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ALGEBRA
  Imaginary and complex numbers
 :: Imaginary numbers
  Any number of the form  y i   where, is the imaginary unit and y is any real number, 
we call imaginary number.   
Powers of the imaginary unit
By using the definition derived are powers of the imaginary unit i,
note that the powers of i repeat in a cycle.
Examples: Reduce,  a)  i15,   b)  i26   and  c)  i149.
Solutions: a)  i15 = i4 3 i3 = i3 = -i,    b)  i26 = i4 6 i2 = i2 -1  and  c)  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 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 | z | of a complex number is defined as the distance from z to the origin in the complex plane.
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
Example: Given are complex numbers,  z1 = -3 + 2i  and  z2 = 4 + 3i, find z1 + z2 and  z1 - z2.
Solution:    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
Multiplication and division of complex numbers
multiplication: z1 z2 = (a + bi) (c + di) = ac + bci + adi + bdi2 = (ac - bd) + (ad + bc)i
division:
Example: Given are complex numbers,  z1 = -3 + 2i  and  z2 = 4 + 3i, find  z1   z2  and  z1 / z2.
Solution:   z1   z2 = (-3 + 2i) (4 + 3i) = -3 4 + 2 4i + (-3) 3i +  2 3 i2 = -18 - i
and   
Example:  For what real number a the real part of the complex number equals 1.
Solution:
    
Example:  Evaluate the expression  where  z = 1 - i.
Solution:
 
 
 
 
 
 
 
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