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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. , the natural logarithm , 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:
• (this is the formal definition)
• • • • • • • • • Continued fraction: e = [2;1,2,1,1,4,1,1,6,1,1,8,...,1,1,2k,...]
• Infinite series: • ## 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:

• (most useful in calculus)
• is that function such that (useful in differential equations)
• • • • • • • • (Euler's formula, angle is to be measured in radians)
• 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 times the initial principal, leave it in there for years. Intuitively, compounding an initial account will yield e times the initial principal after one year.

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