possible to write the greatest common factor of two
numbers a and b in the form:
xa + yb
for some integers x and y. This fact is useful for solving
the famous JUG-FILLING PROBLEM, for example.
See also FUNDAMENTAL THEOREM OF ARITHMETIC.
Euclidean geometry The GEOMETRY based on the
definitions and AXIOMs set out in Euclid’s famous work
THE ELEMENTS is called Euclidean geometry. The
salient feature of this geometry is that the fifth postulate, the PARALLEL POSTULATE, holds. It follows from
this that through any point in the plane there is precisely one line through that point parallel to any given
direction, that all angles in a triangle sum to precisely
180°, and that the ratio of the circumference of any circle to its diameter is always the same value π.
Two-dimensional Euclidean geometry is called
plane geometry, and the three-dimensional Euclidean
geometry is called solid geometry. In 1899 German
mathematician DAVID HILBERT (1862–1943) proved
that the theory of Euclidean geometry is free from
CONTRADICTION.
See also EUCLID; EUCLID’S POSTULATES; HISTORY OF
GEOMETRY (essay); NON-EUCLIDEAN GEOMETRY.
Euclidean space (Cartesian space, n-space) The VECTOR SPACE of all n-TUPLES (x 1 ,x 2 ,…,x n ) of real numbers
x 1 , x 2 ,…,x n with the operations of addition and scalar
multiplication given by:
(x 1 , x 2 ,…, x n ) + (y 1 ,y 2 ,…,y n ) = (x 1 , + y 1 , x 2 + y 2 ,…,
x n + y n )
k(x 1 , x 2 ,…,x n ) = (kx 1 , kx 2 ,…,kx n )
and equipped with the notion of distance between
points x = (x 1 , x 2 ,…,x n ) and y = (y 1 , y 2 ,…,y n ) as given
by the DISTANCE FORMULA:
d(x,y) =
is called a Euclidean space.
Elements of a two-dimensional Euclidean space can
be identified with points in a plane relative to a set of
CARTESIAN COORDINATE axes. The vector space of all
n-tuples of COMPLEX NUMBERS under an analogous set
of operations is called a complex Euclidean space.
Euclid’s postulates EUCLID of Alexandria (ca.
300–260 B.C.E.) began his famous 13-volume piece
THE ELEMENTS with 23 definitions (of the ilk, “a point
is that which has no part” and “a line is that which has
no breadth”) followed by 10 AXIOMs divided into two
types: five common notions and five postulates. His
common notions were:
1. Things that are equal to the same thing are equal to
one another.
2. If equal things are added to equals, then the wholes
are equal.
3. If equal things are subtracted from equals, then the
remainders are equal.
4. Things that coincide with one another are equal to
one another.
5. The whole is greater than the part.
Euclid’s postulates were:
1. A straight line can be drawn to join any two points.
2. Any straight line segment can be extended to a
straight line of any length.
3. Given any straight line segment, it is possible to
draw a circle with center one endpoint and with the
straight line segment as the radius.
4. All right angles are equal to one another.
5. If two straight lines emanating from the endpoints of
a given line segment have interior angles on one given
side of the line segment summing to less than two
right angles, then the two lines, if extended, meet to
form a triangle on that side of the line segment.
(His fourth postulate is a statement about the homogeneity of space, that it is possible to translate figures to different locations without changing their basic structure.)
It is worth noting that Euclid deliberately avoided
any direct mention of the notion of infinity. His wording of the second postulate, for instance, avoids the
need to state that straight lines can be extended indefinitely, and his fifth postulate, also known as the PARALLEL POSTULATE, avoids direct mention of parallel lines,
that is, lines that never meet when extended indefinitely.
From these basic assumptions Euclid deduced, by
pure logical reasoning, 465 statements of truth (THEO√(x 1 – y 1 )
2 + (x 2 + y 2 )
2
+…+(x n – y n )
2
170 Euclidean geometry
numbers a and b in the form:
xa + yb
for some integers x and y. This fact is useful for solving
the famous JUG-FILLING PROBLEM, for example.
