nyuuzyou/wikis huggingface dataset:
Multiple Angle Formula for Sine

== Theorem ==
:$\ds \frac {\map \sin {n z} } {\sin z} = 2^{n - 1} \prod_{k \mathop = 1}^{n - 1} \paren {\cos z - \cos \frac {k \pi} n}$
for $\sin z \ne 0$.
== Proof ==
We have:
Consider the equation:
:$x^n - 1 = 0$
whose solutions are the complex roots of unity:
:$1, e^{-2 \pi i / n}, e^{-4 \pi i / n}, e^{-6 \pi i / n}, \ldots, e^{-2 \paren {n - 1} \pi i / n}$
Then:
Let $x = e^{2 i z}$.
Then:
Plugging $(2)$ into $(1)$, we have:
We now notice:
And also:
Substituting into $\paren {3}$ above, we obtain:
From $(2)$ above, we have:
Therefore:
== Also see ==
* Product Formula for Sine
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $161 \ \text{(a)}$
Category:Sine Function
Category:Complex Roots of Unity
Multiple Angle Formula for Tangent

== Theorem ==
:$\ds \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\floor {\frac n 2} } \paren {-1}^i \binom n {2 i} \tan^{2 i}\theta}$
== Proof ==
Proof by induction:
For all $n \in \N_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\floor {\frac n 2} } \paren {-1}^i \binom n {2 i} \tan^{2 i}\theta}$
=== Basis for the Induction ===
$\map P 0$ is the case:
and so can be seen to hold.
$\map P 1$ is the case:
and so is also seen to hold.
These two cases together form the basis for the induction.
=== Induction Hypothesis ===
Now we need to show that, if $\map P {k - 2}$ and $\map P {k - 1}$ is true, where $k > 2$ is an even number, then it logically follows that $\map P k$ and $\map P {k + 1}$ is true.
So this is our induction hypothesis:
:$\ds \map \tan {\paren {k - 2} \theta} = \frac {\ds \sum_{i \mathop = 0}^{\frac k 2 - 2} \paren {-1}^i \binom {k - 2} {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\frac k 2 - 1} \paren {-1}^i \binom {k - 2} {2 i} \tan^{2 i}\theta}$
:$\ds \map \tan {\paren {k - 1} \theta} = \frac {\ds \sum_{i \mathop = 0}^{\frac k 2 - 1} \paren {-1}^i \binom {k - 1} {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\frac k 2 - 1} \paren {-1}^i \binom {k - 1} {2 i} \tan^{2 i}\theta}$
Then we need to show:
:$\ds \map \tan {k \theta} = \frac {\ds \sum_{i \mathop = 0}^{\frac k 2 - 1} \paren {-1}^i \binom k {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\frac k 2} \paren {-1}^i \binom k {2 i} \tan^{2 i}\theta}$
:$\ds \map \tan {\paren {k + 1} \theta} = \frac {\ds \sum_{i \mathop = 0}^{\frac k 2} \paren {-1}^i \binom {k + 1} {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\frac k 2} \paren {-1}^i \binom {k + 1} {2 i} \tan^{2 i}\theta}$
=== Induction Step ===
This is our induction step:
For the first part:
For the second part:
So $\map P {k - 2} \land \map P {k - 1} \implies \map P k \land \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\ds \forall n \in \N: \map \tan {n \theta} = \frac {\ds \sum_{i \mathop = 0}^{\floor{\frac {n - 1} 2} } \paren {-1}^i \binom n {2 i + 1} \tan^{2 i + 1}\theta} {\ds \sum_{i \mathop = 0}^{\floor {\frac n 2} } \paren {-1}^i \binom n {2 i} \tan^{2 i}\theta}$
== Also see ==
* $n = 2$: Double Angle Formula for Tangent
* $n = 3$: Triple Angle Formula for Tangent
* $n = 4$: Quadruple Angle Formula for Tangent
* $n = 5$: Quintuple Angle Formula for Tangent
Category:Tangent Function
Multiple Angle Formulas

