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)^nMore frequent compounding approaches e
Starting with one unit and applying 100% annual growth in n equal installments gives (1 + 1/n)^n. Increasing n approaches, but never defines e by a rounded decimal.
| n | (1 + 1/n)^n | Interpretation |
|---|---|---|
| 1 | 2 | Annual compounding |
| 2 | 2.25 | Semiannual compounding |
| 12 | about 2.6130 | Monthly compounding |
| 365 | about 2.7146 | Daily compounding |
| n tends to infinity | e about 2.718281828 | Continuous limit |
The limit is one route to e. In calculus, e is also the unique positive base whose exponential function has derivative equal to itself.
The 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^xWorked example 2
Estimate e from compounding
Use n=10.
(1+1/10)^101.1^10
Result: 2.59374, an approximation that approaches e as n increases
Connection 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.
Why e appears in models
- Continuous population, decay, and interest models.
- Natural logarithms and exponential equations.
- Calculus, probability distributions, and complex analysis.
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.