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. = a − ib.

Note:

• z + = 2 Re(z)
• z − = 2i Im(z)
• z = a² + b² which is real
• If z lies in the 1^st quadrant then lies in the 4^th quadrant and 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| =
  
  θ = tan⁻¹
  (angle made by OP with positive x−axis)
  Note:
  1. |z| is always non-negative. Unlike real numbers 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 ; = r (cos θ − i sin θ)
  Note: cos θ + i sin θ is also written as CiS θ.
  Also
• Exponential Representation:
  z = re^iθ ; | z | = r ; arg z = θ ; = re^-iθ

Important Properties of Conjugate/ Moduli/ Amplitude | Complex Numbers

If z , z₁ , z₂ ∈ C then ;

• z + = 2 Re (z) ; z − = 2 i Im (z) ; ; ;
  ; ; ; 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 = 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, & || = |z|

Note:

• If = z = re^iθ then = z₁ = re^{i(θ + φ)} = z . e^iφ. If and are of unequal magnitude then φ
• If A, B, C & D are four points representing the complex numbers z₁, z₂, z₃ & z₄ then is purely real ;
  is purely imaginary ]
  
• If z₁, z₂, z₃ are the vertices of an equilateral triangle where z0 is its circumcentre then (a) (b)

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,
• 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:
  
• 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) &
• 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:

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 =
  (b)  Orthocentre of the ∆ ABC = OR
  (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
  
  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
  which on manipulating takes the form as 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
  which is of the form , r is real centre − α & radius
  . Circle will be real if .
• The equation of the circle described on the line segment joining z₁ & z₂ as diameter is:
  
• Condition for four given points z₁, z₂, z₃ & z₄ to be concyclic is, the number .  is real. Hence the equation of a circle through 3 non collinear points z₁, z₂ & z₃ can be taken as    is real ⇒

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 if and only if; 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
if and only if .

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 where n ∈ I.
• iⁱ represents a set of positive real numbers given by