The modulus of a Complex Number is here. Know the example problems of modules and various forms involved in them. Follow cartesian form, trigonometric or polar form, exponential form, modulus properties, the principal value of the argument of LPA. Refer to the important topics and terminology used in complex values. Check the below sections to know the absolute value and the modulus of the complex number.
Modulus of a Complex Number – Definition
The modulus of a complex number gives you the distance of the complex numbers from the origin point in the argand plane. the conjugate of the complex number gives the reflection of that number about the real axis in the same argand plane. The modulus of the complex number is always positive which is |z| > 0. Also, the complex values have a similar module that lies on a circle.
Cartesian Form
z¡ = x + iy is the complex number that is in the cartesian form. The modulus can be represented as |z|. The value of |z| = √x2 + y2
Trigonometric or Polar Form
The complex number in the polar form is represented with z. The equation of z = r cisθ or z = r∠θ, r(cosθ + isinθ) where r represents the distance of the point z from the origin or the modulus. θ is the subtended angle by z from the positive x-axis.
Here, r = √x2 + y2 & θ represents its argument.
Exponential Form
The complex number “z” in the form of exponential can be expressed as z = rei∅ or |z|eiarg(z) represents the complex number modulus and ∅ is the argument value.
Properties of Modulus
- If |z| = 0, then z = 0 + i0
- |z| = |-z| = |\(\overline{z}\)| = |iz|
- -|z| ≤ Re(z) ≤ |z|
- -|z| ≤ Im(z) ≤ |z|
- \(\overline{z}\)z = |z|², if |\(\overline{z}\)| = 1
These type of modulus are called as unimodulars
then \(\overline{z}\) = 1/z
-
- |z1z2| = |z1||z2|, is true for n complex numbers
- |zn|= |z|n
- |z1/z2| = |z1|/|z2|
- |z1 + z2|² + |z1 – z2|² = 2(|z1|² + |z2|²)
Definition and Meaning of Arg
Argument or Arg in complex numbers resembles the angle which is subtended by any of the complex points on the argand plane from the +(positive) x-axis. For the complex number z, the argument is represented with arg(z) which gives us the measurement of the angle between the + (positive) x-axis and also the
For any complex number z, its argument is represented by arg(z). It gives us the measurement of the angle between the positive x-axis and the line joining origin and the point. There are three ways to express an argument of a complex number. The three ways are as follows.
General Argument
For all the complex numbers z, the general argument is z = x + iy or r(cosθ + isinθ)
Im(z) = r sinØ and Re(z) = r cosØ which shows the real and imaginary parts of the complex numbers and also are the functions of sin and cosine. Hence, they are periodic with the period 2Πwhich means that there exist infinite complex numbers which have the same argument or angle. The same case follows for modulus too.
Assume the complex numbers in the first quadrant (z) = x + iy (x,y >0)
The general argument will be defined as Ø = 2nΠ+ tan-1y/x where n ∈l.
Here, tan-1y/x is known as the argument principle value and hence its value depends upon the quadrant in that point lies.
Also, Read: Properties of Complex Numbers
Prinicipal Value of Argument (PA) or Amplitude (amp)
A general argument or a well-defined complex number cannot be expressed. We used the principle value or amplitude instead of the general argument in the case where the well-defined complex function is required.
In the principle argument, Φ value is restricted to be in interval ( -Π< Φ ≤ Π) or (-Π, Π]. The range of principle argument represents the half-circled range from the +(positive) x-axis in either direction.
| Quadrant | Sign of x & y | Arg (z) |
|---|---|---|
| I | x,y>0 | tan-1y/x |
| II | x<0,y>0 | Î – tan-1|y/x| |
| III | x,y<0 | -Î + tan-1|y/x| |
| IV | x>0,y<0 | -tan-1|y/x| |
LPA (Least Positive Argument)
LPA is another type of argument, where the angle range Ø is kept as (0< Ø ≤ 2Π) or (0,2Π]. This type of argument has only anticlockwise rotation is considered. To find the complex number in any of the quadrants, the angle has to be calculated from the positive x-axis in the counterclockwise or anticlockwise direction only.
The native way of calculating the angle to the point (a,b) can be done by using arctan (b/a) but arctan takes the value in the range of [-Î /2, Î /2], which gives the wrong result for the negative x component coordinates. We can fix it by adding or subtracting Î and it depends on the quadrant of the Argand diagram in which the point lies in
- 1st quadrant (Θ) = arctan (b/a)
- 2nd quadrant (Θ) = arctan (b/a) + Î
- 3rd quadrant (Θ) = arctan (b/a) – Î
- 4th quadrant (Θ) = arctan (b/a)
Modulus of a Complex Number Examples
Problem 1:
If |z-(4/z)| = 2. Find the maximum value of |z|?
Solution:
As given in the question,
|z-(4/z)|
The equation is
|z-(4/z)| ≥ |z| -|4/z|
2 ≥ |z| – 4/|z|
2|z| ≥ |z|² – 4
|z|² – 2|z| – 4 ≤ 0
|z| ≤ √5 + 1
The maximum value of z,
|z| ≤ √5 + 1
Problem 2:
If z + |z| = 1 + 4i. Find the value of |z|?
Solution:
The given equation is
z= x + iy
z + |z| = 1 + 4i
x + iy + √(x² + y²) = 1 + 4i
y=4 and x + √(x² + y²) = 1
x + √(x² + 4²) = 1
√(x² + 4²) = 1-x
Squaring on both the sides, we get
x² + 4² = 1 + x² – 2x
2x = -15
or x = -15/2
|z| = √x² + y²
|z| = √(-15/2)² + 4²
|z| = √(225/4 + 16)
|z| = √(225 + 64/4
|z| = √(289/4)
|z| = 17/2
The value of |z| = 17/2
Problem 3:
For the equation, z = (3-2i)/2i. Find the complex number modulus?
Solution:
As given in the question,
z = (3-2i)/2i
z = (3)/2i – 2i/2i
z= 3/2i – 1
z= 3i/(2i)² – 1
z = -3i/2 – 1
|z| = √(-3/2)² + (-1)²
|z| = √(9/4 + 1)
|z| = √(9+4)/4
|z| = √13/4
|z| = √13/2
Therefore, the complex modules is |z| = √13/2