Calculated Results
The result block appears here after submission.
Plotly Graph
The graph compares component magnitudes and selected norm values.
Calculator Inputs
Use one or many complex entries. Each row supports label, real part, and imaginary part.
Example Data Table
These worked examples show how several norms compare for different complex vectors.
| Case | Vector | p | 1-Norm | 2-Norm | ∞-Norm | p-Norm |
|---|---|---|---|---|---|---|
| Example A | [3 + 4i, 1 - 1i, -2 + 2i] | 3 | 9.2426 | 5.9161 | 5.0000 | 5.3187 |
| Example B | [2 + 3i, -4 + 1i] | 3 | 7.7287 | 5.4772 | 4.1231 | 4.8905 |
| Example C | [0 + 5i, -1 - 2i, 2 + 2i, 3 + 0i] | 3 | 13.0645 | 6.8557 | 5.0000 | 5.7063 |
| Example D | [-3 - 4i] | 3 | 5.0000 | 5.0000 | 5.0000 | 5.0000 |
Formula Used
Single Complex Number Modulus
z = a + bi, so |z| = √(a² + b²).
Complex Vector 1-Norm
‖z‖₁ = Σ |zi|
Complex Vector 2-Norm
‖z‖₂ = √(Σ |zi|²)
Infinity Norm
‖z‖∞ = max(|zi|)
Custom p-Norm
‖z‖p = (Σ |zi|p)1/p, where p ≥ 1.
The calculator also reports phase angles, squared magnitudes, mean magnitude, RMS magnitude, and normalized vector components when the 2-norm is nonzero.
How to Use This Calculator
- Enter one or more complex components using real and imaginary values.
- Keep or edit the labels so the graph and tables stay easy to read.
- Choose the custom p value, decimal precision, and angle unit.
- Click the calculate button to show results above the form.
- Review the summary cards, detail table, normalized vector, and graph.
- Use the CSV or PDF buttons to download a report copy.
FAQs
What does this calculator measure?
It measures the size of one complex number or an entire complex vector. You can compare modulus, 1-norm, 2-norm, infinity norm, and any custom p-norm in one place.
What is the modulus of a complex number?
For z = a + bi, the modulus is |z| = √(a² + b²). It represents the distance from the origin on the complex plane.
Why are several norms shown together?
Different norms describe size differently. The 1-norm totals magnitudes, the 2-norm gives Euclidean length, the infinity norm finds the largest component, and the p-norm lets you test custom sensitivity.
When should I use the infinity norm?
Use the infinity norm when the largest component controls behavior. It is useful for worst-case bounds, stability checks, and maximum-amplitude comparisons.
Can I enter a single complex number only?
Yes. Keep one row only. Then the modulus, 1-norm, 2-norm, infinity norm, and custom p-norm all match that single magnitude.
What happens if all values are zero?
Every norm becomes zero. The calculator still works, but no normalized vector is shown because a zero vector has no valid unit direction.
What does the graph display?
The graph compares each component magnitude with overall norm values. It helps you see whether total length is spread out or dominated by a few entries.
Are the exports useful for reports?
Yes. CSV works well for spreadsheets and data review. PDF is better for printing, sharing summaries, and keeping a clean report copy.