Enter an exponent x and this Euler's Number Calculator computes e^x — the value of Euler's number raised to your chosen power. You'll see the result alongside the constant e ≈ 2.71828, the natural log base used across mathematics, finance, and science. Adjust the number of terms to explore how the factorial series converges to e.
e^x
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Euler's Number (e)
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e Approximated via Series (n terms)
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ln(e^x) — should equal x
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Euler's number, denoted e, is a fundamental mathematical constant approximately equal to 2.71828182845904523536…. It is an irrational and transcendental number, meaning its decimal expansion is infinite and non-repeating. It arises naturally in problems involving continuous growth, compound interest, and calculus.
On most scientific calculators, e appears as the base of the natural exponential function. The 'e^x' or 'exp(x)' button raises e to any power you enter. The constant itself is accessible via a dedicated 'e' key, usually returning approximately 2.718281828.
You can approximate e^x using the Taylor series: e^x = 1 + x + x²/2! + x³/3! + x⁴/4! + …. Adding more terms gives a closer approximation. For example, e^1 ≈ 1 + 1 + 0.5 + 0.1667 + 0.0417 + … ≈ 2.71828. This calculator shows exactly how many terms are needed to converge.
'exp(x)' is simply shorthand for e^x — the exponential function with base e. When you see exp(2), it means e raised to the power 2, which equals approximately 7.389. It is used interchangeably with the e^x notation in mathematics and programming.
As x approaches negative infinity, e^x approaches 0. This is because raising any number greater than 1 to increasingly large negative exponents drives the result toward zero. In notation: lim(x→-∞) e^x = 0.
The derivative of e^x with respect to x is simply e^x itself — it is unchanged by differentiation. This unique property makes e^x the eigenfunction of the differentiation operator and is one reason Euler's number appears so frequently in differential equations and natural growth models.
Euler's number can be defined as the infinite sum: e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + …, where n! denotes the factorial of n. This series converges rapidly — just 20 terms are enough to match e to more than 18 decimal places.
Euler's identity states that e^(iπ) + 1 = 0, connecting five fundamental mathematical constants: e, i (imaginary unit), π, 1, and 0. It is widely regarded as the most beautiful equation in mathematics because it links seemingly unrelated areas — exponential functions, complex numbers, and geometry — in a single elegant statement.