Definition:Prime number
A
natural number
is called a prime number if it is only
divisible
by
and by
.
Definition:Empty set
The set which does not contain any element is called the empty set, denoted by
-
Definition:Subset
Let
and
denote sets. is called a
subset
of
if every element of
is also an element of
.
Definition:Intersection
For sets
und , we call
-
the
intersection
of the two sets.
Definition:Union
For sets
und , we call
-
the
union
of the sets.
Definition:Cartesian product
Suppose that two sets
and
are given. Then the set
-
is called the
product set of the sets.
Definition:Inverse mapping
Let
denote a
bijective mapping.
Then the mapping
-
which sends every element
to the uniquely determined element
with
,
is called the
inverse mapping of
.
Definition:Composition
Let
and
denote sets, let
-
and
-
be
mappings.
Then the mapping
-
is called the composition of the mappings
and
.
Definition:Field
A set is called a field if there are two
binary operations
(called addition and multiplication)
-
and two different elements
,
which fulfill the following properties.
- Axioms for the addition:
- Law of associativity:
holds for all
.
- Law of commutativity:
holds for all
.
- is the neutral element of the addition, i.e.
holds for all
.
- Existence of the negative: For every
,
there exists an element
with
.
- Axioms of the multiplication:
- Law of associativity:
holds for all
.
- Law of commutativity:
holds for all
.
- is the neutral element for the multiplication, i.e.
holds for all
.
- Existence of the inverse: For every
with
,
there exists an element
such that
.
- Law of distributivity:
holds for all
.
Definition:Factorial
For a natural number , one puts
-
and calls this
factorial.
Definition:Binomial coefficient
Definition:Ordered field
A
field
is called an ordered field, if there is a relation
(larger than)
between the elements of , fulfilling the following properties
(
means
or
).
- For two elements
,
we have either
or
or
.
- From
and
,
one may deduce
(for any
).
-
implies
(for any
).
- From
and
,
one may deduce
(for any
).
Definition:Archimedean ordered field
Definition:Real intervals
For real numbers
, ,
we call
-
the closed interval.
-
the open interval.
-
the half-open interval (closed on the right).
-
the half-open interval (closed on the left).
Definition:Modulus of a real number
For a real number
,
the modulus is defined in the following way.
-
Definition:Increasing function
Definition:Decreasing function
Definition:Strictly increasing function
Definition:Strictly decreasing function
Definition:Complex numbers
The set
with
and ,
with componentwise addition and the multiplication defined by
-
is called the field of complex numbers. We denote it by
-
Definition:Real part, imaginary part
For a
complex number
-
we call
-
the real part of and
-
the
imaginary part of
.
Definition:Complex conjugation
The
mapping
-
is called
complex conjugation.
Definition:Modulus of a complex number
For a
complex number
-
the modulus is defined by
-
Definition:Polynomial in one variable
Let be a
field.
An expression of the form
-
with
and
,
is called a
polynomial in one variable over
.
Definition:Degree of a polynomial
The degree of a nonzero polynomial
-
with
is
.
Definition:Rational function
For
polynomials
, ,
the
function
-
where is the
complement
of the
zeroes
of
, is called a
rational function.
Definition:Real sequence
A real sequence is a
mapping
-
Definition:Heron sequence
Definition:Convergent sequence
Let denote a
real sequence,
and let
.
We say that the sequence converges to , if the following property holds.
For every positive
, ,
there exists some
,
such that for all
,
the estimate
-
holds.
If this condition is fulfilled, then is called the limit of the sequence. For this we write
-
If the sequence converges to a limit, we just say that the sequence converges, otherwise, that the sequence
diverges.
Definition:Bounded subset
A subset
of the real numbers is called
bounded,
if there exist real numbers
such that
.
Definition:Increasing sequence
A
real sequence
is called increasing, if
holds for all
.
Definition:Decreasing sequence
A
real sequence
is called decreasing, if
holds for all
.
Definition:Cauchy sequence
Definition:Subsequence
Let be a
real sequence.
For any
strictly increasing
mapping
,
the sequence
-
is called a
subsequence of the sequence.
Definition:Completely ordered field
An
ordered field
is called complete or completely ordered, if every
Cauchy sequence
in
converges.
Definition:Nested intervals
A sequence of
closed intervals
-
in is called
(a sequence of)
nested intervals, if
holds for all
,
and if the sequence of the lengths of the intervals, i.e.
-
converges
to
.
Definition:Tending to
Definition:Tending to
Definition:Series
Let be a
sequence
of
real numbers.
The series is the sequence of the partial sums
-
If the sequence
converges,
then we say that the series converges. In this case, we write also
-
for its
limit,
and this limit is called the
sum of the series.
Definition:Absolute convergence of a series
A
series
-
of
real numbers
is called absolutely convergent, if the series
-
converges.
Definition:Geometric series
For every
,
the
series
-
is called the
geometric series in
.
Definition:Continuous function
Definition:Limit of a function
Definition:Power series
Let be a sequence of
real numbers
and another real number. Then the
series
-
is called the
power series in
for the coefficients
.
Definition:Cauchy product
For two
series
and
of
real numbers,
the series
-
is called the
Cauchy-product of the series.
Definition:Exponential series
For every
,
the
series
-
is called the
exponential series in
.
Definition:Exponential function
The
function
-
is called the (real)
exponential function.
Definition:Euler's number
The real number
-
is called
Euler's number.
Definition:Natural logarithm
The natural logarithm
-
is defined as the
inverse function
of the
real exponential function.
Definition:Exponential function to base
Definition:Logarithm to base
Definition:Hyperbolic sine
The function defined for
by
-
is called
hyperbolic sine.
Definition:Hyperbolic cosine
The function defined for
by
-
is called
hyperbolic cosine.
Definition:Hyperbolic tangent
The function
-
is called
hyperbolic tangent.
Definition:Even function
A
function
is called even, if for all
,
the identity
-
holds.
Definition:Odd function
A
function
is called odd, if for all
,
the identity
-
holds.