Teacher Guide: The Natural Base e

Discover where the number e comes from, why it is about 2.71828, and how the natural exponential function y = e^x behaves.

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10 questions on Exponential Functions. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Explain how the number e arises as the limit of (1 + 1/n)^n
Compute compound interest for different compounding frequencies and describe the limit they approach
Evaluate expressions with e using the approximation e ≈ 2.71828
Describe the graph of y = e^x and compare it with y = 2^x and y = 3^x
Recognize that e is irrational and use approximations correctly

Prerequisites

• Introduction to Exponential Functions: the form f(x) = a · b^x, initial value and base
• Exponential Growth: growth factor and the formula A(t) = A_0(1 + r)^t
• Compound interest formula A = P(1 + r/n)^(nt)
• Negative and zero exponents

Discussion Starters

1. If interest were added every second, would you become infinitely rich? Why or why not?
2. Why do you think (1 + 1/n)^n does not simply shrink to 1 even though 1 + 1/n gets closer to 1?
3. Both e and pi are irrational constants. What other famous constants do you know, and where do they come from?
4. Without a calculator, how could you show that e^x must lie between 2^x and 3^x?

Common Misconceptions

The sequence (1 + 1/n)^n grows forever because the exponent grows forever

Remediation: Have students compute the values for n = 10, 100, 1000 and 1000000 and look at how little each new value adds. Ask whether a sum of ever smaller increases must become infinite.

e is an unknown to solve for, like x

Remediation: Compare e with pi: both are fixed numbers named with a letter because their decimals never end. Rewrite e^x as 2.71828^x in a few problems to make the point.

Continuous compounding gives a huge amount of extra money

Remediation: Show the table for 12% on 1000 dollars: daily gives 1127.47 dollars and continuous 1127.50 dollars. The difference is only a few cents.

Differentiation Ideas

For Struggling Students:

• Start with the table of (1 + 1/n)^n and let students fill in only n = 1 and n = 2 by hand
• Provide e ≈ 2.71828 and the values e^2 ≈ 7.389 and e^-1 ≈ 0.368 on a reference card
• Use the interest calculator to switch compounding periods instead of computing powers by hand

For On-Level Students:

• Compute compound interest for yearly, quarterly, monthly and daily compounding and compare
• Evaluate expressions like 3e^2 and e^0.5 with the calculator
• Sketch y = 2^x, y = e^x and y = 3^x on the same axes from a table of values

For Advanced Students:

• Explore the series e = 1 + 1 + 1/2 + 1/6 + 1/24 + ... and compare how fast it approaches e
• Investigate (1 + r/n)^n for r = 2 and r = 0.5 and guess the limit (e^r)
• Research why 1/e appears in the gift exchange (derangement) problem

Standards Alignment

HSF-IF.C.7e (CCSS.MATH.CONTENT.HSF.IF.C.7.E)
Graph exponential and logarithmic functions, showing intercepts and end behavior
HSF-LE.A.2 (CCSS.MATH.CONTENT.HSF.LE.A.2)
Construct linear and exponential functions given a graph, a description of a relationship, or two input-output pairs
HSA-SSE.B.3c (CCSS.MATH.CONTENT.HSA.SSE.B.3.C)
Use the properties of exponents to transform expressions for exponential functions

Lesson Resources

visualCompounding Frequency Explorer
Set 1000 dollars at 100% for one year and switch from yearly to daily compounding to watch the total approach 2718.28
activityHunting for e
Groups compute (1 + 1/n)^n for n = 1, 2, 10, 100, 1000, 1000000 and plot the results
worksheetEvaluating e^x
Practice evaluating expressions with e and comparing e^x with 2^x and 3^x

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The natural base

e

is a special number, about

2.71828

. It appears when you look at the expression

\left(1 + \frac{1}{n}\right)^n

for larger and larger values of

n

$n$	$\left(1 + \frac{1}{n}\right)^n$
$1$	$2$
$2$	$2.25$
$12$	$\approx 2.61304$
$365$	$\approx 2.71457$
$1\,000\,000$	$\approx 2.71828$

The values keep growing, but they never pass a certain limit. That limit is the number

e

e \approx 2.718281828\ldots

Key facts:

$e$ is irrational: its decimals never end and never repeat, so $2.71828$ is only an approximation
$e$ lies between $2$ and $3$ , so the graph of the natural exponential function $y = e^x$ lies between the graphs of $y = 2^x$ and $y = 3^x$
Like every function $y = b^x$ , it passes through $(0, 1)$ , because $e^0 = 1$
Using $e$ as the base of an exponential function is so common that $e^x$ is often called simply the exponential function

Worked Examples

Compute $\left(1 + \frac{1}{n}\right)^n$ for $n = 1$ , $n = 2$ and $n = 12$ . What do you notice?