== Trigonometric Identities ==
=== Double Angle Formulas ===
==== Double Angle Formula for Sine ====
==== Double Angle Formula for Cosine ====
==== Double Angle Formula for Tangent ====
=== Triple Angle Formulas ===
==== Triple Angle Formula for Sine ====
==== Triple Angle Formula for Cosine ====
==== Triple Angle Formula for Tangent ====
=== Quadruple Angle Formulas ===
==== Quadruple Angle Formula for Sine ====
==== Quadruple Angle Formula for Cosine ====
==== Quadruple Angle Formula for Tangent ====
=== Quintuple Angle Formulas ===
==== Quintuple Angle Formula for Sine ====
==== Quintuple Angle Formula for Cosine ====
==== Quintuple Angle Formula for Tangent ====
=== Sextuple Angle Formulas ===
==== Sextuple Angle Formula for Sine ====
==== Sextuple Angle Formula for Cosine ====
==== Sextuple Angle Formula for Tangent ====
=== Septuple Angle Formulas ===
==== Septuple Angle Formula for Cosine ====
</onlyinclude>
== Also known as ==
== Sources ==
* 
* 
Category:Multiple Angle Formulas
Category:Trigonometric Identities
Multiple Correlation Coefficient has Highest Correlation

== Theorem ==
Let $\expect y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 \cdots$
Let $R$ denote the multiple correlation coefficient.
No other linear function of the $x_i$ has a higher correlation with the $y_i$ than $R$.
== Proof ==
== Sources ==
* 
* 
Category:Multiple Correlation Coefficients
Multiple Function on Ring is Homomorphism

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring.
Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:
:$\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$
where $\cdot$ denotes the multiple operation.
Then:
:$g_a$ is a group homomorphism from $\struct {\Z, +}$ to $\struct {R, +}$.
== Proof ==
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 1^\circ$
Category:Group Homomorphisms
Category:Integers
Category:Ring Theory
Multiple Function on Ring is Zero iff Characteristic is Divisor

== Theorem ==
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.
Let the characteristic of $R$ be $p$.
Let $a \in R$ such that $a$ is not a zero divisor of $R$.
Then:
:$n \cdot a = 0_R$
:
:$p \divides n$
where $\cdot$ denotes the multiple operation.
== Proof ==
Let $g_a: \Z \to R$ be the mapping from the integers into $R$ defined as:
:$\forall n \in \Z:\forall a \in R: \map {g_a} n = n \cdot a$
Then from Kernel of Non-Zero Divisor Multiple Function is Primary Ideal of Characteristic:
:$\map \ker {g_a} = \ideal p$
where:
:$\map \ker {g_a}$ is the kernel of $g_a$
:$\ideal p$ is the principal ideal of $\Z$ generated by $p$.
We have by definition of kernel:
:$n \in \map \ker {g_a} \iff n \cdot a = 0_R$
and by definition of principal ideal:
:$n \in \ideal p \iff p \divides n$
The result follows.
== Sources ==
* : Chapter $\text {IV}$: Rings and Fields: $24$. The Division Algorithm: Theorem $24.8 \ 2^\circ$
Category:Homomorphism from Integers into Ring with Unity
Multiple Root/Examples/Arbitrary Example 1

== Example of Multiple Root ==
Consider the cubic equation $\EE$:
:$\EE: \quad x^3 - 3 x^2 + 4 = 0$
$\EE$ has two solutions: $-1$ and $2$.
$2$ is a multiple root of $\EE$ with multiplicity $2$.
== Proof ==
We have:
Hence $2$ appears twice.
Thus $2$ by definition has a multiplicity $2$, that is, $2$ is a double root of $\EE$.
== Sources ==
* 
* 
Category:Examples of Multiple Roots
Multiple Root is Root of First Derivative of Equation

