COMPLEX NUMBER - 2 (Square Root and Polar Form)
Square root of a complex number
1. If the square root of $a+i b$ is to evaluated let $\sqrt{a+i b}=x+i y, x, y \in R$
2. Square both sides and equate real and imaginary part which will give value of $\left(x^{2}-y^{2}\right)$ and $2 x y$.
3. Find $x^{2}+y^{2}$ by $\left(x^{2}+y^{2}\right)^{2}=\left(x^{2}-y^{2}\right)^{2}+4 x^{2} y^{2}$
4. From $x^{2}-y^{2} \& x^{2}+y^{2}$, we get the value of $x$.
5. Put $x$ in $x y$, we obtained corresponding value of $y$.
6. Now $\sqrt{a+i b}=x+i y$
Direct Formula
The square root of $\mathrm{z}=\mathrm{a}+\mathrm{ib}$ are
$\pm\left[\sqrt{\frac{|z|+a}{2}}+i \sqrt{\frac{|z|-a}{2}}\right]$ for $b>0$
$\& \pm\left[\sqrt{\frac{|z|+a}{2}}-i \sqrt{\frac{|z|-a}{2}}\right]$ for $b<0$
Note:
i. $\sqrt{\mathrm{a}} \sqrt{\mathrm{b}}=\sqrt{\mathrm{ab}}$ is ture only when at least-one of $\mathrm{a} & \mathrm{~b}$ is non-negative.
ii. The square root of $\omega$ are $\pm \omega^{2}$
iii. The square root of $\omega^{2}$ are $\pm \omega$
iv. The square root of $i$ are $\pm\left(\frac{1+i}{\sqrt{2}}\right)$
v. The square root of $-i$ are $\pm\left(\frac{1-i}{\sqrt{2}}\right)$
Example: Find the square root of $-7-24 \mathrm{i}$
Show Answer
Solution: Here $\mathrm{a}=-7 \& \mathrm{~b}=-24<0$
$|z|=\sqrt{(-7)^{2}+(-24)^{2}}=25$
Now by using formula,
$ \sqrt{-7-24 \mathrm{i}}= \pm\left[\sqrt{\frac{|\mathrm{z}|+\mathrm{a}}{2}}-\mathrm{i} \sqrt{\frac{|\mathrm{z}|-\mathrm{a}}{2}}\right]= \pm\left[\sqrt{\frac{25-7}{2}}-\mathrm{i} \sqrt{\frac{25+7}{2}}\right]= \pm(3-4 \mathrm{i}) $
Geometrical Representation of a Point is
Modulus
The modulus of a complex number $x+i y$ is denoted by $|x+i y|=\sqrt{x^{2}+y^{2}}=$ non negative square root of $\mathrm{x}^{2}+\mathrm{y}^{2}$
e.g. $z=3-4 i$, then $|z|=\sqrt{(3)^{2}+(-4)^{2}}=5$
Argument or Amplitude of a complex number
$z=x+i y$
$\cos \theta=\frac{x}{\sqrt{x^{2}+y^{2}}} \& \sin \theta=\frac{y}{\sqrt{x^{2}+y^{2}}}$
$\cos \theta=\frac{\mathrm{x}}{\mathrm{r}} \& \sin \theta=\frac{\mathrm{x}}{\mathrm{r}}$
$x=r \cos \theta \& x=r \sin \theta$
The argument of a complex number $z=x+i y$ is the value of $\theta$ which satisfies the two equations $\cos \theta=\frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}$ and $\sin \theta=\frac{\mathrm{y}}{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}$
Argument of $z$ is denoted by argument $z$ or amplitude $z$.
