Euler's number


 * Not to be confused with Euler's constant. See E for other e's.

e is a number commonly used as base in logarithmic and exponential functions.

The letter e in mathematics usually stands for the so-called "natural base" for logarithms and exponential functions.
 * $$\log_{e}x \equiv \ln x$$, the natural logarithm
 * $$e^{x}$$, the exponential function with base e

Value
Euler's number is an irrational number (and a transcendental number), but it can be approximated as 2.71828 18284 59045 23536...


 * Decimal: 2.71828182845904523536... (non-repeating, non-terminating)
 * Limits:
 * $$\lim_{n \to \infty} (1 + {1\over n})^n = e$$ (this is the formal definition)
 * $$\lim_{n \to -\infty} (1 + {1\over n})^n = e$$
 * $$\lim_{n \to \pm\infty} (1 - {1\over n})^{-n} = e$$
 * $$\lim_{n \to \pm\infty} (1 - {1\over n})^n = \frac{1}{e}$$
 * $$\lim_{n \to \pm\infty} (1 + {1\over n})^{-n} = \frac{1}{e}$$
 * $$\lim_{n \to 0} (1 + n)^{1/n} = e$$
 * $$\lim_{n \to 0} (1 - n)^{-1/n} = e$$
 * $$\lim_{n \to 0} (1 - n)^{1/n} = \frac{1}{e}$$
 * $$\lim_{n \to 0} (1 + n)^{-1/n} = \frac{1}{e}$$
 * Continued fraction: e = [2;1,2,1,1,4,1,1,6,1,1,8,...,1,1,2k,...]
 * Infinite series: $$e = \sum_{n=0}^\infty \frac{1}{n!}$$
 * $$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$

Applications
Euler's number has many practical uses, particularly in higher level mathematics such as calculus, differential equations, discrete mathematics, trigonometry, complex analysis, statistics, among others.

Properties
The reason Euler's number is such an important constant is that is has unique properties that simplify many equations and patterns.

Some of the defining relationships include:


 * $$\frac{d}{dx} e^x = e^x$$ (most useful in calculus)
 * $$y = e^x$$ is that function such that $$y'=y$$ (useful in differential equations)
 * $$\frac{d}{dx} e^x |_{x=0} = 1$$
 * $$\int e^x\,dx = e^x + C$$
 * $$\int_{-\infty}^0 e^x\,dx = 1$$
 * $$\frac{d}{dx} \ln(x) = \frac{1}{x}$$
 * $$\frac{d}{dx} \ln(x) |_{x=1} = 1$$
 * $$\int \frac{1}{x} \,dx = \ln(x) + C$$
 * $$\int_{1}^{e} \frac{1}{x} \,dx = 1$$


 * $$\cos\theta + i \sin\theta = e^{i\theta}$$ (Euler's formula, angle $$\theta$$ is to be measured in radians)
 * $$\ln(-1) = i\pi$$

One of the original defining attributes of e is the fact any bank account having a 100% APR interest rate which is compounded continuously, will grow at the exponential rate et, where t is time in years, discovered by Jacob Bernoulli. To get $$x$$ times the initial principal, leave it in there for $$\ln x$$ years. Intuitively, compounding an initial account will yield e times the initial principal after one year.

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