Skip to main content

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 \(z(b=\operatorname{Im} z)\)

Definition of Complex number

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.




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 \(z+\bar{z}=0\)




3. REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS:

(a) Cartesian Form (Geometrical Representation):

Every complex number \(z=x+\) iy can be

REPRESENTATION OF A COMPLEX NUMBER IN VARIOUS FORMS

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 \(z=r \cdot e^{i \theta}\)




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 \(f (\alpha- i \beta)= x - i y\).




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 \(\max |z|=\frac{a+\sqrt{a^{2}+4}}{2}\)

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




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




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.




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

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

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

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




9. SQUARE ROOT OF COMPLEX NUMBER :

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




10. ROTATION:

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

Rotation of Complex number

Take \(\theta\) in anticlockwise direction




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 \(\triangle ABC\)\(=\dfrac{\left(z_{1} \sin 2 A+z_{2} \sin 2 B+z_{3} \sin 2 C\right)}{(\sin 2 A+\sin 2 B+\sin 2 C)}\)





11. RESULT RELATED WITH TRIANGLE :

(a) Equilateral triangle :

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

RESULT RELATED WITH TRIANGLE IN COMPLEX NUMBER- EQUILATERAL TRIANGLE

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 :

RESULT RELATED WITH TRIANGLE IN COMPLEX NUMBER- 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 \(\frac{1}{4}\left|\begin{array}{lll}z_{1} & \bar{z}_{1} & 1 \\ z_{2} & \bar{z}_{2} & 1 \\ z_{3} & \bar{z}_{3} & 1\end{array}\right|\)




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 \(\dfrac{|\bar{a} \alpha+a \bar{\alpha}+b|}{2|a|}\).




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

EQUATION OF CIRCLE IN COMPLEX FORM
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 \(A \left( z _{1}\right) \& B \left( z _{2}\right)\)




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



Comments

Post a Comment

Popular posts from this blog

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...

Logarithm - Notes, Concept and All Important Formula

LOGARITHM LOGARITHM OF A NUMBER : The logarithm of the number \(\mathrm{N}\) to the base ' \(\mathrm{a}\) ' is the exponent indicating the power to which the base 'a' must be raised to obtain the number \(\mathrm{N}\) . This number is designated as \(\log _{\mathrm{a}} \mathrm{N}\) . (a) \(\log _{a} \mathrm{~N}=\mathrm{x}\) , read as \(\log\) of \(\mathrm{N}\) to the base \(\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}\) . If \(a=10\) then we write \(\log N\) or \(\log _{10} \mathrm{~N}\) and if \(\mathrm{a}=e\) we write \(\ln N\) or \(\log _{e} \mathrm{~N}\) (Natural log) (b) Necessary conditions : \(\mathrm{N}> \,\,0 ; \,\, \mathrm{a}> \,\,0 ; \,\, \mathrm{a} \neq 1\) (c) \(\log _{a} 1=0\) (d) \(\log _{a} a=1\) (e) \(\log _{1 / a} a=-1\) (f) \(\log _{a}(x . y)=\log _{a} x+\log _{a} y ; \,\, x, y> \,\,0\) (g) \(\log _{a}\left(\dfrac{\mathrm{x}}{y}\right)=\log _{\mathrm{a}} \mathrm{x}-\log _{\mathrm{a}} \mathrm{y} ; \,\, \mathrm{...

Trigonometry Equation - Notes, Concept and All Important Formula

TRIGONOMETRIC EQUATION 1. TRIGONOMETRIC EQUATION : An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometric equation. All Chapter Notes, Concept and Important Formula 2. SOLUTION OF TRIGONOMETRIC EQUATION : A value of the unknown angle which satisfies the given equations is called a solution of the trigonometric equation. (a) Principal solution :- The solution of the trigonometric equation lying in the interval \([0,2 \pi]\) . (b) General solution :- Since all the trigonometric functions are many one & periodic, hence there are infinite values of \(\theta\) for which trigonometric functions have the same value. All such possible values of \(\theta\) for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solutions of trigonometric equation. 3. GENERAL SOLUTIONS OF SOME TRIGONOMETRICE EQUATIONS (TO BE REMEMBERED) :   (a) If \(\sin \theta=0\) , then \(\theta=...