The letter comes from Leonhard Euler, who used it in a letter to Christian Goldbach in 1731 and in print in his Mechanica of 1736. Nobody knows for certain why he picked e. The claim that he chose it for his own name is not supported; one suggestion is that e was simply the next free vowel after a, which he was already using.

What is e to the power of 1?

e to the power of 1 is e itself, about 2.71828, because any number raised to the first power is unchanged. For comparison, e^0 = 1, e^-1 is about 0.3679 (which is 1/e), and e^2 is about 7.389.

How do I type e on a calculator?

Look for a key marked e^x. On many scientific calculators it is the second function of the ln key, because e^x and ln x undo each other. Enter e^1 to display e itself, about 2.718281828. Typing 2.718 by hand gives a rounded value, so keep the calculator's e for multi-step work.

Is Euler's number the same as Euler's constant?

No. Euler's number e is about 2.71828, the base of the natural logarithm. Euler's constant, also called the Euler-Mascheroni constant and written γ, is about 0.5772. It measures how far the harmonic sum 1 + 1/2 + 1/3 + ... + 1/n stays above ln n as n grows, and nobody yet knows whether it is irrational.

Euler's Number e: What It Is and Why It Equals 2.718

Type "e to the power of 1" into a calculator and it answers 2.718281828. That number is Euler's number, written e, and it is one of the few constants that every branch of mathematics eventually runs into: compound interest, population growth, radioactive decay, probability, and the whole of calculus.

This guide gives the short answer first, then shows where the value comes from, two ways to compute it, and why calculus treats e as the natural base for exponents. If exponents themselves feel shaky, start with understanding exponents intuitively and come back.

The short answer: what e is and its key values

The value of e is about 2.71828. To more places, it is 2.718281828459045..., and the decimals never end or repeat, because e is irrational (MathWorld). It is the base of the natural logarithm and of the natural exponential function $e^{x}$ . You will also see it called Napier's constant, after John Napier (Wikipedia).

Most searches about e are really asking for one of a handful of values. Here they are, computed to six or more significant digits:

Expression	Value	Why
$e^{1}$	2.718282	Any number to the first power is itself
$e^{0}$	1	Any nonzero number to the power 0 is 1
$e^{-1}$	0.367879	Same as $1/e$
$e^{2}$	7.389056	$e \times e$
$e^{1/2}$	1.648721	Same as $\sqrt{e}$
$\ln e$	1	The natural log undoes $e^{x}$

So e to the power of 1 is just e, about 2.71828. Nothing special happens at the first power; the special part is the base itself.

Where e comes from: compound interest

Jacob Bernoulli met this number in 1683 while studying continuously compounded interest, and showed it had to lie between 2 and 3 (MacTutor). His question is easy to state.

Put 1 dollar in an account that pays 100% interest per year. Paid once at the end of the year, you have 2 dollars. Now split the same 100% into n smaller payments, each adding $1/n$ of the balance, and let each payment earn interest on the ones before it. After one year you have

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

dollars. More frequent compounding pays more, but not without limit:

Compounding	n	Balance after one year
Yearly	1	2.000000
Twice a year	2	2.250000
Quarterly	4	2.441406
Monthly	12	2.613035
Daily	365	2.714567
Hourly	8,760	2.718127
Continuously	n grows without limit	2.718282 = e

Going from monthly to daily adds about ten cents. Going from daily to hourly adds less than half a cent. The balance creeps toward a ceiling it never passes, and that ceiling is e. This is exactly how the OpenStax calculus text defines it: as m grows, $\left(1 + 1/m\right)^{m}$ approaches a number, and that number is called e (OpenStax Calculus Volume 1, section 1.5). For a rate r over t years the same idea gives the continuous growth formula $A = Pe^{rt}$ .

Two definitions that agree, and how fast each gets there

There is a second road to the same number. Add 1 divided by each factorial:

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

MathWorld lists both forms as definitions of e, and Euler set them side by side in his Introductio in analysin infinitorum of 1748, where he also gave e to 18 decimal places (MacTutor). They produce the same number, but at very different speeds. Here is the convergence race, computed directly; the target is 2.71828, the first five decimals of e:

Convergence race: limit vs series

Limit $\left(1 + 1/n\right)^{n}$	Value	Series, terms used	Partial sum
n = 1	2.000000	2 terms (up to 1/1!)	2.000000
n = 10	2.593742	4 terms (up to 1/3!)	2.666667
n = 100	2.704814	6 terms (up to 1/5!)	2.716667
n = 1,000	2.716924	8 terms (up to 1/7!)	2.718254
n = 100,000	2.718268	9 terms (up to 1/8!)	2.718279
n = 1,000,000	2.718280	10 terms (up to 1/9!)	2.718282

The series locks in 2.71828 after 10 terms. The limit needs n of about 743,000 before it first shows 2.71828, and even at a million steps it is still wrong in the sixth decimal. The reason is in the sizes of the errors. Each new factorial term is far smaller than the last, so the series' error shrinks faster than any power. The limit's error shrinks roughly like $e/(2n)$ , so ten times more steps buys only about one more correct digit.

