closure property A BINARY OPERATION on a set S is
said to be closed if the combination of two elements in
that set yields another member of that set. For example, the set of positive whole numbers is closed under
addition, since the sum of any two positive integers is
itself a positive integer. This set is also closed under
multiplication (the product of two positive whole numbers is a positive integer), but not subtraction: if n and
m are positive whole numbers, then n – m could be
zero or negative and thus no longer in the set of positive whole numbers. (For instance, 3 – 7 = –4.)
For a more unusual example, consider the set S of all
whole numbers that can be expressed as the sum of two
SQUARE NUMBERS. We have S = {0, 1, 2, 4, 5, 8, 9, 10, 13,
16, 17, 20, 25,…}. (For instance, 0 = 0
2 + 0
2
, 5 = 1
2 + 2
2
,
20 = 2
2 + 4
2
, and 25 = 0
2 + 5
2 = 3
2 + 4
2
.) Surprisingly,
this set is closed under multiplication. For example, both
5 and 8 belong to S, and so does 5 × 8 = 40. (We have
40 = 2
2 + 6
2 .) Also, 10 and 13 belong to S, and so too
does 130. (We have 130 = 3
2 + 11
2 .) This general
observation follows from the algebraic identity that if
N = a 2 + b
2 and M = c
2 + d
2 ; then N × M = (ac + bd)
2 +
(ad – bc)
2 .
This set S is also closed under exponentiation: if N
and M are each a sum of two squares, then so is N
M .
For example, 5
13 = (12,625)
2 + (31,250)
2 .
See also SQUARE.
coefficient A numerical or constant multiplier of
the variables in a term of an algebraic expression is
called the coefficient of that term. For example, consider the equation 5x
3 – 2x + 7 = 0, where x is the
variable, the coefficient of x
3 is 5, the coefficient of x
is –2, and the coefficient of x
2 is zero. In the equation
3 cos y – 4xy
2 = 7, the coefficients of cos y and xy
2
are 3 and –4, respectively.
Sometimes the value of a coefficient is not known
and a symbol is used in its stead. For instance, in the
expression ax
2 + bx + c with x the variable, the numbers a and b are coefficients (and c is a constant term).
Although the values of a, b, and c are not specified, it is
understood that their values do not change even as the
value of x varies.
In a more general context, the term coefficient is
used for any number that serves as a measure of some
property or characteristic of a set of data or a physical
property. For instance, a CORRELATION COEFFICIENT in
STATISTICS gives a measure of the extent to which two
data sets are interdependent, while the heat coefficient
in physics gives a measure as to how well a material
conducts heat.
See also BINOMIAL COEFFICIENT; COMBINATORIAL COEFFICIENT; CONSTANT; LEADING COEFFICIENT;
POLYNOMIAL.
Collatz’s conjecture (“3n + 1” mapping problem)
Consider the following process:
Select a positive integer. If it is odd, triple it
and add one; otherwise, divide the number by
two. Now perform the same operation again
on the result. Repeat this process indefinitely
to produce a sequence of numbers.
The number 7, for example, yields the sequence:
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16,
8, 4, 2, 1, 4, 2, 1, 4, 2, 1,…
Notice that this sequence finally falls into a 4-2-1 cycle.
In 1937 German mathematician Lothar Collatz
conjectured that, no matter the starting integer selected,
all sequences lead to the same 4-2-1 cycle. Collatz was
unable to prove this claim, but he was also unable to
find an example of a starting number that does not
behave this way. To this day, no one knows whether or
not Collatz’s conjecture is true. All integers up to 2.702
× 10
16 have been checked.
As a first step toward understanding this problem,
mathematicians have proved that 4-2-1 is the only cycle
of reasonable size that could possibly appear; it has
been established that any other cycle that might appear
would be at least 275,000 numbers long.
collinear Any number of points are said to be
collinear if they all lie on the same straight line. Two
points are always collinear. Three points in a plane A =
(a 1 ,a 2 ), B = (b 1 ,b 2 ), and C = (c 1 ,c 2 ) are collinear only if
the lines connecting points A and B and the points connecting A and C have the same SLOPE. This means that
the following relationship must hold:
b a
b a
c a
c a
2
2
1
1
2
2
1
1
−
−
=
−
−
collinear 79
said to be closed if the combination of two elements in
that set yields another member of that set. For example, the set of positive whole numbers is closed under
addition, since the sum of any two positive integers is
itself a positive integer. This set is also closed under
multiplication (the product of two positive whole numbers is a positive integer), but not subtraction: if n and
m are positive whole numbers, then n – m could be
zero or negative and thus no longer in the set of positive whole numbers. (For instance, 3 – 7 = –4.)