There will be infinite number of values of $\theta$ satisfying the above equations and all these values will be the argument of $z$ but usually we take only that value of $\theta$ to which $0 \leq \theta<2 \pi$
Example: $\mathrm{z}=-1-\mathrm{i}$ here $\mathrm{x}=-1, \mathrm{y}=-1$
Show Answer
Solution:
$\cos \theta=\frac{x}{\sqrt{x^{2}+y^{2}}}=-\frac{1}{\sqrt{2}} \quad \theta=\frac{3 \pi}{4}, \frac{5 \pi}{4}$
$\sin \theta=\frac{y}{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}=-\frac{1}{\sqrt{2}} \quad \theta=\frac{5 \pi}{4}, \frac{7 \pi}{4}$
$\therefore \quad \operatorname{since} \theta=\frac{5 \pi}{4}$ satisfies both the equation
$\therefore \quad \operatorname{argument} \mathrm{z}=\frac{5 \pi}{4}$ and general value of argument $\mathrm{z}=2 \mathrm{n} \pi+\frac{5 \pi}{4}$, where $\mathrm{n}=0, \pm 1, \pm 2 \ldots$.
Another way of finding argument of a complex number
Working Rule
i. Take $\tan \theta=\left|\frac{y}{x}\right|$ and from this find the value of $\theta$ lying between 0 and $\frac{\pi}{2}$
ii. Then find in which quadrant the point $z$ lies.
iii. Argument of $z$ will be $\theta, \pi-\theta, \pi+\theta$ or $2 \pi+\theta$ according as the point $z$ lies in the $1^{\text {st }}, 2^{\text {nd }}$, $3^{\text {rd }}$ or $4^{\text {th }}$ quadrants.
Exercise: Let $\mathrm{z}=-1-\mathrm{i}$ here $\mathrm{x}=-1, \mathrm{y}=-1$
Show Answer
Solution:
$\tan \theta=\left|\frac{\mathrm{y}}{\mathrm{x}}\right|=\left|\frac{-1}{-1}\right|=1=\tan \frac{\pi}{4}$
$\Rightarrow \quad \theta=\frac{\pi}{4}$ (between 0 and $\frac{\pi}{2}$ )
Since the point $\mathrm{z}=-1-\mathrm{i} \equiv(-1,-1)$ lies in 3rd quadrant
$\therefore \quad$ argument $=\pi+\theta=\theta=\frac{\pi}{4}=\frac{5 \pi}{4}$
Principal vaule of the argument: There are infinite values of $\theta$ satisfying the equation
$\cos \theta=\frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}$ and $\sin \theta=\frac{\mathrm{y}}{\sqrt{\mathrm{x}^{2}+\mathrm{y}^{2}}}$
But there will be a unique value of $\theta$ such that $-\pi<\theta \leq \pi$. The value of argument $\theta$ satisfying the inequality $-\pi<\theta \leq \pi$ is called principal value of argument.
For above example
Principal value of argument $=\frac{5 \pi}{4}-2 \pi=\frac{-3 \pi}{4}$
Note: If argument $>\pi$, subtract $2 \pi$ from it to get the principal value of argument and if argument $\leq-\pi$, add $2 \pi$ to it, to get the principal value of argument.
Polar form of a complex Number
$\mathrm{z}=\mathrm{x}+\mathrm{iy}$
$=r(\cos \theta+i \sin \theta)$ is called the polar form of complex Number.