Let $n = 1$

$\left(1 + \frac{1}{1}\right)^1 = 2^1 = 2$

Let $n = 2$

$\left(1 + \frac{1}{2}\right)^2 = 1.5^2 = 2.25$

Let $n = 12$ (use a calculator)

$\left(1 + \frac{1}{12}\right)^{12} \approx 1.083333^{12} \approx 2.61304$

Compare with larger $n$

$n = 365$ gives $\approx 2.71457$ , $n = 1\,000\,000$ gives $\approx 2.71828$ → The values grow more and more slowly

Answer: The results $2$ , $2.25$ , $2.61304$ , ... keep increasing, but they level off near $e \approx 2.71828$ and never go past it.

Common Mistakes

Thinking $\left(1 + \frac{1}{n}\right)^n$ grows without limit, or that it shrinks to $1$

Why it's wrong: Two effects compete: $1 + \frac{1}{n}$ gets closer to $1$ , but the exponent $n$ gets bigger. Neither effect wins completely.

Correct: Check with a table: $2$ , $2.25$ , $2.613$ , $2.7146$ , $2.71828$ . The values increase but level off at $e \approx 2.71828$ .

Treating $e$ as a variable, like $x$

Why it's wrong: Letters in algebra usually stand for unknowns, so students try to solve for $e$ or substitute other numbers for it.

Correct: $e$ is a fixed constant, just like $\pi$ . In $y = e^x$ the variable is $x$ ; $e \approx 2.71828$ never changes.

Computing $e^2$ as $2e \approx 5.437$

Why it's wrong: An exponent means repeated multiplication, not multiplication by the exponent.

Correct: $e^2 = e \cdot e \approx 2.71828 \cdot 2.71828 \approx 7.389$ .

Writing $e = 2.71828$ or $e = 2.72$ as if it were exact

Why it's wrong: $e$ is irrational, so no finite decimal and no fraction is exactly equal to it.

Correct: Write $e \approx 2.71828$ and round only at the end of a calculation, to the accuracy the question asks for.

Why It Matters

The number

e

shows up wherever something grows or shrinks in proportion to its current size:

Banking: when interest is added more and more often (monthly, daily, every second), the result approaches a limit built on $e$ . This is called continuous compounding
Biology: bacteria and cell cultures that grow without pauses are modeled with $e^x$
Physics and chemistry: radioactive decay, cooling and charging a capacitor all use powers of $e$
Probability: if many people draw names for a gift exchange, the chance that nobody draws their own name is about $\frac{1}{e} \approx 0.368$
Calculus: $e^x$ is the one exponential function whose rate of change at every point equals its own value, which makes it the natural choice for science

That is why scientific calculators have an

e^x

key, and why later lessons (continuous growth, population models) are written with base

e

Real World Applications

Banks and Compounding Frequency

Banks state an annual rate, but they may add interest yearly, monthly or daily. The more often interest is added, the closer the result gets to the continuous limit based on $e$.

Example:

At 12% per year, 1000 dollars grows to 1120.00 dollars with yearly compounding and to about 1127.50 dollars with continuous compounding ( $1000 \cdot e^{0.12}$ ).

1Try It Yourself

You deposit 1000 dollars at 12% per year, compounded monthly, for one year.

How much is in the account after one year, to the nearest cent?

Step 1: Write the mathematical expression

Calculate $1000 \cdot \left(1 + \frac{0.12}{12}\right)^{12}$ :

Growing Cell Cultures

Cells in a lab culture divide at all times, not once a day, so biologists describe their growth with base $e$: $N = N_0 e^{kt}$.

