e

e ≈ 2.7182818284.... is a transcendental number commonly encountered when working with exponential models (growth, decay,and logistic models, and continuously compounded interest, for example) and exponential functions. e is also the base of the natural logarithm.

Four definitions of e≈2.718: limit as n→∞ of (1+1/n)^n, limit as x→0 of (1+x)^(1/x), and sum 1/n! from n=0 to ∞

See also

Euler's formula e^iπ + 1 = 0

Key Formula

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

Where:

$n$ = A positive integer that grows without bound; as n increases, the expression approaches the exact value of e
$e$ = Euler's number, approximately 2.7182818284…, an irrational constant

Worked Example

Problem: An investment of $1,000 earns 5% annual interest, compounded continuously. What is the value after 6 years?

Step 1: Write the continuously compounded interest formula.

A = P \cdot e^{rt}

Step 2: Identify the values: P = 1000, r = 0.05, t = 6.

A = 1000 \cdot e^{(0.05)(6)}

Step 3: Simplify the exponent.

A = 1000 \cdot e^{0.3}

Step 4: Evaluate e^{0.3} using a calculator.

e^{0.3} \approx 1.34986

Step 5: Multiply to find the final amount.

A \approx 1000 \times 1.34986 = 1349.86

Answer: The investment is worth approximately $1,349.86 after 6 years.

Another Example

This example demonstrates the limit definition of e numerically, rather than applying e in a formula. It helps you understand where the value 2.71828… comes from.

Problem: Approximate the value of e by evaluating (1 + 1/n)^n for n = 1, 10, 100, and 1000.

Step 1: For n = 1, compute the expression.

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

Step 2: For n = 10, compute the expression.

\left(1 + \frac{1}{10}\right)^{10} = (1.1)^{10} \approx 2.59374

Step 3: For n = 100, compute the expression.

\left(1 + \frac{1}{100}\right)^{100} = (1.01)^{100} \approx 2.70481

Step 4: For n = 1000, compute the expression.

\left(1 + \frac{1}{1000}\right)^{1000} = (1.001)^{1000} \approx 2.71692

Step 5: As n grows larger, the result gets closer to e ≈ 2.71828. This illustrates the limit definition of e.

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

Answer: The values 2, 2.594, 2.705, 2.717 converge toward e ≈ 2.71828 as n increases.

Frequently Asked Questions

Why is e important in math?

The number e is the unique base for which the exponential function f(x) = e^x has the special property that its derivative equals itself: d/dx(e^x) = e^x. This makes e the natural choice for modeling continuous growth and decay in calculus, physics, biology, and finance. It also appears in probability, complex analysis (Euler's formula), and many other branches of mathematics.

Is e rational or irrational?

e is irrational, meaning it cannot be written as a fraction of two integers. Its decimal expansion, 2.7182818284…, goes on forever without repeating. Furthermore, e is transcendental, which means it is not a root of any polynomial equation with integer coefficients—placing it in an even more exclusive category than most irrational numbers.

What is the difference between e and π?

Both e and π are irrational, transcendental constants, but they arise in different contexts. π relates to circles and geometry (the ratio of a circle's circumference to its diameter), while e relates to continuous growth, calculus, and the natural logarithm. They are connected through Euler's formula: e^(iπ) + 1 = 0.

e (Euler's number) vs. π (pi)

	e (Euler's number)	π (pi)
Approximate value	2.71828…	3.14159…
Primary context	Exponential growth/decay and calculus	Circles, geometry, and trigonometry
Key property	d/dx(eˣ) = eˣ	Circumference = 2πr
Irrational?	Yes	Yes
Transcendental?	Yes	Yes
Connected by	Euler's formula: e^(iπ) + 1 = 0	Euler's formula: e^(iπ) + 1 = 0

Why It Matters

You encounter e whenever a quantity grows or decays at a rate proportional to its current size—radioactive decay, population growth, and bank interest compounded continuously all use e in their formulas. In calculus, e is essential because the function f(x) = e^x is its own derivative, making it the foundation for solving differential equations. Understanding e is a prerequisite for precalculus, AP Calculus, and virtually every STEM field beyond high school.

Common Mistakes

Mistake: Treating e as a variable instead of a constant.

Correction: The letter e in expressions like e^x is not an unknown to solve for—it is a fixed constant equal to approximately 2.71828. On your calculator, use the dedicated eˣ button rather than substituting a variable.

Mistake: Rounding e to 2.7 or 2.72 in problems that require precision.

Correction: Rounding too early can cause significant error, especially when e is raised to a large exponent. Use at least 2.71828 or, better yet, the exact e key on your calculator and round only at the final step.

Related Terms

Natural Logarithm — ln is the logarithm with base e
Exponential Function — f(x) = eˣ is the fundamental exponential function
Exponential Growth — Models continuous growth using base e
Exponential Decay — Models continuous decay using base e
Continuously Compounded Interest — Uses A = Peʳᵗ with base e
Transcendental Numbers — e belongs to this category of numbers
Logistic Growth — Uses e in a bounded growth model
Euler's Formula — Connects e, i, and π via e^(iθ)