2.7182818285
0.3678794412
1.7182818285
171.828183
%
2.7083333333
0.0099484951
2.7182815256
0.0000003029
2.7182818285
2.7182818285
0.3678794412
1.7182818285
171.828183
%
2.7083333333
0.0099484951
2.7182815256
0.0000003029
2.7182818285
The e Calculator computes $$e^x$$ (the natural exponential function) for any real number $$x$$, where $$e \approx 2.718281828$$ is Euler's number. This calculator provides the exact value, verification via $$\ln(e^x) = x$$, the reciprocal $$e^{-x}$$, and Taylor series approximations to illustrate how the function is computed.
The exponential function $$e^x$$ is arguably the most important function in mathematics. It is its own derivative, appears in compound interest, probability distributions (normal, Poisson), differential equations, quantum mechanics, and signal processing. Understanding $$e^x$$ is foundational to calculus and applied mathematics.
Euler's number $$e$$ is defined as:
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828459045$$
The exponential function $$e^x$$ can be computed using the Taylor series:
$$e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$
This series converges for all real $$x$$. The calculator shows 5-term and 10-term partial sums to demonstrate convergence. The unique property of $$e^x$$ is that its derivative equals itself:
$$\frac{d}{dx}e^x = e^x$$
The reciprocal relationship $$\frac{1}{e^x} = e^{-x}$$ connects growth and decay. The natural logarithm $$\ln$$ is the inverse: $$\ln(e^x) = x$$.
e^x is the primary result. The Verification confirms that $$\ln(e^x) = x$$. The Reciprocal $$e^{-x}$$ represents the corresponding decay factor. The Derivative equals $$e^x$$ itself — this unique self-replicating property is why $$e^x$$ is fundamental to calculus. The Series approximations show how accurately 5 and 10 terms of the Taylor expansion approximate the true value.
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e^1 = e ≈ 2.71828. The 5-term series gives 2.7083 (99.6% accurate), the 10-term series gives 2.71828153 (99.99999% accurate).
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e^(-2) ≈ 0.1353. This is 1/e² ≈ 1/7.389. The series converges more slowly for negative values.
$$e \approx 2.71828$$ is an irrational, transcendental number. It is the base of natural logarithms and arises naturally as $$\lim_{n \to \infty}(1 + 1/n)^n$$. It appears whenever growth rates are proportional to current values.
This is the defining property of $$e$$. It is the unique base for which $$\frac{d}{dx}b^x = b^x$$. For any other base, $$\frac{d}{dx}b^x = b^x \ln(b)$$, which equals $$b^x$$ only when $$\ln(b) = 1$$, i.e., $$b = e$$.
The series $$e^x = 1 + x + x^2/2! + x^3/3! + \cdots$$ sums increasingly smaller terms. Each term $$x^k/k!$$ shrinks because $$k!$$ grows much faster than $$x^k$$. For $$x$$ near 0, just a few terms give excellent accuracy.
They are inverse functions: $$\ln(e^x) = x$$ and $$e^{\ln(x)} = x$$. The natural log "undoes" the exponential and vice versa. On a graph, they are reflections across the line $$y = x$$.
Continuous compounding uses $$A = Pe^{rt}$$. As compounding frequency approaches infinity (daily → hourly → continuous), the growth formula converges to $$e^{rt}$$. This is why $$e$$ naturally appears in financial mathematics.
This is Euler's identity, often called the most beautiful equation in mathematics. It connects five fundamental constants: $$e$$, $$i$$ (imaginary unit), $$\pi$$, 1, and 0. It follows from Euler's formula $$e^{i\theta} = \cos\theta + i\sin\theta$$ with $$\theta = \pi$$.
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