For above example
$\mathrm{r}=|\mathrm{z}|=\sqrt{2}$
Polar form of $z$ is $\sqrt{2}\left[\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right]$
or $\quad \sqrt{2}\left[\cos \left(\frac{-3 \pi}{4}\right)+\mathrm{i} \sin \left(\frac{-3 \pi}{4}\right)\right]$
Complex Number
-
Cartesian Representation
- $\mathrm{z}=\mathrm{a}+\mathrm{ib}$
-
Polar Repersentation
- $\mathrm{z}=\mathrm{r} \cos \theta+\mathrm{i} \sin \theta$
- $=\mathrm{r}(\cos \theta+\mathrm{i} \sin \theta)$
- $=\mathrm{re}^{\mathrm{i} \theta}$ (using euler’s formula)
| Algebraic operations | Cartesian form | Polar form |
|---|---|---|
| Complex No. | $\mathrm{z}=\mathrm{a}+\mathrm{ib}, \mathrm{w}=\mathrm{c}+\mathrm{id}$ | $\mathrm{z}=\mathrm{re}^{\mathrm{i} \theta}, \mathrm{w}=\mathrm{se}^{\mathrm{i} \varphi}$ |
| Addition | $\mathrm{z}+\mathrm{w}=(\mathrm{a}+\mathrm{c})+\mathrm{i}(\mathrm{b}+\mathrm{d})$ | $\mathrm{z}+\mathrm{w}=(\mathrm{r} \cos \theta+\mathrm{s} \cos \varphi)+$ $\mathrm{i}(\mathrm{r} \sin \theta+\mathrm{s} \sin \varphi)$ |
| Subtraction | $\mathrm{z}-\mathrm{w}=(\mathrm{a}-\mathrm{c})+\mathrm{i}(\mathrm{b}-\mathrm{d})$ | $z-w=(r \cos \theta-s \cos \varphi)+$ $i(r \sin \theta-s \sin \varphi)$ |
| Multiplicaiton | $z W=(a c-b d)+i$ $(a d+b c)$ | $\mathrm{zW}=\operatorname{rse}^{\mathrm{i}(\theta+\phi)}$ |
| Division | $\frac{\mathrm{z}}{\omega}=\frac{(\mathrm{a}+\mathrm{id})(\mathrm{c}-\mathrm{id})}{(\mathrm{c}+\mathrm{id})(\mathrm{c}-\mathrm{id})}$ $=\frac{\mathrm{ac}+\mathrm{bd}}{\mathrm{c}^{2}+\mathrm{d}^{2}}+\mathrm{i} \frac{\mathrm{bc}-\mathrm{ad}}{\mathrm{c}^{2}+\mathrm{d}^{2}}$ | $\frac{\mathrm{z}}{\mathrm{w}}=\frac{\mathrm{r}}{\mathrm{S}} \mathrm{e}^{\mathrm{i}(\theta-\varphi)}$ |
Conjugate of a Complex Number
The conjugate of a complex number $\mathrm{z}=\mathrm{x}+$ iy is denoted by $\bar{z}=\overline{x+i y}$ and is defined as $\bar{z}=x-i y$ and if $z=r e^{i \theta}$ (Eular’s form), then $\bar{z}=r e^{-i \theta}$
Equality of Complex Number
Two complex numbers $z _{1}=x _{1}+$ iy and $z _{2}=x _{2}+i y _{2}$ are said to be equal if and only if $x _{1}=x _{2} \& y _{1}=$ $+\mathrm{y} _{2}$
i.e. $z _{1}=z _{2}$, then $\operatorname{Re}\left(z _{1}\right)=\operatorname{Re}\left(z _{2}\right) \& \operatorname{Im}\left(z _{1}\right)=\operatorname{Im}\left(z _{2}\right)$
Properties of Conjugate
1. $|z|=|-z|=|\bar{z}|=|-\bar{z}|$
2. $\overline{\overline{\mathrm{Z}}}=\mathrm{z}$
3. $[\arg z-\arg \bar{z}]=\arg (-z)-\arg (-\bar{z})=\frac{\pi}{2}$
4. $\mathrm{z} \overline{\mathrm{z}}=|\overline{\mathrm{z}}|^{2}$
5. $\overline{\mathrm{z} _{1} \pm \mathrm{z} _{2}}=\overline{\mathrm{z} _{1}} \pm \overline{\mathrm{z} _{2}}$
6. $\left(\overline{z^{\mathrm{n}}}\right)=(\bar{z})^{\mathrm{n}}$
7. $\overline{\mathrm{z} _{1} \cdot \mathrm{z} _{2}}=\overline{\mathrm{z} _{1}} \cdot \overline{\mathrm{z} _{2}}$
8. $\overline{\left(\frac{z _{1}}{z _{2}}\right)}=\frac{\overline{z _{1}}}{\overline{z _{2}}}$
