Euler's Number (e)

Middle School

Definition

e is the base of natural logarithm, appearing naturally in continuous compound interest. It plays a central role in calculus.

e = lim_n → ∈fty (1 + (1)/(n))ⁿ ≈ 2.71828...

Limit definition of e

e = ∑_n=0^∈fty (1)/(n!) = 1 + 1 + (1)/(2) + (1)/(6) + ...

Series definition of e

(d)/(dx) e^x = e^x

Self-derivative property

Example 1

Deposit $100 at 100% annual rate compounded continuously for 1 year.

Discovered by: Leonhard Euler (1727)

Jacob Bernoulli found it in compound interest; Euler systematized it.

Continuous compounding, growth models

Radioactive decay, thermodynamics

Poisson distribution