we can conclude, by deductive reasoning, that Daisy
eats grass.
Deductive reasoning does not rely on the premises
that are made necessarily being true. For example,
“Sydney and Boston are planets, therefore Boston is a
planet” is a valid argument, whereas “Either Boston or
Venus is a planet, therefore Venus is a planet” is invalid.
Mathematicians are not satisfied with conclusions
drawn via inductive reasoning only. They always seek
logical proof to conjectures made. But this certainly
does not bar mathematicians from making conjectures.
For instance, GOLDBACH’S CONJECTURE is an example
of an outstanding conjecture still awaiting mathematical proof (or disproof).
deformation In TOPOLOGY, any geometric transformation that stretches, shrinks, or twists a shape, but
does not tear or break apart any lines or surfaces that
make the shape, is called a (continuous) deformation.
For example, it is possible to mold a solid spherical ball
made of clay into the shape of a cube without tearing
any portions of the clay. In this sense, a cube may be
considered a deformation of a sphere. It is not possible,
however, to mold a sphere into the shape of a TORUS
(donut) without creating a tear. Topologists consequently regard a sphere and a torus as distinct shapes
(but a cube and a sphere as the “same” surface).
The notion of a deformation can be made mathematically precise. If, for a fixed set S, one object A is
the image of a map f, and a second object B is the
image of a second map g:
f : S → A
g : S → B
then B is a deformation of A if there is a continuous
function H(s,t) where s ∈S and 0 ≤ t ≤ 1, so that H(s,0)
is the map f, and H(s,1) is the map g. One also says
that the map H “deforms A into B.”
For example, the function H(x,t) = t cos x + (1 – t)
sin x continuously transforms a sine curve into a
cosine curve.
degree measure See ANGLE.
degree of a polynomial The highest power of the
variable that appears (with nonzero COEFFICIENT) in a
POLYNOMIAL is called the degree of that polynomial. For
instance, the polynomial 4x
3 – 2x + 7 has degree three,
and the polynomial 7w
57 – 154w
18 + w
5 – 73w
4 + πw
2
has degree 57. Any nonzero constant can be thought
of as a polynomial of degree zero. In some mathematical problems it is convenient to regard the constant 0
as a polynomial of degree “negative infinity.” A
POWER SERIES, in some sense, is a polynomial of positive infinite degree.
degree of a vertex (valence) In any GRAPH, the
number of edges meeting at a particular vertex is called
the degree of that vertex. Summing all the degrees of
vertices in a graph counts the total number of edges
twice. The famous HANDSHAKE LEMMA from GRAPH
THEORY is an amusing consequence of this result.
degrees of freedom The number of independent
variables needed to specify completely the solution set
of a SYSTEM OF EQUATIONS is called the number of
degrees of freedom of the system. For example, the
mathematical system described by the equations:
3x + 2y – z = 7
x + 4y – 3z = 6
has just one degree of freedom: if the value of z is specified, then x and y are given by x = (8–z)/5 and
y = (11+8z)/10.
In physics, the number of degrees of freedom of a
mechanical system is the minimum number of coordinates required to describe the state of the system at any
instant relative to a fixed frame of reference. For
instance, a particle moving in a circle has one degree of
freedom: its position is completely specified by the
angle between a fixed line of reference and the line connecting the center of the circle to the particle. A particle
moving in a PLANE, or on the surface of a SPHERE, has
two degrees of freedom.
See also INDETERMINATE EQUATION.
De Moivre, Abraham (1667–1754) French Geometry, Statistics Born on May 26, 1667, French scholar
Abraham De Moivre is remembered for his pioneering
work in the development of analytic geometry and the
theory of probability. He was the first to introduce
De Moivre, Abraham 121
eats grass.
Deductive reasoning does not rely on the premises
that are made necessarily being true. For example,
“Sydney and Boston are planets, therefore Boston is a
planet” is a valid argument, whereas “Either Boston or
Venus is a planet, therefore Venus is a planet” is invalid.
Mathematicians are not satisfied with conclusions
drawn via inductive reasoning only. They always seek
logical proof to conjectures made. But this certainly
does not bar mathematicians from making conjectures.
For instance, GOLDBACH’S CONJECTURE is an example
of an outstanding conjecture still awaiting mathematical proof (or disproof).
deformation In TOPOLOGY, any geometric transformation that stretches, shrinks, or twists a shape, but
does not tear or break apart any lines or surfaces that
make the shape, is called a (continuous) deformation.
For example, it is possible to mold a solid spherical ball
made of clay into the shape of a cube without tearing
any portions of the clay. In this sense, a cube may be
considered a deformation of a sphere. It is not possible,
however, to mold a sphere into the shape of a TORUS
(donut) without creating a tear. Topologists consequently regard a sphere and a torus as distinct shapes
(but a cube and a sphere as the “same” surface).
The notion of a deformation can be made mathematically precise. If, for a fixed set S, one object A is
the image of a map f, and a second object B is the
image of a second map g:
f : S → A
g : S → B
then B is a deformation of A if there is a continuous
function H(s,t) where s ∈S and 0 ≤ t ≤ 1, so that H(s,0)
is the map f, and H(s,1) is the map g. One also says
that the map H “deforms A into B.”
For example, the function H(x,t) = t cos x + (1 – t)
sin x continuously transforms a sine curve into a
cosine curve.
degree measure See ANGLE.
degree of a polynomial The highest power of the
variable that appears (with nonzero COEFFICIENT) in a
POLYNOMIAL is called the degree of that polynomial. For
instance, the polynomial 4x
3 – 2x + 7 has degree three,
and the polynomial 7w
57 – 154w
18 + w
5 – 73w
4 + πw
2
has degree 57. Any nonzero constant can be thought
of as a polynomial of degree zero. In some mathematical problems it is convenient to regard the constant 0
as a polynomial of degree “negative infinity.” A
POWER SERIES, in some sense, is a polynomial of positive infinite degree.
degree of a vertex (valence) In any GRAPH, the
number of edges meeting at a particular vertex is called
the degree of that vertex. Summing all the degrees of
vertices in a graph counts the total number of edges
twice. The famous HANDSHAKE LEMMA from GRAPH
THEORY is an amusing consequence of this result.
degrees of freedom The number of independent
variables needed to specify completely the solution set
of a SYSTEM OF EQUATIONS is called the number of
degrees of freedom of the system. For example, the
mathematical system described by the equations:
3x + 2y – z = 7
x + 4y – 3z = 6
has just one degree of freedom: if the value of z is specified, then x and y are given by x = (8–z)/5 and
y = (11+8z)/10.
In physics, the number of degrees of freedom of a
mechanical system is the minimum number of coordinates required to describe the state of the system at any
instant relative to a fixed frame of reference. For
instance, a particle moving in a circle has one degree of
freedom: its position is completely specified by the
angle between a fixed line of reference and the line connecting the center of the circle to the particle. A particle
moving in a PLANE, or on the surface of a SPHERE, has
two degrees of freedom.
See also INDETERMINATE EQUATION.
De Moivre, Abraham (1667–1754) French Geometry, Statistics Born on May 26, 1667, French scholar
Abraham De Moivre is remembered for his pioneering
work in the development of analytic geometry and the
theory of probability. He was the first to introduce
De Moivre, Abraham 121