Extending this idea to three-dimensions, points in
space can be specified by a triple of numbers (x,y,z)
representing the distances along three mutually perpendicular number lines. The coordinate axes are usually
called the x-, y-, and z-axes. They intersect at a point
O, called the origin, which is zero on all three number
lines. The axes could be oriented to either form a lefthanded or a right-handed system.
Coordinate Geometry
The advent of a coordinate system allowed mathematicians, for the first time, to bring the power of algebra
to the study of geometry. For example, straight lines
are represented as sets of points (x,y) that satisfy equations of the form y = mx + b. Multiplying the SLOPE m
of one line with the slope of another quickly ascertains
whether or not those two lines are perpendicular, for
example. (The product of the slopes of two perpendicular lines is –1.)
French mathematician NICOLE ORESME (1323–82)
was the first to describe a way of graphing the relationship between an independent variable and a dependent
one, and thus the first to make steps toward uniting
geometry and algebra. The explicit construction of
what we would call a coordinate system first appeared
with the work of French lawyer and amateur mathematician PIERRE DE FERMAT (1601–65). Starting with
some horizontal reference line to represent an independent variable x, Fermat would graphically depict the
relationship of a second variable y to it as a line segment, held at a fixed angle to the reference line, whose
length would vary according to the variable y as it
slides along the x-axis. Fermat did not think in terms,
however, of identifying a second axis, nor did he
require the line segment representing y to be perpendicular to the x-axis.
In his famous 1637 text La géométrie (Geometry),
René Descartes independently described similar methods for representing algebraic relationships graphically.
Because the work of Fermat was not published until
after his death, the discovery of coordinate geometry
was attributed to Descartes.
Because Fermat and Descartes interpreted the
unknown variable y in an algebraic relationship as a
physical length, both scholars only ever considered positive coordinates. English mathematician JOHN WALLIS
(1616–1703) was the first to introduce the possibility of
negative coordinates. The idea of setting a fixed second
axis, the y-axis, perpendicular to the x-axis was not
popular until the mid 1700s. It was an idea that seemed
to evolve gradually. SIR ISAAC NEWTON (1642–1727) is
considered the originator of POLAR COORDINATES.
See also COORDINATES; GRAPH OF A FUNCTION.
Cartesian product (cross product, external direct
product, product set, set direct product) Given two
sets A and B, their Cartesian product, denoted A × B, is
the set of all ordered pairs (a,b), where a ∈ A and b ∈
B. For example, if A = {1,2,3} and B = {α,β}, then:
A × B = { (1,α), (2,α), (3,α), (1,β), (2,β), (3,β) }
This is different from the set B × A.
If sets A and B are both finite, with n and m elements, respectively, then A × B is a finite set with nm
elements. German mathematician GEORG CANTOR
(1845–1918) showed that if A and B are both infinite
COUNTABLE sets, then their Cartesian product A × B is
again countable.
The Cartesian product of three sets A, B, and C,
denoted A × B × C, is defined as the set of all ordered
triples (a,b,c), with a ∈ A, b ∈ B, and c ∈ C. The
Cartesian product of any finite collection of sets is
defined similarly. Any SEQUENCE can be thought of as
an element of the Cartesian product of a countable
number of sets.
If two sets A and B have a particular structure
(they might both be GROUPs or VECTOR SPACEs, for
instance), then it is usually possible to give the Cartesian product A × B the same structure. For example, if
A and B are groups with group operations * and •,
respectively, then A × B has the structure of a group
with group operation given by:
(a 1 , b 1 ) · (a 2 , b 2 ) = (a 1 * a 2 , b 1 • b 2 )
The Klein four-group is the Cartesian product of the twoelement group Z 2 = {0,1} with itself. (The group operation for Z 2 is addition in mod 2 MODULAR ARITHMETIC.)
