Complex Number - Notes, Concept and All Important Formula

COMPLEX NUMBER

1. DEFINITION :

Complex numbers are defined as expressions of the form $a + ib$ where $a, b \in R \quad \& i=\sqrt{-1}$. It is denoted by $z$ i.e. $z=a+i b$. 'a' is called real part of $z(a=R e z)$ and ' $b$ ' is called imaginary part of $z(b=\operatorname{Im} z)$

Note :

(i) The set $R$ of real numbers is a proper subset of the Complex Numbers. Hence the Complex Number system is $N \subset W \subset I \subset Q \subset R \subset C$

(ii) Zero is both purely real as well as purely imaginary but not imaginary.

(iii) $i =\sqrt{-1}$ is called the imaginary unit. Also $i ^{2}=-1 ;\, i ^{3}=- i$; $i ^{4}=1$ etc.

(iv) $\sqrt{a} \sqrt{b}=\sqrt{a b}$ only if atleast one of a or $b$ is non-negative.

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All Chapter Notes, Concept and Important Formula

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2. CONJUGATE COMPLEX :

If $z=a+i b$ then its conjugate complex is obtained by changing the sign of its imaginary part $\&$ is denoted by $\bar{z}$. i.e. $\bar{z}=a-i b$. Note that:

(i) $\quad z+\bar{z}=2 \operatorname{Re}(z)$

(ii) $\quad z-\bar{z}=2 i \operatorname{Im}(z)$

(iii) $z \bar{z}=a^{2}+b^{2}$ which is real

(iv) If $z$ is purely real then $z-\bar{z}=0$

(v) If $z$ is purely imaginary then $z+\bar{z}=0$

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3. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS:

(a) Cartesian Form (Geometrical Representation):

Every complex number $z=x+$ iy can be

represented by a point on the cartesian plane known as complex plane (Argand diagram) by the ordered pair $( x , y )$.

Length OP is called modulus of the complex number denoted by $|z|$  & $\theta$ is called the principal argument or amplitude, $(\theta \in(-\pi, \pi])$.

e.g. $|z|=\sqrt{x^{2}+y^{2}}$ & $\theta=\tan ^{-1} \frac{y}{x}$ (angle made by OP with positive $x$ -axis), $x >0$

Geometrically $|z|$ represents the distance of point $P$ from origin. $(|z| \geq 0)$

(b) Trigonometric / Polar Representation :

$z=r(\cos \theta+i \sin \theta)$ where $|z|=r$; $\arg z=\theta$; $\bar{z}=r(\cos \theta-i \sin \theta)$

Note : $\cos \theta+ i \sin \theta$ is also written as $\operatorname{CiS} \theta$.

Euler's formula :

The formula $e^{ ix }=\cos x + i \sin x$ is called Euler's formula. Also $\cos x=\frac{e^{ ix }+e^{- ix }}{2} \, \& \, \sin x =\frac{e^{ ix }-e^{- ix }}{2 i }$ are known as Euler's identities.

(c) Exponential Representation :

Let $z$ be a complex number such that $|z|=r \, \&$ arg $z=\theta$, then $z=r \cdot e^{i \theta}$

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4. IMPORTANT PROPERTIES OF CONJUGATE :

(a) $\overline{(\bar{z})}=z$

(b) $\overline{z_{1}+z_{2}}=\bar{z}_{1}+\bar{z}_{2}$

(c) $\overline{z_{1}-z_{2}}=\bar{z}_{1}-\bar{z}_{2}$

(d) $\overline{z_{1} z_{2}}=\bar{z}_{1} \cdot \bar{z}_{2}$

(e) $\overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\bar{z}_{1}}{\bar{z}_{2}} ; \quad z_{2} \neq 0$

(f) If $f$ is a polynomial with real coefficient such that $f (\alpha+ i \beta)= x + i y$, then $f (\alpha- i \beta)= x - i y$.

