Modulus of Complex Number Calculator

Formula first

Overview

The modulus of a complex number represents the geometric distance between the origin and the point in the complex plane. It is calculated by taking the square root of the sum of the squares of the real and imaginary components, effectively applying the Pythagorean theorem.

Symbols

Variables

a = Real Part, b = Imaginary Part, |z| = Modulus

Apply it well

When To Use

When to use: Apply this formula when you need to determine the magnitude of a complex value or convert coordinates from rectangular to polar form. It is a prerequisite for finding the argument of a complex number and for normalizing vectors in complex space.

Why it matters: In electrical engineering, the modulus represents the peak amplitude of alternating current signals. In quantum mechanics, the squared modulus of a wave function determines the probability density of finding a particle in a specific state.

Avoid these traps

Common Mistakes

• Including i in the square (b² not (bi)²).
• Sign errors.

One free problem

Practice Problem

Calculate the modulus of the complex number z = 3 + 4i, where 3 is the real component and 4 is the imaginary component.

Solve for: $m$

Hint: Apply the Pythagorean theorem using the real and imaginary parts as the legs of a triangle.

The full worked solution stays in the interactive walkthrough.

Keep going

Related Formulas

References

Sources

1. Wikipedia: Complex number
2. Britannica: Complex number
3. Stewart, James. Calculus: Early Transcendentals, 8th Edition
4. Wikipedia: Modulus of a complex number
5. Edexcel Further Mathematics — Core Pure (Complex Numbers)