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Eulers Number

Q: What is the value of e to 10 decimal places?

To 10 decimal places, the value of Euler's number is $2.7182818284$. It's easy to remember the first few digits with the mnemonic "2.7 1828 1828," which looks like it repeats, but it doesn't.

Q: What is the difference between log(x) and ln(x)?

The only difference is the base. The common logarithm, $\log(x)$, assumes a base of $10$. The natural logarithm, $\ln(x)$, uses a base of $e$. You use $\ln(x)$ to solve equations involving $e$.

Q: Is it okay to just use 2.72 for e in my calculations?

For a quick estimate, $2.72$ is fine. However, for homework or test problems, you should always use the $e^x$ button on your calculator. Using a rounded value early in a problem can make your final answer inaccurate.

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Have you ever met a number as famous as $π$ ? Meet Euler's number, $e$ ! It’s a special mathematical constant, approximately $2.718$ , that pops up everywhere from calculating bank interest to describing population growth. It’s the secret ingredient behind anything that grows or decays continuously.

Eulers Number — an original Algebra911 reference diagram defining eulers number with its key formula and a worked example. — Euler's Number (e): A Guide to the Constant of Natural Growth

What is Euler's Number ( $e$ )?

Euler's number, denoted by the letter $e$ , is a fundamental mathematical constant approximately equal to $2.71828$ . It is an irrational number, which means its decimal representation goes on forever without repeating, just like its famous cousin, $π$ (pi). While $π$ is the star of circles and geometry, $e$ is the champion of growth, change, and processes that happen continuously.

This special number is named after the brilliant Swiss mathematician Leonhard Euler, who made extensive discoveries about its properties in the 18th century. You'll find $e$ appearing in finance when calculating interest, in biology when modeling population growth, in physics to describe radioactive decay, and even in probability. It's a cornerstone of calculus and is often called the "natural" number because it describes so many phenomena in the natural world.

Where Does $e$ Come From? The Idea of Continuous Growth

To truly understand $e$ , let's explore a story about money and growth. Imagine you have exactly $$ 1$ to invest for one year in a bank that offers an incredible $100 %$ annual interest rate. How much money you'll have at the end of the year depends on how often the bank calculates and adds the interest to your account—a process called compounding.

Let's look at a few scenarios:

Compounded Annually: The bank calculates your $100 %$ interest just once at the end of the year. You earn $$ 1 \times 100 % = $ 1$ in interest. Your total is $$ 1 + $ 1 = $ 2$ .
Compounded Semi-Annually (Twice a year): The bank gives you half the interest ( $50 %$ ) two times. After 6 months, you have $$ 1 \times (1 + 0.50) = $ 1.50$ . For the next 6 months, you earn $50 %$ on this new amount: $$ 1.50 \times (1 + 0.50) = $ 2.25$ .
Compounded Quarterly (Four times a year): You get one-quarter of the interest ( $25 %$ ) four times. This would be $$ 1 \times (1 + 0.25)^{4} \approx $ 2.44$ .

Notice a pattern? The more frequently the interest is compounded, the more money you end up with. But does it grow infinitely? Let's organize this in a table and see what happens as we increase the number of compounding periods, which we'll call $n$ .

Compounding Frequency	$n$ (Periods per Year)	Calculation	Final Amount
Annually	$1$	$(1 + \frac{1}{1})^{1}$	$$ 2.00$
Semi-Annually	$2$	$(1 + \frac{1}{2})^{2}$	$$ 2.25$
Quarterly	$4$	$(1 + \frac{1}{4})^{4}$	$$ 2.4414 . . .$
Monthly	$12$	$(1 + \frac{1}{12})^{12}$	$$ 2.6130 . . .$
Daily	$365$	$(1 + \frac{1}{365})^{365}$	$$ 2.7145 . . .$
Hourly	$8, 760$	$(1 + \frac{1}{8760})^{8760}$	$$ 2.7181 . . .$
Every Second	$31, 536, 000$	$(1 + \frac{1}{31536000})^{31536000}$	$$ 2.71828 . . .$

As we compound more and more frequently—approaching an infinite number of tiny moments—the final amount doesn't shoot off to infinity. Instead, it gets closer and closer to a specific, magical number: $2.71828 . . .$ . This number is Euler's number, $e$ . This process of compounding at every possible instant is called continuous growth, and $e$ is its fundamental constant.

How Do You Calculate with Continuous Growth?

The story of compounding interest reveals the origin of $e$ , but we need a more practical tool for everyday problems. Fortunately, there is a simple and powerful formula for calculating the final amount ( $A$ ) when an initial principal ( $P$ ) grows continuously.

