Modulus of Complex Number Equation, Derivation & Rearrangements

Core idea

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.

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.

Symbols

Variables

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

a

Real Part

Variable

b

Imaginary Part

Variable

|z|

Modulus

Variable

Walkthrough

Derivation

Understanding the Modulus of a Complex Number

The modulus |z| is the distance of a complex number from the origin on an Argand diagram.

z=x+iy with real x and y.

1

Use Pythagoras on the Argand Diagram:

x and y form perpendicular components, so distance squared is x²+y².

∣ z ∣^{2} = x^{2} + y^{2}

2

Take the Square Root:

This gives the magnitude (distance from the origin).

∣ z ∣ = x^{2} + y^{2}

Result

∣ z ∣ = x^{2} + y^{2}

Source: Edexcel Further Mathematics — Core Pure (Complex Numbers)

Free formulas

Rearrangements

Solve for |z|

Make m the subject

∣ z ∣ = a^{2} + b^{2}

m is already the subject of the formula.

Difficulty: 1/5

Solve for $a$

Make a the subject

a = \pm ∣ z ∣^{2} - b^{2}

Rearrange the formula for the modulus of a complex number, |z| = $a^{2} + b^{2}$ , to make the real part, a, the subject.

Difficulty: 2/5

Solve for $b$

Make b the subject

b = \pm ∣ z ∣^{2} - a^{2}

Rearrange the formula for the modulus of a complex number, |z| = $a^{2} + b^{2}$ , to isolate the imaginary part, b.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph follows a square root function shape, representing the upper half of a hyperbola that starts at a minimum value determined by the constant b and increases as the real part a grows. For a student, this shape shows that as the real part a becomes very large, the modulus m approaches the value of a, whereas small values of a indicate that the imaginary part b dominates the magnitude. The most important feature is that the curve never drops below the value of b, meaning the modulus can never be smaller than t

Graph type: other

Why it behaves this way

Intuition

The modulus represents the length of the hypotenuse of a right-angled triangle in the complex plane, where the real and imaginary parts form the other two sides.

|z|

The magnitude or length of the complex number z.

Represents how 'large' the complex number is, or its distance from the origin in the complex plane.

a

The real part of the complex number z = a + bi.

Corresponds to the horizontal position on the real axis in the complex plane.

b

The imaginary part of the complex number z = a + bi.

Corresponds to the vertical position on the imaginary axis in the complex plane.

Signs and relationships

a^2 + b^2: Squaring the real and imaginary parts ensures that their contributions to the magnitude are always positive, regardless of their original signs, effectively calculating the squared distance from the origin.
√(...): The square root operation converts the squared distance ( $a^{2}$ + $b^{2}$ ) back into a linear distance, ensuring the modulus is a non-negative real number representing a length.

Free study cues

Insight

Canonical usage

The modulus of a complex number will have the same unit and dimension as its real and imaginary components.

Common confusion

A common mistake is to assign different units to the real and imaginary parts of a complex number, or to assume the modulus is always dimensionless, even when it represents a physical quantity with units (e.g., voltage

Dimension note

If the real and imaginary parts (a and b) are dimensionless quantities (e.g., pure numbers, ratios), then the modulus |z| will also be dimensionless.

Unit systems

$z$ Unit of the represented quantity · The complex number z, its real part a, and its imaginary part b must all share the same dimension and unit.

$a$ Unit of the represented quantity · The real part 'a' must have the same unit and dimension as the imaginary part 'b'.

$b$ Unit of the represented quantity · The imaginary part 'b' must have the same unit and dimension as the real part 'a'.

|z|Same as a and b · The modulus |z| will inherit the unit and dimension from its real and imaginary components.

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.

Real Part3

Imaginary Part4

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.

Where it shows up

Real-World Context

Signal amplitude.

Study smarter

Tips

The imaginary unit 'i' is never included inside the square root; only use the real coefficient 'b'.
Because the components are squared, the modulus of a complex number and its conjugate are always identical.
The result 'm' is always a non-negative real number, regardless of whether 'a' or 'b' are negative.

Avoid these traps

Common Mistakes

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

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The modulus |z| is the distance of a complex number from the origin on an Argand diagram.

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.

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.

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

Signal amplitude.

The imaginary unit 'i' is never included inside the square root; only use the real coefficient 'b'. Because the components are squared, the modulus of a complex number and its conjugate are always identical. The result 'm' is always a non-negative real number, regardless of whether 'a' or 'b' are negative.

References

Sources

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

Modulus of Complex Number

Overview

Variables

Derivation

Use Pythagoras on the Argand Diagram:

Take the Square Root:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Argument of Complex Number

Frequently Asked Questions

Sources