Euler's Identity and Formula
Euler's identity e^(iπ) + 1 = 0 and the general formula e^(ix) = cos(x) + i·sin(x) explained with examples.
The Identity
Euler's identity is often called the most beautiful equation in mathematics. It links five fundamental constants in a single, elegant statement: e (the base of natural logarithms), i (the imaginary unit), π (the ratio of a circle's circumference to its diameter), 1 (the multiplicative identity), and 0 (the additive identity).
The General Formula
Euler's formula connects the complex exponential function to trigonometry. When you substitute x = π, you get cos(π) + i · sin(π) = −1 + 0 = −1, which leads directly to the identity above.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| e | Euler's number (≈ 2.71828) | dimensionless |
| i | Imaginary unit (√−1) | dimensionless |
| x | Angle in the complex plane | radians |
| π | Pi (≈ 3.14159) | radians |
Example 1
Evaluate eiπ/2 using Euler's formula
Apply eix = cos(x) + i · sin(x) with x = π/2
cos(π/2) = 0
sin(π/2) = 1
eiπ/2 = 0 + i · 1 = i
Example 2
Evaluate eiπ/3
Apply eix = cos(x) + i · sin(x) with x = π/3
cos(π/3) = 1/2 = 0.5
sin(π/3) = √3/2 ≈ 0.8660
eiπ/3 = 0.5 + 0.8660i
Example 3
Verify Euler's identity by substituting x = π
eiπ = cos(π) + i · sin(π)
cos(π) = −1, sin(π) = 0
eiπ = −1 + 0i = −1
eiπ + 1 = −1 + 1 = 0 ✓
When to Use It
- Converting between polar and rectangular forms of complex numbers
- Solving differential equations involving oscillations and waves
- Signal processing and Fourier analysis
- Electrical engineering for AC circuit analysis using phasors
- Quantum mechanics where complex exponentials describe wave functions