The Euler’s Identity Calculator evaluates one of the most elegant and celebrated equations in mathematics: e^(iπ) + 1 = 0. This identity bridges five fundamental mathematical constants—e, i, π, 1, and 0—into a single expression that is both simple and deeply meaningful. The calculator confirms the result of Euler’s identity and supports learners, educators, and engineers in exploring its applications in fields such as complex analysis, signal processing, and electrical engineering.
The calculator not only verifies the identity but can also be extended to evaluate Euler’s formula e^(ix) = cos(x) + i·sin(x) for different values of x, providing insights into the relationships between exponential and trigonometric functions in the complex plane.
Formula of Euler’s Identity Calculator
Euler’s Identity:
e^(iπ) + 1 = 0
Where:
- e ≈ 2.71828 (Euler’s number, the base of the natural logarithm)
- i = √(−1) (the imaginary unit)
- π ≈ 3.14159 (pi, the ratio of a circle’s circumference to its diameter)
Origin from Euler’s Formula:
e^(ix) = cos(x) + i·sin(x)
Substitute x = π:
e^(iπ) = cos(π) + i·sin(π) = −1 + 0i = −1
So:
e^(iπ) + 1 = −1 + 1 = 0
This result is not only mathematically true but also demonstrates the deep unity among different branches of mathematics.
Helpful Reference Table
Here is a reference for evaluating Euler’s formula e^(ix) = cos(x) + i·sin(x) for various angles x (in radians):
| x (radians) | e^(ix) Expression | Result (Complex Form) |
|---|---|---|
| 0 | cos(0) + i·sin(0) | 1 + 0i |
| π/2 | cos(π/2) + i·sin(π/2) | 0 + i |
| π | cos(π) + i·sin(π) | −1 + 0i |
| 3π/2 | cos(3π/2) + i·sin(3π/2) | 0 − i |
| 2π | cos(2π) + i·sin(2π) | 1 + 0i |
This table illustrates how Euler’s formula traces the unit circle in the complex plane, making it a foundational concept in both theoretical and applied mathematics.
Example of Euler’s Identity Calculator
Let’s verify Euler’s identity step-by-step using Euler’s formula:
- Start with Euler’s formula:
e^(ix) = cos(x) + i·sin(x) - Set x = π:
e^(iπ) = cos(π) + i·sin(π)
e^(iπ) = −1 + 0i - Add 1:
e^(iπ) + 1 = −1 + 1 = 0
Result: Euler’s identity is verified:
e^(iπ) + 1 = 0
This confirms the elegant unification of exponential, trigonometric, and complex number concepts.
Most Common FAQs
Euler’s identity is celebrated because it connects five of the most important mathematical constants in a single, elegant equation. It shows the beauty and unity of mathematics by linking algebra, geometry, and complex numbers.
Yes. It is used in electrical engineering, quantum mechanics, signal processing, and vibration analysis, particularly in calculations involving sinusoidal signals and phasor analysis.
Euler’s formula e^(ix) = cos(x) + i·sin(x) is the general expression for complex exponentials. Euler’s identity is a specific case where x = π, yielding e^(iπ) + 1 = 0.