Euler Calculator - Free Online Tool
Calculate Euler's number e with animated convergence demonstration.
Watch Taylor series, limit definition, and continued fractions converge.
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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
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