- , the natural logarithm
- , the exponential function with base e
- Decimal: 2.71828182845904523536... (non-repeating, non-terminating)
- (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:
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.
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)
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.