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

Labels:

Comments

Post a Comment

Popular posts from this blog

Circle- Notes, Concept and All Important Formula

CIRCLE 1. DEFINITION : A circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle. All Chapter Notes, Concept and Important Formula 2. STANDARD EQUATIONS OF THE CIRCLE : (a) Central Form: If \((\mathrm{h}, \mathrm{k})\) is the centre and \(\mathrm{r}\) is the radius of the circle then its equation is \((\mathbf{x}-\mathbf{h})^{2}+(\mathbf{y}-\mathbf{k})^{2}=\mathbf{r}^{2}\) (b) General equation of circle : \(\mathbf{x}^{2}+\mathbf{y}^{2}+\mathbf{2 g x}+\mathbf{2 f y}+\mathbf{c}=\mathbf{0}\) , where \(g, \mathrm{f}, c\) are constants and centre is \((-g,-f)\) i.e. \(\left(-\frac{\text { coefficient of } \mathrm{x}}{2},-\frac{\text { coefficient of } \mathrm{y}}{2}\right)\) and radius \(r=\sqrt{g^{2}+f^{2}-c}\) Note : The general quadratic equation in \(\mathrm{x}\) and \(\mathrm{y}\) , \(a x^{2}+b y^{2}+2 ...
Post a Comment
Read more

Determinant - Notes, Concept and All Important Formula

DETERMINANT 1. MINORS : The minor of a given element of determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands. For example, the minor of \(a _{1}\) in \(\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\ a _{2} & b _{2} & c _{2} \\ a _{3} & b _{3} & c _{3}\end{array}\right|\) is \(\left|\begin{array}{ll} b _{2} & c _{2} \\ b _{3} & c _{3}\end{array}\right| \&\) the minor of \(b_{2}\) is \(\quad\left|\begin{array}{ll}a_{1} & c_{1} \\ a_{3} & c_{3}\end{array}\right|\) . Hence a determinant of order three will have " 9 minors". 2. COFACTORS : If \(M _{ ij }\) represents the minor of the element belonging to \(i ^{\text {th }}\) row and \(j ^{\text {th }}\) column then the cofactor of that element : \(C_{i j}=(-1)^{i+j} \cdot M_{i j}\) Important Note : Consider \(\Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} ...
Post a Comment
Read more

Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...
Post a Comment
Read more