Example- Write the following in Polar form
(1) $\mathrm{z}=1+\sqrt{3} \mathrm{i}$
(2) $\mathrm{z}=-1+\sqrt{3} i$
(3) $z=-1-\sqrt{3} i$
(4) $z=1-\sqrt{3} i$
Show Answer
Solution : $z=1+\sqrt{3} i \equiv(1, \sqrt{3})$
$\mathrm{r}=|z|=\sqrt{(1)^{2}+(\sqrt{3})^{2}}=2$
Then $\tan \alpha=\left|\frac{y}{x}\right|=\left|\frac{\sqrt{3}}{1}\right|=\sqrt{3}=\tan \frac{\pi}{3}$
(1) As $z$ lies in 1st quadrant
so, $\operatorname{\arg z}=\theta=\alpha=\frac{\pi}{3}$
Therefore polar form of $z$ is
$2\left(\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}\right)$
(2) $z$ lies in 2nd quadrant so $\operatorname{\arg z}=\theta=\pi-\alpha=\frac{2 \pi}{3}$
Therefore polar form of $z$ is
$2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) \Rightarrow 2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$
(3) $\mathrm{z}$ lies in 3rd quadrant so $\operatorname{argz}=\theta=\pi+\alpha=\frac{4 \pi}{3}$
Therefore polar form of $z$ is
$2\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)$
(4) lies in 4th quadrant so $\operatorname{argz}=\theta=-\alpha=\frac{-\pi}{3}$
Therefore polar form of $\mathrm{z}$ is
$2\left(\cos \left(\frac{-\pi}{3}\right)+i \sin \left(\frac{-\pi}{3}\right)\right)$
Practice questions
1. $\operatorname{Im}(\mathrm{z})$ is equal to
(a). $\frac{1}{2}(\mathrm{z}+\overline{\mathrm{z}}) \mathrm{i}$
(b). $\frac{(z-\bar{z})}{2 i}$
(c). $ \frac{1}{2}(\bar{z}-z) i$
(d). none of these
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Answer: (c)2. If $z _{1}=9 y^{2}-4-10 i x, z _{2}=8 y^{2}-20 i$ where $z _{1}=\bar{z} _{2}$ then $z=x+i y$ is equal to
(a). $-2+2 \mathrm{i}$
(b). $-2 \pm 2 \mathrm{i}$
(c). $-2-i$
(d). none of these
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Answer: (b)3. If $z$ is a complex number satisfying the relation $|z+1|=z+2(1+i)$, then $z$ is
(a). $\frac{1}{2}(1+4 \mathrm{i})$
(b). $\frac{1}{2}(3+4 \mathrm{i})$
(c). $\frac{1}{2}(1-4 \mathrm{i})$
(d). $\frac{1}{2}(3-4 i)$
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Answer: (c)4. For a complex number $z$, the minimum value of $|z|+|2-z|$ is
(a). $1$
(b). $2$
(c). $3$
(d). none of these
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Answer: (b)5. If $|z|=1$, then $\frac{1+\bar{z}}{1+\bar{z}}$ is equal to
(a). $\mathrm{z}$
(b). $\overline{\mathrm{Z}}$
(c). $\mathrm{z}^{+} \overline{\mathrm{z}}$
(d). none of these
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Answer: (a)6. If $\left|z _{1}-1\right|<1,\left|z _{2}-2\right|<2,\left|z _{3}-3\right|<3$, then $\left|z _{1}+z _{2}+z _{3}\right|$
(a). is less than 6
(b). is more than 3
(c). is less than 12
(d). is between 6 and 12
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Answer: (c)7. If $z _{1}, z _{2}$ are two non-zero complex numbers such that $\left|z _{1}+z _{2}\right|=\left|z _{1}\right|+\left|z _{2}\right|$, then complex $\arg \left(\frac{z _{1}}{z _{2}}\right)$ is equal to
(a). $\pi$
(b). $-\pi$
(c). $0$
(d). $\frac{\pi}{2}$
