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Equality of Two Complex Numbers

Two complex numbers z1 = a1 + ib1 and z2 = a2 + ib2 are equal if a1 = a2 and b1 = b2, i.e., Re(z1) = Re(z2) and Im(z1) = Im(z2).
 
Complex numbers do not possess the property of order, i.e., (a + ib) < (or) > (c + id) is not defined.

Square root of a complex number

Let 60071.png = 60065.png, for b > 0
= 60059.png, for b < 0
 
To find the square root of a ib, replace i by –i in the above result.

Geometrical representation of a complex number on argand plane

Distance of any complex number from the origin is called the modulus of complex number and is denoted by |z|, i.e., |z| = 62349.png.
 
Angle of any complex number with positive direction of the x-axis is called amplitude or argument of z. i.e., amp (z) = arg(z) = tan–1 62371.png (here z lies in the first quadrant)
 
60053.png
 
Argument or amplitude of a complex number z = x + iy for different signs of x and y
  1. z = x + i y when x > 0 and y > 0
     
    60029.png
  2. z = x + i y when x < 0 and y > 0
     
    60017.png
  3. z = x + i y when x < 0 and y < 0
     
    60023.png
  4. z = x + i y when x > 0 and y < 0
     
    60011.png

Polar form of a complex number

z = x + iy = 59998.png
= 59992.png
 
This is a polar form of the complex number.
Here, 
|z| = distance of z from origin and θ is argument.
 
Properties of modulus
 
If z, z1, z2 C, then
  1. |z| = 0 z = 0, i.e., Re(z) = Im(z) = 0
  2. |z| = |z| = |–z| = |–z|
  3. –|z| ≤ Re(z) ≤ |z|; –|z| ≤ Im(z) ≤ |z|
  4. zz = |z|2
  5. |z1z2| = |z1| |z2|
  6. 59986.png; z2 ≠ 0
  7. |z1 + z2|2 = |z1|2 + |z2|2 + 2 Re(z1z2)
  8. |z1z2|2 = |z1|2 + |z2|2 – 2 Re(z1z2)
  9. |z1 + z2|2 + |z1z2|2 = 2(|z1|2 + |z2|2)
  10. |az1bz2|2 + |bz1 + az2|2 = (a2 + b2) (|z1|2 + |z2|2), where a, b R
  11. |zn| = |z|n, where n Q
  12. |z1 ± z2| ≤ |z1| + |z2|
  13. |z1 ± z2| ≥ ||z1| – |z2||

Conjugate of a complex number

If there exists a complex number z = a + ib, (a, b) R, then its conjugate is defined as  
= a – ib.
 
59980.png
 
Hence, we have Re(z) = 59974.png.
 
Geometrically, the conjugate of z is the reflection or point image of z in the real axis.
 
Properties of conjugate
 
If z, z1, z2 are complex numbers, then
  1. (z) = z
  2. z + z = 2 Re(z)
  3. z + z = 2i Im(z)
  4. z = z z is purely real
  5. z + z = 0 is purely imaginary
  6. 59968.png
  7. 59962.png
  8. 59955.png
  9. 59949.png z2 ≠ 0
Properties of arguments
  1. arg(z1z2) = arg(z1) + arg(z2)
     
    In general, arg(z1z2z3zn) = arg(z1) + arg(z2) + arg(z3) + … + arg(zn)
  2. arg(z1z2) = arg(z1) – arg(z2)
  3. arg59943.png = arg z1 – arg z2
  4. arg59937.png = 2 arg z
  5. arg(zn) = n arg z
  6. arg z = –arg z = arg
  7. z1z2 + z1z2 = 2 |z1| |z2| cos (θ1θ2), where θ1 = arg (z1) and θ2 = arg(z2)
  8. If z is purely imaginary then arg(z) = 59925.png
  9. If z is purely real then arg (z) = 0 or π
  10. Angle between two line segments joining z1 and z2 and joining z3 and z4 is θ = ±arg59919.png
  11. 59913.png
     
    59907.png
  12. If z1, z2, z3 be the vertices of an equilateral triangle, then
     
    59901.png
     
    or 59895.png




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