Complex Numbers Definition, Examples, Formulas, Polar Form, Amplitude and Application

Conjugate Complex | Complex Numbers

If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by z. i.e. \(\bar { z } \) = a − ib.

Note:

• z + \(\bar { z } \)= 2 Re(z)
• z − \(\bar { z } \)= 2i Im(z)
• z \(\bar { z } \) = a² + b² which is real
• If z lies in the 1^st quadrant then \(\bar { z } \) lies in the 4^th quadrant and \(\bar {-z } \) lies in the 2^nd quadrant.

Algebraic Operations | Complex Numbers

The algebraic operations on complex numbers are similar to those on real numbers treating i as a polynomial. Inequalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.
e.g. z > 0, 4 + 2i < 2 + 4 i are meaningless .
However in real numbers if a² + b² = 0 then a = 0 = b but in complex numbers,
z₁² + z₂² = 0 does not imply z₁ = z₂ = 0.

Equality in Complex Number

Two complex numbers z₁ = a₁ + ib₁ & z₂ = a₂ + ib₂ are equal if and only if their real & imaginary parts coincide.

Representation of Complex Number in Various Forms

• Cartesian Form (Geometric Representation):
  Every complex number z = x + i y 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| & θ is called the argument or amplitude
  e.g. |z| = \(\sqrt {x^{2} + Y^{2}}\)
  
  θ = tan⁻¹\(\frac {y}{x}\)
  (angle made by OP with positive x−axis)
  Note:
  1. |z| is always non-negative. Unlike real numbers \(\left| z \right| = \left[ \begin{array}{ccc}{\mathbf{z}} & {\text { if }} & {\mathrm{z}>0} \\ {-\mathbf{z}} & {\text { if }} & {\mathbf{z}<0}\end{array}\right.\) is not correct
  2. Argument of a complex number is a many valued function . If θ is the argument of a complex number then 2 nπ + θ ; n ∈ I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nπ.
  3. The unique value of θ such that – π < θ ≤ π is called the principal value of the argument.
  4. Unless otherwise stated, amp z implies principal value of the argument.
  5. By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus.
  6. There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.
• Trignometric / Polar Representation:
  z = r (cos θ + i sin θ) where | z | = r ; arg z = θ ; \(\bar {z}\) = r (cos θ − i sin θ)
  Note: cos θ + i sin θ is also written as CiS θ.
  Also \(\cos x=\frac{e^{i x}+e^{-i x}}{2} \& \sin x=\frac{e^{i x}-e^{-i x}}{2}\)
• Exponential Representation:
  z = re^iθ ; | z | = r ; arg z = θ ; \(\bar {z}\) = re^-iθ

Important Properties of Conjugate/ Moduli/ Amplitude | Complex Numbers

If z , z₁ , z₂ ∈ C then ;

• z + \(\bar {z}\) = 2 Re (z) ; z − \(\bar {z}\) = 2 i Im (z) ; \(\overline{(\overline{z})}=\mathbf{z}\) ; \(\overline{z_{1}+z_{2}}=\overline{z}_{1}+\overline{z}_{2}\) ;
  \(\overline{z_{1}-z_{2}}=\overline{z}_{1}-\overline{z}_{2}\) ; \(\overline{z_{1} z_{2}}=\overline{z}_{1} \cdot \overline{z}_{2}\) ; \(\overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\overline{z}_{1}}{\overline{z}_{2}}\) ; z₂ ≠ 0
• |z₁ + z₂|² + |z₁ – z₂|² = 2 [|z₁|² + |z₂|²]
  ||z₁| − |z₂|| ≤ |z₁ + z₂| ≤ |z₁| + |z₂|
• (i) amp (z₁ . z₂) = amp z₁ + amp z₂ + 2 kπ. k ∈ I
  (ii) amp \(\frac {z_{1}}{z_{2}}\) = amp z₁ – amp z₂ + 2kπ; k ∈ I
  (iii) amp(zⁿ) = n amp(z) + 2kπ .
  where proper value of k must be chosen so that RHS lies in (− π , π ].

Vectorial Representation Of A Complex Number

Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then, \(\overrightarrow{\mathrm{OP}} = z\) & |\(\overrightarrow{\mathrm{OP}}\)| = |z|

Note:

• If \(\overrightarrow { OP }\)= z = re^iθ then \(\overrightarrow { OQ }\) = z₁ = re^{i(θ + φ)} = z . e^iφ. If \(\overrightarrow { OP }\) and \(\overrightarrow { OQ }\) are of unequal magnitude then φ \(\hat{\mathrm{OQ}}=\hat{\mathrm{OP}} \mathrm{e}^{\mathrm{i} \theta}\)
• If A, B, C & D are four points representing the complex numbers z₁, z₂, z₃ & z₄ then \(\mathrm{AB}| | \mathrm{CD} \quad \text { if } \quad \frac{\mathrm{z}_{4}-\mathrm{z}_{3}}{\mathrm{z}_{2}-\mathrm{z}_{1}}\) is purely real ;
  \(A B \perp C D \text { if } \frac{z_{4}-z_{3}}{z_{2}-z_{1}}\) is purely imaginary ]
  
• If z₁, z₂, z₃ are the vertices of an equilateral triangle where z0 is its circumcentre then (a) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}-z_{1} z_{2}-z_{2} z_{3}-z_{3} z_{1}=0\) (b) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=3 z_{0}^{2}\)

Demoivre’S Theorem

Statement: cos nθ + i sin nθ is the value or one of the values of (cos θ + i sin θ)ⁿ ¥ n ∈ Q. The theorem is very useful in determining the roots of any complex quantity
Note: Continued product of the roots of a complex quantity should be determined using theory of equations.