== Theorem ==
Let $\EE$ be the equation:
:$\map f x = 0$
where $f: \R \to \R$ is a real function.
Let $\xi$ be a multiple root of $\EE$.
Then $\xi$ is also a root of the equation:
:$\map {f'} x = 0$
where $f': \R \to \R$ is the first derivative of $f$ $x$.
== Proof ==
== Sources ==
* 
* 
Category:Multiple Roots
Category:Differential Calculus
Multiple Rule for Sequence in Normed Vector Space

== Theorem ==
Let $\Bbb F$ be a subfield of $\C$. 
Let $\struct {X, \norm \cdot}$ be a normed vector space over $\Bbb F$. 
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$ converging to $x$. 
Let $\lambda \in \Bbb F$. 
Then: 
:$\lambda x_n \to \lambda x$
== Proof ==
The case $\lambda = 0$ follows from Constant Sequence in Normed Vector Space Converges.
Now take $\lambda \ne 0$. 
Since $\sequence {x_n}_{n \mathop \in \N}$ converges to $x$, for each $\epsilon > 0$ there exists $N \in \N$ such that: 
:$\ds \norm {x_n - x} < \frac \epsilon {\cmod \lambda}$ for all $n \ge N$.
Noting that: 
:$\ds \cmod \lambda \norm {x_n - x} = \norm {\lambda x_n - \lambda x}$
from the norm axioms, we have:
:$\ds \norm {\lambda x_n - \lambda x} < \epsilon$ for all $n \ge N$. 
Since $\epsilon$ was arbitrary, it follows that: 
:$\lambda x_n \to \lambda x$
Category:Convergent Sequences (Normed Vector Spaces)
Multiple of 6 is Semiperfect

== Theorem ==
Let $n \in \Z_{>0}$ be a multiple of $6$.
Then $n$ is semiperfect.
== Proof ==
Let $n = 6 k$.
Then:
:$n = 2 \times 3 k$
and so $3 k$ is a factor of $n$.
:$n = 3 \times 2 k$
and so $2 k$ is a factor of $n$.
:$n = 6 \times k$
and so $k$ is a factor of $n$.
But:
:$n = k + 2 k + 3 k$
and so is the sum of a subset of its factors
Hence the result by definition of semiperfect.
Category:Semiperfect Numbers
Category:6
Multiple of 999 can be Split into Groups of 3 Digits which Add to 999/Mistake

== Source Work ==
:
:The Dictionary
::$999$
:
:The Dictionary
::$999$
== Mistake ==
:In fact, any multiple at all of $999$ can be separated into groups of $3$ digits from the unit position, which when added will total $999$. The same principle applies to multiples of $9 \quad 99 \quad 9999$ and so on.
== Correction ==
Not every multiple of $999$, $9$, $99$ and so on.
The simplest counterexamples are:
This mistake is repeated in a slightly different form on the page $142,857$.
== Sources ==
* : $999$
* : $999$
Category:Mistakes/Curious and Interesting Numbers
Category:Integer whose Digits when Grouped in 3s add to Multiple of 999 is Divisible by 999
Multiple of Absolutely Continuous Function is Absolutely Continuous

== Theorem ==
Let $k$ be a real number.
Let $I \subseteq \R$ be a real interval.
Let $f : I \to \R$ be an absolutely continuous real function.
Then $k f$ is absolutely continuous.
== Proof ==
Note that if $k = 0$, then $k f$ is constant. 
Hence, by Constant Real Function is Absolutely Continuous: 
:$k f$ is absolutely continuous if $k = 0$.
Take now $k \ne 0$.
Let $\epsilon$ be a positive real number.
Since $f$ is absolutely continuous, there exists real $\delta > 0$ such that for every set of disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ with: 
:$\ds \sum_{i \mathop = 1}^n \paren {b_i - a_i} < \delta$
we have: 
:$\ds \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \frac \epsilon {\size k}$
Then:
whenever:
:$\ds \sum_{i \mathop = 1}^n \paren {b_i - a_i} < \delta$
Since $\epsilon$ was arbitrary: 
:$k f$ is absolutely continuous if $k \ne 0$.
Therefore: 
:$k f$ is absolutely continuous for all $k \in \R$.
Category:Absolutely Continuous Real Functions
Multiple of Abundant Number is Abundant

