equal to an 8 or a 9, and equal to 8 if
b 2 is any other number; set α 3 equal
to 1 if c 3 is equal to an 8 or a 9, and
equal to 8 if c 3 is any other number;
and so forth.
Then the number x does not appear on the list.
It is not the first number on the list, since x and
0.a 1 a 2 a 3 a 4 … differ in the first decimal place;
nor is it the second number in the list, since x
and 0. b 1 b 2 b 3 … differ in the second decimal
place; nor is it the third, fourth, or 107th number in the list. Thus from any list of real numbers, it is possible to construct another real
number that fails to be on the list.
Even if one were to include the number x constructed
above on a new list of real numbers, one can repeat the
diagonal argument again to produce a new real number
y that 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 cardinality than the set of rational numbers. (It is worth
commenting that, at first, it seems easier to simply construct the real number x = 0.α 1 α 2 α 3 … by selecting α 1
to be any digit different from α 1 , α 2 any digit different
from b 2 , α 3 any digit different from c 3 , 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 maximal distance apart. For example, the diameter of a circle 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 homeland 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 permission to purchase only as much land as could be surrounded 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 perimeter, 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
b 2 is any other number; set α 3 equal
to 1 if c 3 is equal to an 8 or a 9, and
equal to 8 if c 3 is any other number;
and so forth.
Then the number x does not appear on the list.
It is not the first number on the list, since x and
0.a 1 a 2 a 3 a 4 … differ in the first decimal place;
nor is it the second number in the list, since x
and 0. b 1 b 2 b 3 … differ in the second decimal
place; nor is it the third, fourth, or 107th number in the list. Thus from any list of real numbers, it is possible to construct another real
number that fails to be on the list.
Even if one were to include the number x constructed
above on a new list of real numbers, one can repeat the
diagonal argument again to produce a new real number
y that 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 cardinality than the set of rational numbers. (It is worth
commenting that, at first, it seems easier to simply construct the real number x = 0.α 1 α 2 α 3 … by selecting α 1
to be any digit different from α 1 , α 2 any digit different
from b 2 , α 3 any digit different from c 3 , 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 maximal distance apart. For example, the diameter of a circle 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 homeland 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 permission to purchase only as much land as could be surrounded 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 perimeter, 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