equal to an 8 or a 9, and equal to 8 if
b2is any other number; set α3equal
to 1 if c3is equal to an 8 or a 9, and
equal to 8 if c3is any other number;
and so forth.
Then the number xdoes not appear on the list.
It is not the first number on the list, since xand
0.a1a2a3a4… differ in the first decimal place;
nor is it the second number in the list, since x
and 0. b1b2b3… differ in the second decimal
place; nor is it the third, fourth, or 107th num-
ber in the list. Thus from any list of real num-
bers, it is possible to construct another real
number that fails to be on the list.
Even if one were to include the number xconstructed
above on a new list of real numbers, one can repeat the
diagonal argument again to produce a new real number
ythat fails to be on the list. In this way, one can argue
that there are always “more” real numbers than can be
listed. The set of real numbers is thus of greater cardi-
nality than the set of rational numbers. (It is worth
commenting that, at first, it seems easier to simply con-
struct the real number x= 0.α1α2α3… by selecting α1
to be any digit different from α1, α2any digit different
from b2, α3any digit different from c3, and so forth.
Arbitrary choices, however, could lead to ambiguity
and damage the argument. For instance, the number x
produced could be 0.50000 … which already appears
on the list as 0.49999… The approach taken above
carefully obviates this concern.)
Cantor also proved that there is an infinitude of
infinite sets all larger than the infinite set of natural
numbers.
diameter The furthest distance between two points
on the boundary of a geometric figure is called the
diameter of the figure. For example, the diameter of a
square of side-length 1 is the distance between two
opposite corners of the square. This distance is √
–
2. An
equilateral triangle of side-length 1 has diameter equal
to 1. In this context, the diameter of an object is always
a number.
Sometimes the term diameter also refers to the line
segment itself connecting two boundary points of max-
imal distance apart. For example, the diameter of a cir-
cle is any line segment through the center of the circle
connecting two boundary points. A diameter of a
SPHERE
also passes through its center.
Circles and spheres are figures of
CONSTANT WIDTH
.
See also
DIAGONAL
.
Dido’s problem According to legend, in the year 800
B
.
C
.
E
., Princess Dido of Tyre fled her Phoenecian home-
land to free herself of the tyranny of her murderous
king brother. She crossed the Mediterranean and
sought to purchase land for a new city upon the shores
of northern Africa. Confronted with only prejudice and
distrust by the local inhabitants, she was given permis-
sion to purchase only as much land as could be sur-
rounded by a bull’s hide. The challenge to accept these
terms and still enclose enough land to found a city
became known as Dido’s problem.
The Roman poet Virgil (70–19
B
.
C
.
E
.), in his epic
work Aeneid, refers to the legend of Dido and her
clever solution to the problem. He claims that Dido cut
the hide into very thin strips and pieced them together
to form one very long strand, which she then used to
enclose a proportion of land of maximal area, as given
by the shape of a circle. (More precisely, with coastline
as part of the boundary, Dido formed a semicircle with
bull-hide strips.) The portion of land she consequently
purchased for a minimal price was indeed large enough
to build a city. According to Virgil, this story represents
the founding of the city of Carthage, which is now a
residential suburb of the city of Tunis.
In this story, Princess Dido solved the famous
ISOPERIMETRIC PROBLEM
:
Of all figures in the plane with a given perime-
ter, which encloses the largest area?
Mathematical analysis of this problem is difficult. It was
not until the late 19th century that mathematicians were
finally able to prove that the solution presented in this
ancient tale—namely, the circle—is the correct shape.
See also
ISOPERIMETRIC PROBLEM
.
difference In
ARITHMETIC
, the result of subtracting
one quantity from another is called the difference of the
two quantities. For example, the difference of 105 and
83 is 22. The minus sign is used to denote differences.
For instance, we write: 105 – 83 = 22. The difference
130 diameter