nyuuzyou/wikis huggingface dataset:
Euler's Equations of Motion for Rotation of Rigid Body/Also presented as

== Euler's Equations of Motion for Rotation of Rigid Body: Also known as == 
Euler's Equations of Motion for Rotation of Rigid Body can also be seen presented in the form:
where:
:$I_1$, $I_2$ and $I_1$ are the components of the torque applied about the principal axes
:$I_1$, $I_2$ and $I_1$ are the moments of inertia at fixed point $O$
:$\omega_1$, $\omega_2$ and $\omega_3$ are the components of angular velocity along the principal axis.
== Sources ==
* 
* 
Category:Euler's Equations of Motion for Rotation of Rigid Body
Euler's Formula

== Theorem ==
Let $z \in \C$ be a complex number.
Then:
:$e^{i z} = \cos z + i \sin z$
where:
:$e^{i z}$ denotes the complex exponential function
:$\cos z$ denotes the complex cosine function
:$\sin z$ denotes complex sine function
:$i$ denotes the imaginary unit.
=== Real Domain ===
This result is often presented and proved separately for arguments in the real domain:
=== Corollary ===
== Proof ==
== Examples ==
== Also known as ==
== Also see ==
* Euler's Identity
* Sum of Hyperbolic Sine and Cosine equals Exponential
== Historical Note ==
== Sources ==
* 
* 
* 
* 
* 
* : Appendix $6$: Basic Algebra
Category:Euler's Formula
Category:Euler's Identities
Category:Exponential Function
Category:Sine Function
Category:Cosine Function
Euler's Formula/Also known as

== Euler's Formula: Also known as == 
Euler's Formula in this and its corollary form, along with Euler's Sine Identity and Euler's Cosine Identity are also found referred to as Euler's Identities.
However, this allows for confusion with Euler's Identity:
:$e^{i \pi} + 1 = 0$
It is wise when referring to it by name, therefore, to ensure that the equation itself is also specified.
== Sources ==
* 
* 
Category:Euler's Formula
Euler's Formula/Corollary

== Corollary to Euler's Formula ==
Let $z \in \C$ be a complex number.
Then:
:$e^{-i z} = \cos z - i \sin z$
where:
:$e^{-i z}$ denotes the complex exponential function
:$\cos z$ denotes the complex cosine function
:$\sin z$ denotes complex sine function
:$i$ denotes the imaginary unit.
=== Corollary ===
This result is often presented and proved separately for arguments in the real domain:
== Proof ==
== Also known as ==
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.16)$
Category:Euler's Formula
Euler's Formula/Examples

== Examples of Use of Euler's Formula ==
=== Example: $e^{i \pi / 4}$ ===
=== Example: $e^{i \pi / 2}$ ===
=== Example: $e^{-i \pi / 2}$ ===
=== Example: $e^{i \pi}$ ===
=== Example: $e^{2 i \pi}$ ===
=== Example: $e^{2 k i \pi}$ ===
Category:Examples of Euler's Formula
Euler's Formula/Examples/e^2 i pi

== Example of Use of Euler's Formula ==
:$e^{2 i \pi} = 1$
== Proof ==
== Sources ==
* : $\S 2$. Geometrical Representations: $(2.19)$
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$
Category:Examples of Euler's Formula
Euler's Formula/Historical Note

== Historical Note on Euler's Formula ==
 famously published what is now known as Euler's Formula in $1748$.
However, it needs to be noted that first introduced it in $1714$, in the form:
:$\map \ln {\cos \theta + i \sin \theta} = i \theta$
== Sources ==
* : $0$ The origins of complex analysis, and a modern viewpoint: $1$. The origins of complex numbers
Category:Historical Notes
Euler's Formula/Proof

== Theorem ==
== Proof ==
As Complex Sine Function is Absolutely Convergent and Complex Cosine Function is Absolutely Convergent, we have:
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.14)$
* : $\S 1.5$
* 
Category:Euler's Formula
Category:Direct Proofs
Euler's Formula/Real Domain