For a more unusual example, consider the set S of all
whole numbers that can be expressed as the sum of two
SQUARE NUMBERS. We have S = {0, 1, 2, 4, 5, 8, 9, 10, 13,
16, 17, 20, 25,…}. (For instance, 0 = 0
2 + 0
2
, 5 = 1
2 + 2
2
,
20 = 2
2 + 4
2
, and 25 = 0
2 + 5
2 = 3
2 + 4
2
.) Surprisingly,
this set is closed under multiplication. For example, both
5 and 8 belong to S, and so does 5 × 8 = 40. (We have
40 = 2
2 + 6
2 .) Also, 10 and 13 belong to S, and so too
does 130. (We have 130 = 3
2 + 11
2 .) This general
observation follows from the algebraic identity that if
N = a 2 + b
2 and M = c
2 + d
2 ; then N × M = (ac + bd)
2 +
(ad – bc)
2 .
This set S is also closed under exponentiation: if N
and M are each a sum of two squares, then so is N
M .
For example, 5
13 = (12,625)
2 + (31,250)
2 .
See also SQUARE.
coefficient A numerical or constant multiplier of
the variables in a term of an algebraic expression is
called the coefficient of that term. For example, consider the equation 5x
3 – 2x + 7 = 0, where x is the
variable, the coefficient of x
3 is 5, the coefficient of x
is –2, and the coefficient of x
2 is zero. In the equation
3 cos y – 4xy
2 = 7, the coefficients of cos y and xy
2
are 3 and –4, respectively.
Sometimes the value of a coefficient is not known
and a symbol is used in its stead. For instance, in the
expression ax
2 + bx + c with x the variable, the numbers a and b are coefficients (and c is a constant term).
Although the values of a, b, and c are not specified, it is
understood that their values do not change even as the
value of x varies.
In a more general context, the term coefficient is
used for any number that serves as a measure of some
property or characteristic of a set of data or a physical
property. For instance, a CORRELATION COEFFICIENT in
STATISTICS gives a measure of the extent to which two
data sets are interdependent, while the heat coefficient
in physics gives a measure as to how well a material
conducts heat.
See also BINOMIAL COEFFICIENT; COMBINATORIAL COEFFICIENT; CONSTANT; LEADING COEFFICIENT;
POLYNOMIAL.
Collatz’s conjecture (“3n + 1” mapping problem)
Consider the following process:
Select a positive integer. If it is odd, triple it
and add one; otherwise, divide the number by
two. Now perform the same operation again
on the result. Repeat this process indefinitely
to produce a sequence of numbers.
The number 7, for example, yields the sequence:
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16,
8, 4, 2, 1, 4, 2, 1, 4, 2, 1,…
Notice that this sequence finally falls into a 4-2-1 cycle.
In 1937 German mathematician Lothar Collatz
conjectured that, no matter the starting integer selected,
all sequences lead to the same 4-2-1 cycle. Collatz was
unable to prove this claim, but he was also unable to
find an example of a starting number that does not
behave this way. To this day, no one knows whether or
not Collatz’s conjecture is true. All integers up to 2.702
× 10
16 have been checked.
As a first step toward understanding this problem,
mathematicians have proved that 4-2-1 is the only cycle
of reasonable size that could possibly appear; it has
been established that any other cycle that might appear
would be at least 275,000 numbers long.
collinear Any number of points are said to be
collinear if they all lie on the same straight line. Two
points are always collinear. Three points in a plane A =
(a 1 ,a 2 ), B = (b 1 ,b 2 ), and C = (c 1 ,c 2 ) are collinear only if
the lines connecting points A and B and the points connecting A and C have the same SLOPE. This means that
the following relationship must hold:
b a
b a
c a
c a
2
2
1
1
2
2
1
1
−
−
=
−
−
collinear 79