Euler Calculator - Free Online Tool
Calculate Euler's number e with animated convergence demonstration.
Watch Taylor series, limit definition, and continued fractions converge.
Show this tool on your website
Animation Controls
Algorithms
Convergence Error (Lower is Better)
What is Euler's Number (e)?
Euler's number e ā 2.71828... is one of the most important constants in mathematics. It's the unique number where the function f(x) = eĖ£ is its own derivative - the rate of growth equals the current value. This makes it fundamental to calculus, compound interest, probability, and physics.
Taylor Series
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ...
This is the most common formula for calculating e. Since factorials grow extremely fast (4! = 24, 10! = 3,628,800, 20! ā 2.4 quintillion), each term shrinks rapidly. Just 10 terms give you accuracy to 7 decimal places! This formula comes from the Taylor series expansion of eĖ£ evaluated at x = 1.
Limit Definition
e = lim(nāā) (1 + 1/n)āæ
This is the original definition of e, arising from compound interest. If you invest $1 at 100% annual interest, compounded n times per year, your balance after one year approaches $e as n approaches infinity. With monthly compounding (n=12) you get $2.61; with daily compounding (n=365) you get $2.71456. The limit is e ā $2.71828.
Continued Fraction
e = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...]
A continued fraction represents a number as a sequence of nested fractions: 2 + 1/(1 + 1/(2 + 1/(1 + ...))). Euler discovered that e has a beautiful pattern in its continued fraction: [2; 1, 2k, 1] repeating. This representation often converges faster than series and is useful for proving e is irrational.
Brothers' Formula
e = Ī£ (2n + 2) / (2n + 1)!
This variant of the factorial series converges slightly faster than the standard Taylor series. Each term contributes more information because we're computing (2n+2) in the numerator rather than just 1. It's named after the Brothers who discovered that small modifications to classical formulas can improve convergence.
Where Does e Appear?
- Compound interest: Continuous compounding at rate r for time t multiplies your money by e^(rt)
- Population growth: Continuous exponential growth/decay follows e^(kt)
- Probability: The probability of no events in a Poisson process involves e^(-Ī»)
- Calculus: The derivative of e^x is e^x - the only function that is its own derivative
- Euler's identity: e^(iĻ) + 1 = 0 connects e, i, Ļ, 1, and 0 - often called the most beautiful equation in mathematics
Frequently Asked Questions
Common questions about the Euler Calculator - Free Online Tool
Explore Number Convert
Free unit conversion tools