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 dis-
tinct elements of Ato subsets of Aafter all.
Given an infinite set A, Cantor had thus shown
that the sets P(A), P(P(A)), P(P(P(A))),… form a never-
ending chain of increasingly larger infinite sets. Thus,
in a very definite sense, there are infinitely many differ-
ent 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 infi-
nite 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 Nand P(N) = R.
One might suppose that P(R), the power set of the
set of all points on the real number line, is R2, 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.x1x2x3… and y= 0.y1y2y3…,
the real number r= 0.x1y1x2y2x3y3… in the
interval [0,1], and, conversely, match each real
number r= 0.r1r2r3r4r5r6… with the point
(0.r1r3r5…, 0.r2r4r6…) in the unit square.
(Again there is a technical difficulty caused by those
real numbers that have two different decimal represen-
tations. 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
; P
EANO
’
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 equa-
tion of the form r= a(1– cosθ) where ais the common
radii of the circles, and in C
ARTESIAN COORDINATES
by
(x2+ y2– ax)2= a2(x2+ y2). The
PARAMETRIC EQUA
-
TIONS
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, rect-
angular coordinates) One of the biggest break-
throughs 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
R
ENÉ
D
ESCARTES
(whose name Latinized reads Carte-
sius), Cartesian coordinates provide a means of repre-
senting each point in the plane via a pair of numbers.
One begins by selecting a fixed point Oin the plane,
called the origin, and drawing through it two perpendic-
ular number lines, called axes, one horizontal and one
vertical, and both with the point Oat the zero position
on the line. It has become the convention to set the posi-
tive 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 ver-
tical one the y-axis. The Cartesian coordinates of a point
Pin 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 hori-
zontal distance of the point from Oalong the
horizontal axis. (A positive distance repre-
sents a point to the right of the vertical axis; a
negative distance one to the left.) The y-coor-
dinate, or “ordinate,” is the vertical distance
of the point from Oalong the vertical axis. (A
positive distance represents a point located
above the horizontal axis, and a negative dis-
tance one located below.)
For example, if the bottom left corner of this page is
the origin of a Cartesian coordinate system, with x-
and 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 bot-
tom of the page.
62 cardioid