nyuuzyou/wikis huggingface dataset:
Arc-Connected Space is not necessarily Locally Arc-Connected

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space which is arc-connected.
Then it is not necessarily the case that $T$ is also locally arc-connected.
== Proof ==
Let $T$ be the extended topologist's sine curve.
From Extended Topologist's Sine Curve is Arc-Connected, $T$ is an arc-connected space.
From Extended Topologist's Sine Curve is not Locally Arc-Connected, $T$ is not a locally arc-connected space.
Hence the result.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $4$: Connectedness
Category:Locally Arc-Connected Spaces
Category:Arc-Connected Spaces
Arc-Connectedness in Uncountable Finite Complement Space

== Theorem ==
Let $T = \struct {S, \tau}$ be a finite complement topology on a uncountable set $S$.
If the Continuum Hypothesis is accepted as true, then:
:$T$ is arc-connected
:$T$ is locally arc-connected.
== Proof ==
Let $a, b \in S$ such that $a \ne b$.
Let us assume the truth of the Continuum Hypothesis.
Then $a$ and $b$ are contained in a subset $X \subseteq S$ whose cardinality is the same as that of $\closedint 0 1$.
So by definition of cardinality we can set up a bijection $f: \closedint 0 1 \leftrightarrow X$ such that $\map f 0 = a$ and $\map f 1 = b$.
Each element of $S$ is closed from Finite Complement Space is $T_1$.
Hence $f$ is continuous and therefore an arc in $T$.
As $a$ and $b$ are arbitrary, $T$ is arc-connected.
Now let $\BB$ be a basis for $T$.
Let $B \in \BB$ and let $a, b \in B$.
By definition of a finite complement topology on a uncountable set, $B$ is uncountable.
So, by the same argument as above, we can set up an arc $f$ between any points of $B$.
So $T$ is locally arc-connected.
== Sources ==
* : Part $\text {II}$: Counterexamples: $19$. Finite Complement Topology on an Uncountable Space: $11$
Category:Continuum Hypothesis
Category:Arc-Connected Spaces
Category:Locally Arc-Connected Spaces
Category:Finite Complement Topology
Arc-Connectedness is Equivalence Relation

== Theorem ==
Let $T = \struct {S, \tau}$ be a topological space.
Let $a \sim b $ denote the relation:
:$a \sim b \iff a$ is arc-connected to $b$
where $a, b \in S$.
Then $\sim$ is an equivalence relation.
== Proof ==
Checking in turn each of the criteria for equivalence:
=== Reflexivity ===
The definition of arc-connected specifically makes provision that:
:$\forall a \in S: a \sim a$
specifically to ensure that $\sim$ be reflexive.
=== Symmetry ===
If $x \sim y$ then $x$ is is arc-connected to $y$ by definition.
In the trivial case of $x = y$, $y \sim x$ by reflexivity.
Suppose that $x \ne y$.
We form the mapping $g: \R \to \R$:
:$\map g x = 1 - x$
which is trivially continuous and injective.
By Composite of Continuous Mappings is Continuous, $g \circ f$ is continuous.
By Composite of Injections is Injection, $g \circ f$ is also injective.
Putting it together we see that $g \circ f$ is a continuous injection which maps $0$ to $y$ and $1$ to $x$.
So $y \sim x$ and $\sim$ has been shown to be symmetric.
=== Transitivity ===
Follows directly from Joining Arcs makes Another Arc.
$\sim$ has been shown to be reflexive, symmetric and transitive.
Hence by definition it is an equivalence relation.
== Sources ==
* : Part $\text I$: Basic Definitions: Section $4$: Connectedness
Category:Examples of Equivalence Relations
Category:Arc-Connected Spaces
Arc Length for Parametric Equations

== Theorem ==
Let $x = \map f t$ and $y = \map g t$ be real functions of a parameter $t$. 
Let these equations describe a curve $\CC$ that is continuous for all $t \in \closedint a b$ and continuously differentiable for all $t \in \openint a b$.
Suppose that the graph of the curve does not intersect itself for any $t \in \openint a b$.
Then the arc length of $\CC$ between $a$ and $b$ is given by:
:$\ds s = \int_a^b \sqrt {\paren {\frac {\d x} {\d t} }^2 + \paren {\frac {\d y} {\d t} }^2} \rd t$
for $\dfrac {\d x} {\d t} \ne 0$.
== Proof ==
== Also see == 
* Continuously Differentiable Curve has Finite Arc Length
* Length of Arc of Cycloid
== Sources ==
* 
* : $\S 10.3$
* 
* 
Category:Integral Calculus
Arc Length for Polar Curve

== Theorem ==
Let $a$ and $b$ be real numbers.
Let $\CC$ be a simple curve continuous on $\closedint a b$ and continuously differentiable on $\openint a b$.
Let $\CC$ be described by the parametric equations:
:$\begin {cases} x & = r \cos \theta \\ y & = r \sin \theta \end {cases}$
where:
:$r$ is a function of $\theta$
:$\theta \in \closedint a b$.
Then the length $s$ of $\CC$ is given by:
:$\ds s = \int_a^b \sqrt {r^2 + \paren {\frac {\d r} {\d \theta} }^2} \rd \theta$
== Theorem ==
Note that $\CC$ satisfies the conditions of Arc Length for Parametric Equations.
So: 
:$\ds s = \int_a^b \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
We have: 
and:
We then have:
== Also see ==
* Arc Length for Parametric Equations
Category:Integral Calculus
Arc Length of Curve in Polar Coordinates/Function of Angle

