angle 13

the perpendicular bisectors of any triangle are

CONCURRENT

, that is, meet at a point. But the

perpendicular bisectors of triangle DEF are

precisely the altitudes of triangle ABC.

One can also show that the three altitudes of a tri-

angle satisfy:

where Ris the radius of the largest circle that sits inside

the triangle.

See also E

ULER LINE

.

amicable numbers (friendly numbers) Two whole

numbers aand bare said to be amicable if the sum of

the

FACTOR

s of a, excluding aitself, equals b, and the

sum of the factors of b, excluding bitself, equals a. For

example, the numbers 220 and 284 are amicable:

284 has factors 1, 2, 4, 71, and 142, and their sum

is 220

220 has factors 1, 2, 4, 5, 10, 11, 20, 22, 44, 55,

and 110, and their sum is 284

The pair (220, 284) is the smallest amicable pair. For

many centuries it was believed that this pair was the only

pair of amicable numbers. In 1636, however, P

IERRE DE

F

ERMAT

discovered a second pair, (17296, 18416), and in

1638, R

ENÉ

D

ESCARTES

discovered the pair (9363584,

9437056). Both these pairs were also known to Arab

mathematicians, perhaps at an earlier date.

By 1750, L

EONHARD

E

ULER

had collated 60 more

amicable pairs. In 1866, 16-year-old Nicolò Paganini

found the small pair (1184, 1210) missed by all the

scholars of preceding centuries. Today more than 5,000

different amicable pairs are known. The largest pair

known has numbers each 4,829 digits long.

See also

PERFECT NUMBER

.

analysis Any topic in mathematics that makes use of

the notion of a

LIMIT

in its study is called analysis.

CAL

-

CULUS

comes under this heading, as does the summa-

tion of infinite

SERIES

, and the study of

REAL NUMBERS

.

Greek mathematician P

APPUS OF

A

LEXANDRIA

(ca.

320

C

.

E

.) called the process of discovering a proof or a

solution to a problem “analysis.” He wrote about “a

method of analysis” somewhat vaguely in his geometry

text Collection, which left mathematicians centuries

later wondering whether there was a secret method hid-

den behind all of Greek geometry.

The great R

ENÉ

D

ESCARTES

(1596–1650) devel-

oped a powerful method of using algebra to solve geo-

metric problems. His approach became known as

analytic geometry.

See also

ANALYTIC NUMBER THEORY

; C

ARTESIAN

COORDINATES

.

analytic number theory The branch of

NUMBER THE

-

ORY

that uses the notion of a

LIMIT

to study the proper-

ties of numbers is called analytic number theory. This

branch of mathematics typically deals with the “aver-

age” behavior of numbers. For example, to answer:

On average, how many square factors does a

number possess?

one notes that all numbers have 1 as a factor, one-

quarter of all numbers have 4 as a factor, one-ninth

have the factor 9, one-sixteenth the factor 16, and

so on. Thus, on average, a number possesses

square factors. This par-

ticular argument can be made mathematically precise.

See also

ANALYSIS

;

ZETA FUNCTION

.

angle Given the configuration of two intersecting

LINE

s, line segments, or

RAY

s, the amount of

ROTATION

about the point of intersection required to bring one

line coincident with the other is called the angle

between the lines. Simply put, an angle is a measure of

“an amount of turning.” In any diagram representing

an angle, the lengths of the lines drawn is irrelevant.

For example, an angle corresponding to one-eighth of a

full turn can be represented by rays of length 2 in., 20

in., or 200 in.

The image of a lighthouse with a rotating beam of

light helps clarify the concept of an angle: each ray or

line segment in a diagram represents the starting or end-

ing position of the light beam after a given amount of

turning. For instance, angles corresponding to a quarter

of a turn, half a turn, and a full turn appear as follows: