z=r(\cos\theta+i\sin\theta)=re^{i\theta}Variables
- r: nonnegative modulus
- θ: argument in radians
- i: imaginary unit
How to use this formula
Expresses a complex number by magnitude and argument.
Important notes
- The argument is defined modulo 2π.
- At z=0 the argument is not unique.
Quick example
The number 1+i has r=√2 and θ=π/4.
Applicability, worked calculation, and verification
Domain and applicability
Apply Complex Number Polar Form 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 argument is defined modulo 2π.
- The angle unit and the domain or branch of any inverse trigonometric function must be explicit.
- For the Complex Number Polar Form, 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 Polar Form when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Complex Number Polar Form, check zero, negative, and extreme input values before relying on the result.
- When using Complex Number Polar Form, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form z=r(\cos\theta+i\sin\theta)=re^{i\theta} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
For z=1+i, r=√(1²+1²)=√2 and θ=atan2(1,1)=π/4.
- Compute the modulus r=√2.
- Determine the first-quadrant argument θ=π/4.
- Write z=√2(cosπ/4+i sinπ/4)=√2e^(iπ/4).
Independent verification
Converting back gives real and imaginary parts equal to 1.
Common mistakes
- Before substituting values into Complex Number Polar Form, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Complex Number Polar Form, 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 Polar Form 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=r(\cos\theta+i\sin\theta)=re^{i\theta}. - 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 Polar Form.
Verified against: OpenStax Algebra and Trigonometry
Automated quality check: Passed the core formula indexing gate.
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 Polar Form used for?
Expresses a complex number by magnitude and argument.
Can I copy this formula as LaTeX?
Yes. Copy z=r(\cos\theta+i\sin\theta)=re^{i\theta} 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.