== Theorem ==
Let $n$ be an abundant number.
Let $m$ be a positive integer such that $m > 1$.
Then $m n$ is abundant.
== Proof ==
We have by definition of divisor sum function and abundant number that:
:$\dfrac {\map {\sigma_1} n} n > 2$
But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
:$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$
Hence the result by definition of abundant.
== Sources ==
* : $12$
* : $12$
Category:Abundant Numbers
Multiple of Chi-Squared Random Variable has Gamma Distribution

== Theorem == 
Let $n$ be a strictly positive integer. 
Let $k > 0$ be a real number. 
Let $X \sim \chi^2_n$ where $\chi^2_n$ is the chi-squared distribution with $n$ degrees of freedom. 
Then: 
:$k X \sim \map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$
where $\map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$ is the gamma distribution with parameters $\dfrac n 2$ and $\dfrac 1 {2 k}$.
== Proof ==
Let:
:$Y \sim \map \Gamma {\dfrac n 2, \dfrac 1 {2 k}}$
We aim to show that: 
:$\map \Pr {Y < k x} = \map \Pr {X < x}$
for all real $x \ge 0$.
We have: 
Category:Chi-Squared Distribution
Category:Gamma Distribution
Multiple of Codeword in Linear Code/Examples/V(3,3)

== Example of Multiple of Codeword in $V \paren {3, 3}$ ==
In the master code $V \paren {3, 3}$, the codeword $102$ is multiplied by $\eqclass 2 3$ thus:
:$2 \paren {102} = 201$
== Proof ==
Taking the elements of $201$ in turn, we have:
Hence the result.
== Sources ==
* : Chapter $6$: Error-correcting codes
Category:Examples of Linear Codes
Multiple of Column Added to Column of Determinant

== Theorem ==
Let $\mathbf A = \begin {bmatrix}
a_{1 1} & \cdots & a_{1 r} & \cdots & a_{1 s} & \cdots & a_{1 n} \\
a_{2 1} & \cdots & a_{2 r} & \cdots & a_{2 s} & \cdots & a_{2 n} \\
 \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n 1} & \cdots & a_{n r} & \cdots & a_{n s} & \cdots & a_{n n} \\
\end {bmatrix}$ be a square matrix of order $n$.
Let $\map \det {\mathbf A}$ denote the determinant of $\mathbf A$.
Let $\mathbf B = \begin{bmatrix}
a_{1 1} & \cdots & a_{1 r} + \lambda a_{1 s} & \cdots & a_{1 s} & \cdots & a_{1 n} \\
a_{2 1} & \cdots & a_{2 r} + \lambda a_{2 s} & \cdots & a_{2 s} & \cdots & a_{2 n} \\
 \vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{n 1} & \cdots & a_{n r} + \lambda a_{n s} & \cdots & a_{n s} & \cdots & a_{n n} \\
\end{bmatrix}$.
Then $\map \det {\mathbf B} = \map \det {\mathbf A}$.
That is, the value of a determinant remains unchanged if a constant multiple of any column is added to any other column.
== Proof ==
We have that:
:$\mathbf A^\intercal = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{r 1} & a_{r 2} & \cdots & a_{r n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {bmatrix}$
where $\mathbf A^\intercal$ denotes the transpose of $\mathbf A$.
Similarly, we have that:
:$\mathbf B^\intercal = \begin{bmatrix}
a_{1 1} & a_{1 2} & \ldots & a_{1 n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{r 1} + \lambda a_{s 1} & a_{r 2} + \lambda a_{s 2} & \cdots & a_{r n} + \lambda a_{s n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{s 1} & a_{s 2} & \cdots & a_{s n} \\
 \vdots & \vdots & \ddots & \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {bmatrix}$
From Multiple of Row Added to Row of Determinant:
:$\map \det {\mathbf B^\intercal} = \map \det {\mathbf A^\intercal}$
From from Determinant of Transpose:
:$\map \det {\mathbf B^\intercal} = \map \det {\mathbf B}$
:$\map \det {\mathbf A^\intercal} = \map \det {\mathbf A}$
and the result follows.
== Also see ==
* Multiple of Row Added to Row of Determinant
== Sources ==
* : $\text{A}.2$: Linear algebra and determinants: Theorem $\text{A}.10 \ (4)$
* : $\S 1.2.3$: Sums and Products: Exercises -- Second Set
* : Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace: Proposition $1.13$
* 
* 
* 
Category:Determinants
Multiple of Divisor Divides Multiple

