The Number e: Continuous Growth

The number e is a special constant, about 2.71828, that shows up whenever something grows continuously. It is the value that $\left(1 + \frac{1}{n}\right)^n$ gets closer to as $n$ gets very large. It is used in continuous compound interest, $A = Pe^{rt}$ , and in growth and decay models of the form $y = ae^{kt}$ .

Before this lesson: You should know Exponential Functions, Exponential Growth and Decay, Compound Interest and Exponent Rules.

Where e Comes From

Imagine putting $1 in an account that pays 100% interest per year. Compounded once a year, you end with $2. Compounded twice, each half-year pays 50%, so you get $1.5^2 = 2.25$ . Compounding more often helps, but by less and less each time.

Times compounded per year ( $n$ )	Amount $\left(1 + \frac{1}{n}\right)^n$
1	2
2	2.25
4	2.4414
12 (monthly)	2.6130
365 (daily)	2.7146
Continuously	$e \approx 2.71828$

The amounts level off at a number called $e$ . Like $\pi$ , it is irrational: its decimals never end or repeat.

Graph of the amount (1 + 1/n)^n for n = 1, 2, 4, 12 and 365. The points rise from 2 to 2.25, 2.44, 2.61 and 2.71, flattening out just below a dashed horizontal line labeled e ≈ 2.718. — Compounding more often helps less and less. The amounts level off at e ≈ 2.718.

Formulas with e

A = Pe^{rt} \qquad y = ae^{kt}

In $A = Pe^{rt}$ , $P$ is the starting amount, $r$ is the yearly rate as a decimal, and $t$ is the time in years. In $y = ae^{kt}$ , the model shows growth when $k \gt 0$ and decay when $k \lt 0$ .

Calculator note: on a TI-84, press 2nd, then LN, to get $e^x$ . On most other calculators, look for an $e^x$ key. Round money to the nearest cent and other answers to the nearest hundredth.

Worked Examples

Example 1: Evaluate powers of e

Evaluate (a) $e^3$ and (b) $e^{-0.5}$ to the nearest hundredth.

(a) Enter $e^3$ on a calculator: $20.0855\ldots$
Round to the nearest hundredth: $20.09$
(b) Enter $e^{-0.5}$ : $0.6065\ldots$
Round: $0.61$ . A negative exponent gives a number less than 1 (see zero and negative exponents).

Answer: (a) about 20.09 (b) about 0.61

Example 2: Simplify with exponent rules

Simplify (a) $e^2 \cdot e^3$ , (b) $\left(e^2\right)^3$ and (c) $\frac{e^5}{e^2}$ .

(a) Same base, so add the exponents (see exponent rules): $e^{2 + 3} = e^5$
(b) Power of a power, so multiply the exponents: $e^{2 \cdot 3} = e^6$
(c) Same base, so subtract the exponents: $e^{5 - 2} = e^3$

Answer: (a) $e^5$ (b) $e^6$ (c) $e^3$ . The letter $e$ is just a number, so every exponent rule works.

Example 3: Continuous compound interest

You invest $2,000 at 5% interest compounded continuously. How much is in the account after 10 years?

Write the formula: $A = Pe^{rt}$
Substitute $P = 2000$ , $r = 0.05$ and $t = 10$ : $A = 2000e^{0.05 \cdot 10}$
Multiply in the exponent: $A = 2000e^{0.5}$
Evaluate with a calculator and round to the nearest cent: $A \approx 3297.44$

Answer: about $3,297.44

Example 4: Continuous versus monthly

How much more does Example 3 earn than monthly compounding at the same rate (see compound interest)?

Write the monthly formula: $A = P\left(1 + \frac{r}{n}\right)^{nt}$
Substitute $n = 12$ : $A = 2000\left(1 + \frac{0.05}{12}\right)^{120}$
Evaluate and round: $A \approx 3294.02$
Subtract: $3297.44 - 3294.02 = 3.42$

Answer: Continuous compounding earns only about $3.42 more. That matches the table: compounding more often helps less and less.

Example 5: Word problem with decay

The amount of a medicine in a patient’s blood is modeled by $A = 50e^{-0.2t}$ milligrams, where $t$ is hours after the dose. Is this growth or decay? How much is left after 3 hours?

