|z|=\sqrt{a^2+b^2},\quad z=a+biVariables
- z: complex number
- a: real part
- b: imaginary part
How to use this formula
Calculates the distance of a complex number from the origin in the complex plane.
Important notes
- The modulus is always nonnegative.
- It obeys |zw|=|z||w|.
Quick example
For z=3+4i, |z|=5.
Applicability, worked calculation, and verification
Domain and applicability
Apply Complex Number Modulus only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- The modulus is always nonnegative.
- For a real-valued result, each even-root radicand must be nonnegative unless complex values are intended.
- For the Complex Number Modulus, all variables must lie in the stated domain, and every denominator or inverse operation must be defined.
Do not use this formula when
- Do not use Complex Number Modulus when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Complex Number Modulus, check zero, negative, and extreme input values before relying on the result.
- When using Complex Number Modulus, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form |z|=\sqrt{a^2+b^2},\quad z=a+bi for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For z=3+4i, |z|=√(3²+4²)=5.
- Identify the real part a=3 and imaginary coefficient b=4.
- Compute a²+b²=9+16=25.
- Take the nonnegative square root to obtain 5.
Independent verification
Multiplying z by its conjugate gives 25, so |z|²=25.
Common mistakes
- Do not drop the radical boundary in Complex Number Modulus; verify exactly which terms are inside the root and whether the chosen root is valid.
- For the Complex Number Modulus, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Complex Number Modulus in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
|z|=\sqrt{a^2+b^2},\quad z=a+bi. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Complex Number Modulus.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
Frequently asked questions
What is the Complex Number Modulus used for?
Calculates the distance of a complex number from the origin in the complex plane.
Can I copy this formula as LaTeX?
Yes. Copy |z|=\sqrt{a^2+b^2},\quad z=a+bi or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.