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5. IMPORTANT PROPERTIES OF MODULUS :

(a) $|z| \geq 0$

(b) $|z| \geq \operatorname{Re}(z)$

(c) $|z| \geq \operatorname{Im}(z)$

(d) $|z|=|\bar{z}|=|-z|=|-\bar{z}|$

(e) $z \bar{z}=|z|^{2}$

(f) $\left|z_{1} z_{2}\right|=\left|z_{1}\right| \cdot\left|z_{2}\right|$

(g) $\left|\frac{z_{1}}{z_{2}}\right|=\frac{\left|z_{1}\right|}{\left|z_{2}\right|}, \quad z_{2} \neq 0$

(h) $\left|z^{n}\right|=|z|^{n}$

(i) $\left|z_{1}+z_{2}\right|^{2}=\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+2 \operatorname{Re}\left(z_{1} \bar{z}_{2}\right)$

or $\left|z_{1}+z_{2}\right|^{2}=\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+2\left|z_{1}\right|\left|z_{2}\right| \cos \left(\theta_{1}-\theta_{2}\right)$

(j) $\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}-z_{2}\right|^{2}=2\left[\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right]$

(k) ||$z_{1}|-| z_{2}|| \leq\left|z_{1}+z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right| \quad$ [Triangular Inequality]

(I) ||$z_{1}|-| z_{2}|| \leq\left|z_{1}-z_{2}\right| \leq\left|z_{1}\right|+\left|z_{2}\right| \quad[$ Triangular Inequality $]$

(m) If $\left|z+\frac{1}{z}\right|=a(a>0)$, then $\max |z|=\frac{a+\sqrt{a^{2}+4}}{2}$

$\& \min |z|=\frac{1}{2}\left(\sqrt{a^{2}+4}-a\right)$

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6. IMPORTANT PROPERTIES OF AMPLITUDE:

(a) (i) $\operatorname{amp}\left(z_{1} \cdot z_{2}\right)=\operatorname{amp} z_{1}+\operatorname{amp} z_{2}+2 k \pi ; k \in I$

(ii) $\operatorname{amp}\left(\frac{z_{1}}{z_{2}}\right)=\operatorname{amp} z_{1}-\operatorname{amp} z_{2}+2 k \pi ; \quad k \in I$

(iii) $\operatorname{amp}\left(z^{n}\right)=n \operatorname{amp}(z)+2 k \pi$,

where proper value of $k$ must be chosen so that RHS lies in $(-\pi, \pi]$.

(b) $\log (z)=\log \left(r e^{i \theta}\right)=\log r+i \theta=\log |z|+i \operatorname{amp}(z)$

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7. DE'MOIVER'S THEOREM :

The value of $(\cos \theta+i \sin \theta)^{n}$ is $\cos \theta+i \sin n \theta$ if 'n' is integer $\&$ it is one of the values of $(\cos \theta+i \sin \theta)^{n}$ if $n$ is a rational number of the form $p / q$, where $p$  & $q$ are co-prime.

Note : Continued product of roots of a complex quantity should be determined using theory of equation.

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8. CUBE ROOT OF UNITY :

(a) The cube roots of unity are $1, \omega=\frac{-1+ i \sqrt{3}}{2}=e^{ i 2 \pi / 3}$ & $\omega^{2}=\frac{-1- i \sqrt{3}}{2}=e^{ i 4 \pi / 3}$

(b) $1+\omega+\omega^{2}=0, \omega^{3}=1$, in general

$1+\omega^{ r }+\omega^{2 r }=\left[\begin{array}{l}0, r \text { is not integral multiple of } 3 \\ 3, r \text { is multiple of } 3\end{array}\right.$

(c) $a^{2}+b^{2}+c^{2}-a b-b c-c a$$=\left(a+b \omega+c \omega^{2}\right)\left(a+b \omega^{2}+c \omega\right)$

$a^{3}+b^{3}=(a+b)(a+\omega b)\left(a+\omega^{2} b\right)$

$a^{3}-b^{3}=(a-b)(a-\omega b)\left(a-\omega^{2} b\right)$

$x ^{2}+ x +1=( x -\omega)\left( x -\omega^{2}\right)$

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9. SQUARE ROOT OF COMPLEX NUMBER :

