152

E

e(Euler’s number) Swiss mathematician L

EONHARD

E

ULER

(1707–83) introduced a number, today denoted e,

that plays a fundamental role in studies of compound

INTEREST

,

TRIGONOMETRY

,

LOGARITHMS

, and

CALCULUS

,

and that unites these disparate fields. (E

ULER

’

S FORMULA

,

for instance, illustrates this.) The number ehas approxi-

mate value 2.718281828459045 … and can be defined

in any of the following different ways:

1. The number eis the limit value of the expression

raised to the nth power, as nincreases

indefinitely:

2. If L(a) denotes the area under the curve y= 1/x

above the interval [1,a], then eis the location on the

x-axis for which L(e) = 1.

3. If f(x) is a function that equals its own

DERIVATIVE

,

that is, f(x) = f(x), then f(x) is an

EXPONENTIAL

FUNCTION

with base value e: f(x) = ex.

4. eis the value of the infinite sum 1 + + +

+ ….

Definition 1 is linked to the problem of computing

compound interest. As we show below, definition 2

defines the natural logarithm, and definition 3 arises

from studies of natural growth and decay, and conse-

quently the consideration of

EXPONENTIAL FUNCTION

s.

The fourth definition arises from the study of T

AYLOR

SERIES

. One proves that all four definitions are equiva-

lent as follows:

First consider the curve y= 1/x. It has the remarkable

property that rectangles touching the curve and just under

it have the same area if the endpoints of the rectangles are

in the same ratio r. For example, in the first diagram on

the opposite page, the rectangles above the intervals [a,ra]

and [b,rb] each have area . By taking narrower and

narrower rectangles, all in the same ratio r, it then fol-

lows that the area under the curve above any two inter-

vals of the form [a,ra] and [b,rb] are equal.

Following definition 2, let L(x) denote the area

under this from position 1 to position x. (If xis less

than 1, deem the area negative.) Notice that L(1) = 0.

Also, set eto be the location on the x-axis where the

area under the curve is 1: L(e) = 1.

Notice that the area under the curve from 1 to

position ab, L(ab),is the sum of the areas under the

curve above the intervals [1,a] and [a,ab]. The first area

is L(a) and the second, by the property above, equals

L(b).We thus have:

L(ab) = L(a) + L(b)

This shows that Lis a function that converts multipli-

cation into addition, which is enough to prove that it is

the

LOGARITHMIC FUNCTION

base e. We have L(x) =

loge(x).This function is called the natural logarithm

function and is usually written ln(x).

r – 1

–––

r

1

–

3!

1

–

2!

1

–

1!

d

––

dx