Example:

A culture of 200 cells modeled by $N = 200e^{t}$ ( $t$ in hours) has $200e^{1} \approx 544$ cells after 1 hour.

2Try It Yourself

A culture starts with 200 cells and is modeled by $N = 200e^{t}$ , with $t$ in hours.

About how many cells are there after 2 hours?

Step 1: Write the mathematical expression

Calculate $200 \cdot e^2$ :

Gift Exchanges and 1/e

In a gift exchange where everyone draws a name from a hat, the chance that nobody draws their own name is almost exactly $\frac{1}{e}$ once the group has more than a few people.

Example:

For a class of 25 students, the chance that nobody draws their own name is about $0.368$ , so a redraw is needed in about 63% of cases.

3Try It Yourself

A large group draws names for a gift exchange. The chance that nobody gets their own name is $\frac{1}{e}$ .

What is the chance, in percent, that at least one person draws their own name?

Step 1: Write the mathematical expression

Calculate $1 - \frac{1}{e}$ :

Key Takeaways

1As $n$ grows, $\left(1 + \frac{1}{n}\right)^n$ gets closer and closer to the number $e \approx 2.71828$
2 $e$ is irrational: $2.71828$ is an approximation, not the exact value
3Compounding interest more often (yearly, monthly, daily) gives more money, but the amount approaches a limit: continuous compounding
4The natural exponential function $y = e^x$ passes through $(0, 1)$ and has the x-axis as horizontal asymptote
5Because $2 < e < 3$ , the graph of $y = e^x$ lies between the graphs of $y = 2^x$ and $y = 3^x$
6Continuous growth uses $A = Pe^{rt}$ , which you will study in Continuous Growth and Decay

Frequently Asked Questions

Why is e called the natural base?

Because it appears on its own in nature and finance whenever growth happens all the time instead of in steps. In calculus,

e^x

is the only exponential function whose slope at every point equals its own value, which makes formulas with base

e

the simplest.

Who discovered e?

Jacob Bernoulli found it around 1683 while studying compound interest, exactly as in this lesson. Leonhard Euler later named it

e

, proved many of its properties and computed it to many decimals, which is why it is called Euler's number.

Is e the same as Euler's constant?

No. Euler's number

e \approx 2.71828

is the base of natural exponential functions. Euler's constant

\gamma \approx 0.5772

is a different number from number theory. Only

e

is used in this topic.

How do I use e on a calculator?

Most scientific calculators have an

e^x

key. The calculator in this lesson has

e

too: type e^0.5 for

e^{0.5}

, which gives about

1.6487

. Without an

e

key, type 2.71828 instead, for example 2.71828^0.5.

Glossary

Euler's number e: The irrational constant $e \approx 2.71828$ , the limit of $\left(1 + \frac{1}{n}\right)^n$ as $n$ grows
Natural exponential function: The function $f(x) = e^x$ , an exponential function with base $e$
Irrational number: A number that cannot be written as a fraction; its decimals never end and never repeat (examples: $e$ , $\pi$ , $\sqrt{2}$ )
Compounding frequency: How many times per year interest is added to the balance, written $n$ (yearly $n = 1$ , monthly $n = 12$ , daily $n = 365$ )
Continuous compounding: The limit of adding interest more and more often; the amount is $A = Pe^{rt}$

Formula Card

Definition of e

e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n \approx 2.71828

For very large $n$, $\left(1 + \frac{1}{n}\right)^n$ is very close to $e$.

Natural exponential function

f(x) = e^x

Passes through $(0, 1)$, always positive, horizontal asymptote $y = 0$, lies between $2^x$ and $3^x$.

Compound interest

A = P\left(1 + \frac{r}{n}\right)^{nt}

$P$ = principal, $r$ = annual rate as a decimal, $n$ = compoundings per year, $t$ = years.

Continuous compounding

A = Pe^{rt}

The limit of compound interest as $n$ grows without bound.

Teacher Guide: The Natural Base e

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Banks and Compounding Frequency

Growing Cell Cultures

Gift Exchanges and 1/e

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Definition of e

Natural exponential function

Compound interest

Continuous compounding

More in This Topic

Introduction to Exponential Functions

Exponential Growth

Graphing Exponential Functions

Exponential Decay