$\begin{array}{l}\sqrt{a+i b}=\pm\left\{\frac{\sqrt{|z|+a}}{2}+i \frac{\sqrt{|z|-a}}{2}\right\} \text { for } b>0 \\\& \pm\left\{\frac{\sqrt{|z|+a}}{2}-i \frac{\sqrt{|z|-a}}{2}\right\} \text { for } b<0 \end{array}$$\text{where} |z|=\sqrt{a^{2}+b^{2}} .$

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10. ROTATION:

$\dfrac{z_{2}-z_{0}}{\left|z_{2} z_{0}\right|}=\dfrac{z_{1}-z_{0}}{\left|z_{1}-z_{0}\right|} e^{i \theta}$

Take $\theta$ in anticlockwise direction

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11. GEOMETRY IN COMPLEX NUMBER :

(a) Distance formula $:\left|z_{1}-z_{2}\right|=$ distance between the points $z _{1}\,  \& \, z _{2}$ on the Argand plane.

(b) Section formula : If $z_{1}\,  \& \, z_{2}$ are two complex numbers then the complex number $z =\dfrac{ nz _{1}+ mz _{2}}{ m + n }$ divides the join of $z _{1}\, \& \, z _{2}$ in the ratio $m : n$

(c) If the vertices $A, B, C$ of a triangle represent the complex numbers $z _{1}, z _{2}, z _{3}$ respectively, then :

• Centroid of the $\Delta ABC =\dfrac{ z _{1}+ z _{2}+ z _{3}}{3}$
• Orthocentre of the $\triangle ABC$ $\begin{array}{l}=\dfrac{(a \sec A) z_{1}+(b \sec B) z_{2}+(c \sec C) z_{3}}{a \sec A+b \sec B+c \sec C} \\\text { or } \dfrac{z_{1} \tan A+z_{2} \tan B+z_{3}\tan C}{\tan A+\tan B+\tan C}\end{array}$
• Incentre of the $\Delta ABC =\dfrac{\left( az _{1}+ bz _{2}+ cz _{3}\right)}{( a + b + c )}$
• Circumcentre of the $\triangle ABC$$=\dfrac{\left(z_{1} \sin 2 A+z_{2} \sin 2 B+z_{3} \sin 2 C\right)}{(\sin 2 A+\sin 2 B+\sin 2 C)}$

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11. RESULT RELATED WITH TRIANGLE :

(a) Equilateral triangle :

$\begin{array}{l}\dfrac{z_{1}-z_{2}}{\ell}=\dfrac{z_{3}-z_{2}}{\ell} e^{i \pi / 3} \\\text { Also } \dfrac{z_{2}-z_{3}}{\ell}=\dfrac{z_{1}-z_{3}}{\ell} \cdot e^{i \pi / 3} \quad \ldots \ldots \text { (ii) }\end{array}$

from (i) & (ii)

$\Rightarrow z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}$

or $\dfrac{1}{z_{1}-z_{2}}+\dfrac{1}{z_{2}-z_{3}}+\dfrac{1}{z_{3}-z_{1}}=0$

(b) Isosceles triangle :

If then $4 \cos ^{2} \alpha\left(z_{1}-z_{2}\right)\left(z_{3}-z_{1}\right)=\left(z_{3}-z_{2}\right)^{2}$

(c) Area of triangle $\triangle ABC$ given by modulus of $\frac{1}{4}\left|\begin{array}{lll}z_{1} & \bar{z}_{1} & 1 \\ z_{2} & \bar{z}_{2} & 1 \\ z_{3} & \bar{z}_{3} & 1\end{array}\right|$

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12. EQUATION OF LINE THROUGH POINTS $\mathbf{z_{1} \, \&\, z_{2}}$ :

