]]>
LearnNext
Get a free home demo of LearnNext

Available for CBSE, ICSE and State Board syllabus.
Call our LearnNext Expert on 1800 419 1234 (tollfree)
OR submit details below for a call back

clear

Modulus and Conjugate of a Complex Number

6,282 Views
Have a doubt? Clear it now.
live_help Have a doubt, Ask our Expert Ask Now
format_list_bulleted Take this Lesson Test Start Test

Modulus and Conjugate of a Complex Number - Lesson Summary

Modulus of a complex number
 
Modulus of a real number is its absolute value.
 
The modulus of a number is the value of the number excluding its sign.
 
|7| = 7, |– 21| = 21, | – ½ | = ½
 
Consider a complex number z = a + ib, where a is the real part and b the imaginary part of z.
 
a = Re z, b = Im z

Modulus or absolute value of z = |z|
 
|z| = a 2 + b 2
 
Since a and b are real, the modulus of the complex number will also be real.
 
Ex: Find the modulus of z = 3 – 4i.
 
|z| = |3 – 4i| = 3 2 + (-4) 2

                    = 25
 
                     = 5
 
Comparison of complex numbers
 
Consider two complex numbers z 1 = 2 + 3i, z 2 = 4 + 2i.
 
Since these complex numbers have imaginary parts, it is not possible to find out the greater complex number between them.
 
Thus, the ordering relation (greater than or less than) of complex numbers, that is greater than or less than, is meaningless.
 
Ordering relations can be established for the modulus of complex numbers, because they are real numbers.
 
Some important properties of the modulus of complex numbers
  • |Z| = 0 ⇔ z = 0 i.e, Re(z) = 0 and Im (z) = 0
  • |z| = | z _ | = |–z|
  • – |z| ≤ Re(z) ≤ |z| and -|z| ≤ Im(z) ≤ |z|
  • z-1 = 1 z   =  z _ z 2 , z ≠ 0    
  • z. z _ = z 2   ,  z 2   =  z _ 2   
  • z 1 . z 2 = z 1 | z 2 |
  •  | z 1 z 2 |= z 1 z 2
  •  z ! + z 2 2 = z 1 2 + z 2 2 + 2 Re ( z 1 z 2 _ )
  •  z !   –  z 2 2 = z 1 2 + z 2 2 – 2 Re ( z 1 z 2 _ )
  • z 1 + z 1   ≤  z 1 + z 1
  • z 1 –z 1   ≥  z 1 – z 1
  • z 1 + z 2 2 + z 1 -z 2 2 = 2( z 1 2 + z 2 2 )
  • b z 1 + a z 2 2 + a z 1 - b z 2 2 = ( a 2 + b 2 )( z 1 2 + z 2 2 ) , where a and b are real

Comments(0)

Feel the LearnNext Experience on App

Download app, watch sample animated video lessons and get a free trial.

Desktop Download Now
Tablet
Mobile
Try LearnNext at home

Get a free home demo. Book an appointment now!

GET DEMO AT HOME