Cube Root Of Unity | Complex Numbers

• The cube roots of unity are 1, \(\frac{-1 + i\sqrt {3}}{2}, \frac{-1 – i\sqrt{3}}{2}\)
• If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general 1 + w^r + w²^r = 0 ; where r ∈ I but is not the multiple of 3.
• In polar form the cube roots of unity are:
  \(\cos 0+i \sin 0 ; \cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}, \quad \cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\)
• The three cube roots of unity when plotted on the Argand plane constitute the vertices of an equilateral triangle.
• The following factorisation should be remembered:
  (a, b, c ∈ R & ω is the cube root of unity)

a³ − b³ = (a − b) (a − ωb) (a − ω^²b); x² + x + 1 = (x − ω) (x − ω²);
a³ + b³ = (a + b) (a + ωb) (a + ω²b);
a³ + b³ + c³ − 3abc = (a + b + c)(a + ωb + ω²c)(a + ω²b + ωc)

n^th Roots Of Unity | Complex Numbers

If 1 ,₁ ,α₂ , α₃ ….. α_{n − 1} are the n, nth root of unity then:

• They are in G.P. with common ratio e^i(2π/n) &
• \(1^{\mathrm{p}}+\alpha_{1}^{\mathrm{p}}+\alpha_{2}^{\mathrm{p}}+\ldots\ldots+\alpha_{\mathrm{n}-1}^{\mathrm{p}}=0\) if p is not an integral multiple of n
  = n if p is an integral multiple of n
• (1 − α₁) (1 − α₂) …… (1 − α_{n – 1}) = n &
  (1 + α₁) (1 + α₂) ……. (1 + α_n − 1) = 0 if n is even and 1 if n is odd.
• 1 . α₁ . α₂ . α₃ ……… α_{n − 1} = 1 or −1 according as n is odd or even.

The Sum Of The Following Series Should Be Remembered:

• \(\cos \theta+\cos 2 \theta+\cos 3 \theta+\ldots \ldots+\cos n \theta=\frac{\sin (n \theta / 2)}{\sin (\theta / 2)} \cos \left(\frac{n+1}{2}\right) \theta\)
• \(\sin \theta+\sin 2 \theta+\sin 3 \theta+\ldots \ldots+\sin n \theta=\frac{\sin (n \theta / 2)}{\sin (\theta / 2)} \sin \left(\frac{n+1}{2}\right) \theta\)

Straight Lines & Circles In Terms Of Complex Numbers:

