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# E (Mathematical Constant)

# Colorized Definition

number_e

\newcommand{\growth}{\color{c1}}
\newcommand{\unitQuantity}{\color{c2}}
\newcommand{\unitInterest}{\color{c3}}
\newcommand{\unitTime}{\color{c4}}
\newcommand{\perfectly}{\color{c5}}
\newcommand{\compounded}{\color{c6}}

$$\growth e
\plain =
\perfectly \lim_{n\to\infty}
\plain \left(
\unitQuantity 1 + \unitInterest \frac{1}{\compounded n}
\plain \right)
\unitTime^{1 \cdot \compounded n}
$$

\growth       The base for continuous growth
\plain        is
\\
\unitQuantity the unit quantity 
\unitInterest earning unit interest
\unitTime     for unit time, 
\\
\compounded   compounded
\perfectly    as fast as possible

# Plain English

  • What is e? A constant (2.718...) representing continuous unit growth: the unit quantity (1.0), continuously growing the unit rate (100%), for unit time (1 period).

  • Why's e special? All circles are the unit circle, scaled up. All continuously growing systems are e^{rt}, scaled to some rate and time.

  • When should I use e? Use e^{rt} for things that change constantly (radioactive decay, populations). For growth based on discrete intervals (interest payments, combinatorics), (1 + rate)^{time} is a better model.

# Read More

← E (Mathematical Constant 2) →