Equations · Algebra and geometry
Euler's formula
The complex exponential written as a point on the unit circle, joining exponentials and trigonometry.
e^(iθ) = cos θ + i sin θ
Aethel says
e to the power of i theta, equals cos theta plus i sin theta
What it says
As θ grows, e to the power of i theta travels around the circle of radius one in the complex plane: its real part is the cosine of θ and its imaginary part the sine.
This is why oscillations and waves are written as complex exponentials in physics and engineering: multiplying exponentials is easier than combining sines and cosines.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| e | e | Euler's number, about 2.71828 |
| i | i | the imaginary unit, whose square is −1 |
| θ | theta | an angle, in radians |
More onθTheta
Where you meet it
Complex numbers, signal processing, alternating-current circuits and quantum mechanics.
Questions
How do you say Euler's formula out loud?
e^(iθ) = cos θ + i sin θ is read: “e to the power of i theta, equals cos theta plus i sin theta.” That is exactly what Aethel says when it meets the formula in a book.
What does e mean in Euler's formula?
e (said “e”) is Euler's number, about 2.71828.
What does i mean in Euler's formula?
i (said “i”) is the imaginary unit, whose square is −1.
Can I listen to the equations in my own textbook like this?
Yes. Upload a PDF, EPUB or text file and Aethel reads it aloud with every equation spoken as maths, the way it reads this one, and highlights each sentence as it goes. You can start free.