== Theorem ==
Let $a, b, c \in \Z$.
Let:
:$a \divides b$
where $\divides$ denotes divisibility.
Then:
:$a c \divides b c$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : $\text I$: The Series of Primes: $1.1$ Divisibility of integers
* : Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $2 \ \text{(c)}$
Category:Divisors
Category:Multiple of Divisor Divides Multiple
Multiple of Divisor Divides Multiple/Proof 2

== Theorem ==
== Proof ==
By definition, if $a \divides b$ then $\exists d \in \Z: a d = b$.
Then:
:$\paren {a d} c = b c$
that is:
:$\paren {a c} d = b c$
which follows because Integer Multiplication is Commutative and Integer Multiplication is Associative.
Hence the result.
== Sources ==
* : Chapter $3$: The Integers: $\S 10$. Divisibility: Theorem $16 \ \text{(ii)}$
Category:Multiple of Divisor Divides Multiple
Multiple of Divisor in Integral Domain Divides Multiple

== Theorem ==
Let $\struct {D, +, \times}$ be an integral domain.
Let $a, b, c \in D$.
Let $a \divides b$, where $\divides$ denotes divisibility.
Then $a \times c$ is a divisor of $b \times c$.
=== Corollary ===
== Proof ==
By definition, if $a \divides b$ then $\exists d \in D: a \times d = b$.
Then $\paren {a \times d} \times c = b \times c$, that is:
:$\paren {a \times c} \times d = b \times c$
which follows because $\times$ is commutative and associative in an integral domain.
Hence the result.
== Sources ==
* : Chapter $6$: Polynomials and Euclidean Rings: $\S 26$. Divisibility: Theorem $49 \ \text{(ii)}$
Category:Divisibility
Category:Integral Domains
Multiple of Exponential Random Variable has Exponential Distribution

== Theorem ==
Let $\beta, k$ be real numbers with $\beta > 0$.
Let $X$ be a random variable.
Let $X \sim \Exponential \beta$, where $\Exponential \beta$ is the exponential distribution with parameter $\beta$. 
Then:
:$k X \sim \Exponential {k \beta}$
== Proof ==
Let:
:$Y \sim k X$
We aim to show that: 
:$\ds \map \Pr {Y \le y} = 1 - \map \exp {-\frac y {k \beta} }$
for each $y > 0$. 
We have: 
Category:Exponential Distribution
Multiple of Function of Bounded Variation is of Bounded Variation

