be in U, and by being in U, u cannot be in U.
This absurdity shows that there cannot be a
meaningful correspondence that assigns distinct elements of A to subsets of A after all.
Given an infinite set A, Cantor had thus shown
that the sets P(A), P(P(A)), P(P(P(A))), … form a neverending chain of increasingly larger infinite sets. Thus,
in a very definite sense, there are infinitely many different types of infinity. At the other end of the spectrum,
the study of denumerable sets shows that every infinite
set contains a denumerable subset. Thus of all the infinite sets, denumerable sets are the “smallest” type of
infinite sets. The CONTINUUM HYPOTHESIS asks whether
or not there is an infinite set with cardinality that lies
somewhere between that of N and P(N) = R.
One might suppose that P(R), the power set of the
set of all points on the real number line, is R
2
, the set
of all points in the plane, or, equivalently, that the
power set of the set of all points in the unit interval
[0,1] is the set of all points inside the unit square [0,1]
× [0,1]. Surprisingly, this is not the case: there are just
as many points in the unit square as there are in a unit
interval. This is seen as follows:
Associate to each point (x,y) with 0 ≤ x ≤ 1 and
0 ≤ y ≤ 1, each written as an infinite decimal
expansion, x = 0.x 1 x 2 x 3 … and y = 0.y 1 y 2 y 3 …,
the real number r = 0.x 1 y 1 x 2 y 2 x 3 y 3 … in the
interval [0,1], and, conversely, match each real
number r = 0.r 1 r 2 r 3 r 4 r 5 r 6 … with the point
(0.r 1 r 3 r 5 …, 0.r 2 r 4 r 6 …) in the unit square.
(Again there is a technical difficulty caused by those
real numbers that have two different decimal representations. For instance, one-half can be written both as
0.5000… and 0.4999… Mathematicians have shown
that this difficulty can be obviated.) It turns out that
P(R) corresponds to the set of all possible real-valued
functions y = f(x).
See also INFINITY; PEANO’S CURVE.
cardioid The heart-shaped curve traced by a point on
the circumference of one circle as it rolls around
another circle of equal size is called a cardioid. In
POLAR COORDINATES, the cardioid is given by an equation of the form r = a(1– cosθ) where a is the common
radii of the circles, and in CARTESIAN COORDINATES by
(x
2 + y
2 – ax)
2 = a
2 (x
2 + y
2 ). The PARAMETRIC EQUATIONS of the curve are x = acosθ(1 + cosθ) and y =
asinθ(1 + cosθ). The curve has area one-and-a-half
times the area of either generating circle, and perimeter
eight times the radius.
The cardioid was first studied extensively by Italian
mathematician Johann Castillon, who also coined its
name in 1741.
See also CYCLOID.
Cartesian coordinates (orthogonal coordinates, rectangular coordinates) One of the biggest breakthroughs in the development of mathematics occurred
when geometry and algebra were united through the
invention of the Cartesian coordinate system. Credited
to 17th-century French mathematician and philosopher
RENÉ DESCARTES (whose name Latinized reads Cartesius), Cartesian coordinates provide a means of representing each point in the plane via a pair of numbers.
One begins by selecting a fixed point O in the plane,
called the origin, and drawing through it two perpendicular number lines, called axes, one horizontal and one
vertical, and both with the point O at the zero position
on the line. It has become the convention to set the positive side of the horizontal number line to the right of O,
and the positive side of the vertical number line above
O, and to call the horizontal axis the x-axis, and the vertical one the y-axis. The Cartesian coordinates of a point
P in the plane is a pair of numbers (x,y) which then
describes the location of that point as follows:
The x-coordinate, or “abscissa,” is the horizontal distance of the point from O along the
horizontal axis. (A positive distance represents a point to the right of the vertical axis; a
negative distance one to the left.) The y-coordinate, or “ordinate,” is the vertical distance
of the point from O along the vertical axis. (A
positive distance represents a point located
above the horizontal axis, and a negative distance one located below.)
For example, if the bottom left corner of this page is
the origin of a Cartesian coordinate system, with xand y-axes marked in units of inches, then the point
with coordinates (4, 1) lies four inches to the right of
the left edge of the page, and one inch above the bottom of the page.
