nyuuzyou/wikis huggingface dataset:
Properties of Integers

== Additive Group of Integers is Countably Infinite Abelian Group ==
=== Integer Addition is Closed ===
=== Integer Addition is Associative ===
=== Integer Addition Identity is Zero ===
=== Inverse for Integer Addition ===
=== Integer Addition is Commutative ===
=== Integers are Countably Infinite ===
== Integers under Addition form Totally Ordered Group ==
== Integers under Multiplication form Semigroup ==
=== Integer Multiplication is Closed ===
=== Integer Multiplication is Associative ===
== Integers under Multiplication form Monoid ==
=== Integer Multiplication Identity is One ===
== Integers under Multiplication form Countably Infinite Commutative Monoid ==
=== Integer Multiplication is Commutative ===
== Integers form Commutative Ring ==
=== Integers under Addition form Abelian Group ===
=== Integer Multiplication Distributes over Addition ===
=== Integer Multiplication has Zero ===
== Integers form Commutative Ring with Unity ==
=== Integer Multiplication has Identity Element ===
== Integers under Addition form Infinite Cyclic Group ==
== Integers form Integral Domain ==
=== Integers form Commutative Ring with Unity ===
=== Ring of Integers has no Zero Divisors ===
== Integers form Totally Ordered Ring ==
== Integers form Ordered Integral Domain ==
== Integers are Euclidean Domain ==
== Substructures and Superstructures ==
=== Additive Group of Integers is Normal Subgroup of Rationals ===
=== Additive Group of Integers is Normal Subgroup of Reals ===
=== Additive Group of Integers is Normal Subgroup of Complex ===
=== Integers form Subdomain of Rationals ===
=== Integers form Subdomain of Reals ===
== Also see ==
* Properties of Natural Numbers
* Properties of Rational Numbers
* Properties of Real Numbers
* Properties of Complex Numbers
* Properties of Hamiltonian Numbers
Category:Integers
Properties of Inverses of Commuting Elements

== Theorems ==
Let $\struct {S, \circ}$ be a monoid with identity element $e_S$.
=== Commutation with Inverse in Monoid ===
=== Commutation of Inverses in Monoid ===
=== Inverse of Commuting Pair ===
=== Conjugate of Commuting Elements ===
=== Product of Commuting Elements with Inverses ===
Category:Commutativity
Properties of Join

== Theorem ==
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\eta, \gamma$ be finite partitions of $\Omega$.
Then:
:$\map \sigma {\eta \vee \gamma} = \map \sigma \eta \vee \map \sigma \gamma$
where:
:$\map \sigma \cdot$ denotes the generated $\sigma$-algebra
:$\vee$ on the denotes the join of finite partitions
:$\vee$ on the denotes the join of finite sub-$\sigma$-algebras
== Proof ==
== Sources ==
* $4.1$: Partitions and Subalgebras
Category:Probability Theory
Category:Ergodic Theory
Properties of Legendre Symbol

== Theorem ==
Let $p$ be an odd prime.
Let $a \in \Z$.
Let $\paren {\dfrac a p}$ be the Legendre symbol::
:$\paren {\dfrac a p} := a^{\paren {\frac {p - 1} 2} } \pmod p$
=== Quadratic Character ===
:$\paren {\dfrac a p} = 0$ $a \equiv 0 \pmod p$
:$\paren {\dfrac a p} = 1$ $a$ is a quadratic residue mod $p$
:$\paren {\dfrac a p} = -1$ $a$ is a quadratic non-residue mod $p$.
=== Congruent Integers ===
If $a \equiv b \pmod p$, then:
:$\paren {\dfrac a p} = \paren {\dfrac b p}$
=== Multiplicative Nature ===
:$\paren {\dfrac {a b} p} = \paren {\dfrac a p} \paren {\dfrac b p}$
=== Square is Quadratic Residue ===
:$\paren {\dfrac {a^2} p} = 1$
== Proofs ==
=== Proof of Quadratic Character ===
::$\paren {\dfrac a p} = 0$ $a \equiv 0 \pmod p$:
Follows from Euler's Criterion for Quadratic Residue.
:$\paren {\dfrac a p} = 1$ $a$ is a quadratic residue mod $p$:
This follows directly from the definition of quadratic residue and Euler's Criterion for Quadratic Residue.
:$\paren {\dfrac a p} = -1$ $a$ is a quadratic non-residue mod $p$:
This follows directly from the definition of quadratic non-residue and Euler's Criterion for Quadratic Residue.
=== Proof of Congruent Integers ===
If $a \equiv b \pmod p$, then $\paren {\dfrac a p} = \paren {\dfrac b p}$:
This is just a statement of the quadratic character of congruent integers.
=== Proof of Multiplicative Nature ===
:$\paren {\dfrac {a b} p} = \paren {\dfrac a p} \paren {\dfrac b p}$:
Follows directly from the identity:
:$\paren {a b}^{\paren {\frac {p - 1} 2} } = a^{\paren {\frac {p - 1} 2} } b^{\paren {\frac {p - 1} 2} }$
=== Proof that Square is Quadratic Residue ===
:$\paren {\dfrac {a^2} p} = 1$:
Follows directly from the definition.
Alternatively, it also follows from the fact that the Legendre symbol is multiplicative.
Category:Legendre Symbol
Properties of Limit at Infinity of Real Function/Combined Sum Rule