$\left|\begin{array}{lll}z & \bar{z} & 1 \\ z_{1} & \bar{z}_{1} & 1 \\ z_{2} & \bar{z}_{2} & 1\end{array}\right|=0$ $\Rightarrow z\left(\bar{z}_{1}-\bar{z}_{2}\right)+\bar{z}\left(z_{2}-z_{1}\right)+z_{1} \bar{z}_{2}-\bar{z}_{1} z_{2}=0$

$\Rightarrow z\left(\bar{z}_{1}-\bar{z}_{2}\right) i+\bar{z}\left(z_{2}-z_{1}\right) i+i\left(z_{1} \bar{z}_{2}-\bar{z}_{1} z_{2}\right)=0$

Let $\left(z_{2}-z_{1}\right) i=a$, then equation of line is $\boxed{\bar{a} z+a \bar{z}+b=0}$ where a $C \, \& \, b \in R$

Note:

(i) Complex slope of line joining points $z_{1}\, \& \, z_{2}$ is $\dfrac{\left(z_{2}-z_{1}\right)}{\left(\overline{z_{2}-z_{1}}\right)}$. Also note that slope of a line in Cartesian plane is different from complex slope of a line in Argand plane.

(ii) Complex slope of line $\bar{a} z+a \bar{z}+b=0$ is $-\dfrac{a}{\bar{a}}, b \in R$

(iii) Two lines with complex slope $\mu_{1} \, \& \, \mu_{2}$ are parallel or perpendicular if $\mu_{1}=\mu_{2}$ or $\mu_{1}+\mu_{2}=0$.

(iv) Length of perpendicular from point $A (\alpha)$ to line $\overline{ a } z + a \overline{ z }+ b =0$ is $\dfrac{|\bar{a} \alpha+a \bar{\alpha}+b|}{2|a|}$.

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13. EQUATION OF CIRCLE :

(a) Circle whose centre is $z_{0}$ &  radius $=r$

       $\left|z-z_{0}\right|=r$

(b) General equation of circle is

      $z \bar{z}+a \bar{z}+\bar{a} z+b=0$

      centre '-a' & radius $=\sqrt{| a |^{2}- b }$

(c) Diameter form $\left(z-z_{1}\right)\left(\bar{z}-\bar{z}_{2}\right)+\left(z-z_{2}\right)\left(\bar{z}-\bar{z}_{1}\right)=0$

or $\quad \arg \left(\frac{z-z_{1}}{z-z_{2}}\right)=\pm \frac{\pi}{2}$

(d) Equation $\left|\frac{z-z_{1}}{z-z_{2}}\right|=k$ represent a circle if $k \neq 1$ and a straight line if $k =1$

(e) Equation $\left|z-z_{1}\right|^{2}+\left|z-z_{2}\right|^{2}=k$

represent circle if $k \geq \frac{1}{2}\left|z_{1}- z _{2}\right|^{2}$

(f) $\arg \left(\frac{z-z_{1}}{z-z_{2}}\right)=\alpha \quad 0<\alpha<\pi, \alpha \neq \frac{\pi}{2}$

represent a segment of circle passing through $A \left( z _{1}\right) \& B \left( z _{2}\right)$

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14. STANDARD LOCI :

(a) $\left|z-z_{1}\right|+\left|z-z_{2}\right|=2 k$ (a constant) represent

•  If $2 k >\left| z _{1}- z _{2}\right| \quad \Rightarrow$ An ellipse
•  If $2 k =\left| z _{1}- z _{2}\right| \quad \Rightarrow A$ line segment
•  If $2 k <\left|z_{1}-z_{2}\right| \Rightarrow$ No solution

(b) Equation ||$z-z_{1}|-| z-z_{2}||=2 k$ (a constant) represent

•  If $2 k <\left|z_{1}-z_{2}\right| \Rightarrow$ A hyperbola
•  If $2 k =\left| z _{1}- z _{2}\right| \quad \Rightarrow A$ line ray
•  $2 k >\left| z _{1}- z _{2}\right| \quad \Rightarrow$ No solution

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