Definition:Prime number
A
natural number
is called a prime number if it is only
divisible
by
![{\displaystyle {}1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e1b23db5ab6d3c02520ce17d0a7fdabddc8f0f)
and by
![{\displaystyle {}n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89f1551c84124ced21a5db61d4add476ed93e589)
.
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
![{\displaystyle {}M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/721df7cbe87df695d471a9ee60ad739e3614e51c)
if every element of
![{\displaystyle {}T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b187f8e4d3fd9d0ab9bb26cdb70587a158e5851c)
is also an element of
![{\displaystyle {}M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/721df7cbe87df695d471a9ee60ad739e3614e51c)
.
Definition:Intersection
For sets
und
, we call
-
![{\displaystyle {}L\cap M={\left\{x\mid x\in L{\text{ and }}x\in M\right\}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d86f8a969dc6c2ab11b7ca1e99141fcb6326e1e)
the
intersection
of the two sets.
Definition:Union
For sets
und
, we call
-
![{\displaystyle {}L\cup M={\left\{x\mid x\in L{\text{ or }}x\in M\right\}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b19bdf85026010cf1cf172cae95e0d45e03316)
the
union
of the sets.
Definition:Cartesian product
Suppose that two sets
and
are given. Then the set
-
![{\displaystyle {}L\times M={\left\{(x,y)\mid x\in L,\,y\in M\right\}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10d5c2beafa55e8a77710fd3241d5893e2b1b6f8)
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
![{\displaystyle {}F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93572d890f704d0126d032045604fd170e0fba8f)
.
Definition:Composition
Let
and
denote sets, let
-
and
-
be
mappings.
Then the mapping
-
is called the composition of the mappings
![{\displaystyle {}F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93572d890f704d0126d032045604fd170e0fba8f)
and
![{\displaystyle {}G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aade5636c348f4cb8dd8221e2b587abaeb68bfa9)
.
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
-
![{\displaystyle {}n!:=n(n-1)(n-2)\cdots 3\cdot 2\cdot 1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9cac001f197c59d7476e015b672c77ca265cd0)
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.
-
![{\displaystyle {}\vert {x}\vert ={\begin{cases}x\,,{\text{ if }}x\geq 0\,,\\-x,\,{\text{ if }}x<0\,.\end{cases}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc72946d153a061714d315f145c6b86daddc084)
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
-
![{\displaystyle {}(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4db10a2c8918727b2f4942eaf17e01dcf130c34)
is called the field of complex numbers. We denote it by
-
Definition:Real part, imaginary part
For a
complex number
-
![{\displaystyle {}z=a+b{\mathrm {i} }\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18259c503c13bdcd7c278516ee4b91428927e910)
we call
-
![{\displaystyle {}\operatorname {Re} \,{\left(z\right)}=a\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a45233d1cdba651e4f289ae7e55ca540f26f59)
the real part of
and
-
![{\displaystyle {}\operatorname {Im} \,{\left(z\right)}=b\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e2ee32eea0e9bb4ee8d39fcb3f455697f04f9d)
the
imaginary part of
![{\displaystyle {}z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caf5321978681d2ca477978ee8c4d5e0e08b795f)
.
Definition:Complex conjugation
The
mapping
-
is called
complex conjugation.
Definition:Modulus of a complex number
For a
complex number
-
![{\displaystyle {}z=a+b{\mathrm {i} }\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18259c503c13bdcd7c278516ee4b91428927e910)
the modulus is defined by
-
![{\displaystyle {}\vert {z}\vert ={\sqrt {a^{2}+b^{2}}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81d50244c8a92f8c291bb059d209043c4e043561)
Definition:Polynomial in one variable
Let
be a
field.
An expression of the form
-
![{\displaystyle {}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a88a7744cc357b4e12a75640a708d171300bf63)
with
and
,
is called a
polynomial in one variable over
![{\displaystyle {}K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/812534e3ff684d765e18cc3438e50dc85b2a8796)
.
Definition:Degree of a polynomial
The degree of a nonzero polynomial
-
![{\displaystyle {}P=a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{n}X^{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1081f30196097cc70793fd2ef2012efb478545e)
with
is
![{\displaystyle {}n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89f1551c84124ced21a5db61d4add476ed93e589)
.
Definition:Rational function
For
polynomials
,
,
the
function
-
where
is the
complement
of the
zeroes
of
![{\displaystyle {}Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3925649ad5dc923b22e56355a5d6a466a39cb7a)
, 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
-
![{\displaystyle {}\vert {x_{n}-x}\vert \leq \epsilon \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c779a96c4f076e413c32182fde8ca7d6bee693)
holds.
If this condition is fulfilled, then
is called the limit of the sequence. For this we write
-
![{\displaystyle {}\lim _{n\rightarrow \infty }x_{n}:=x\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/681f36f4a2211a4ac6dd93de0797980e3b6eca60)
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
![{\displaystyle {}M\subseteq [s,S]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2da93df0ec2dd28ea51d22e2df040f8587d61771)
.
Definition:Increasing sequence
A
real sequence
is called increasing, if
holds for all
![{\displaystyle {}n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b24d1aed8fb8b6c63d76258c4bf62224094ceec4)
.
Definition:Decreasing sequence
A
real sequence
is called decreasing, if
holds for all
![{\displaystyle {}n\in \mathbb {N} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b24d1aed8fb8b6c63d76258c4bf62224094ceec4)
.
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
![{\displaystyle {}0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5428e3b06006771c083bd17ed8fce8f3be334b2)
.
Definition:Tending to
Definition:Tending to
Definition:Series
Let
be a
sequence
of
real numbers.
The series
is the sequence
of the partial sums
-
![{\displaystyle {}s_{n}:=\sum _{k=0}^{n}a_{k}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83794115acdd506db184d9d181afc4b8bc53ebdb)
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
![{\displaystyle {}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cef7035ba3d8882f7b3f26329ab9fb9641f5ab)
.
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
![{\displaystyle {}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cef7035ba3d8882f7b3f26329ab9fb9641f5ab)
for the coefficients
![{\displaystyle {}{\left(c_{n}\right)}_{n\in \mathbb {N} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b9926f9a511b73bd1c1b1e26dc02fe15f12a9ed)
.
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
![{\displaystyle {}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cef7035ba3d8882f7b3f26329ab9fb9641f5ab)
.
Definition:Exponential function
The
function
-
is called the (real)
exponential function.
Definition:Euler's number
The real number
-
![{\displaystyle {}e:=\sum _{k=0}^{\infty }{\frac {1}{k!}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9b96b430efb722a9aafc7d3da172a9e1d49542)
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
-
![{\displaystyle {}\sinh x:={\frac {1}{2}}{\left(e^{x}-e^{-x}\right)}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e5ce4cd2ae8cd94c09058668e91fdfdfc3ac70)
is called
hyperbolic sine.
Definition:Hyperbolic cosine
The function defined for
by
-
![{\displaystyle {}\cosh x:={\frac {1}{2}}{\left(e^{x}+e^{-x}\right)}\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebcdd8a666104619030833265e4e3242c88e38f)
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
-
![{\displaystyle {}f(x)=f(-x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2a785d34f9ab073c9108aa5697435020379084)
holds.
Definition:Odd function
A
function
is called odd, if for all
,
the identity
-
![{\displaystyle {}f(x)=-f(-x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a429f0028891e4ed606f69716ee46d369cf78e)
holds.