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Modulus and the Conjugate of a Complex Number

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Introduction

modulusandconjugateofcomplex1
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
modulusandconjugateofcomplex3

Properties of Modulus and conjugate of complex numbers

For any two complex numbers z1 and z2, we have
modulusandconjugateofcomplex2


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, modulusandconjugateofcomplex4
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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