The exponent’s coefficient is −0.2, which is negative, so this is decay (see exponential growth and decay).
Substitute $t = 3$ : $A = 50e^{-0.2 \cdot 3}$
Multiply in the exponent: $A = 50e^{-0.6}$
Evaluate and round to the nearest hundredth: $A \approx 27.44$

Answer: Decay; about 27.44 mg remain after 3 hours.

Common Mistakes

Mistake	Example of the error	Fix
Treating e as a variable	Trying to solve for $e$	$e$ is a constant, about 2.71828, like $\pi$ .
Using the percent, not the decimal	$A = 2000e^{5 \cdot 10}$	Write 5% as 0.05: $2000e^{0.5}$ .
Multiplying instead of using the exponent	$e^{0.5}$ typed as $e \times 0.5$	Use the $e^x$ key so 0.5 is the exponent.
Rounding too early	Rounding $e^{0.5}$ to 1.6 first	Keep full calculator precision until the last step.
Mixing up growth and decay	$50e^{-0.2t}$ is growth	A negative $k$ means decay.

Practice Problems

Evaluate $e^4$ to the nearest hundredth.
Evaluate $e^{-1}$ to the nearest hundredth.
Simplify $e^3 \cdot e^4$ .
Simplify $\left(e^{2x}\right)^3$ .
Simplify $\frac{e^7}{e^2}$ .
Does $y = 3e^{0.4t}$ show growth or decay? What about $y = 10e^{-1.2t}$ ?
Find the value of $1,000 invested at 4% compounded continuously for 5 years.
Find the value of $5,000 invested at 3% compounded continuously for 20 years.
A town’s population is modeled by $P = 1200e^{0.03t}$ , where $t$ is years from now. Estimate the population in 10 years.
Compare $1,000 at 6% for 2 years compounded once a year and compounded continuously. How much more does continuous compounding earn?

Show answers

Calculator: $e^4 = 54.598\ldots$
Round: about 54.60
Calculator: $e^{-1} = 0.3678\ldots$
Round: about 0.37
Add the exponents: $e^{3 + 4}$
Answer: $e^7$
Multiply the exponents: $e^{2x \cdot 3}$
Answer: $e^{6x}$
Subtract the exponents: $e^{7 - 2}$
Answer: $e^5$
First: $k = 0.4 \gt 0$ , so growth
Second: $k = -1.2 \lt 0$ , so decay
Substitute: $A = 1000e^{0.04 \cdot 5}$
Multiply the exponent: $A = 1000e^{0.2}$
Evaluate: $A \approx 1221.40$
Answer: about $1,221.40
Substitute: $A = 5000e^{0.03 \cdot 20}$
Multiply the exponent: $A = 5000e^{0.6}$
Evaluate: $A \approx 9110.59$
Answer: about $9,110.59
Substitute: $P = 1200e^{0.03 \cdot 10}$
Multiply the exponent: $P = 1200e^{0.3}$
Evaluate: $P \approx 1619.83$
Answer: about 1,620 people
Yearly: $1000(1.06)^2$
Square: $1000(1.1236)$
Multiply: $1123.60$
Continuous: $1000e^{0.06 \cdot 2}$
Multiply the exponent: $1000e^{0.12}$
Evaluate: about $1127.50$
Subtract: $1127.50 - 1123.60 = 3.90$
Answer: about $3.90 more

FAQ

What is the number e?

It is an irrational constant, about 2.71828. It is the limit of $\left(1 + \frac{1}{n}\right)^n$ as $n$ grows, and it is the base used for continuous growth.

What is the formula for continuous compound interest?

$A = Pe^{rt}$ , where $P$ is the principal, $r$ is the rate as a decimal, and $t$ is the time in years.

Why is e used instead of other bases?

Growth that happens at every instant leads to $e$ naturally. It also makes calculus formulas simpler, which is why scientists use it so often.

How do you type e on a calculator?

On a TI-84, press 2nd, then LN, to get $e^x$ . Type the exponent and close the parenthesis. For $e$ itself, enter $e^1$ .

Summary

$e \approx 2.71828$ is the limit of $\left(1 + \frac{1}{n}\right)^n$ .
Continuous compounding: $A = Pe^{rt}$ .
$y = ae^{kt}$ shows growth when $k \gt 0$ and decay when $k \lt 0$ .
All exponent rules work with base $e$ .
Use the $e^x$ key and round only at the end.

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