== Theorem ==
Let $\theta \in \R$ be a real number.
Then:
:$e^{i \theta} = \cos \theta + i \sin \theta$
where:
:$e^{i \theta}$ denotes the complex exponential function
:$\cos \theta$ denotes the real cosine function
:$\sin \theta$ denotes the real sine function
:$i$ denotes the imaginary unit.
=== Corollary ===
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Proof 4 ==
== Proof 5 ==
== Proof 6 ==
== Examples ==
== Also known as ==
== Also see ==
* Euler's Identity
== Sources ==
* : $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.16$
* : $\S 3.17$: The Homogeneous Equation with Constant Coefficients
* : $\S 3$: Appendix $\text A$: Euler
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : Chapter $\text {A}.21$: Euler ()
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
Category:Euler's Formula
Category:Exponential Function
Category:Trigonometric Functions
Euler's Formula/Real Domain/Corollary

== Corollary to Euler's Formula: Real Domain ==
Let $\theta \in \R$ be a real number.
Then:
:$e^{-i \theta} = \cos \theta - i \sin \theta$
where:
:$e^{-i \theta}$ denotes the complex exponential function
:$\cos \theta$ denotes the real cosine function
:$\sin \theta$ denotes the real sine function
:$i$ denotes the imaginary unit.
== Proof ==
== Sources ==
* : $\S 2$. Geometrical Representations
* : $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.16$
Category:Euler's Formula
Euler's Formula/Real Domain/Proof 1

== Theorem ==
== Proof ==
Consider the differential equation:
:$D_z \map f z = i \cdot \map f z$
=== Step 1 ===
We will prove that $z = \cos \theta + i \sin \theta$ is a solution.
=== Step 2 ===
We will prove that $y = e^{i\theta}$ is a solution.
=== Step 3 ===
Consider the initial condition $\map f 0 = 1$.
So $y$ and $z$ are both particular solutions.
But a particular solution to a differential equation is unique.
Therefore $y = z$, that is, $e^{i \theta} = \cos \theta + i \sin \theta$.
Category:Euler's Formula
Category:Direct Proofs
Euler's Formula/Real Domain/Proof 2

== Theorem ==
== Proof ==
This:
:$e^{i \theta} = \cos \theta + i \sin \theta$
is logically equivalent to this:
:$\dfrac {\cos \theta + i \sin \theta} {e^{i \theta} } = 1$
for every $\theta$.
Note that the left expression is nowhere undefined.
Taking the derivative of this:
Thus the expression, as a function of $\theta$, is constant and so yields the same value for every $\theta$.
We know the value at at least one point, that is, when $\theta = 0$:
:$\dfrac {\cos 0 + i \sin 0} {e^{0 i}} = \dfrac {1 + 0} 1 = 1$
Thus it is $1$ for every $\theta$, which verifies the above.
Hence the result.
Category:Euler's Formula
Category:Direct Proofs
Euler's Formula/Real Domain/Proof 4

== Theorem ==
== Proof ==
Note that the following proof, as written, only holds for real $\theta$.
Define:
:$\map x \theta = e^{i \theta}$
:$\map y \theta = \cos \theta + i \sin \theta$
Consider first $\theta \ge 0$.
Taking Laplace transforms:
So $x$ and $y$ have the same Laplace transform for $\theta \ge 0$.
Now define $\tau = -\theta, \sigma = -s$, and consider $\theta < 0$ so that $\tau > 0$.
Taking Laplace transforms of $\map x \tau$ and $\map y \tau$: 
So $\map x \theta$ and $\map y \theta$ have the same Laplace transforms for $\theta < 0$.
The result follows from Injectivity of Laplace Transform.
Category:Euler's Formula
Category:Laplace Transforms
Euler's Formula/Real Domain/Proof 5

== Theorem ==
== Proof ==
As Sine Function is Absolutely Convergent and Cosine Function is Absolutely Convergent, we have:
== Sources ==
* : $\S 2$. Geometrical Representations: $(2.18)$
* : $1$: Complex Numbers: Euler's Formula
* 
* : $\S 1.5$
Category:Euler's Formula
Category:Direct Proofs
Euler's Homogeneous Function Theorem

Let $\map f{x,y}$ be a homogeneous function of order $n$ so that
:$\map f{tx,ty}=t^n\map f{x,y}$
Then:
:$x\dfrac{\partial f}{\partial x}+y\dfrac{\partial f}{\partial y}=n\map f{x,y}$
==Proof==
Define $x'=xt$ and $y'=yt$. By chain rule
Let $t=1$, then
:$x\dfrac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=n\map f{x,y}. 	$
==Generalization==
This can be generalized to an arbitrary number of variables
:$\ds\sum_i x_i\frac{\partial f}{\partial x_i}=n\map f {\ldots,x_i,\ldots}	$
== Source ==
* 
Category:Calculus
Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid

