(Copyright © 2000-2001 --
this item revised 26 Nov 2000)

e is the limit value of the expression (1 + 1/n)^n, as n increases without bound:

e = lim[ n -> oo, (1 + 1/n)^n ] e ~ 2.71828

Euler (1707-1783) was first to use the symbol e for this number, perhaps as the first letter of "exponential".

Like pi, e is not just irrational but transcendental, meaning that there is no polynomial equation with integer coefficients that has e as a solution. Like pi, e can only compute approximations to e.

The limit definition of e corresponds to interest compounded continuously. In other words, a dollar deposited at 100% annual interest compounded continuously will grow to almost $2.72 in a year.

There is a simple and beautiful series expansion for e:

e = 1 + 1 + 1/2! + 1/3! + 1/4! + ... = sum[ n=0 to oo, 1/n! ]

This converges very rapidly indeed, because the factorial denominators grow very rapidly.

The decimal expansion of e starts off very regularly, unlike that of pi: 2.718281828. It would be nice if that pattern continued; but e is irrational, so we know it does not. Curiously, what comes next also exhibits a sort of pattern:

e ~ 2.7 1828 1828 45 90 45

(after Jan Gullberg in Mathematics from the Birth of Numbers)

If you graph the simple function f(x) = 1/x, for x > 0, you have half of a hyperbola, with the positive x and y axes as asymptotes. Draw a vertical line at x = 1, and another vertical line at any positive x value, say t. The natural logarithm of t is defined as the area bounded by the x axis, the two vertical lines, and the curve. (If t < 1, the natural logarithm is negative.) In symbols,

ln(t) = INT[x=1 to t, 1/x, dx]

What makes it "natural" is the simplicity of that function 1/x. One way to define e is that it is the base of such logarithms, or that it is the number whose natural logarithm is 1.

The slope of the curve y = ln(x) at any point is simply 1/x. The slope of y = e^x at any point is simply e^x. If you chose a different base of logarithms, you would have to introduce proportionality constants. It's simpler -- more "natural" -- just to do without them.

Nothing done in base-e logarithms is impossible in any other base, but the calculations are usually easier in base e.

Processes of growth and decay usually follow exponential functions,

y = a*e^(bt)

If b>0, the quantity is growing over time; if b<0, the quantity is decaying. For instance, radioactive decay follows this curve.

There's an excellent book about e, natural and common logarithms, and applications: e: Story of a Number by Eli Maor.

- Stan Brown
- Haran Pilpel

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