A = Pe^{rt}

Let's break down this elegant formula:

$A$ is the final amount of money (or whatever is growing).
$P$ is the principal, or the initial amount you start with.
$e$ is Euler's number, the base of natural growth.
$r$ is the annual growth rate, expressed as a decimal (so $5 %$ becomes $0.05$ ).
$t$ is the time in years.

This formula is incredibly useful for modeling anything that experiences continuous growth, from investments to bacterial colonies. All you need is a calculator with an $e^{x}$ button to use it.

Example 1

You invest $$ 5, 000$ in an account that earns $4 %$ annual interest, compounded continuously. How much money will be in the account after $8$ years?

Step 1: Identify your variables.
Principal $P = $ 5, 000$
Rate $r = 4 % = 0.04$
Time $t = 8$ years

Step 2: Plug the variables into the continuous compounding formula.
$A = P e^{r t}$
$A = 5000 \cdot e^{(0.04 \cdot 8)}$

Step 3: Simplify the exponent.
$A = 5000 \cdot e^{0.32}$

Step 4: Use a calculator to find the value of $e^{0.32}$ .
$e^{0.32} \approx 1.3771277$ (It's best to use the full value your calculator gives you).

Step 5: Calculate the final amount.
$A \approx 5000 \cdot 1.3771277$
$A \approx 6885.64$

After 8 years, you will have approximately $$ 6, 885.64$ in the account.

What is the Formal Mathematical Definition of $e$ ?

Our story about compound interest gives us the intuition behind $e$ , but in mathematics, we need precise definitions. There are two primary ways to formally define Euler's number.

1. The Limit Definition

This definition directly captures the idea of compounding more and more frequently. It states that $e$ is the value that the expression $(1 + \frac{1}{n})^{n}$ approaches as $n$ becomes infinitely large. In mathematical notation, we write this using a limit:

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

This is exactly what we saw in our table! As $n$ (the number of compounding periods) got bigger, the result got closer and closer to $e$ . The limit notation is just a formal way of saying "the value we approach as we head towards infinity."

2. The Infinite Series Definition

Another fascinating way to define $e$ is as the sum of an infinite series. This method involves factorials. A factorial, written as $n!$ , is the product of all positive integers up to $n$ (for example, $4! = 4 \times 3 \times 2 \times 1 = 24$ ). By definition, $0!$ is equal to $1$ .

e = \sum_{n=0}^{\infty} \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \dots

This means if you keep adding these fractions forever, the sum will be exactly $e$ . Even adding just the first few terms gives a remarkably close approximation.

Example 2

Approximate the value of $e$ by calculating the sum of the first six terms of its infinite series (from $n = 0$ to $n = 5$ ).

Step 1: Write out the terms we need to sum.
$S = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!}$

Step 2: Calculate the value of each factorial.
$0! = 1$
$1! = 1$
$2! = 2 \times 1 = 2$
$3! = 3 \times 2 \times 1 = 6$
$4! = 4 \times 3 \times 2 \times 1 = 24$
$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

Step 3: Calculate the value of each fraction.
$S = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120}$

Step 4: Convert to decimals and add them up.
$S = 1 + 1 + 0.5 + 0.16666 . . . + 0.04166 . . . + 0.00833 . . .$
$S \approx 2.71666 . . .$

As you can see, with just the first six terms, we are already very close to the actual value of $e \approx 2.71828$ . Adding more terms would get us even closer.

What is the Natural Logarithm ( $\ln$ )?

Every great mathematical operation has an inverse. Addition has subtraction. Multiplication has division. Squaring a number has taking the square root. For exponential functions involving $e$ , the inverse operation is the natural logarithm, written as $\ln (x)$ .

A logarithm answers the question: "What exponent do I need to raise a specific base to in order to get a certain number?"

The common logarithm ( $\log$ ) uses base $10$ . For example, $\log (100) = 2$ because $10^{2} = 100$ .
The natural logarithm ( $\ln$ ) uses base $e$ . So, $\ln (x)$ asks: " $e$ to what power equals $x$ ?"

This relationship is key:

\ln(y) = x \quad \iff \quad e^x = y

This inverse relationship is incredibly useful for solving equations where the variable you're looking for is in the exponent. By taking the natural logarithm of both sides of an equation, you can "bring the exponent down" and solve for it. Two important properties to remember are $\ln (e) = 1$ (since $e^{1} = e$ ) and $\ln (e^{x}) = x$ .

Example 3

A population of bacteria starts with $400$ cells and grows continuously at a rate that causes it to reach $1200$ cells in $6$ hours. What is the growth rate, $r$ ?

