What is Euler's formula?
e^(iθ) = cos θ + i sin θ. It says that raising e to an imaginary power traces a circle in the complex plane.
// maths › Famous Equations
Euler's Formula
Euler's formula drawn as a helix that lands exactly on -1 at θ = π - Euler's identity, e^(iπ) + 1 = 0.
e^(iθ) = cos θ + i sin θ
A mind behind this: Leonhard Euler 1707–1783
Drag to rotate · scroll to zoom · right-drag to pan
Frequently asked questions
Why is the helix shape?
The angle θ runs along one axis while cosine and sine spin around it, so a circle stretched over time becomes a helix.
What is Euler's identity?
Setting θ = π gives e^(iπ) + 1 = 0, linking five fundamental constants - e, i, π, 1 and 0 - in one line.
Where is it used?
Electrical engineering, signal processing and physics all use it to handle waves as rotating arrows (phasors).
Do I need imaginary numbers for real life?
Yes, indirectly - the maths of AC electricity, radio and audio is far simpler written with i, even though the final answers are real.