Complex Modulus and Argument Analyzer

Math Algebra • Complex Numbers

Written by STEM Calculators Team Published December 20, 2025 Updated February 24, 2026

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7. Complex Modulus And Argument Analyzer

Compute \(|z|\), \(\arg(z)\) (principal value) and visualize points/regions like \(|z-a|

Inputs

Mode

Angle unit

\(z=\) (expression)

Supported: \(+\), \(-\), \(\cdot\), \(/\), parentheses, \(i\), \(\pi\). Use 2*pi. (Powers ^ allowed only with real integer exponent.)

Show modulus circle \(|w|=|z|\) Show argument ray Click plane sets \(z\)

Points (one per line)

Each line is parsed as a complex expression. Blank lines are ignored.

Click plane adds a point Label points

Center \(a=\)

Comparator

Radius \(R\)

Test point \(z=\)

The plot shades the region and checks whether the test point satisfies the inequality.

Shade region Click plane sets test \(z\)

Ready

Complex plane (interactive)

Drag to pan • wheel/pinch to zoom • click to pick points • use Auto fit if it goes off screen

Re: 0, Im: 0, zoom: 1

Principal argument uses \(-\pi < \arg(z) \le \pi\). Note the branch cut along the negative real axis.

Results

Click Calculate.

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Theory: 7. Complex Modulus And Argument Analyzer

A complex number is \(z=a+bi\), where \(a=\Re(z)\) and \(b=\Im(z)\). Two key geometric quantities are the modulus \(|z|\) (distance from the origin) and the argument \(\arg(z)\) (the angle from the positive real axis).

1) Modulus \(|z|\)

\[ |z|=\sqrt{a^2+b^2}. \]

This comes from the Pythagorean theorem: the point \((a,b)\) in the complex plane is at distance \(\sqrt{a^2+b^2}\) from \((0,0)\).

2) Argument \(\arg(z)\) and the principal value

The argument is an angle \(\theta\) such that \(z=|z|(\cos\theta+i\sin\theta)\) (for \(z\neq 0\)). A robust way to compute it is \(\theta=\operatorname{atan2}(b,a)\), which automatically chooses the correct quadrant.

\[ \arg(z)\in(-\pi,\pi]\quad\text{(principal value).} \]

Important: \(\arg(z)\) is multi-valued in general because adding full turns does not change the direction: \[ \arg(z)=\theta+2\pi k,\quad k\in\mathbb{Z}. \] The calculator reports the principal value in \((-\pi,\pi]\), which has a discontinuity (branch cut) along the negative real axis.

3) Disks and circles from \(|z-a| \,\square\, R\)

Expressions of the form \(|z-a|\) measure distance from a point \(a\) (not necessarily the origin).

\[ |z-a|=R \;\;\text{is a circle centered at } a \text{ with radius } R. \]

\(|z-a|
\(|z-a|\le R\): interior + boundary (closed disk)
\(|z-a|>R\): exterior region (outside the circle)

Solved example

Example: \(z=3+4i\).

\[ |z|=\sqrt{3^2+4^2}=5. \]

\[ \arg(z)=\operatorname{atan2}(4,3)\approx 0.9273\text{ rad}\quad(\text{Quadrant I}). \]

Tip: If you drag the plane and click, you can choose \(z\) directly and instantly see \(|z|\) and \(\arg(z)\) update.

Frequently Asked Questions

What is the modulus of a complex number and how is it calculated?

For z = a + bi, the modulus |z| is the distance from the origin to (a, b) on the complex plane. It is computed as |z| = sqrt(a^2 + b^2).

What does the principal argument arg(z) mean in this calculator?

The principal argument is the angle of z measured from the positive real axis, reported in the range (-pi, pi]. The value depends on the quadrant, so a quadrant-aware angle (like atan2(Im(z), Re(z))) is used.

Why can arg(z) change suddenly near the negative real axis?

The principal argument has a branch cut along the negative real axis because it is restricted to (-pi, pi]. Crossing that line can shift the reported angle by about 2pi even though the direction changes continuously.

How do I interpret an inequality like |z-a| ≤ R on the complex plane?

|z-a| measures the distance from z to the center a. The condition |z-a| ≤ R describes the closed disk centered at a with radius R, while |z-a| = R is the boundary circle and |z-a| > R is the exterior region.

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Frequently Asked Questions

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