Master

e (the Number)

Master e (the number) with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

e (the Number)

Study Guide

Key Definition

The number, $e$, is an irrational number defined as the limit $\lim_{{x \to \infty}} \left(1 + \frac{1}{x}\right)^x$.

Important Notes

$e$ is approximately 2.71828
The function $f(x) = e^x$ has a tangent slope of 1 at $(0, 1)$
$e^x$ is the natural exponential function
Used in continuous growth models
Essential in calculus due to its derivative properties

Mathematical Notation

$e^x$ is the exponential function

$\lim$ represents the limit

$\ln(x)$ represents the natural logarithm

Remember to use proper notation when solving problems

Why It Works

The function $f(x) = e^x$ is unique because its rate of change, or derivative, is the same as the function itself.

Remember

The number $e$ is fundamental in calculus and continuous growth models.

Quick Reference

Limit Definition:$\lim_{{x \to \infty}} \left(1 + \frac{1}{x}\right)^x = e$

Understanding e (the Number)

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

The number $e$ is an irrational constant approximately equal to 2.71828. It arises naturally as the base of the exponential function $f(x)=e^x$. You can approximate it by computing $(1+1/n)^n$ for large values of $n$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the approximate value of $e$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A bank offers a principal of $P = 1000$ at a continuously compounded annual interest rate of $r = 0.05$. Calculate the amount $A$ after one year using $A = Pe^{rt}$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explore how the limit definition of $e$ can be derived from the expression $\lim_{{x \to \infty}} \left(1 + \frac{1}{x}\right)^x$.

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following is the derivative of $f(x) = e^x$?

Recap

Watch & Learn

Review key concepts and takeaways