Math
e Calculator
Euler's number e ≈ 2.718281828 is the base of natural logarithms and continuous growth.
Show e to as many digits as you need (computed exactly from its series), evaluate e^x or e^(1/x), and watch both the limit (1 + 1/n)^n and the factorial series approach it.
Up to 1,000 exact decimal places.
Try an example
Result
Result
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- Expression
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- Limit (1 + 1/n)^n at n = 1,000,000
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- Series after 10 terms
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What is Euler's number?
e is the unique The number being raised to a power, or the side a shape's height is measured from. for which the exponential function e^x grows at a rate equal to its own value. That property makes it the natural choice for continuous growth, compound interest in the limit, radioactive decay, and the natural The exponent you would need to raise a base to in order to get a number. ln. Like π, it is A number that cannot be written as a fraction of two whole numbers, like π or √2. and transcendental.
Two ways to define e
| n | (1 + 1/n)^n | Terms | Series sum |
|---|---|---|---|
| 1 | 2 | 1 | 1 |
| 10 | 2.593742 | 3 | 2.5 |
| 100 | 2.704814 | 5 | 2.708333 |
| 1,000 | 2.716924 | 7 | 2.718056 |
| 1,000,000 | 2.718280 | 10 | 2.718282 |
Where e shows up
- Continuous compounding: A = P e^(rt).
- Exponential growth and decay: y = a e^(kt).
- Probability: the chance of zero events in a Poisson process is e^-λ.
- Euler's identity: e^(iπ) + 1 = 0.
How to use it
- Choose e to n decimal places to see the constant itself.
- Choose e^x to raise e to a power, or x-th root of e for e^(1/x).
- Open the work to see the series partial sums and the limit approximations approach e.
How to read the answer
e is irrational, so its decimals never repeat; the digits shown are exact to the requested place. The series converges fast (10 terms give 7 correct digits) while the limit converges slowly (a million steps give only 6 digits), which is why the series is used to compute e.
Common mistakes and edge cases
- Using 2.7 or 2.718 in calculations where precision matters; use 2.718281828 or the e key.
- Confusing e with the calculator's E notation for powers of 10 (1E5 = 100,000).
- Thinking (1 + 1/n)^n reaches e for some finite n. It only approaches e.
- e^0 = 1 and e^1 = e; e^-1 = 1/e ≈ 0.3679.
Worked examples
e to 15 places
The classic value
Result
2.718281828459045
e to 50 places
More digits, computed exactly
Result
2.71828182845904523536028747135266249775724709369996
e^2
e squared
Result
7.3890560989
e^-1
1/e
Result
0.3678794412
Square root of e
e^(1/2)
Result
1.6487212707
e to 0 places
Rounds to 3
Result
3
Undefined root
Root index 0
Result
Error
Frequently asked questions
What is the value of e?+
e ≈ 2.718281828459045. It is irrational, so the decimals go on forever without repeating.
Why is e called Euler's number?+
Leonhard Euler introduced the letter e for it in the 1730s and proved many of its properties, including the series 1 + 1/1! + 1/2! + ... .
How is e defined?+
As the limit of (1 + 1/n)^n as n grows without bound, or equivalently as the sum of 1/k! for k = 0, 1, 2, ... . Both give 2.71828...
What is e^0, e^1, and e^-1?+
e^0 = 1, e^1 = e ≈ 2.71828, and e^-1 = 1/e ≈ 0.36788.
What is the square root of e?+
e^(1/2) ≈ 1.64872. Any root of e is e raised to one over the root index.
Where does e come from in compound interest?+
Compounding $1 at 100% interest n times per year gives (1 + 1/n)^n. As n grows, the balance approaches e ≈ $2.72, the continuous-compounding limit.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026