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

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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 1st quadrant then (bar { z } ) lies in the 4th quadrant and (bar {-z } ) lies in the 2nd 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
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}})
    Complex Numbers
    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}
    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
    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 ; | 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 then (overrightarrow { OQ }) = z1 = rei(θ + φ) = z . e. 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 ]
    Complex Numbers Formulas
  • 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 + isqrt {3}}{2}, frac{-1 – isqrt{3}}{2})
  • If w is one of the imaginary cube roots of unity then 1 + w + w² = 0. In general 1 + wr + w²r = 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 − ω²b); 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)

nth Roots Of Unity | Complex Numbers

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

  • They are in G.P. with common ratio ei(2π/n) &
  • (1^{mathrm{p}}+alpha_{1}^{mathrm{p}}+alpha_{2}^{mathrm{p}}+ldotsldots+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 − αn – 1) = n &
    (1 + α1) (1 + α2) ……. (1 + αn − 1) = 0 if n is even and 1 if n is odd.
  • 1 . α1 . α2 . α3 ……… αn − 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
    (zleft(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}}cdotfrac{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

  • Loge(α + iβ) = ½Loge(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)})

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