== Theorem ==
Let $a, \alpha, \beta \in \R$. 
Let $f, g : \hointr a \infty \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then: 
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x + \beta \map g x} = \alpha L_1 + \beta L_2$
== Proof ==
From Properties of Limit at Positive Infinity of Real Function: Multiple Rule, we have: 
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x} = \alpha L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \paren {\beta \map f x} = \beta L_2$
From Properties of Limit at Positive Infinity of Real Function: Sum Rule, we have: 
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x + \beta \map g x} = \alpha L_1 + \beta L_2$
Category:Properties of Limit at Infinity of Real Function
Properties of Limit at Infinity of Real Function/Difference Rule

== Theorem ==
Let $a \in \R$. 
Let $f, g : \hointr a \infty \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x - \map g x} = L_1 - L_2$
== Proof ==
This follows immediately by plugging $\alpha = 1$ and $\beta = -1$ into the combined sum rule.
Category:Properties of Limit at Infinity of Real Function
Properties of Limit at Infinity of Real Function/Multiple Rule

== Theorem ==
Let $a, \alpha \in \R$. 
Let $f : \hointr a \infty \to \R$ be a real function such that: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then: 
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x} = \alpha L$
== Proof ==
If $\alpha = 0$, the result follows from Limit of Constant Function: Limit at $\infty$.
Otherwise, since: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L$
given $\epsilon > 0$ we can find a real number $M \ge 0$ such that: 
:$\ds \size {\map f x - L} < \frac \epsilon {\size \alpha}$ for $x \ge M$.
Then: 
:$\ds \size {\alpha \map f x - \alpha L} < \epsilon$ for $x \ge M$.
Since $\epsilon$ was arbitrary, we have:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f x} = \alpha L$
Category:Properties of Limit at Infinity of Real Function
Properties of Limit at Infinity of Real Function/Product Rule

== Theorem ==
Let $a \in \R$. 
Let $f, g : \hointr a \infty \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x \map g x} = L_1 L_2$
== Proof ==
Write: 
Since: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
there exists a real number $M_1 \ge 0$ such that: 
:$\size {\map f x - {L_1} } < 1$
for $x \ge M_1$.
By the Reverse Triangle Inequality, we then have: 
:$\size {\size {\map f x} - \size {L_1} } < 1$
so that:
:$\size {L_1} - 1 < \size {\map f x} < \size {L_1} + 1$
So we have, for $x \ge M_1$: 
:$\size {\map f x} \size {\map g x - L_2} + \size {L_2} \size {\map f x - L_1} < \paren {\size {L_1} + 1} \size {\map g x - L_2} + \size {L_2} \size {\map f x - L_1}$
Since: 
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
given $\epsilon > 0$ we can select a real number $M_2 \ge 0$ such that: 
:$\ds \size {\map g x - L_2} < \frac \epsilon {2 \paren {\size {L_1} + 1} }$ for $x \ge M_2$.
If $L_2 = 0$, we are done, since if $x \ge \max \set {M_1, M_2}$, we have: 
Then since $\epsilon$ was arbitrary, we then have: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x \map g x} = 0$
Now suppose that $L_2 \ne 0$. 
Since: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
given $\epsilon > 0$ we can pick a real number $M_3 \ge 0$ such that: 
:$\ds \size {\map f x - L_1} < \frac \epsilon {2 \size {L_2} }$
Let:
:$M = \max \set {M_1, M_2, M_3}$
Then, for $x \ge M$, we have: 
Category:Properties of Limit at Infinity of Real Function
Properties of Limit at Infinity of Real Function/Sum Rule