62 cardioid
This absurdity shows that there cannot be a
meaningful correspondence that assigns distinct elements of A to subsets of A after all.
Given an infinite set A, Cantor had thus shown
that the sets P(A), P(P(A)), P(P(P(A))), … form a neverending chain of increasingly larger infinite sets. Thus,
in a very definite sense, there are infinitely many different types of infinity. At the other end of the spectrum,
the study of denumerable sets shows that every infinite
set contains a denumerable subset. Thus of all the infinite sets, denumerable sets are the “smallest” type of
infinite sets. The CONTINUUM HYPOTHESIS asks whether
or not there is an infinite set with cardinality that lies
somewhere between that of N and P(N) = R.
One might suppose that P(R), the power set of the
set of all points on the real number line, is R
2
, the set
of all points in the plane, or, equivalently, that the
power set of the set of all points in the unit interval
[0,1] is the set of all points inside the unit square [0,1]
× [0,1]. Surprisingly, this is not the case: there are just
as many points in the unit square as there are in a unit
interval. This is seen as follows:
Associate to each point (x,y) with 0 ≤ x ≤ 1 and
0 ≤ y ≤ 1, each written as an infinite decimal
expansion, x = 0.x 1 x 2 x 3 … and y = 0.y 1 y 2 y 3 …,
the real number r = 0.x 1 y 1 x 2 y 2 x 3 y 3 … in the
interval [0,1], and, conversely, match each real
number r = 0.r 1 r 2 r 3 r 4 r 5 r 6 … with the point
(0.r 1 r 3 r 5 …, 0.r 2 r 4 r 6 …) in the unit square.
(Again there is a technical difficulty caused by those
real numbers that have two different decimal representations. For instance, one-half can be written both as
0.5000… and 0.4999… Mathematicians have shown
that this difficulty can be obviated.) It turns out that
P(R) corresponds to the set of all possible real-valued
functions y = f(x).
See also INFINITY; PEANO’S CURVE.
cardioid The heart-shaped curve traced by a point on
the circumference of one circle as it rolls around
another circle of equal size is called a cardioid. In
POLAR COORDINATES, the cardioid is given by an equation of the form r = a(1– cosθ) where a is the common
radii of the circles, and in CARTESIAN COORDINATES by
(x
2 + y
2 – ax)
2 = a
2 (x
2 + y
2 ). The PARAMETRIC EQUATIONS of the curve are x = acosθ(1 + cosθ) and y =
asinθ(1 + cosθ). The curve has area one-and-a-half
times the area of either generating circle, and perimeter
eight times the radius.
The cardioid was first studied extensively by Italian
mathematician Johann Castillon, who also coined its
name in 1741.
See also CYCLOID.
Cartesian coordinates (orthogonal coordinates, rectangular coordinates) One of the biggest breakthroughs in the development of mathematics occurred
when geometry and algebra were united through the
invention of the Cartesian coordinate system. Credited
to 17th-century French mathematician and philosopher
RENÉ DESCARTES (whose name Latinized reads Cartesius), Cartesian coordinates provide a means of representing each point in the plane via a pair of numbers.
One begins by selecting a fixed point O in the plane,
called the origin, and drawing through it two perpendicular number lines, called axes, one horizontal and one
vertical, and both with the point O at the zero position
on the line. It has become the convention to set the positive side of the horizontal number line to the right of O,
and the positive side of the vertical number line above
O, and to call the horizontal axis the x-axis, and the vertical one the y-axis. The Cartesian coordinates of a point
P in the plane is a pair of numbers (x,y) which then
describes the location of that point as follows:
The x-coordinate, or “abscissa,” is the horizontal distance of the point from O along the
horizontal axis. (A positive distance represents a point to the right of the vertical axis; a
negative distance one to the left.) The y-coordinate, or “ordinate,” is the vertical distance
of the point from O along the vertical axis. (A
positive distance represents a point located
above the horizontal axis, and a negative distance one located below.)
For example, if the bottom left corner of this page is
the origin of a Cartesian coordinate system, with xand y-axes marked in units of inches, then the point
with coordinates (4, 1) lies four inches to the right of
the left edge of the page, and one inch above the bottom of the page.
62 cardioid