# # E (Mathematical Constant)

## # Colorized Definition

```
\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.