The practical lesson: the limit explains why e exists (it is the ceiling of compound growth), and the series is how you actually compute it.

Why calculus prefers e

Every exponential function $b^{x}$ with $b > 0$ is a curve that rises (or falls) faster as it goes. Its slope at any point is proportional to its height. For most bases the proportion is an awkward constant: for $2^{x}$ the slope is about 0.693 times the height; for $3^{x}$ it is about 1.099 times the height.

There is exactly one base where that constant is 1, so the slope at every point equals the height. That base is e. OpenStax defines e this way in its derivatives chapter: the unique base whose exponential has slope 1 at $x = 0$ , from which

\frac{d}{dx}\, e^{x} = e^{x}

follows (OpenStax Calculus Volume 1, section 3.9). A function that is its own derivative makes calculus with growth almost free: differentiate $e^{x}$ and nothing changes, integrate it and nothing changes. If the idea of a derivative as slope is new, derivatives from scratch builds it up.

The natural logarithm, written ln, is the inverse of $e^{x}$ : $\ln(e^{x}) = x$ and $e^{\ln x} = x$ for $x > 0$ . Its derivative is just as clean, $\frac{d}{dx}\ln x = \frac{1}{x}$ . That is why scientists call base e "natural": it is the base that makes the formulas shortest. Our guide to understanding logarithms intuitively covers ln alongside base 10 and base 2.

The series from the previous section is not a coincidence either. The Maclaurin series of $e^{x}$ is $\sum_{n=0}^{\infty} x^{n}/n!$ (OpenStax Calculus Volume 2, section 6.3); set $x = 1$ and you get the factorial sum for e.

Euler's number vs Euler's constant vs Euler's identity

Euler's name is attached to so many results that three different things get mixed up in search:

Name	Symbol and value	What it is
Euler's number	e ≈ 2.71828	Base of the natural logarithm; the limit and series above
Euler's constant (Euler-Mascheroni)	γ ≈ 0.57722	The limit of $1 + \tfrac{1}{2} + \cdots + \tfrac{1}{n} - \ln n$ as n grows
Euler's identity	$e^{i\pi} + 1 = 0$	An equation linking e, i, π, 1 and 0

They differ in what is known about them, too. e has been proved irrational and, by Charles Hermite in 1873, transcendental (MathWorld). For γ, it is still not known whether it is even irrational (MathWorld, Euler-Mascheroni constant). And Euler's identity is not a number at all but a statement about e raised to an imaginary power; our walkthrough of Euler's identity shows why $e^{i\pi}$ lands on -1.

Common mistakes with e

Treating e as a variable. In $e^{x}$ , x varies and e does not. Solving $e^{x} = 5$ means taking ln of both sides: $x = \ln 5 \approx 1.609$ .

Rounding too early. Using 2.7 instead of e in $e^{3}$ gives 19.683 instead of 20.0855. Keep the calculator's e until the last step.

Confusing $e^{-1}$ with $-e$ . A negative exponent means a reciprocal: $e^{-1} = 1/e \approx 0.3679$ , which is positive. $-e \approx -2.718$ is a different number.

Mixing up ln and log. On most calculators "log" means base 10 and "ln" means base e. $\ln e = 1$ , but $\log e \approx 0.4343$ .

Practice with e

Try these before checking:

Without a calculator, what are $e^{0}$ , $e^{1}$ and $\ln 1$ ?
Simplify $e^{2} \cdot e^{3}$ and $\ln\left(e^{7}\right)$ .
Solve $e^{x} = 1$ .

Answers: 1, e and 0; $e^{5}$ and 7; $x = 0$ .

Math Zen's logarithms practice drills ln and change of base, and the functions practice includes exponential functions, so you can turn these rules into habits one problem at a time.

Frequently asked questions

Is e a rational number?

No. e is irrational and its decimals never end or repeat. Hermite proved in 1873 that it is also transcendental: it is not the root of any nonzero polynomial with rational coefficients.

Why is it called e?

Leonhard Euler used the letter in a 1731 letter to Christian Goldbach and in print in 1736. Nobody knows for certain why he picked e; the story that he chose his own initial is not supported by historians.

What is e^2?

About 7.389. It is e multiplied by itself, $2.71828 \times 2.71828$ .

What is e to the power of 0?

Exactly 1, like any nonzero number raised to the power 0.