Euler's Identity and Formula
Reference for Euler's identity e^(i*pi)+1=0 and Euler's formula e^(ix)=cos(x)+i*sin(x).
Explains the link between complex exponentials and trigonometry.
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
Key Notes
- Euler's identity: e^(iπ) + 1 = 0: A special case of Euler's formula e^(iθ) = cosθ + i sinθ evaluated at θ = π. Since cos π = −1 and sin π = 0: e^(iπ) = −1 + 0i = −1. Adding 1 to both sides gives the celebrated form.
- Five fundamental constants in one equation: e (Euler's number, base of natural logarithm ≈ 2.718), i (imaginary unit √−1), π (ratio of circumference to diameter ≈ 3.14159), 1 (multiplicative identity), and 0 (additive identity). Their appearance together is non-trivial — it reveals deep structure connecting analysis, algebra, and geometry.
- Derived from Taylor series: eˣ = 1 + x + x²/2! + x³/3! + …; substituting x = iπ and using the fact that i² = −1, i³ = −i, i⁴ = 1 (cycle of period 4) yields the real part cosπ = −1 and imaginary part sinπ = 0.
- Beyond beauty — practical importance of Euler's formula: The underlying e^(iθ) = cosθ + i sinθ allows all oscillatory phenomena (waves, AC circuits, quantum states) to be written compactly as complex exponentials. Differentiation and integration of e^(iθ) are far simpler than directly manipulating sines and cosines.
- Applications: Euler's identity and the broader Euler's formula are foundational in signal processing (Fourier transforms decompose signals into e^(iωt) terms), quantum mechanics (wave functions), electrical engineering (phasor notation), and number theory (connecting the distribution of prime numbers to the complex zeros of the Riemann zeta function).