Complex modulus calculator
The Complex modulus calculator computes complex modulus from the relation |z| = sqrt(a^2 + b^2). It takes 2 inputs (real part a, imag part b) and returns the modulus |z|. These calculators are for students and tutors working through algebra, and for anyone in a technical or trades role who needs to solve an equation, work with a sequence or series, or handle an expression involving logarithms, complex numbers, matrices or vectors before carrying the result into a longer piece of working. Enter your values below and the result updates instantly, and you can share a permalink that pre-fills the exact calculation. Enter each value in the form the calculator expects, whether that is a list of coefficients, the terms of a sequence, or the entries of a matrix or vector, keeping every minus sign attached to the quantity it belongs to and checking the order of entry wherever the operation is not symmetric; where a result has an exact form, carry that form into any further steps rather than a rounded decimal, since rounding early compounds through the rest of the working. For example, with real part a = 5, imag part b = 12, the modulus |z| works out to 13, and the worked example further down the page shows every step so you can follow the arithmetic and reproduce it by hand. The method is the standard form documented by NIST DLMF, and the figure above each result carries the date it was last verified. Algebraic results hold only where the operation is defined, so watch for inputs that give no real solution, force a division by zero, leave a matrix without an inverse, or fall outside the permitted domain, and read any decimal on screen as a rounded rendering of an exact value rather than the value itself.
With Real part a = 5, Imag part b = 12, the result is 13.
Applies to: any numeric inputs. Method source: NIST DLMF, checked 2026-07-09.
The formula
|z| = sqrt(a^2 + b^2)
Worked example
With Real part a = 5, Imag part b = 12:
- |z| = sqrt(a^2 + b^2)
- Modulus |z| = 13
This worked example is one of the automated golden-value tests this calculator must pass before it can publish.
What this assumes
- Inputs are real numbers in the units shown.
- The result is the exact value of |z| = sqrt(a^2 + b^2); general information, not professional advice.
Frequently asked questions
What formula does this use?
|z| = sqrt(a^2 + b^2), the standard form documented by NIST DLMF.
Does the result ever change over time?
No. This is a pure formula with no external rate, so the same inputs always give the same result.
Official sources and verification
- Method: NIST DLMF, checked 2026-07-09.
Reviewed by the CalculatorHub team, edited by James Graham, 2026-07-09. See our methodology. General information, not professional advice.