Step 1: Set up the continuous growth formula with the known information.
We have $A = 1200$ , $P = 400$ , and $t = 6$ . We need to find $r$ .
$1200 = 400 e^{r \cdot 6}$

Step 2: Isolate the exponential term.
Divide both sides by $400$ .
$\frac{1200}{400} = e^{6 r}$
$3 = e^{6 r}$

Step 3: Use the natural logarithm to solve for the exponent.
Take the natural log ( $\ln$ ) of both sides.
$\ln (3) = \ln (e^{6 r})$

Step 4: Apply the logarithm property $\ln (e^{x}) = x$ .
The $\ln$ and $e$ on the right side "cancel" each other out.
$\ln (3) = 6 r$

Step 5: Solve for $r$ .
Divide both sides by $6$ .
$r = \frac{\ln (3)}{6}$
Using a calculator, $\ln (3) \approx 1.0986$ .
$r \approx \frac{1.0986}{6} \approx 0.1831$

The growth rate is approximately $0.1831$ , or $18.31 %$ per hour.

Key formulas for eulers number by Algebra911.

What Are Common Mistakes When Using $e$ ?

Euler's number is a powerful tool, but like any tool, it requires careful handling. Here are some common mistakes that students often make when working with $e$ and the natural logarithm.

Treating $e$ as a Variable. Remember, $e$ is not a variable like $x$ or $y$ . It is a specific, constant number, just like $π$ . You can't "solve for $e$ ." Its value is always approximately $2.71828$ .
Using an Overly Rounded Value. While $2.72$ is a decent quick estimate for $e$ , using it in multi-step calculations can lead to significant rounding errors. Always use the $e^{x}$ button on your calculator for the most accurate results.
Confusing Natural Log ( $\ln$ ) and Common Log ( $\log$ ). The $\log$ button on most calculators refers to the base-10 logarithm. The $\ln$ button is the base- $e$ logarithm. Using the wrong one will give you incorrect answers when solving equations with $e$ . Remember: $\ln$ goes with $e$ .
Incorrectly Applying the Compounding Formula. A common error is to forget to express the interest rate $r$ as a decimal. An interest rate of $7 %$ must be entered into the formula as $0.07$ . Another mistake is misplacing the variables in $A = P e^{r t}$ .
Forgetting Logarithm Rules. When solving an equation like $3 = e^{6 r}$ , you can't just divide by $e$ . The variable is in the exponent, which means you must use a logarithm to solve it. Taking the natural log of both sides is the correct step.

Quick Summary and Key Formulas

Feeling overwhelmed? Don't be. Here's a quick rundown of the most important concepts and formulas related to Euler's number.

Key Concepts

What is $e$ ? A fundamental irrational constant, approximately $2.71828$ , that represents the base of continuous or "natural" growth.
Where does it come from? It's the limit of $(1 + 1 / n)^{n}$ as $n$ approaches infinity, which models the idea of compounding interest continuously.
What is $\ln (x)$ ? The natural logarithm, which is the inverse of $e^{x}$ . It answers the question, " $e$ to what power equals $x$ ?"

Key Formulas

The Limit Definition of $e$ :

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

Continuous Growth / Compounding Formula:

A = Pe^{rt}

The Inverse Relationship with the Natural Logarithm:

\ln(y) = x \quad \iff \quad e^x = y

Frequently Asked Questions

Who discovered Euler's number?

While Leonhard Euler is who the number is named after and who studied it extensively, it was first discovered by the Swiss mathematician Jacob Bernoulli in 1683. He encountered the constant while studying compound interest, the very same problem we explored in this lesson.

Is e a rational or irrational number?

Euler's number (e) is an irrational number. This means its decimal representation goes on forever without ever repeating a pattern, much like $π$ .

What is the value of e to 10 decimal places?

To 10 decimal places, the value of Euler's number is $2.7182818284$ . It's easy to remember the first few digits with the mnemonic "2.7 1828 1828," which looks like it repeats, but it doesn't.

Why is e often called the 'natural' number?

It's called the natural number because it appears in many formulas that describe natural phenomena. Processes involving continuous growth or decay, like population sizes, radioactive decay, and even the shape of a hanging cable, are all modeled using the base $e$ .

What is the difference between log(x) and ln(x)?

The only difference is the base. The common logarithm, $\log (x)$ , assumes a base of $10$ . The natural logarithm, $\ln (x)$ , uses a base of $e$ . You use $\ln (x)$ to solve equations involving $e$ .

Where is Euler's number used besides finance?

Euler's number is essential in many fields. In biology, it's used for population modeling. In physics, it describes radioactive decay and the cooling of objects. In computer science and statistics, it appears in probability distributions and complex number theory.

Is it okay to just use 2.72 for e in my calculations?

For a quick estimate, $2.72$ is fine. However, for homework or test problems, you should always use the $e^{x}$ button on your calculator. Using a rounded value early in a problem can make your final answer inaccurate.