== Theorem ==
Let $a \in \R$. 
Let $f, g : \hointr a \infty \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
and:
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
where $\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$.
Then: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x + \map g x}$ exists
with:
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x + \map g x} = L_1 + L_2$
== Proof ==
Since: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L_1$
given $\epsilon > 0$, we can find $M_1 \ge 0$ such that: 
:$\ds \size {\map f x - L_1} < \frac \epsilon 2$ for $x \ge M_1$
Since: 
:$\ds \lim_{x \mathop \to \infty} \map g x = L_2$
we can find $M_2 \ge 0$ such that: 
:$\ds \size {\map f x - L_2} < \frac \epsilon 2$ for $x \ge M_2$.
Let:
:$M = \max \set {M_1, M_2}$
Then, for $x \ge M$, we have: 
Since $\epsilon$ was arbitrary, we have:
:$\ds \lim_{x \mathop \to \infty} \paren {\map f x + \map g x} = L_1 + L_2$
Category:Properties of Limit at Infinity of Real Function
Properties of Limit at Minus Infinity of Real Function/Combined Sum Rule

== Theorem ==
Let $a, \alpha, \beta \in \R$. 
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x + \beta \map g x}$ exists
with: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x + \beta \map g x} = \alpha \lim_{x \mathop \to \infty} \map f x + \beta \lim_{x \mathop \to \infty} \map g x$
== Proof ==
From Properties of Limit at Minus Infinity of Real Function: Multiple Rule:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x} = \alpha \lim_{x \mathop \to -\infty} \map f x$
and:
:$\ds \lim_{x \mathop \to -\infty} \paren {\beta \map g x}$ exists
with:
:$\ds \lim_{x \mathop \to -\infty} \paren {\beta \map g x} = \beta \lim_{x \mathop \to -\infty} \map g x$
From Properties of Limit at Minus Infinity of Real Function: Sum Rule, we then have: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x + \beta \map g x}$ exists
with: 
Category:Properties of Limit at Minus Infinity of Real Function
Properties of Limit at Minus Infinity of Real Function/Difference Rule

== Theorem ==
Let $a, \alpha, \beta \in \R$. 
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x - \map g x}$ exists
with: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x - \map g x} = \lim_{x \mathop \to \infty} \map f x - \lim_{x \mathop \to \infty} \map g x$
== Proof ==
This is the combined sum rule with $\alpha = 1$, $\beta = -1$.
Category:Properties of Limit at Minus Infinity of Real Function
Properties of Limit at Minus Infinity of Real Function/Multiple Rule

== Theorem ==
Let $a, \alpha \in \R$. 
Let $f : \hointl {-\infty} a \to \R$ be a real function such that: 
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x}$ exists
with:
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x} = \alpha \lim_{x \mathop \to -\infty} \map f x$
== Proof ==
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have: 
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists.
From Properties of Limit at Infinity of Real Function: Multiple Rule, we then have: 
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f {-x} }$ exists
with:
:$\ds \lim_{x \mathop \to \infty} \paren {\alpha \map f {-x} } = \alpha \lim_{x \mathop \to \infty} \map f {-x}$
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, this gives: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\alpha \map f x} = \alpha \lim_{x \mathop \to -\infty} \map f x$
Category:Properties of Limit at Minus Infinity of Real Function
Properties of Limit at Minus Infinity of Real Function/Product Rule

== Theorem ==
Let $a \in \R$. 
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x \map g x}$ exists
with: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x \map g x} = \paren {\lim_{x \mathop \to \infty} \map f x} \paren {\lim_{x \mathop \to \infty} \map g x}$
== Proof ==
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have: 
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ and $\ds \lim_{x \mathop \to \infty} \map g {-x}$ exist.
From Properties of Limit at Infinity of Real Function: Product Rule, we then have: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f {-x} \map g {-x} }$ exists
with: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f {-x} \map g {-x} } = \paren {\lim_{x \mathop \to \infty} \map f {-x} } \paren {\lim_{x \mathop \to \infty} \map g {-x} }$
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, this gives: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x \map g x} = \paren {\lim_{x \mathop \to \infty} \map f x} \paren {\lim_{x \mathop \to \infty} \map g x}$
Category:Properties of Limit at Minus Infinity of Real Function
Properties of Limit at Minus Infinity of Real Function/Relation with Limit at Infinity

