Equations · Algebra and geometry
Euler's identity
One equation that links the five most important constants in mathematics: e, i, π, 1 and 0.
e^(iπ) + 1 = 0
Aethel says
e to the power of i pi, plus 1 equals 0
What it says
Raising e to the power of i times pi gives exactly minus one, so adding one gives zero. It is Euler's formula evaluated at an angle of pi, half a turn around the unit circle.
It brings together analysis (e), algebra (i, the square root of minus one), geometry (π) and arithmetic (0 and 1) in a single line.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| e | e | Euler's number, about 2.71828, the base of the natural logarithm |
| i | i | the imaginary unit, whose square is −1 |
| π | pi | the ratio of a circle's circumference to its diameter, about 3.14159 |
More onπPi
Where you meet it
Complex analysis and courses on complex numbers.
Questions
How do you say Euler's identity out loud?
e^(iπ) + 1 = 0 is read: “e to the power of i pi, plus 1 equals 0.” That is exactly what Aethel says when it meets the formula in a book.
What does e mean in Euler's identity?
e (said “e”) is Euler's number, about 2.71828, the base of the natural logarithm.
What does i mean in Euler's identity?
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.