Imaginary and complex numbers
      Imaginary numbers basic definitions
         Imaginary unit
      Complex numbers
         Real and imaginary parts
         The complex plane
         The set of all complex numbers C
         Absolute value, modulus of a complex number
         Complex conjugates
Imaginary numbers basic definitions
Imaginary numbers are introduced to enable us to take the square root of negative numbers.
Thus, for example
so, the square roots of negative numbers are called imaginary numbers since they do not lie on the real number line. 
Therefore, the square of any imaginary number (except 0) is a negative number.
That is, any number of the form
  yi, where - the imaginary unit and y is any real number,
we call imaginary number.
By using the definition derived are powers of the imaginary unit i:
note that the powers of i repeat in a cycle, so that
Examples:  Reduce,  i15i26  and  i149.
Solutions:       i15 = i4 3 i3 = i3 = -i,    i26 = i4 6 i2 = i2 -1  and  i149 = i4 37 i = i.
Complex numbers
A complex number is the sum of a real number and an imaginary number.
A complex number z is written in the form of z = x + yi, where x and y are real numbers, and
The real number x is called the real part of the complex number, and the real number y is the imaginary part.
The real part of z is denoted  Re(z) = and the imaginary part is denoted  Im(z) = y.
Hence, an imaginary number is a complex number whose real part is zero, while real numbers may be considered to be complex numbers with an imaginary part of zero.
That is, the real number x is equivalent to the complex number x + 0i.
Equality of complex numbers
Two complex numbers are equal if their real parts are equal and their imaginary parts are equal.
The complex plane
Complex numbers are represented by points or position vectors in the coordinate plane called the complex plane (or the Gauss plane). Where, the x-axis is called the real axis and the y-axis is called the imaginary axis.
The representation of a complex number by Cartesian coordinates is called the rectangular form or algebraic form of the complex number.
The standard symbol for the set of all complex numbers is C
Absolute value, modulus of a complex number
The absolute value of a complex number z is defined as the distance from z to the origin in the complex
 plane, i.e.,   
Complex conjugates
The complex conjugate of the complex number  z = x + yi  is  x - yi  that has the same real part x, but differ in the sign of the imaginary part.
That is, the conjugate is the reflection of z about the real axis, as is shown in the above figure.
Addition and subtraction of complex numbers
To add or subtract two complex numbers z1 = a + bi and z2 = c + di, we add or subtract the real parts and the imaginary parts.
Addition:       z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i  
Subtraction:       z1 - z2 = (a + bi) - (c + di) = (a - c) + (b - d)i  
Examples:  Given are complex numbers,  z1 = -3 + 2i  and  z2 = 4 + 3i, find z1 + z2 and  z1 - z2.
Solutions:    z1 + z2 = (-3 + 2i) + (4 + 3i) = (-3 + 4) + (2 + 3)i = 1 + 5i
  and             z1 - z2 = (4 + 3i) - (1 + 5i) = (4 - 1) + (3 - 5)i = 3 - 2i
Given addition and subtraction are shown in the complex plane in the figures below.
z1 + z2 = (-3 + 2i) + (4 + 3i) = 1 + 5i   z1 - z2 = (4 + 3i) - (1 + 5i) = 3 - 2i
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