Number Systems
What Is Euler's Number e?
Euler's number e is an irrational constant approximately equal to 2.718281828. It is the natural base for continuous growth and the exponential function whose rate of change equals its value.
Reviewed
A limit from increasingly frequent compounding
Compounding a 100 percent increase n times during one period produces (1+1/n)^n. As n grows without bound, this expression approaches e.
The limiting value is finite even though the number of compounding intervals increases indefinitely.
e = limit as n approaches infinity of (1 + 1/n)^nThe natural exponential function
The function e^x is distinguished in calculus because its derivative is itself. Its instantaneous growth rate at every point equals its current value.
This property makes e the simplest base for differential equations describing growth, decay, charging, cooling, and probability models.
d/dx e^x = e^xConnection to the natural logarithm
The natural logarithm ln(x) is the inverse of e^x. It converts multiplicative continuous growth into additive time or rate.
The constant can also be characterized by ln(e)=1.
Where this idea is used
- Continuous population, decay, and interest models.
- Natural logarithms and exponential equations.
- Calculus, probability distributions, and complex analysis.
Worked examples
Continuous growth
A quantity starts at 500 and grows continuously at 4 percent per year for 3 years.
Use A=Pe^(rt)A=500e^(0.04 x 3)A=500e^0.12
Result: A is approximately 563.75
Estimate e from compounding
Use n=10.
(1+1/10)^101.1^10
Result: 2.59374, an approximation that approaches e as n increases
Common mistakes
- Treating e as a variable rather than a fixed constant.
- Using e when a model specifies discrete rather than continuous compounding.
- Rounding e too early in a multi-step calculation.
Questions about euler's number e
Is e exactly 2.71828?
No. That decimal is an approximation; e has infinitely many nonrepeating decimal digits.
Is e related to pi?
They are different constants, but both appear together in identities such as Euler's formula.
Why not use base 10 for continuous growth?
Any positive base can model exponentials, but base e removes conversion factors from derivatives and integrals.