What it is
e is a number, e = 2.718281828… It is irrational, so its decimals never end and never repeat in a period. There are three routes to it, and all three give the same number.
Compound interest. Put in 1 at 100% interest a year. If the interest is split into n equal parts of the year, the amount after one year is (1 + 1/n)ⁿ.
| n | Interest added | (1 + 1/n)ⁿ |
|---|---|---|
| 1 | once a year | 2 |
| 2 | twice a year | 2.25 |
| 12 | every month | 2.613 |
| 365 | every day | 2.7146 |
| 1,000,000 | about every 32 seconds | 2.7182805 |
The values approach e as n grows. So e is the limit of (1 + 1/n)ⁿ as n tends to infinity.
The base with slope 1. The graph of aˣ has slope ln a where it crosses the y-axis. For a = 2 the slope is ln 2 ≈ 0.693, and for a = 3 it is ln 3 ≈ 1.099. e is the base between 2 and 3 where the slope is exactly 1. That is why the derivative of eˣ is eˣ itself: the slope at every point equals the value of the function there. The limit of (eˣ − 1)/x says the same thing.
A sum. e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + …, where k! = 1 · 2 · … · k. The first partial sums are 2, 2.5, 2.667, 2.708 and 2.717. This is the Maclaurin expansion of eˣ at x = 1.
e is also the base of the natural logarithm: ln e = 1, and e^(ln x) = x for x > 0.
Not the number e:
- E on a calculator. 3.5E4 means 3.5 · 10⁴ = 35,000, a power of ten.
- Euler’s constant γ ≈ 0.5772. A different number, named after the same mathematician. e is called Euler’s number.
Why it matters
- Change that happens all the time. y = Ce^(kx) has derivative k · y and so solves the differential equation y′ = ky. Growth, cooling and decay are therefore often written with powers of e.
- Continuous compounding. Interest added all the time gives A = P · e^(rt). 1,000 at 5% for 10 years becomes 1,000 · e^(0.5) ≈ 1,648.72, against 1,000 · 1.05¹⁰ ≈ 1,628.89 when the interest is added once a year. See continuous compounding.
- Every exponential function can be written with e. aˣ = e^(x · ln a). Since ln 1.05 ≈ 0.0488, 1.05ˣ ≈ e^(0.0488x). See exponential functions.
- Differentiating and integrating. (eˣ)′ = eˣ and ∫ eˣ dx = eˣ + C; see the antiderivative of eˣ.
- Complex numbers. Euler’s identity e^(iπ) + 1 = 0 is a special case of the formula on the complex exponential form page.
The scientific calculator has keys for e and eˣ, and the graphing calculator shows that y = eˣ has slope 1 where it crosses the y-axis.
Common misconceptions
- “e is about 2.7, so I can use 2.7.” Small errors grow in powers. e¹⁰ ≈ 22,026, but 2.7¹⁰ ≈ 20,589, about 6.5% less. Use the calculator’s e.
- “eˣ is e times x.” eˣ means e to the power x. e³ ≈ 20.09, while 3e ≈ 8.15.
- “The derivative of eˣ is x · e^(x−1).” The power rule applies when x is the base, as in x³ with derivative 3x². In eˣ, x is the exponent, and the derivative is eˣ.
- “An exponential function has to be written with e.” 2ˣ works just as well, and 2ˣ ≈ e^(0.693x). The base e is convenient because the slope comes out as 1, but it is not required.
Common questions
Why e in particular?
With base e the exponential function has slope 1 at x = 0 and is therefore its own derivative. That simplifies
work with derivatives, integrals and differential equations, and the logarithm with base e, ln x, is the inverse
of eˣ.
Is e a rational number?
No, e cannot be written as a fraction of two integers. The digits 1828 appear twice in a row near the start,
2.718281828…, but that is a coincidence, not a period.
Where does the name come from?
It is called Euler’s number after Leonhard Euler, who began using the letter e in the 1700s. The limit
of (1 + 1/n)ⁿ is usually credited to Jacob Bernoulli, who studied compound interest.