Modulus of a complex number

Modulus of a Complex Number

The modulus (or absolute value) of a complex number gives its distance from the origin in the complex plane. If a complex number is given by \( z = a + ib \), where \( a \) is the real part and \( b \) is the imaginary part, then the modulus of \( z \), denoted by \( |z| \), is defined as:

\( |z| = \sqrt{a^2 + b^2} \)

This expression is derived using the Pythagorean theorem by treating the complex number as the coordinate point \( (a, b) \) in the Cartesian plane.

Example 1:

Consider the complex number \( z = 3 + 4i \). Then: \( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \) Hence, the modulus of \( 3 + 4i \) is 5.

Example 2:

Consider \( z = -2 - i \). Then: \( |z| = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \)

Properties of Modulus:

1. Non-Negativity:
The modulus of any complex number \( z \) is always non-negative, that is, \( |z| \geq 0 \), and it equals zero only when \( z = 0 \).

2. Multiplicative Property:
Consider any two complex numbers \( z_1 \) and \( z_2 \): \( |z_1 z_2| = |z_1| \cdot |z_2| \)

3. Quotient Property:
If z₂ ≠ 0, then: \( \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \)

4. Conjugate Relation:
A complex number and its conjugate share the same modulus: \( |z| = |\overline{z}| \) since \( \overline{z} = a - ib \), and \( |\overline{z}| = \sqrt{a^2 + (-b)^2} = \sqrt{a^2 + b^2} = |z| \)

5. Triangle Inequality:
Given any two complex numbers \( z_1 \) and \( z_2 \), the following holds: \( |z_1 + z_2| \leq |z_1| + |z_2| \)

Geometric Interpretation:

In the complex plane, the modulus \( |z| \) represents the length of the line segment from the origin to the point \( (a, b) \). Therefore, the modulus always results in a real number that is zero or positive.

The modulus of a complex number is a fundamental concept that provides insight into its magnitude. It is widely used in mathematics, physics, and engineering to understand the behavior of complex-valued functions and quantities.