== Theorem ==
:$\begin{cases} \dfrac {D \mathbf u} {D t} & = & - \nabla w + \mathbf g \\
\nabla \cdot \mathbf u & = & 0
\end{cases}$
where:
:$\mathbf u$ denotes the flow velocity vector, with components in an $N$-dimensional space $u_1, u_2, \dots, u_N$
:$\dfrac D {D t}$ denotes the material derivative in time
:$\cdot$ denotes the dot product
:$\nabla$ denotes the nabla operator, used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation)
:$\mathbf u \cdot \nabla$ is the convective derivative
:$w$ is the thermodynamic work per unit mass, the internal source term
:$\mathbf g$ denotes body acceleration per unit mass acting on the continuum.
== Historical Note ==
== Sources ==
* : $\S 3$: Appendix $\text A$: Euler
* : Chapter $\text {A}.21$: Euler ()
Category:Hydrodynamics
Euler's Identities

== Theorem ==
The following identities are referred to as Euler's Identities:
=== Euler's Formula ===
=== Euler's Sine Identity ===
=== Euler's Cosine Identity ===
On , the following are also classified as members of the set of Euler's Identities:
=== Euler's Tangent Identity ===
=== Euler's Cotangent Identity ===
=== Euler's Secant Identity ===
=== Euler's Cosecant Identity ===
where:
:$z \in \C$ is a complex number
:$i$ denotes the imaginary unit: $i^2 = -1$
== Historical Note ==
== Sources ==
* 
* 
Category:Euler's Identities
Category:Sine Function
Category:Cosine Function
Category:Exponential Function
Euler's Identity

== Theorem ==
:$e^{i \pi} + 1 = 0$
== Proof ==
Follows directly from Euler's Formula $e^{i z} = \cos z + i \sin z$, by plugging in $z = \pi$:
:$e^{i \pi} + 1 = \cos \pi + i \sin \pi + 1 = -1 + i \times 0 + 1 = 0$
== Also presented as ==
== Also see ==
* Euler's Formula
* Euler's Identities
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(i)}$
* : $\S 3$: Appendix $\text A$: Euler
* 
* 
Category:Euler's Identity
Category:Euler's Number
Category:Pi
Category:Complex Analysis
Euler's Identity/Also presented as

== Euler's Identity: Also presented as == 
Euler's Identity can also be presented as:
:$e^{i \pi} = -1$
or:
:$e^{\pi i} = -1$
== Sources ==
* : $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(i)}$
* : $0$ The origins of complex analysis, and a modern viewpoint: $1$. The origins of complex numbers
* : $-1$ and $i$
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
* : Chapter $\text {A}.21$: Euler ()
* : $-1$ and $i$
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
* : $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
* 
* 
Category:Euler's Identity
Euler's Integral Representation of Hypergeometric Function

== Theorem ==
Let $a, b, c \in \C$.
Let $\size x < 1$
Let $\map \Re c > \map \Re b > 0$.
Then:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c } {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{- a} \rd t$
in the $x$ plane cut along the real axis from $1$ to $\infty$. 
where:
:$\map \arg t = \map \arg {1 - t} = 0$
:$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$:
::$\map F {a, b; c; x} := \ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}$
:$x^{\overline k}$ denotes the $k$th rising factorial power of $x$
:$\map \Gamma {n + 1} = n!$ is the Gamma function.
== Proof ==
Letting $\size x < 1$ and expanding the product of $\paren {1 - x t}^{-a}$:
Therefore:
Therefore:
:$\ds \map F {a, b; c; x} = \dfrac {\map \Gamma c} {\map \Gamma b \map \Gamma {c - b} } \int_0^1 t^{b - 1} \paren {1 - t}^{c - b - 1} \paren {1 - x t}^{-a} \rd t$
== Also see ==
* Gauss's Hypergeometric Theorem
== Sources ==
* : $\S 31$: Hypergeometric Functions: Miscellaneous Properties: $31.16$
* : Chapter $\text {2}$. The Hypergeometric Functions
* Chapter $\text {1}$. The hypergeometric series
* 
Category:Beta Function
Category:Euler's Integral Representation of Hypergeometric Function
Category:Gaussian Hypergeometric Function
Category:Gamma Function
Euler's Integral Theorem