• If z₁ & z₂ are two complex numbers then the complex number z =mn
  divides the joins of z₁ & z₂ in the ratio m : n.
  Note:
  (i) If a , b , c are three real numbers such that az₁ + bz₂ + cz₃ = 0 ; where a + b + c = 0 and a,b,c are not all simultaneously zero, then the complex numbers z₁, z₂ & z₃ are collinear.
  (ii) If the vertices A, B, C of a ∆ represent the complex nos. z₁, z₂, z₃ respectively, then:
  (a) Centroid of the ∆ ABC = \(\frac{z_{1}+z_{2}+z_{3}}{3}\)
  (b)  Orthocentre of the ∆ ABC = \(\frac{(a \sec A) z_{1}+(b \sec B) z_{2}+(c \sec C) z_{3}}{a \sec A+b \sec B+c \sec C}\) OR \(\frac{\mathrm{z}_{1} \tan \mathrm{A}+\mathrm{z}_{2} \tan \mathrm{B}+\mathrm{z}_{3} \tan \mathrm{C}}{\tan \mathrm{A}+\tan \mathrm{B}+\tan \mathrm{C}}\)
  (c)  Incentre of the ∆ ABC = (az₁ + bz₂ + cz₃) ÷ (a + b + c)
  (d) Circumcentre of the ∆ ABC = :
  (Z₁ sin 2A + Z₂ sin 2B + Z₃ sin 2C) ÷ (sin 2A + sin 2B + sin 2C)
• amp(z) = θ is a ray emanating from the origin inclined at an angle θ to the x− axis.
• |z − a| = |z − b| is the perpendicular bisector of the line joining a to b.
• The equation of a line joining z₁ & z₂ is given by;
  z = z₁ + t (z₁ − z₂) where t is a parameter.
•  z = z₁ (1 + it) where t is a real parameter is a line through the point z₁ & perpendicular to oz₁.
• The equation of a line passing through z₁ & z₂ can be expressed in the determinant form as
  \(\left| \begin{matrix} z & \bar { z } & 1 \\ { z }_{ 1 } & \bar { { z }_{ 1 } } & 1 \\ { z }_{ 2 } & \bar { { z }_{ 2 } } & 1 \end{matrix} \right| = 0\)
  This is also the condition for three complex numbers to be collinear.
• Complex equation of a straight line through two given points z₁ & z₂ can be written as
  \(z\left(\overline{z}_{1}-\overline{z}_{2}\right)-\overline{z}\left(z_{1}-z_{2}\right)+\left(z_{1} \overline{z}_{2}-\overline{z}_{1} z_{2}\right)=0\) which on manipulating takes the form as \(\overline{\alpha} \mathrm{z}+\alpha \overline{\mathrm{z}}+\mathrm{r}=0\) where r is real and α is a non zero complex constant.
• The equation of circle having center z₀ & radius ρ is: |z − z₀| = ρ or
  \(z \overline{z}-z_{0} \overline{z}-\overline{z}_{0} z+\overline{z}_{0} z_{0}-\rho^{2}=0\) which is of the form \(\mathrm{zz}+\overline{\alpha} z+\alpha \overline{z}+r=0\) , r is real centre − α & radius
  \(\sqrt{\alpha \overline{\alpha}-r}\). Circle will be real if \(\alpha \overline{\alpha}-r \geq 0\).
• The equation of the circle described on the line segment joining z₁ & z₂ as diameter is:
  \(\arg \frac{\mathrm{z}-\mathrm{z}_{2}}{\mathrm{z}-\mathrm{z}_{1}}=\pm \frac{\pi}{2} \quad \text { or }\left(\mathrm{z}-\mathrm{z}_{1}\right)\left(\overline{\mathrm{z}}-\overline{\mathrm{z}}_{2}\right)+\left(\mathrm{z}-\mathrm{z}_{2}\right)\left(\overline{\mathrm{z}}-\overline{\mathrm{z}}_{1}\right)=0\)
• Condition for four given points z₁, z₂, z₃ & z₄ to be concyclic is, the number \(\frac{\mathrm{z}_{3}-\mathrm{z}_{1}}{\mathrm{z}_{3}-\mathrm{z}_{2}}\cdot\frac{\mathrm{z}_{4}-\mathrm{z}_{2}}{\mathrm{z}_{4}-\mathrm{z}_{1}}\).  is real. Hence the equation of a circle through 3 non collinear points z₁, z₂ & z₃ can be taken as  \(\frac {(\mathrm{z}-\mathrm{z}_{2})(\mathrm{z}_{3}-\mathrm{z}_{1})}{(\mathrm{z}-\mathrm{z}_{1})(\mathrm{z}_{3}-\mathrm{z}_{2})}\)  is real ⇒ \(\frac {(\mathrm{z}-\mathrm{z}_{2})(\mathrm{z}_{3}-\mathrm{z}_{1})}{(\mathrm{z}-\mathrm{z}_{1})(\mathrm{z}{3}-\mathrm{z}_{2})} = \frac { \left( \bar { z } -\bar { { z }_{ 2 } } \right) \left( \bar { { z }_{ 3 } } -\bar { { z }_{ 1 } } \right) }{ \left( \bar { z } -\bar { { z }_{ 1 } } \right) \left( \bar { { z }_{ 3 } } -\bar { { z }_{ 2 } } \right) } \)

Reflection points for a straight line: Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z₁ & z₂ will be the reflection points for the straight line \(\overline{\alpha} z+\alpha \overline{z}+r=0\) if and only if; \(\overline{\alpha} z_{1}+\alpha \overline{z}_{2}+r=0\) where r is real and α is non zero complex constant.

Inverse points w.r.t. a circle:
Two points P & Q are said to be inverse w.r.t. a circle with center ‘O’ and radius ρ, if :

• the point O, P, Q are collinear and on the same side of O.
• OP. OQ = ρ².

Note that the two points z₁ & z₂ will be the inverse points w.r.t. the circle
\(z \overline{z}+\overline{\alpha} z+\alpha \overline{z}+r=0\) if and only if \(z_{1} \overline{z}_{2}+\overline{\alpha} z_{1}+\alpha \overline{z}_{2}+r=0\).

Ptolemy’s Theorem:

It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides.
i.e. |z₁ − z₃| |z₂ − z₄| = |z₁ − z₂| |z₃ − z₄| + |z₁ − z₄| |z₂ − z₃|.

Logarithm Of A Complex Quantity

• Log_e(α + iβ) = ½Log_e(α² + β²) + i( 2nπ + tan⁻¹\(\frac {\beta}{\alpha}\) where n ∈ I.
• iⁱ represents a set of positive real numbers given by  \(\mathrm{e}^{-\left(2 \mathrm{n} \pi+\frac{\pi}{2}\right)}\)