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Complex Numbers
Origin of Complex Numbers
 The concept of complex numbers was first identified by the Greek mathematician,
Leonhard Euler (17071783), 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 z1z2 = (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 (17771855) 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 zPlane. The Complex Plane is also known as the Argand Plane or Argand
diagram,named after JeanRobert 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.
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
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