== Theorem ==
:$H_n = \ln n + \gamma + \map \OO {\dfrac 1 n}$
where:
:$H_n$ denotes the $n$th harmonic number
:$\gamma$ denotes the Euler-Mascheroni constant
:$\map \OO {\dfrac 1 n}$ denotes big-$\OO$ of $\dfrac 1 n$.
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Preface to first edition: Prerequisites: $\text {(iii)}$
Category:Euler's Integral Theorem
Category:Integral Calculus
Category:Analytic Number Theory
Euler's Integral Theorem/Proof 1

== Theorem ==
== Proof ==
Recall the definition of the floor function:
For all $n \in \N_{>0}$:
Let $N \ge n \ge 1$.
By Sum of Integrals on Adjacent Intervals for Integrable Functions:
:$\ds (1): \quad \paren {H_n - \ln n} - \paren {H_N - \ln N} = \int_n^N \dfrac {u - \floor u} {u^2} \rd u$
On the other hand, from follows:
:$\forall u \in \R_{\ge 1} : 0 \le \dfrac {u - \floor u} {u^2} \le \dfrac 1 {u^2}$
In view of Integral Operator is Positive, integrating the above inequality on $\closedint n N$:
From $(1)$ and $(2)$, it follows:
:$(3): \quad 0 \le \paren {H_n - \ln n} - \paren {H_N - \ln N} \le \dfrac 1 n$
In particular, $\sequence {H_n - \ln n}$ is a Cauchy sequence.
Thus the limit $\gamma$, the Euler-Mascheroni constant, exists by Cauchy's Convergence Criterion.
In $(3)$, for each $n \in \N$, let $N \to \infty$.
Then:
:$\forall n \in \N : 0 \le H_n - \ln n - \gamma \le \dfrac 1 n$
Category:Euler's Integral Theorem
Euler's Integral Theorem/Proof 2

== Theorem ==
== Proof ==
Recall the definition of the floor function:
Hence:
:$0 \le x - \floor x < 1$
For all $n \in \N_{>0}$:
From Existence of Euler-Mascheroni Constant Proof 1, we have:
:$\ds \Delta_n = \sum_{k \mathop = 1}^n \dfrac 1 k - \int_1^n \dfrac 1 x \rd x$
is decreasing and bounded below by zero.
Therefore:
:$H_n - \ln n \ge 0$
Therefore:
:$\forall n \in \N_{>0} : 0 \le \size {H_n - \ln n - \gamma} < \dfrac 1 n$
Category:Euler's Integral Theorem
Euler's Number: Limit of Sequence implies Base of Logarithm

== Theorem ==
Let $e$ be Euler's number defined by:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then $e$ is the unique solution to the equation $\map \ln x = 1$.
That is:
:$\map \ln x = 1 \iff x = e$
== Proof ==
First we prove that $e$ is a solution to $\map \ln x = 1$:
From Logarithm is Strictly Increasing, $\ln$ is strictly monotone.
By Strictly Monotone Mapping with Totally Ordered Domain is Injective it follows that $\ln$ is an injection.
So the solution to $\map \ln x = 1$ is unique.
Hence the result.
Category:Logarithms
Category:Euler's Number
Euler's Number: Limit of Sequence implies Limit of Series

== Theorem ==
Let Euler's number $e$ be defined as:
:$\ds e := \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n$
Then:
:$\ds e = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
That is:
:$e = \dfrac 1 {0!} + \dfrac 1 {1!} + \dfrac 1 {2!} + \dfrac 1 {3!} + \dfrac 1 {4!} \cdots$
== Proof 1 ==
== Proof 2 ==
Category:Euler's Number
Euler's Number: Limit of Sequence implies Limit of Series/Proof 1

== Theorem ==
== Proof ==
We expand $\paren {1 + \dfrac 1 n}^n$ by the Binomial Theorem, using that $\dfrac {n - k} n = 1 - \dfrac k n$:
Take one of the terms in the above:
:$x = \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \cdots \paren {1 - \dfrac {k - 1} n} \dfrac 1 {k!}$
From Sequence of Powers of Reciprocals is Null Sequence, $\dfrac 1 n \to 0$ as $n \to \infty$.
From the Combination Theorem for Sequences:
:$\forall \lambda \in \R: \dfrac \lambda n \to 0$ as $n \to \infty$
:$\forall \lambda \in \R: 1 - \dfrac \lambda n \to 1$ as $n \to \infty$
:$x = \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \cdots \paren {1 - \dfrac {k - 1} n} \dfrac 1 {k!} \to \dfrac 1 {k!}$ as $n \to \infty$
Hence:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = \frac 1 {0!} + \frac 1 {1!} + \frac 1 {2!} + \frac 1 {3!} + \cdots = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
Category:Euler's Number
Euler's Number: Limit of Sequence implies Limit of Series/Proof 2

