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Complex Numbers


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Complex Numbers

Origin of Complex Numbers


  • The concept of complex numbers was first identified by the Greek mathematician,
    Leonhard Euler (1707-1783), while he was trying to find the square root of the Quadratic
    Equation x + 1 = 0.



Definition of Complex Numbers

  • A complex number is an ordered pair of real numbers. The set of all complex numbers
    is denoted by the symbol 'C'. We have C = {(a, b) / a, b R} = R X R.


  • A Complex Number is a number of the form z = a + ib , where 'a' and 'b' are
    real numbers and 'i' is the imaginary unit, with the property i = (- 1).
    Z = a + ib can also be represented as z = (a , b)


  • The real number a is called the real part [Re(z)] of the complex number
    and the real number 'b'is the imaginary part [Im(z)].


  • Real numbers may be considered to be complex numbers with an imaginary part of Zero;
    that is, the real number a is equivalent to the complex number 'a + i0'



Example

  • Consider the complex number 3 + i5, its real part is 3 and imaginary part is 5 .


  • Consider the complex number 7 ? i2, its real part is 7 and imaginary part is -2.


  • 7 can be considered as a complex number with its imaginary part as zero.



Example

  • Consider the complex number 3 + i5, its real part is 3 and imaginary part is 5 .


  • Consider the complex number 7 ? i2, its real part is 7 and imaginary part is -2.


  • 7 can be considered as a complex number with its imaginary part as zero.



Arithmetic Operations on Complex Numbers

  • All the four operations, addition, subtraction, multiplication and division can be
    performed on complex numbers.



Addition of Complex Numbers

  • If z1 = (a , b) and z2 = (c , d) then z1+ z2 = (a + c , b + d).

  • For Example: z1 = 8 + i5 and z2 = 6 + i2 then z1 + z2 = 14 + i7 = (14 , 7)



Negative of a Complex Number

  • If z = (a, b) then we define negative of a complex number as ? z = (- a , - b) = (- a) + i(- b).

  • For Example: z = 2 + i4, then ? z = (- 2) + i(- 4) = (- 2 , - 4).


Subtraction of Complex Numbers

  • If z1 = (a , b) and z2 = (c , d) then z1-z2 = (a ? c , b ? d).

  • For Example: z1 = 4 + i7 and z2 = 2 + i5 then z1 - z2 = 2 + i2 = (2 , 2).


Multiplication of Complex Numbers

  • If z1 = (a , b) and z2 = (c , d) then z1 . z2 = (a , b) . (c , d) = (ac ? bd , ad + bc)

  • For Example: z1 = 2 + i3 and z2 = 4 + i5 then z1 . z2 = - 7 + i22 = (- 7 , 22).


Division of Complex Numbers

  • If z1 = (a , b) and z2 = (c , d) then =

  • For Example: z1 = 2 + i3 and z2 = 3 + i4 then = =


Conjugate of a Complex number:

  • For any complex number z= a + bi, we define the conjugate of z as a + (-b)i and

  • denote this by and = (a ? bi), that is, = (a ? bi)


Geometrical Representation of Complex Numbers

  • Carl Friedrich Gauss (1777-1855) was one of the mathematicians who first thought
    that complex numbers can be represented on a two.dimensional plane called a Complex
    Plane or a z-Plane. The Complex Plane is also known as the Argand Plane or Argand
    diagram,named after Jean-Robert Argand. The geometrical representation of complex
    number z and its conjugate are shown in the figure given.
    ? ? ? ? ? ? ? ? ? ? ? ?
    ? ? ? ? ? ? ? ? ? ? ? ?
    ? ? ? ? ? ? ? ? ? ? ? ?
    ? ? ? ? ? ? ? ? ?





    The figure shows the representation of z = x + iy.Point z is obtained on the Cartesian
    plane by taking the real part 'x' along the horizontal line/axis (as the x coordinate)
    and then the imaginary part y along the vertical line/axis, (as the y coordinate).
    Hence the Horizontal line/axis is known as the real axis and the vertical line/axis
    is known as the imaginary axis.
    As seen in the figure , the conjugate of z, is the
    reflection of z along the real axis. oz = r, is called the modulus of the complex
    number Z, where r = The angle of inclination of oz, with the positive real axis is
    and is called the amplitude of the complex number, where =tan The complex number
    z can also be represented in terms of r and as z = r(cos + isin ) = x + iy
    where x = r cos and y = r sin . This notation is referred to as the polar form or
    the trigonometric form.





Further reading on Complex Numbers

  • Square root of a complex number

  • Cube root of a complex number

  • nth root of a complex number

  • DeMovires Theorem

  • Application of complex numbers



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Additional Links for Complex Numbers

  • Click here for samples

  • Click here for Mathematics Dictionary




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