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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.