== Theorem ==
== Proof ==
It will be shown that:
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = \sum_{k \mathop = 0}^\infty \frac 1 {k!}$
Let $t_n := \paren {1 + \dfrac 1 n}^n$
Then:
:$t_n = \dfrac 1 {0!} + \dfrac 1 {1!} + \paren {1 - \dfrac 1 n} \dfrac 1 {2!} + \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \dfrac 1 {3!} + \cdots + \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \cdots \paren {1 - \dfrac {n-1} n} \dfrac 1 {n!}$
Now let:
:$\ds s_m := \sum_{k \mathop = 0}^m \frac 1 {k!}$
We have that:
:$\forall n: t_n \le s_n$
Hence:
:$\limsup \paren {t_n} \le e$
Now, for all $m$, for $n \ge m$:
:$t_n \ge \dfrac 1 {0!} + \dfrac 1 {1!} + \paren {1 - \dfrac 1 n} \dfrac 1 {2!} + \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \dfrac 1 {3!} + \cdots + \paren {1 - \dfrac 1 n} \paren {1 - \dfrac 2 n} \cdots \paren {1 - \dfrac {m - 1} n} \dfrac 1 {m!}$
Hence, for all $m$, we have the right side as being a sequence in $n$, and then:
:$\ds \liminf \paren {t_n} \ge \sum_{k \mathop = 0}^m \frac 1 {m!}$
Since this is true for all $m$:
:$\liminf \paren {t_n} \ge e$
So $\ds \lim \paren {t_n}$ exists and is equal to $e$.
== Sources ==
* : Theorem $3.31$
Category:Euler's Number
Euler's Number as Limit of 1 + Reciprocal of n to nth Power

== Theorem ==
:$\ds \lim_{n \mathop \to \infty} \paren {1 + \frac 1 n}^n = e$
where $e$ denotes Euler's number.
== Proof 1 ==
== Proof 2 ==
Category:Euler's Number as Limit of 1 + Reciprocal of n to nth Power
Category:Euler's Number
Euler's Number as Limit of 1 + Reciprocal of n to nth Power/Proof 1

== Theorem ==
== Proof ==
By definition of the real exponential function as the limit of a sequence:
:$(1): \quad \exp x := \ds \lim_{n \mathop \to \infty} \paren {1 + \frac x n}^n$
By definition of Euler's number:
:$e = e^1 = \exp 1$
The result follows by setting $x = 1$ in $(1)$.
Category:Euler's Number as Limit of 1 + Reciprocal of n to nth Power
Euler's Number as Limit of n over nth Root of n Factorial

== Theorem ==
:$\ds e = \lim_{n \mathop \to \infty} \dfrac n {\sqrt [n] {n!} }$
where:
:$e$ denotes Euler's number
:$n!$ denotes $n$ factorial.
== Proof ==
From Limit of Root of Positive Real Number:
:$\ds \lim_{n \mathop \to \infty} \paren {2 \pi}^{1 / 2 n} = 1$
and from Limit of Integer to Reciprocal Power:
:$n^{1 / n} = 1$
Hence the result.
== Sources ==
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ldots$
Category:Euler's Number
Euler's Number as Sum of Egyptian Fractions

== Theorem ==
The reciprocal of Euler's number $e$ can be approximated by the following sequence of Egyptian fractions:
:$\dfrac 1 e = \dfrac 1 3 + \dfrac 1 {29} + \dfrac 1 {15 \, 786} + \dfrac 1 {513 \, 429 \, 610} + \cdots$
== Proof ==
We have by definition of the reciprocal of $e$ that:
:$\dfrac 1 e \approx 0 \cdotp 36787 \, 94411 \, 71442 \, 32159 \, 55237 \, 70161 \, 46086 \, 74458 \, 11131 \, 031 \ldots$
By inspection:
:$\dfrac 1 3 < \dfrac 1 e < \dfrac 1 2$
Thus:
:$\dfrac 1 e - \dfrac 1 3 \approx 0 \cdotp 03454 \, 61078 \, 38109 \, 08826 \, 21904 \, 36828 \, 12753 \, 41124 \, 77797 \, 70 \ldots$
Then:
:$\dfrac 1 {29} = 0 \cdotp 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 931 \ldots$ repeating
and:
:$\dfrac 1 {28} = 0 \cdotp 03571 \, 42857 \, 14285 \ldots$ repeating
and so:
:$\dfrac 1 e - \dfrac 1 3 - \dfrac 1 {29} \approx 0 \cdotp 00006 \, 33482 \, 17499 \, 43308 \, 97766 \, 33724 \, 67925 \, 82504 \, 08832 \, 19 \cdots$
 
