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 triangle satisfy:
where R is the radius of the largest circle that sits inside
the triangle.
See also EULER LINE.
amicable numbers (friendly numbers) Two whole
numbers a and b are said to be amicable if the sum of
the FACTORs of a, excluding a itself, equals b, and the
sum of the factors of b, excluding b itself, 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, PIERRE DE
FERMAT discovered a second pair, (17296, 18416), and in
1638, RENÉ DESCARTES discovered the pair (9363584,
9437056). Both these pairs were also known to Arab
mathematicians, perhaps at an earlier date.
By 1750, LEONHARD EULER 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. CALCULUS comes under this heading, as does the summation of infinite SERIES, and the study of REAL NUMBERS.
Greek mathematician PAPPUS OF ALEXANDRIA (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 hidden behind all of Greek geometry.
The great RENÉ DESCARTES (1596–1650) developed a powerful method of using algebra to solve geometric problems. His approach became known as
analytic geometry.
See also ANALYTIC NUMBER THEORY; CARTESIAN
COORDINATES.
analytic number theory The branch of NUMBER THEORY that uses the notion of a LIMIT to study the properties of numbers is called analytic number theory. This
branch of mathematics typically deals with the “average” 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, onequarter 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 particular argument can be made mathematically precise.
See also ANALYSIS; ZETA FUNCTION.
angle Given the configuration of two intersecting
LINEs, line segments, or RAYs, 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 ending 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:
1
1
4
1
9
1
16
6
1 64
2
+ + +
+ =
≈
L
π
.
1
1
1
1
h
h
h
R
a
b
c
+
+
=
angle 13
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 triangle satisfy:
where R is the radius of the largest circle that sits inside
the triangle.
See also EULER LINE.
amicable numbers (friendly numbers) Two whole
numbers a and b are said to be amicable if the sum of
the FACTORs of a, excluding a itself, equals b, and the
sum of the factors of b, excluding b itself, 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, PIERRE DE
FERMAT discovered a second pair, (17296, 18416), and in
1638, RENÉ DESCARTES discovered the pair (9363584,
9437056). Both these pairs were also known to Arab
mathematicians, perhaps at an earlier date.
By 1750, LEONHARD EULER 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. CALCULUS comes under this heading, as does the summation of infinite SERIES, and the study of REAL NUMBERS.
Greek mathematician PAPPUS OF ALEXANDRIA (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 hidden behind all of Greek geometry.
The great RENÉ DESCARTES (1596–1650) developed a powerful method of using algebra to solve geometric problems. His approach became known as
analytic geometry.
See also ANALYTIC NUMBER THEORY; CARTESIAN
COORDINATES.
analytic number theory The branch of NUMBER THEORY that uses the notion of a LIMIT to study the properties of numbers is called analytic number theory. This
branch of mathematics typically deals with the “average” 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, onequarter 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 particular argument can be made mathematically precise.
See also ANALYSIS; ZETA FUNCTION.
angle Given the configuration of two intersecting
LINEs, line segments, or RAYs, 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 ending 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:
1
1
4
1
9
1
16
6
1 64
2
+ + +
+ =
≈
L
π
.
1
1
1
1
h
h
h
R
a
b
c
+
+
=
angle 13