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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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All practice problems on paper, with a separate answer key.

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Class quiz

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 ee is a special number, about 2.718282.71828. It appears when you look at the expression
(1+1n)n\left(1 + \frac{1}{n}\right)^n
for larger and larger values of nn:
nn(1+1n)n\left(1 + \frac{1}{n}\right)^n
1122
222.252.25
1212≈2.61304\approx 2.61304
365365≈2.71457\approx 2.71457
1 000 0001\,000\,000≈2.71828\approx 2.71828
The values keep growing, but they never pass a certain limit. That limit is the number ee:
e≈2.718281828…e \approx 2.718281828\ldots
Key facts:
  • ee is irrational: its decimals never end and never repeat, so 2.718282.71828 is only an approximation
  • ee lies between 22 and 33, so the graph of the natural exponential function y=exy = e^x lies between the graphs of y=2xy = 2^x and y=3xy = 3^x
  • Like every function y=bxy = b^x, it passes through (0,1)(0, 1), because e0=1e^0 = 1
  • Using ee as the base of an exponential function is so common that exe^x is often called simply the exponential function

Worked Examples

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

Let n=1n = 1

(1+11)1=21=2\left(1 + \frac{1}{1}\right)^1 = 2^1 = 2
2

Let n=2n = 2

(1+12)2=1.52=2.25\left(1 + \frac{1}{2}\right)^2 = 1.5^2 = 2.25
3

Let n=12n = 12 (use a calculator)

(1+112)12≈1.08333312≈2.61304\left(1 + \frac{1}{12}\right)^{12} \approx 1.083333^{12} \approx 2.61304
4

Compare with larger nn

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

Common Mistakes

Thinking (1+1n)n\left(1 + \frac{1}{n}\right)^n grows without limit, or that it shrinks to 11

Why it's wrong: Two effects compete: 1+1n1 + \frac{1}{n} gets closer to 11, but the exponent nn gets bigger. Neither effect wins completely.

Correct: Check with a table: 22, 2.252.25, 2.6132.613, 2.71462.7146, 2.718282.71828. The values increase but level off at e≈2.71828e \approx 2.71828.

Treating ee as a variable, like xx

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

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

Computing e2e^2 as 2e≈5.4372e \approx 5.437

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

Correct: e2=e⋅e≈2.71828⋅2.71828≈7.389e^2 = e \cdot e \approx 2.71828 \cdot 2.71828 \approx 7.389.

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

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

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

Why It Matters

The number ee 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 ee. This is called continuous compounding
  • Biology: bacteria and cell cultures that grow without pauses are modeled with exe^x
  • Physics and chemistry: radioactive decay, cooling and charging a capacitor all use powers of ee
  • Probability: if many people draw names for a gift exchange, the chance that nobody draws their own name is about 1e≈0.368\frac{1}{e} \approx 0.368
  • Calculus: exe^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 exe^x key, and why later lessons (continuous growth, population models) are written with base ee.

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⋅e0.121000 \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⋅(1+0.1212)121000 \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=200etN = 200e^{t} (tt in hours) has 200e1≈544200e^{1} \approx 544 cells after 1 hour.

2Try It Yourself

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

About how many cells are there after 2 hours?

Step 1: Write the mathematical expression

Calculate 200⋅e2200 \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 0.3680.368, 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 1e\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 1−1e1 - \frac{1}{e}:

Key Takeaways

  • 1As nn grows, (1+1n)n\left(1 + \frac{1}{n}\right)^n gets closer and closer to the number e≈2.71828e \approx 2.71828
  • 2ee is irrational: 2.718282.71828 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=exy = e^x passes through (0,1)(0, 1) and has the x-axis as horizontal asymptote
  • 5Because 2<e<32 < e < 3, the graph of y=exy = e^x lies between the graphs of y=2xy = 2^x and y=3xy = 3^x
  • 6Continuous growth uses A=PertA = 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, exe^x is the only exponential function whose slope at every point equals its own value, which makes formulas with base ee 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 ee, 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≈2.71828e \approx 2.71828 is the base of natural exponential functions. Euler's constant γ≈0.5772\gamma \approx 0.5772 is a different number from number theory. Only ee is used in this topic.

How do I use e on a calculator?

Most scientific calculators have an exe^x key. The calculator in this lesson has ee too: type e^0.5 for e0.5e^{0.5}, which gives about 1.64871.6487. Without an ee key, type 2.71828 instead, for example 2.71828^0.5.

Glossary

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

Formula Card

Definition of e

e=lim⁡n→∞(1+1n)n≈2.71828e = \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)=exf(x) = e^x

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

Compound interest

A=P(1+rn)ntA = 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=PertA = Pe^{rt}

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

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