== Theorem ==
Let $a, b, k$ be real numbers with $a < b$.
Let $f : \closedint a b \to \R$ be a functions of bounded variation.
Let the total variation of $f$ on $\closedint a b$ be $\map {V_f} {\closedint a b}$.
Then $k f$ is of bounded variation with: 
:$\map {V_{k f} } {\closedint a b} = \size k \map {V_f} {\closedint a b}$
where $\map {V_{k f} } {\closedint a b}$ is the total variation of $k f$ on $\closedint a b$.
== Proof ==
For each finite subdivision $P$ of $\closedint a b$, write: 
:$P = \set {x_0, x_1, \ldots, x_n }$
with:
:$a = x_0 < x_1 < x_2 < \cdots < x_{n - 1} < x_n = b$
Then:
Since $f$ is of bounded variation, there exists $M \in \R$ such that: 
:$\map {V_f} {P ; \closedint a b} \le M$
for all finite subdivisions $P$. 
So:
:$\map {V_{k f} } {P ; \closedint a b} \le \size k M$
So $k f$ is of bounded variation.
We then have: 
Category:Bounded Variation
Category:Total Variation of Real Function
Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number

== Theorem ==
Let $\struct {A, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\lambda \in \C$.
Then $\lambda {\mathbf 1}_A$ is positive $\lambda \in \R_{\ge 0}$. 
=== Corollary ===
== Proof ==
First we have:
:$\tuple {\lambda {\mathbf 1}_A}^\ast = \overline \lambda {\mathbf 1}_A$
by the definition of an involution and Identity Element in Unital *-Algebra is Hermitian.
Since:
:$\lambda \overline \lambda {\mathbf 1}_A = \overline \lambda \lambda {\mathbf 1}_A$
Hence $\lambda {\mathbf 1}_A$ is normal.
From the Spectral Mapping Theorem for Polynomials and Spectrum of Identity Element of Unital Algebra, we have:
:$\map {\sigma_A} {\lambda {\mathbf 1}_A} = \set \lambda$ 
Hence by Normal Element of C*-Algebra is Hermitian iff Spectrum is Real, we have that $\lambda {\mathbf 1}_A$ is Hermitian if $\set \lambda \subseteq \R$.
That is, $\lambda \in \R$.
From the definition of a positive element, we then have that $\lambda {\mathbf 1}_A$ is positive $\set \lambda \subseteq \hointr 0 \infty$.
That is, $\lambda \in \R_{\ge 0}$. 
Category:Positive Elements of C*-Algebras
Category:Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number
Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number/Corollary

== Theorem ==
Let $\struct {A, \norm {\, \cdot \,} }$ be a unital $\text C^\ast$-algebra.
Let $\lambda, \mu \in \R$.
Let $\le_A$ be the canonical preordering of $A$.
Then $\lambda {\mathbf 1}_A \le_A \mu {\mathbf 1}_A$ $\lambda \le \mu$. 
== Proof ==
We have $\lambda {\mathbf 1}_A \le_A \mu {\mathbf 1}_A$ $\paren {\mu - \lambda} {\mathbf 1}_A$ is positive.
From Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number, we have that $\paren {\mu - \lambda} {\mathbf 1}_A$ is positive $\mu - \lambda \ge 0$.
That is, $\lambda \le \mu$. 
Category:Multiple of Identity Element of Unital C*-Algebra is Positive iff Coefficient is Non-Negative Real Number
Multiple of Infimum

== Theorem ==
Let $T \subseteq \R: T \ne \O$ be a non-empty subset of the set of real numbers $\R$.
Let $T$ be bounded below.
Let $z \in \R: z > 0$ be a (strictly) positive real number.
Then:
:$\ds \map {\inf_{x \mathop \in T} } {z x} = z \map {\inf_{x \mathop \in T} } x$
where $\inf$ denotes infimum.
== Proof ==
From Negative of Infimum is Supremum of Negatives:
:$\ds -\inf_{x \mathop \in T} x = \map {\sup_{x \mathop \in T} } {-x} \implies \inf_{x \mathop \in T} x = -\map {\sup_{x \mathop \in T} } {-x}$
Let $S = \set {x \in \R: -x \in T}$.
From Negative of Infimum is Supremum of Negatives, $S$ is bounded above.
From Multiple of Supremum:
:$\ds \map {\sup_{x \mathop \in S} } {z x} = z \map {\sup_{x \mathop \in S} } x$
Hence:
:$\ds \map {\inf_{x \mathop \in T} } {z x} = -\map {\sup_{x \mathop \in T} } {-z x} = -z \map {\sup_{x \mathop \in T} } {-x} = z \map {\inf_{x \mathop \in T} } x$
== Sources ==
* : $\S 2$: Continuum Property: Exercise $\S 2.13 \ (3)$
Category:Real Analysis
Multiple of Metric forms Metric

