*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 e^{t}, 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.