Max. and Min. Value of Modulus of a Complex Number

The modulus of a complex number is a measure of its size or magnitude. It is denoted by |z|, where z is a complex number. The modulus is always a non-negative real number. In this article, we will explore how to find the maximum and minimum values of the modulus of a complex number under various conditions.

Understanding the Modulus of a Complex Number

A complex number z is usually expressed in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i^2 = -1. The modulus of z is defined as:

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

This is equivalent to the distance of the point (a, b) from the origin (0, 0) in the complex plane.

Maximum and Minimum Values

The maximum and minimum values of the modulus of a complex number depend on the context in which the problem is set. Here are some common scenarios:

1. Modulus with Given Constraints

When a complex number is subject to certain constraints, such as being on a circle or a line, the maximum and minimum values of its modulus can be determined by analyzing the geometry of the situation.

Example 1: Complex Numbers on a Circle

Consider complex numbers that lie on a circle of radius r centered at the origin. The modulus of any such complex number is constant and equal to r. Therefore, both the maximum and minimum values are r.

Example 2: Complex Numbers on a Line

Consider complex numbers that lie on a line ax + by + c = 0. The minimum modulus occurs at the point closest to the origin, and the maximum modulus is unbounded as the line extends infinitely in both directions.

2. Modulus of a Function of a Complex Number

When dealing with a function of a complex number, such as f(z) = z^2 + 1, finding the maximum and minimum modulus may involve calculus or other analytical methods.

Example 3: Modulus of a Polynomial

For the polynomial f(z) = z^2 + 1, the minimum modulus occurs at z = 0, which is |f(0)| = 1. The maximum modulus is unbounded as |z| approaches infinity.

Table of Differences and Important Points

Property	Maximum Modulus	Minimum Modulus
Definition	The largest value of `	z
Depends On	Constraints and context	Constraints and context
Circle Example	Constant (r)	Constant (r)
Line Example	Unbounded	Closest point to origin
Polynomial Example	Unbounded as `	z

Formulas

• Modulus of a complex number: |z| = \sqrt{a^2 + b^2}
• Distance from origin: d = |z|
• Modulus of a function: |f(z)|, where f(z) is a complex function

Examples to Explain Important Points

Example 4: Modulus on a Complex Plane

Consider the set of complex numbers z such that |z - 1| = 2. This represents a circle centered at (1, 0) with radius 2. The maximum and minimum modulus values are:

• Maximum: |z| is max when z is farthest from the origin, which is at (3, 0). So, |z|_max = 3.
• Minimum: |z| is min when z is closest to the origin, which is at (-1, 0). So, |z|_min = 1.

Example 5: Modulus with Inequality

Consider the inequality |z - 2i| ≤ 3. This represents the set of all complex numbers within or on a circle centered at (0, 2) with radius 3. The maximum and minimum modulus values are:

• Maximum: |z| is max when z is on the circle farthest from the origin, which is at (0, 5). So, |z|_max = 5.
• Minimum: |z| is min when z is on the circle closest to the origin, which is at (0, -1). So, |z|_min = 1.

In conclusion, the maximum and minimum values of the modulus of a complex number are determined by the geometric or functional constraints imposed on the number. Understanding these constraints and the geometry of the complex plane is crucial for solving problems related to the modulus of complex numbers.