276 interior angle

Consider now a bank that offers its savings

account customers an annual interest rate of 10 percent

compounded weekly. A $1,000 investment after 1

year would thus yield .

If the interest is instead computed daily, customers

receive a slightly better return:

= $1,105.16, and an even better return if the interest is

compounded hourly, or even every minute. The maxi-

mum possible return is achieved when interest is com-

puted every instant.

In the mid-1700s Swiss mathematician L

EON

-

HARD

E

ULER

tackled the problem of computing con-

tinuously compounded interest and showed that as

nbecomes large, the quantity A= P(1 + )nT

approaches the value PeRT. (A study of the number e

establishes this claim.) The formula:

A= PeRT

thus represents the balance of an investment, or a loan,

after Tyears, under the ideal state of interest com-

pounded continuously. Thus our $1,000 investment,

after 1 year, compounded continuously at a rate of 10

percent per annum, yields a return of 1,000 ×e0.10×1=

$1,105.17. Banks today use this formula to compute

interest on savings accounts.

The practice of charging a fee for the use of money

is an ancient one. Records show that many civiliza-

tions, the Hebrews, the Greeks, the Romans, and even

the Babylonians of 2000

B

.

C

.

E

., for instance, charged

simple interest on loans. The Romans called the prac-

tice usury and often fees were as high as 60 percent.

Some religious orders, including Christianity, Judaism,

and Islam, questioned the ethics of the practice, arguing

that one should only be charged a fee for the use of

something that could be worn out or lose value due to

wear and tear. (At the time, money was not seen to lose

any value during the course of a loan.) For many cen-

turies the practice of charging a fee for the use of

money was forbidden by these orders.

During the growth of industry and trade during the

Middle Ages and the Renaissance, attitudes changed.

More and more people requested cash loans to take

part in new opportunities, and lenders felt it appropri-

ate to be compensated for not taking part in those

opportunities themselves. Lenders again started charg-

ing fees. The church relaxed its attitude toward usury,

and new establishments, called banks, were formed to

handle, store, and loan money. The term interest from

the Latin phrase id quod interest meaning “that which

is between” soon replaced the term usury. Today the

word usury is used only in a negative context of charg-

ing illegally high interest rates.

See also

E

.

interior angle The

ANGLE

formed by two sides of a

POLYGON

lying inside the polygon is called an interior

angle. For example, all four interior angles of a

RECT

-

ANGLE

equal 90°.

If an interior angle is greater than 180°, then that

angle is called a “re-entrant angle” and the polygon is

concave. Any interior angle less than 180°is called

“salient.”

Each interior angle of an n-sided regular polygon

equals × 180°.

See also

CONCAVE

/

CONVEX

;

TRANSVERSAL

.

intermediate-value theorem (Bolzano’s theorem)

Named after the Czech mathematician B

ERNHARD

P

LACIDUS

B

OLZANO

(1781–1848), the intermediate-

value theorem asserts that if f(x) is a

CONTINUOUS

func-

tion defined on a closed

INTERVAL

, then this function

assumes every value between f(a) and f(b);that is, if N

is any number between f(a) and f(b),then there is at

least one point cbetween aand bsuch that f(c) = N.

For example, the function f(x) = x2is continuous on

the interval [3,4] and has f(3) = 9 and f(4) = 16. Since

11 is between 9 and 16, the intermediate value theorem

ensures us of the existence of a number c, between 3

and 4, such that f(c) = c2= 11, that is, it proves that the

square root of 11 exists.

The theorem is intuitively clear if we think of a con-

tinuous function on a closed interval as one whose

graph consists of a single continuous piece with no

gaps, jumps, or holes: in moving a pencil from the left

endpoint (a, f(a)) to the right endpoint (b, f(b)),it seems

obvious that the pencil tip adopts all “heights” between

initial height f(a) and final height f(b).In climbing the

face of a mountain, say, one must indeed pass through

n– 2

––

n

R

–

n