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Answer: (c)8. If $z=\frac{\sqrt{3}+i}{\sqrt{3}-\mathrm{i}}$, then the fundamental argument of $\mathrm{z}$ is
(a). $\frac{-\pi}{3}$
(b). $\frac{\pi}{3}$
(c). $\frac{\pi}{6}$
(d). none of these
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Answer: (b)9. If $z=x+$ iy satisfies $\operatorname{amp}(z-1)=a m p(z+3 i)$ then the value of $(x-1)$ : $y$ is equal to
(a). $2 : 1$
(b). $1 : 3$
(c). $-1 : 3$
(d). none of these
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Answer: (b)10. If $(1+x)^{n}=a _{0}+a _{1} x+a _{2} x^{2}+\ldots .+a _{n} x^{n}$, then $\left(a _{0}-a _{2}+\ldots \ldots\right)^{2}+\left(a _{1}-a _{3}+\ldots . .\right)^{2}$ is equal to
(a). $3^{n}$
(b). $ 2^{\mathrm{n}}$
(c). $\frac{1-2^{\mathrm{n}}}{1+2^{\mathrm{n}}}$
(d). none of these
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Answer: (b)11. If $x=2+5 i$ and $2\left(\frac{1}{1 ! 9 !}+\frac{1}{3 ! 7 !}\right)+\frac{1}{5 !} 5 !=\frac{2}{b !}$, then the value of $x^{3}-5 x^{2}+33 x-19$ is equal to
(a). $ \mathrm{a}$
(b). $\mathrm{b}$
(c). $\mathrm{a}-\mathrm{b}$
(d). none of these
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Answer: (b)12. If $x+$ iy such that $|z+1|=|z-1|$ and complex $\frac{z-1}{z+1}=\frac{\pi}{4}$ then
(a). $\mathrm{x}=\sqrt{2}+1, \mathrm{y}=0$
(b). $x=0, y=\sqrt{2}+1$
(c). $\mathrm{x}=0, \mathrm{y}=\sqrt{2}-1$
(d). $x=\sqrt{2}-1, y=0$
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Answer: (b)13. If the square root of $\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}+\frac{1}{2 i}\left(\frac{y}{x}+\frac{x}{y}\right)+\frac{31}{16}$ is $\pm\left(\frac{\mathrm{x}}{\mathrm{y}}+\frac{\mathrm{y}}{\mathrm{x}}-\frac{\mathrm{i}}{\mathrm{m}}\right)$, then $\mathrm{m}$ is
(a). $2$
(b). $3$
(c). $4$
(d). none of these
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Answer: (c)14. If $\mathrm{i}=\sqrt{-1}$, then $4+5\left[-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{2}\right]^{334}+3\left[-\frac{1}{2}+\mathrm{i} \frac{\sqrt{3}}{3}\right]^{365}$ is equal to
(a). $1-\mathrm{i} \sqrt{3}$
(b). $-1-i \sqrt{3}$
(c). $i \sqrt{3}$
(d). $-i \sqrt{3}$
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Answer: (c)15. If $\mathrm{z} _{1}, \mathrm{z} _{2}$ and $\mathrm{z} _{3}$ are complex numbers such that $\left|\mathrm{z} _{1}\right|=\left|\mathrm{z} _{2}\right|=\left|\mathrm{z} _{3}\right|=\left|\frac{1}{\mathrm{z} _{1}}+\frac{1}{\mathrm{z} _{2}}+\frac{1}{\mathrm{z} _{3}}\right|=1$, then $\left|z _{1}+z _{2}+z _{3}\right|$ is
(a). equal to 1
(b). less than 1
(c). greater than 3
(d). equal to 3
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Answer: (a)16. The complex number $\sin x+i \cos x$ and $\cos x-i \sin x$ are conjugate to each other for
(a). $ \mathrm{x}=\mathrm{n} \pi$
(b). $\mathrm{x}=0$
(c). $ \mathrm{x}=\left(\mathrm{n}+\frac{1}{2} \pi\right)$
(d). $\mathrm{x}=\frac{\pi}{4}$ or $\mathrm{x}=\mathrm{n} \pi+\frac{\pi}{4}$
