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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

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