See also FUNDAMENTAL THEOREM OF ARITHMETIC.
Euclidean geometry The GEOMETRY based on the
definitions and AXIOMs set out in Euclid’s famous work
THE ELEMENTS is called Euclidean geometry. The
salient feature of this geometry is that the fifth postulate, the PARALLEL POSTULATE, holds. It follows from
this that through any point in the plane there is precisely one line through that point parallel to any given
direction, that all angles in a triangle sum to precisely
180°, and that the ratio of the circumference of any circle to its diameter is always the same value π.
Two-dimensional Euclidean geometry is called
plane geometry, and the three-dimensional Euclidean
geometry is called solid geometry. In 1899 German
mathematician DAVID HILBERT (1862–1943) proved
that the theory of Euclidean geometry is free from
CONTRADICTION.
See also EUCLID; EUCLID’S POSTULATES; HISTORY OF
GEOMETRY (essay); NON-EUCLIDEAN GEOMETRY.
Euclidean space (Cartesian space, n-space) The VECTOR SPACE of all n-TUPLES (x 1 ,x 2 ,…,x n ) of real numbers
x 1 , x 2 ,…,x n with the operations of addition and scalar
multiplication given by:
(x 1 , x 2 ,…, x n ) + (y 1 ,y 2 ,…,y n ) = (x 1 , + y 1 , x 2 + y 2 ,…,
x n + y n )
k(x 1 , x 2 ,…,x n ) = (kx 1 , kx 2 ,…,kx n )
and equipped with the notion of distance between
points x = (x 1 , x 2 ,…,x n ) and y = (y 1 , y 2 ,…,y n ) as given
by the DISTANCE FORMULA:
d(x,y) =
is called a Euclidean space.
Elements of a two-dimensional Euclidean space can
be identified with points in a plane relative to a set of
CARTESIAN COORDINATE axes. The vector space of all
n-tuples of COMPLEX NUMBERS under an analogous set
of operations is called a complex Euclidean space.
Euclid’s postulates EUCLID of Alexandria (ca.
300–260 B.C.E.) began his famous 13-volume piece
THE ELEMENTS with 23 definitions (of the ilk, “a point
is that which has no part” and “a line is that which has
no breadth”) followed by 10 AXIOMs divided into two
types: five common notions and five postulates. His
common notions were:
1. Things that are equal to the same thing are equal to
one another.
2. If equal things are added to equals, then the wholes
are equal.
3. If equal things are subtracted from equals, then the
remainders are equal.
4. Things that coincide with one another are equal to
one another.
5. The whole is greater than the part.
Euclid’s postulates were:
1. A straight line can be drawn to join any two points.
2. Any straight line segment can be extended to a
straight line of any length.
3. Given any straight line segment, it is possible to
draw a circle with center one endpoint and with the
straight line segment as the radius.
4. All right angles are equal to one another.
5. If two straight lines emanating from the endpoints of
a given line segment have interior angles on one given
side of the line segment summing to less than two
right angles, then the two lines, if extended, meet to
form a triangle on that side of the line segment.
(His fourth postulate is a statement about the homogeneity of space, that it is possible to translate figures to different locations without changing their basic structure.)
It is worth noting that Euclid deliberately avoided
any direct mention of the notion of infinity. His wording of the second postulate, for instance, avoids the
need to state that straight lines can be extended indefinitely, and his fifth postulate, also known as the PARALLEL POSTULATE, avoids direct mention of parallel lines,
that is, lines that never meet when extended indefinitely.
From these basic assumptions Euclid deduced, by
pure logical reasoning, 465 statements of truth (THEO√(x 1 – y 1 )
2 + (x 2 + y 2 )
2
+…+(x n – y n )
2
170 Euclidean geometry