Properties of Modulus & Argument: Complex Number

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Properties of Modulus:

Modulus of z is the length of vector representing z form origin to the point z.

 1. |z| = |\overline z | = | - z| = | - \overline z |

 If z = x + iy then  |z| = |\overline z | = | - z| = | - \overline z | = \sqrt {{x^2} + y{}^2}


2. |z| \ge {{\rm Re}\nolimits} (z) & |z| \ge {{\rm Im}\nolimits} (z)

 Since,\sqrt {{x^2} + y{}^2} \ge x ( equality follows when y = 0)
and \sqrt {{x^2} + y{}^2} \ge y (equality follows when x = 0)
IMP
3. |z{|^2} = z.\overline z

Since,
{|z{|^2} = {x^2} + y{}^2}

 and, z.\overline z = (x + iy)(x - iy) = {x^2} - {i^2}{y^2} = {x^2} + {y^2}
4. |{z^n}| = |z{|^n}, n \in Q
5. |{z_1}.{z_2}| = |{z_1}|.|{z_2}|
6. \left| {\frac{{{z_1}}}{{{z_2}}}} \right| = \frac{{|{z_1}|}}{{|{z_2}|}}


 7. |{z_1} \pm {z_2}| \le |{z_1}| + |{z_2}|
|{z_1} + {z_2}| = |{z_1}| + |{z_2}| holds when Arg({z_1}) = Arg({z_2}) i.e. both the vectors are in same direction.
|{z_1} - {z_2}| = |{z_1}| + |{z_2}| holds when Arg({z_1}) = \pi + Arg({z_2}) i.e. both the vectors are in opposite direction
In general,
|{z_1} \pm {z_2} \pm {z_3} \pm ....... \pm {z_n}| \le |{z_1}| + |{z_2}| + ........|{z_n}|

8. |{z_1} - {z_2}| \ge \left| {|{z_1}| - |{z_2}|} \right|
IMP
9. |{z_1} + {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2} + 2{{\rm Re}\nolimits} ({z_1}\overline {{z_2}} )

 Explanation:
Since, |z{|^2} = z.\overline z

Therefore, |{z_1} + {z_2}{|^2} = ({z_1} + {z_2})\overline {({z_1} + {z_2})}

or, |{z_1} + {z_2}{|^2} = ({z_1} + {z_2})(\overline {{z_1}} + \overline {{z_2}} )

                       = {z_1}\overline {{z_1}} + {z_2}\overline {{z_2}} + {z_1}\overline {{z_2}} + {z_2}\overline {{z_1}}

                      = |{z_1}{|^2} + |{z_2}{|^2} + 2{{\rm Re}\nolimits} ({z_1}\overline {{z_2}} )   [Since {z_1}\overline {{z_2}} + {z_2}\overline {{z_1}} = 2{{\rm Re}\nolimits} ({z_1}\overline {{z_2}} )  (? click here) ]

 and |{z_1} - {z_2}{|^2} = |{z_1}{|^2} + |{z_2}{|^2} - 2{{\rm Re}\nolimits} ({z_1}\overline {{z_2}} )


10. |{z_1} + {z_2}{|^2} + |{z_1} - {z_2}{|^2} = 2[|{z_1}{|^2} + |{z_2}{|^2}]   ( Property of Parallelogram )



Properties of Argument:
Argument is the angle between the vector representing z from the positive direction of x-axis
The Principal Argument belongs to (-pi, pi]
1. Arg({z^n}) = nArg(z),  n \in Q
2. Arg({z_1}.{z_2}) = Arg({z_1}) + Arg({z_2}) + 2k\pi  , where k = 0, 1 or -1
3. Arg\left( {\frac{{{z_1}}}{{{z_2}}}} \right) = Arg({z_1}) - Arg({z_2}) + 2k\pi  , where k = 0, 1 or -1
4. Arg(\overline z ) = - Arg(z) , z cannot be negative Real Number because for negative Real number Arg(z) is equal to pi and so Arg(\overline z ) will become -pi but it cannot be possible because -pi does not belong to principal argument interval.
5. Arg\left( {\frac{{{z_1}}}{{{z_2}}}} \right) = \theta


then Arg\left( {\frac{{{z_2}}}{{{z_1}}}} \right) = 2k\pi - \theta   , where k = 0, 1 or -1

Comments

  1. Unknown25 March 2014 at 08:00

    In the 2nd and 3rd properties of argument , how do you determine value of k ?

    ReplyDelete
    Replies
    1. Unknown13 November 2016 at 07:04

      K is any integer value that makes your answer lie between 0 and 2pi.

      Delete
  2. Unknown29 December 2015 at 05:52

    Thanks

    ReplyDelete
  3. Unknown5 May 2017 at 14:51

    so it means that there is no property for arg(z1 + z2)?

    ReplyDelete
    Replies
    1. abhijatindia5 May 2017 at 18:14

      You just don't need any new formula. Please go through the magnitude and direction of resultant of two vectors, you would know how to find the argument of sum of two complex numbers.

      from resultant of two vectors , we get the angle which is made by the resultant vector from the x axis as

      arg(z1+z2) =tan inverse {|z2|Sin[arg(z2/z1)]}/ {|z1|+|z2|cos[arg(z2/z1)]}

      Delete
  4. Unknown12 February 2018 at 19:03

    Write the properties of principal argument

    ReplyDelete
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