== Theorem ==
Let $M = \struct {A, d}$ be a metric space.
Let $d_1: A^2 \to \R$ be the mapping defined as:
:$\forall \tuple {x, y} \in A^2: \map {d_1} {x, y} = k \map d {x, y}$
for some strictly positive $k \in \R_{>0}$.
Then $d_1$ is a metric for $A$.
== Proof ==
It is to be demonstrated that $d_1$ satisfies all the metric space axioms.
=== Proof of ===
So holds for $d_1$.
=== Proof of ===
So holds for $d_1$.
=== Proof of ===
So holds for $d_1$.
=== Proof of ===
So holds for $d_1$.
Thus $d_1$ satisfies all the metric space axioms and so is a metric.
== Sources ==
* : $2$: Continuity generalized: metric spaces: Exercise $2.6: 6$
Category:Examples of Metric Spaces
Multiple of Perfect Number is Abundant

== Theorem ==
Let $n$ be a perfect number.
Let $m$ be a positive integer such that $m > 1$.
Then $m n$ is abundant.
== Proof ==
We have by definition of divisor sum function and perfect number that:
:$\dfrac {\map {\sigma_1} n} n = 2$
But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:
:$\dfrac {\map {\sigma_1} {m n} } {m n} > 2$
Hence the result by definition of abundant.
== Sources ==
* : $12$
* : $12$
Category:Abundant Numbers
Category:Perfect Numbers
Multiple of Repdigit Base minus 1

== Theorem ==
Let $b \in \Z_{>1}$ be an integer greater than $1$.
Let $n$ be a repdigit number of $k$ instances of the digit $b - 1$ for some integer $k$ such that $k \ge 1$.
Let $m \in \Z_{>1}$ be an integer such that $1 < m < b$.
Then $m \times n$, when expressed in base $b$, is of the form:
:$m n = \sqbrk {r d d \cdots d s}_b$
where:
:$d = b - 1$
:$r = m - 1$
:$s = b - m$
:there are $k - 1$ occurrences of $d$.
=== Generalization ===
== Proof ==
which is exactly the representation $\sqbrk {r d d \cdots d s}_b$ as defined.
== Examples ==
Category:Repdigit Numbers
Category:Multiple of Repdigit Base minus 1
Multiple of Repdigit Base minus 1/Examples/3 times 6666 base 7

== Example of Multiple of Repdigit Base minus 1 ==
:$3 \times 6666$ (base $7$) is equal to $26664$ (base $7$).
== Proof ==
From Multiple of Repdigit Base minus 1:
:the first digit of $3 \times 6666$ is $3 - 1 = 2$
:the last digit of $3 \times 6666$ is $7 - 3 = 4$
:there are $4 - 1 = 3$ instances of $6$ in between.
Hence:
:$3 \times 6666 = 26664$
Category:Multiple of Repdigit Base minus 1
Multiple of Repdigit Base minus 1/Examples/4 times 999 base 10

== Example of Multiple of Repdigit Base minus 1 ==
:$4 \times 999$ (base $10$) is equal to $3996$.
== Proof ==
From Multiple of Repdigit Base minus 1:
:the first digit of $4 \times 999$ is $4 - 1 = 3$
:the last digit of $4 \times 999$ is $10 - 4 = 6$
:there are $3 - 1 = 2$ instances of $9$ in between.
Hence:
:$4 \times 999 = 3996$
Category:Multiple of Repdigit Base minus 1