== Theorem ==
Let $a \in \R$. 
Let $f : \hointl {-\infty} a \to \R$ be a real function.
Then:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists $\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists
and in this case:
:$\ds \lim_{x \mathop \to -\infty} \map f x = \lim_{x \mathop \to \infty} \map f {-x}$
where:
:$\ds \lim_{x \mathop \to \infty}$ denotes the limit at $+\infty$
:$\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
== Proof ==
Note that: 
:$\ds \lim_{x \mathop \to -\infty} \map f x = L$
 given $\epsilon > 0$ we can find real number $M \le 0$ such that: 
:$\size {\map f x - L} < \epsilon$ for $x \le M$. 
This is equivalent to: 
:given $\epsilon > 0$ we can find real number $M \le 0$ such that $\size {\map f x - L} < \epsilon$ for $x \ge -M$.
Since $\epsilon$ was arbitrary and $-M \ge 0$, we have:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ exists $\ds \lim_{x \mathop \to \infty} \map f {-x}$ exists
with: 
:$\ds \lim_{x \mathop \to \infty} \map f x = L = \lim_{x \mathop -\infty} \map f {-x}$
which was the demand.
Category:Properties of Limit at Minus Infinity of Real Function
Properties of Limit at Minus Infinity of Real Function/Sum Rule

== Theorem ==
Let $a \in \R$. 
Let $f, g : \hointl {-\infty} a \to \R$ be real functions such that: 
:$\ds \lim_{x \mathop \to -\infty} \map f x$ and $\ds \lim_{x \mathop \to -\infty} \map g x$ exist
where $\ds \lim_{x \mathop \to -\infty}$ denotes the limit at $-\infty$.
Then: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x + \map g x}$ exists
with: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x + \map g x} = \lim_{x \mathop \to \infty} \map f x + \lim_{x \mathop \to \infty} \map g x$
 
== Proof ==
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, we have: 
:$\ds \lim_{x \mathop \to \infty} \map f {-x}$ and $\ds \lim_{x \mathop \to \infty} \map g {-x}$ exist.
From Properties of Limit at Infinity of Real Function: Sum Rule, we then have: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f {-x} + \map g {-x} }$ exists
with: 
:$\ds \lim_{x \mathop \to \infty} \paren {\map f {-x} + \map g {-x} } = \lim_{x \mathop \to \infty} \map f {-x} + \lim_{x \mathop \to \infty} \map g {-x}$
From Properties of Limit at Minus Infinity of Real Function: Relation with Limit at Infinity, this gives: 
:$\ds \lim_{x \mathop \to -\infty} \paren {\map f x + \map g x} = \lim_{x \mathop \to \infty} \map f x + \lim_{x \mathop \to \infty} \map g x$
Category:Properties of Limit at Minus Infinity of Real Function
Properties of Linear Subspace of Finite Dimensional Scalar Product Space

== Theorem ==
Let $\struct {V, q}$ be a scalar product space.
Let $S \subseteq V$ be a linear subspace.
Then:
:$\dim S + \dim S^\perp = \dim V$
:$\paren {S^\perp}^\perp = S$
where $\dim$ denotes the dimension of vector space, and $S^\perp$ denotes the vector subspace perpendicular to $S$ with respect to $q$.
== Proof ==
== Sources ==
* : $\S 2$: Riemannian Metrics. Pseudo-Riemannian Metrics
Category:Scalar Product Spaces
Properties of Mapping on Class of All Ordinals

== Theorem ==
Let $\On$ denote the class of all ordinals.
Let $F$ be a mapping on $\On$.
For $\alpha \in \On$, let $F \restriction \alpha$ denote the restriction of $F$ to $\alpha$.
Then:
== Proof ==
== Sources ==
* : Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Remark
Category:Class of All Ordinals
Properties of Matrix Entrywise Addition over Ring

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $S$ over an algebraic structure $\struct {R, +, \circ}$.
Let $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$.
Let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
The operation of matrix entrywise addition satisfies the following properties:
:$+$ is closed on $\map {\MM_R} {m, n}$
:$+$ is associative on $\map {\MM_R} {m, n}$
:$+$ is commutative on $\map {\MM_R} {m, n}$.
== Proof ==
=== Matrix Entrywise Addition over Ring is Closed ===
=== Matrix Entrywise Addition over Ring is Associative ===
=== Matrix Entrywise Addition over Ring is Commutative ===
Category:Matrix Entrywise Addition
Properties of Matrix Exponential

