Lesson 5.3

The Natural Base e

The number is irrational (like ) and appears naturally in continuous growth. It's the base that makes calculus elegant.

Introduction

What happens if you compound interest more and more frequently — hourly, every second, every nanosecond? As compounding becomes continuous, the growth factor converges to . This number is the foundation of all natural growth models.

Past Knowledge

Graphing (5.2), irrational numbers.

Today's Goal

Understand what is and graph .

Future Success

Continuous growth (5.5) uses , and (5.9) is the inverse of .

Key Concepts

Where Does e Come From?

Compound $1 at 100% interest, times per year. As , you get .

n	Result
1	2.000
12	2.613
365	2.7146
∞	2.71828...

Graph of

Same shape as but slightly steeper. Behaves between and .

Worked Examples

Example 1: Calculator Evaluation

Basic

Evaluate and .

Use the button on your calculator (usually 2nd + LN)

Example 2: Comparing Bases

Visual

Compare , , and .

(green) sits between (blue) and (orange)

Example 3: Transformed e^x

Advanced

Graph and identify the asymptote.

Identify each transformation

The negative exponent reflects across the y-axis (decay). The 3 stretches vertically ×3. The +1 shifts up 1.

Key point

y-intercept:

Asymptote: · y-intercept:

Common Pitfalls

e Is Not a Variable

is a fixed constant ≈ 2.71828. It's not a variable — it's a specific number like .

e ≠ 2.72

is irrational — its decimal never terminates or repeats. Use the exact symbol whenever possible; approximate only at the final step.

Real-Life Applications

The number appears everywhere: radioactive decay, population modeling, compound interest, probability, and even the bell curve. In calculus, — it's the only function that is its own derivative.