Introduction

The modulus of a complex number denotes its magnitude and conjugate of a complex number is obtained by changing the sign of imaginary part of a complex number.  If the imaginary part is positive we make it negative and if it is negative, we make it positive.  The sign of real part doesn’t change.
Let z=a + ib be a complex number, then the modulus of ‘z’ is denoted by |z| and is defined as |z| = √a²+b²
For example: 1) |5-i| = √5² + (-1)²
                              = √25 + 1
                              = √26
                   2) |-3+4i| = √(-3)2 + 42
                                 = √9+16
                                 = √25
                                 = 5
The conjugate of a complex number z=a + ib is denoted by z and is defined as z = a-ib

Properties of Modulus and conjugate of complex numbers

For any two complex numbers z1 and z2, we have

Square Root of a negative real number

We know i²=-1 and (–i)²=i² =-1
Hence the square roots of -1 are i, -i but the symbol √-1 mean ‘i’ only.
Now we can see that i and –i both are the solutions of the equation x²+1=0
Similarly, (√5i)² = 5i² = 5 x -1 = -5
              (-√5i)² = 5i² = 5 x -1 = -5
Generally, if ‘a’ is a positive real number, √-a = √a √-1 = √a i
We have already learned that √a x √b = √ab for all positive real numbers ‘a’ and ‘b’
We can examine this result for i2,
                              i² = i x i = √-1 x √-1 = √-1 x -1 = √1 =1, which is a contradiction to the fact that i²=-1
Hence √a x √b ≠ √ab if both ‘a’ and ‘b’ are negative real numbers
If ‘a’ and ‘b’ are zero, then √a x √b = √ab =0

Powers of ‘i’

We know that i² = -1, different powers of ‘i’ are given below
                     i³ = i² x i = -1 x i = -i
                     i⁴ = (i²)2 = (-1)² = 1
                     i⁵ = i⁴ x i = 1 x i = i
                     i⁶ = i⁵ x i = i x i = i² = -1 and so on.
Also we have,
In general, for any integer k, i4k=1, i4k+1=i, i4k+2 = -1, i4k+3 = -i

Identities in Complex numbers

For any two complex numbers z1 and z2 we have the following identities

•    (z₁ + z₂)² = z₁² + z₂² + 2z₁z₂

(z₁ + z₂)² = (z₁ + z₂) (z₁+z₂)
                 = z₁² + z₁z₂ + z₁z₂ + z₂²
                 = z₁² + 2z₁z₂ + z₂²

•   (z₁-z₂)2  = z₁² -2z₁z₂+z₂²
•   (z1+z2)3 = z13+3z12z2+3z1z22+z23
•   (z1-z2)3 = z13-3z12z2+3z1z22+z23
•    z12-z22= (z1+z2)(z1-z2)

Express in the form a + ib
1) i⁹ + i¹⁹
1) i⁹ + i¹⁹ = i⁸ i + i¹⁸ i
               = 1 x i + -1 x i
               = i + -i
               = 0
               = 0 + i0

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Reference Links:

• http://en.wikipedia.org/wiki/Modulus_of_complex_number
• http://tvwiki.tv/wiki/Square_root
• http://www.cut-the-knot.org/arithmetic/algebra/ComplexNumberIdentities.shtml
• http://www.purplemath.com/modules/complex.htm
• http://en.wikipedia.org/wiki/Complex_conjugate