Euler's Number Calculator

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. Also try the Gamma Function Calculator.

Enter any real number x to compute e^x

How many terms of the factorial series to sum when approximating e

Results

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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Results Table

Frequently Asked Questions

What is Euler's number e?

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. See also our calculate Complex Number to Rectangular Form Rectangular Form (a + bi).

What is e on a calculator?

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.

How do you calculate e to the power x without a calculator?

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.

What does 'exp' mean on a calculator?

'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. You might also find our find erf(x) with Error Function Calculator (erf) useful.

What is e to the negative infinity?

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.

What is the derivative of e to the x?

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.

How is e defined using a factorial series?

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.

What is Euler's identity and why is it famous?

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.