(z)>0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{13-5}{2}}+i \sqrt{\frac{13+5}{2}}\right]$

$=\pm[\sqrt{4}+i \sqrt{9}]$

$=\pm(2+3 i)$

(ii) $z=-7-24 i, \quad \operatorname{Re}(z)=-7, \quad|z|=\sqrt{49+576}=25$

Here, $\operatorname{Im}(z)<0$

$\sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{25-7}{2}}-i \sqrt{\frac{25+7}{2}}\right]$

$=\pm(3-4 i)$

(iii) $z=1-i, \operatorname{Re}(z)=1,|z|=\sqrt{1+1}=\sqrt{2}$

Here, $\operatorname{Im}(z)<0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{\sqrt{2}+1}{2}}-i \sqrt{\frac{\sqrt{2}-1}{2}}\right]$

(iv) $z=-8-6 i, \quad \operatorname{Re}(z)=-8, \quad|z|=\sqrt{64+36}=10$

Here, $\operatorname{Im}(z)<0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{10-8}{2}}-i \sqrt{\frac{10+8}{2}}\right]$

$=\pm(1-3 i)$

(v) $z=8-15 i, \quad \operatorname{Re}(z)=8, \quad|z|=\sqrt{64+225}=17$

Here, $\operatorname{Im}(z)<0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{17+8}{2}}-i \sqrt{\frac{17+8}{2}}\right]$

$=\pm \frac{1}{\sqrt{2}}(5-3 i)$

(vi) $-11-60 \sqrt{-1}=-11-60 i, \quad \operatorname{Re}(z)=-11,|z|=\sqrt{121+3600}=61$

Here, $\operatorname{Im}(z)<0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{61-11}{2}}-i \sqrt{\frac{61+11}{2}}\right]$

$=\pm(5-6 i)$

(vii) $z=1+4 \sqrt{3} \sqrt{-1}=1+4 \sqrt{3} i, \operatorname{Re}(z)=1,|z|=\sqrt{1+16 \times 3}=7$

Here, $\operatorname{Im}(z)>0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{7+1}{2}}+i \sqrt{\frac{7-1}{2}}\right]$

$=\pm(2+\sqrt{3} i)$

(viii) $z=0+4 i, \quad \operatorname{Re}(z)=0, \quad|z|=4$

Here, $\operatorname{Im}(z)>0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{4+0}{2}}+i \sqrt{\frac{4-0}{2}}\right]$

$=\pm(\sqrt{2}+i \sqrt{2})$

$=\pm \sqrt{2}(1+i)$

(ix) $z=-i, \quad \operatorname{Re}(z)=0,|z|=1$

Here, $\operatorname{Im}(z)<0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\frac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\frac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\frac{1}{2}}-i \sqrt{\frac{1}{2}}\right]$

$=\pm \frac{1}{\sqrt{2}}(1-i)$

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