e^{i\pi}+1=0Variables
- e: Euler number
- i: imaginary unit
- π: circle constant
How to use this formula
Connects the constants e, i, π, one, and zero in a single complex exponential identity.
Important notes
- It follows from Euler’s formula e^{ix}=cos x+i sin x.
- The exponent uses radians.
Quick example
Setting x=π gives cos π+i sin π=-1.
Applicability, worked calculation, and verification
Domain and applicability
Apply Euler Identity only when the listed variables are defined, denominators are nonzero, and the problem satisfies the assumptions stated on this page.
Assumptions and domain checks
- It follows from Euler’s formula e^{ix}=cos x+i sin x.
- For the Euler Identity, 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 Euler Identity when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Euler Identity, check zero, negative, and extreme input values before relying on the result.
- When using Euler Identity, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form e^{i\pi}+1=0 for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Euler’s formula at x=π gives e^(iπ)=cosπ+i sinπ=-1, hence e^(iπ)+1=0.
- Substitute x=π into e^(ix)=cosx+i sinx.
- Use cosπ=-1 and sinπ=0.
- Add 1 to both sides to obtain zero.
Independent verification
The real part is -1 and the imaginary part is 0, matching the unit-circle point at angle π.
Common mistakes
- Before substituting values into Euler Identity, map each variable to its definition and preserve every sign, exponent, subscript, and grouping mark.
- For the Euler Identity, check the final value in the original equation; algebraic rearrangement can introduce or lose solutions when domains are restricted.
Continue the workflow
Use Euler Identity 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
e^{i\pi}+1=0. - 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 Euler Identity.
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 Euler Identity used for?
Connects the constants e, i, π, one, and zero in a single complex exponential identity.
Can I copy this formula as LaTeX?
Yes. Copy e^{i\pi}+1=0 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.