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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.

• 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'

• 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.
  

• 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.
  

• All the four operations, addition, subtraction, multiplication and division can be
  performed on 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)

• 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).

• 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).

• 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).

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

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

• 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)

• 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.
  ? ? ? ? ? ? ? ? ? ? ? ?
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  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.
  

• 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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