See also SET THEORY.
casting out nines The DIVISIBILITY RULES show that
the remainder of any number, when divided by 9, is the
sum of its digits. For example, 59,432,641 leaves a
casting out nines 63
space can be specified by a triple of numbers (x,y,z)
representing the distances along three mutually perpendicular number lines. The coordinate axes are usually
called the x-, y-, and z-axes. They intersect at a point
O, called the origin, which is zero on all three number
lines. The axes could be oriented to either form a lefthanded or a right-handed system.
Coordinate Geometry
The advent of a coordinate system allowed mathematicians, for the first time, to bring the power of algebra
to the study of geometry. For example, straight lines
are represented as sets of points (x,y) that satisfy equations of the form y = mx + b. Multiplying the SLOPE m
of one line with the slope of another quickly ascertains
whether or not those two lines are perpendicular, for
example. (The product of the slopes of two perpendicular lines is –1.)
French mathematician NICOLE ORESME (1323–82)
was the first to describe a way of graphing the relationship between an independent variable and a dependent
one, and thus the first to make steps toward uniting
geometry and algebra. The explicit construction of
what we would call a coordinate system first appeared
with the work of French lawyer and amateur mathematician PIERRE DE FERMAT (1601–65). Starting with
some horizontal reference line to represent an independent variable x, Fermat would graphically depict the
relationship of a second variable y to it as a line segment, held at a fixed angle to the reference line, whose
length would vary according to the variable y as it
slides along the x-axis. Fermat did not think in terms,
however, of identifying a second axis, nor did he
require the line segment representing y to be perpendicular to the x-axis.
In his famous 1637 text La géométrie (Geometry),
René Descartes independently described similar methods for representing algebraic relationships graphically.
Because the work of Fermat was not published until
after his death, the discovery of coordinate geometry
was attributed to Descartes.
Because Fermat and Descartes interpreted the
unknown variable y in an algebraic relationship as a
physical length, both scholars only ever considered positive coordinates. English mathematician JOHN WALLIS
(1616–1703) was the first to introduce the possibility of
negative coordinates. The idea of setting a fixed second
axis, the y-axis, perpendicular to the x-axis was not
popular until the mid 1700s. It was an idea that seemed
to evolve gradually. SIR ISAAC NEWTON (1642–1727) is
considered the originator of POLAR COORDINATES.
See also COORDINATES; GRAPH OF A FUNCTION.
Cartesian product (cross product, external direct
product, product set, set direct product) Given two
sets A and B, their Cartesian product, denoted A × B, is
the set of all ordered pairs (a,b), where a ∈ A and b ∈
B. For example, if A = {1,2,3} and B = {α,β}, then:
A × B = { (1,α), (2,α), (3,α), (1,β), (2,β), (3,β) }
This is different from the set B × A.
If sets A and B are both finite, with n and m elements, respectively, then A × B is a finite set with nm
elements. German mathematician GEORG CANTOR
(1845–1918) showed that if A and B are both infinite
COUNTABLE sets, then their Cartesian product A × B is
again countable.
The Cartesian product of three sets A, B, and C,
denoted A × B × C, is defined as the set of all ordered
triples (a,b,c), with a ∈ A, b ∈ B, and c ∈ C. The
Cartesian product of any finite collection of sets is
defined similarly. Any SEQUENCE can be thought of as
an element of the Cartesian product of a countable
number of sets.
If two sets A and B have a particular structure
(they might both be GROUPs or VECTOR SPACEs, for
instance), then it is usually possible to give the Cartesian product A × B the same structure. For example, if
A and B are groups with group operations * and •,
respectively, then A × B has the structure of a group
with group operation given by:
(a 1 , b 1 ) · (a 2 , b 2 ) = (a 1 * a 2 , b 1 • b 2 )
The Klein four-group is the Cartesian product of the twoelement group Z 2 = {0,1} with itself. (The group operation for Z 2 is addition in mod 2 MODULAR ARITHMETIC.)
See also SET THEORY.
casting out nines The DIVISIBILITY RULES show that
the remainder of any number, when divided by 9, is the
sum of its digits. For example, 59,432,641 leaves a
casting out nines 63