Complex Numbers



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Chapter 1 : Complex Numbers



Complex Numbers arrow_upward


  • A Complex Number is a number consisting of a real and imaginary part.
  • It is denoted by letter z.
  • It can be written in the form z = a + bi
    • Where a and b are real numbers.
    •  i is the imaginary unit.
  • i is the standard imaginary unit with the property .

  • Imaginary Numbers arrow_upward


  • An imaginary number is one that when squared gives a negative result.
  • Imaginary number is a complex number which has an imaginary part and zero real part.
  • We can split the negative numbers into positive number and –1.
  • We are defining √(-1) = i
  •      i2 = -1

         i3 = -i

         i4 = 1

  • Example: Find the imaginary number of √(-5)?
  • Solution: We know -5 = -1 × 5. So
  •     √(-5) = √(-1 × 5) = √(-1) × √5

        = √5 i

        = 2.23i

  • So the imaginary number is 2.23i.

  • Complex Conjugate arrow_upward


  • Complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs.
  • The complex conjugate of   is .
  • Conjugate of a complex number is also denoted by .
  • Also,
  • A real number and a sum of two squares.

  • Adding and Subtracting Complex Numbers arrow_upward


  • Let  and  be the two complex numbers.
  • The rule is to add or subtract the real and imaginary parts separately:
  •     = a + ib + c + id

        = a + c + i(b + d)

  • So   = a + c + i(b + d).
  •     = a + ib – c – id

        = a – c + i(b - d)

  • So   = a – c + i(b - d).
  • Example 1: Add the complex numbers (1 + i) and (3 + i).
  • Solution:
  • (1 + i) + (3 + i)
  •     = 1 + 3 + i (1 + 1)

  •     = 4 + 2i
  • Example 2: Subtract the complex numbers (2 + 5i) and (1 – 4i).
  • Solution:
  • (2 + 5i) – (1 – 4i)
  •     = 2 + 5i – 1 + 4i

        = 1 + 9i


    Multiplying Complex Numbers arrow_upward


  • We multiply two complex numbers just as we multiply expressions of the form (x + y) together.
  •      z1 × z2

       = (a + ib)(c + id)

       = ac + a(id) + (ib)c + (ib)(id)

       = ac + iad + ibc - bd

       = ac – bd + i(ad + bc)

  • So  z1 × z2 = ac – bd + i(ad + bc)
    •  
  • Example: Multiply the complex numbers (2 + 3i) and (3 + 2i).
  • Solution:
  • (2 + 3i)(3 + 2i)
  •     = 2 × 3 + 2 × 2i + 3i × 3 + 3i × 2i

        (By substituting)

    •     = 6 + 4i + 9i - 6
    •     = 13i

    Dividing Complex Numbers arrow_upward


  • For dividing two complex numbers, multiply both the numerator and denominator by the complex conjugate of the denominator.
  • The denominator is now a real number.
  • Example:
  •        


    The Complex Plane arrow_upward


  • The Complex Plane or z-plane is a geometric representation of the complex numbers established by real axis and orthogonal imaginary axis.
  • It can be modified as a Cartesian plane.
  • The real part of a complex number is represented by a displacement along the x-axis and the imaginary part by a displacement along the y-axis.
  • The multiplication of two complex numbers can be expressed easily in polar coordinates.
  • The magnitude or modulus of the product is the product of the two absolute value, or moduli.
  • The angle or argument of the product is the sum of the two angles, or arguments.
  • In particular, multiplication by a complex number of modulus 1 acts as a rotation.
  • Geometric representation of  and its conjugate   in the complex plane

  • The distance along the red line from the origin to the point z is the modulus or absolute value of z.
  • The angle   is the argument of z.


  • Thank You from Kimavi arrow_upward


  • Please email us at Admin@Kimavi.com and help us improve this tutorial.


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