== Theorem ==
In the following:
:$\mathbf A$ and $\mathbf B$ are constant square matrices of order $m$ for some $m \in \Z_{\ge 1}$
:$\mathbf P$ is a nonsingular square matrix of order $m$
:$t, s \in \R$ are arbitrary real numbers.
The matrix exponential $e^{\mathbf A t}$ has the following properties:
=== Derivative of Matrix Exponential ===
=== Determinant of Matrix Exponential is Non-Zero ===
=== Same-Matrix Product of Matrix Exponentials ===
=== Inverse of Matrix Exponential ===
=== Product with Matrix Exponential of Commutative Matrices ===
=== Matrix Exponential of Sum of Commutative Matrices ===
=== Series Expansion of Matrix Exponential ===
=== Decomposition of Matrix Exponential ===
Category:Matrix Exponential
Properties of Natural Logarithm

== Theorem ==
Let $x \in \R$ be a real number such that $x > 0$.
Let $\ln x$ be the natural logarithm of $x$.
Then:
=== Natural Logarithm of 1 is 0 ===
=== Natural Logarithm of e is 1 ===
=== Logarithm is Continuous ===
=== Derivative of Natural Logarithm Function ===
=== Logarithm is Strictly Increasing ===
=== Logarithm is Strictly Concave ===
=== Logarithm Tends to Infinity ===
=== Logarithm Tends to Negative Infinity ===
Category:Natural Logarithms
Properties of Natural Numbers

== Natural Numbers are Infinite ==
Note that by definition of countability, $\N$ is a countable set.
=== Natural Number Addition is Closed ===
=== Natural Number Addition is Associative ===
== Natural Numbers under Addition form Commutative Monoid ==
=== Natural Number Addition is Commutative ===
=== Identity Element of Natural Number Addition is Zero ===
== Non-Zero Natural Numbers under Multiplication form Commutative Monoid ==
=== Natural Number Multiplication is Closed ===
=== Natural Number Multiplication is Associative ===
=== Natural Number Multiplication is Commutative ===
=== Identity Element of Natural Number Multiplication is One ===
== Natural Numbers form Commutative Semiring ==
=== Natural Number Multiplication Distributes over Addition ===
== Non-Zero Natural Numbers form Commutative Semiring ==
== Natural Numbers form Subsemiring of Integers ==
== Natural Numbers Set Equivalent to Ideals of Integers ==
== Also see ==
* Properties of Integers
* Properties of Rational Numbers
* Properties of Real Numbers
* Properties of Complex Numbers
* Properties of Hamiltonian Numbers
Category:Natural Numbers
Properties of Norm on Division Ring

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x, y \in R$.
Then the following hold:
=== Norm of Unity ===
=== Norm of Negative of Unity ===
 
=== Norm of Negative ===
=== Norm of Difference ===
=== Norm of Inverse ===
=== Norm of Quotient ===
=== Norm of Power Equals Unity ===
=== Norm of Integer ===
== Sources ==
* : $\S 2.2$: Basic Properties: Lemma $2.2.1$
* : $\S 1.2$: Normed Fields: Theorem $1.6$
Category:Normed Division Rings
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Difference

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x, y \in R$.
Then:
:$\norm {x - y} \le \norm x + \norm y$
== Proof ==
Then:
as desired.
== Sources ==
* : $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(d)}$
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Integer

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
For all $n \in \N_{>0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is:
:$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{\text {$n$ times} }$
Then:
:$\norm {n \cdot 1_R} \le n$.
== Proof ==
Let $n \in \N_{>0}$.
Then:
== Sources ==
* : $\S 1.2$: Normed Fields, Theorem $1.6 \,(f)$
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Inverse

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$.
Let $\norm {\, \cdot \,}$ be a norm on $R$.
Let $x \in R$
Then:
:$x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$
== Proof ==
Let $x \ne 0_R$.
By :
:$\norm x \ne 0$
So:
== Sources ==
* : $\S 1.2$: Normed Fields, Theorem $1.6 \ \text{(e)}$
Category:Properties of Norm on Division Ring
Properties of Norm on Division Ring/Norm of Negative

== Theorem ==
Let $\struct {R, +, \circ}$ be a division ring with unity $1_R$.
Let $\norm {\,\cdot\,}$ be a norm on $R$.
Let $x \in R$.
Then:
:$\norm {-x} = \norm x$
== Proof ==
By Norm of Negative of Unity:
:$\norm {-1_R} = 1$
Then:
as desired.
== Sources ==
* : $\S 2.2$: Basic Properties, Lemma $2.2.1 \ \text {iv)}$
* : $\S 1.2$: Normed Fields, Theorem $1.6 \ \text {(b)}$
Category:Properties of Norm on Division Ring