Thus one can generate this sequence of denominators $\sequence {D_n}$ by:
:$D_n = \ceiling {\paren {\dfrac 1 e - \ds \sum_{i \mathop = 0}^{n - 1} \frac 1 {D_i} }^{-1} }$
== Sources ==
* : $0 \cdotp 36787 \, 94411 \, 71442 \, 32159 \, 55237 \, 70161 \, 46086 \, 74458 \, 11131 \, 031$
Category:Euler's Number
Euler's Number is Irrational

== Theorem ==
Euler's number $e$ is irrational.
== Proof ==
 that $e$ is rational.
Then there exist coprime integers $m$ and $n$ (and we can choose $n$ to be positive) such that:
:$\dfrac m n = e = \ds \sum_{i \mathop = 0}^\infty \frac 1 {i!}$ from the definition of Euler's number.
Multiplying both sides by $n!$, observe that:
:$\dfrac m n n! = n! \ds \sum_{i \mathop = 0}^\infty \frac 1 {i!} = \paren {\dfrac {n!} {0!} + \dfrac {n!} {1!} + \dfrac {n!} {2!} + \cdots + \dfrac {n!} {n!} } + \paren {\frac {n!} {\paren {n + 1}!} + \dfrac {n!} {\paren {n + 2}!} + \dfrac {n!} {\paren {n + 3}!} + \cdots}$
Hence:
Observe that the quantity on the must be an integer, as it is composed entirely of sums and differences of integer terms.
It must be strictly positive, as it is equal to:
:$\dfrac 1 {\paren {n + 1} } + \dfrac 1 {\paren {n + 1} \paren {n + 2} } + \dfrac 1 {\paren {n + 1} \paren {n + 2} \paren {n + 3} } + \cdots$
which is strictly positive.
Thus:
:$m \paren {n - 1}! - \paren {\dfrac {n!} {0!} + \dfrac {n!} {1!} + \dfrac {n!} {2!} + \cdots + \dfrac {n!} {n!} }$
must be a strictly positive integer less than $1$.
From this contradiction it follows that $e$ must be irrational.
== Historical Note ==
== Sources ==
* : $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : Chapter $\text {B}.17$: More About Irrational Numbers. $\pi$ is Irrational: Appendix: A Proof that $e$ is Irrational
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
* 
Category:Irrationality Proofs
Category:Euler's Number
Euler's Number is Irrational/Historical Note

== Historical Note on $e$ is Irrational ==
The proof that Euler's number $e$ is irrational was demonstrated by in $1737$.
== Sources ==
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : Chapter $\text {A}.29$: Liouville ()
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
Category:Historical Notes
Euler's Number is Transcendental

== Theorem ==
Euler's Number $e$ is transcendental.
== Proof 1 ==
== Proof 2 ==
== Proof 3 ==
== Historical Note ==
== Sources ==
* : Chapter $8$: Field Extensions: $\S 38$. Simple Algebraic Extensions: Example $77$
* : $\S 1$: Real Numbers: $\S 1.13$: Irrational Numbers
* : $1$: Complex Numbers: Solved Problems: Miscellaneous Problems: $47$
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : Chapter $\text {A}.29$: Liouville ()
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
* 
* 
* 
* 
Category:Euler's Number is Transcendental
Category:Euler's Number
Category:Transcendental Number Theory
Euler's Number is Transcendental/Historical Note

== Historical Note on $e$ is Transcendental ==
The transcendental nature of Euler's number $e$ was conjectured by in $1844$, after he had proved that it was not the root of a quadratic equation with integer coefficients.
The proof that $e$ is transcendental was first achieved by in $1873$.
== Sources ==
* : $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$
* : Chapter $\text {A}.29$: Liouville ()
* : Chapter $\text {A}.30$: Hermite ()
* : Chapter $\text {B}.18$: Algebraic and Transcendental Numbers. $e$ is Transcendental
* : $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$
Category:Historical Notes