== Theorem ==
Let $C$ be a curve embedded in a polar plane.
Let the radial coordinate $r$ of $C$ be defined as a function of the angular coordinate $\theta$:
:$r = \map f \theta$
The arc length $s$ of $C$ between $\theta = \alpha$ and $\theta = \beta$ is defined as: 
:$\ds s := \int_\alpha^\beta \paren {\sqrt {\paren {\frac {\d r} {\d \theta} }^2 + r^2} } \rd \theta$
== Proof ==
== Also see ==
* Arc Length of Curve in Polar Coordinates as Function of Radius
== Sources ==
* 
* 
Category:Arc Length of Curve in Polar Coordinates
Arc Length of Curve in Polar Coordinates/Function of Radius

== Theorem ==
Let $C$ be a curve embedded in a polar plane.
Let the angular coordinate $\theta$ of $C$ be defined as a function of the radial coordinate $r$:
:$\theta = \map f r$
The arc length $s$ of $C$ between $r = u$ and $r = v$ is defined as: 
:$\ds s := \int_u^v \paren {\sqrt {1 + r^2 \paren {\frac {\d \theta} {\d r} }^2} } \rd r$
== Proof ==
== Also see ==
* Arc Length of Curve in Polar Coordinates as Function of Angle
== Sources ==
* 
* 
Category:Arc Length of Curve in Polar Coordinates
Arc Length of Right Parabolic Segment

== Theorem ==
Let $ABC$ be a right parabolic segment where:
:$AB$ is the defining chord $\LL$ of $ABC$
:$C$ is the vertex of the defining parabola $\PP$ of $ABC$.
:
The length $\LL$ of the arc $ACB$ is given by:
where:
:$a$ is the length of the line segment $CF$, where $F$ is the point at which the axis of $\PP$ intersects $AB$
:$b$ is the length of the line segment $AB$.
== Proof ==
== Sources ==
* : $\S 4$: Geometric Formulas: Segment of a Parabola: $4.25$
* : $\S 7$: Geometric Formulas: Segment of a Parabola: $7.25.$
* 
Category:Right Parabolic Segments
Arc Length of Sector

== Theorem ==
Let $\CC = ABC$ be a circle whose center is $A$ and with radii $AB$ and $AC$.
Let $BAC$ be the sector of $\CC$ whose angle between $AB$ and $AC$ is $\theta$.
:
Then the length $s$ of arc $BC$ is given by:
:$s = r \theta$
where:
:$r = AB$ is the length of the radius of the circle
:$\theta$ is measured in radians.
== Proof ==
From Perimeter of Circle, the perimeter of $\CC$ is $2 \pi r$.
From Measurement of Full Angle, the angle within $\CC$ is $2 \pi$.
The fraction of the perimeter of $\CC$ within the sector $BAC$ is therefore $2 \pi r \times \dfrac \theta {2 \pi}$.
Hence the result.
== Sources ==
* : $\S 4$: Geometric Formulas: Sector of Circle of Radius $r$: $4.14$
* : $\S 7$: Geometric Formulas: Sector of Circle of Radius $r$: $7.14.$
Category:Sectors of Circles
Arc Length on Circle forms Metric

== Theorem ==
Let $A \subseteq \R^2$ be the set defined as:
:$A = \set {\tuple {x_1, x_2}: x_1^2 + y_2^2 = 1}$
Thus from Equation of Unit Circle, $A$ is the unit circle embedded in the Cartesian plane.
Let $d: A^2 \to \R$ be the mapping defined as:
:$\forall \tuple {x, y} \in A^2: \map d {x, y} = \begin {cases} 0 & : x = y \\ \pi & : x = -y \\ l & : \text {otherwise} \end {cases}$
where:
:$x$ and $y$ are of the form $\tuple {x_1, x_2}$
:$l$ denotes the length of the minor arc of $A$ between $x$ and $y$.
Then $d$ is a metric for $A$.
== Proof ==
It is to be demonstrated that $d$ satisfies all the metric space axioms.
=== Proof of ===
:$\map d {x, x} = 0$ by definition.
So holds for $d$.
=== Proof of ===
Let $x$, $y$ and $z$ be points on $A$.
Suppose $z$ be such that either:
:it lies on the minor arc from $x$ and $y$
:$x$ and $y$ lie on a diameter of $A$.
Then $x z$ and $z y$ are both minor arcs of $A$.
Hence:
:$\map d {x, z} + \map d {z, y} = \map d {x, y}$
Suppose otherwise, that $z$ lies on the major arc of $A$.
Then by inspection:
:$\map d {x, z} + \map d {z, y} > \map d {x, y}$
So holds for $d$.
=== Proof of ===
By inspection:
:$\map d {x, y} = \map d {y, x}$
for all $x, y \in A$.
So holds for $d$.
=== Proof of ===
By inspection:
:$\map d {x, y} \ge 0$
for all $x, y \in A$.
So holds for $d$.
Thus $d$ satisfies all the metric space axioms and so is a metric.
== Sources ==
* : $2$: Continuity generalized: metric spaces: Exercise $2.6: 8$
Category:Examples of Metric Spaces
Arc in Topological Space is Path

== Theorem ==
Let $T = \struct {X, \tau}$ be a topological space.
Let $f$ be an arc in $T$.
Then $f$ is a path in $T$.
== Proof ==
By definition, an arc from $a$ to $b$ is a path $f: I \to T$ such that $f$ is injective.
Hence the result, by definition.
Category:Topology
Arc of Cycloid is Concave

== Theorem ==
Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane.
Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis.
Consider the cycloid traced out by the point $P$.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.
Then the locus of $P$ is concave.
== Proof ==
From Second Derivative of Locus of Cycloid:
:$y = -\dfrac a {y^2}$
As $y \ge 0$ throughout, then $y < 0$ wherever $y \ne 0$, which is at the cusps.
The result follows from Second Derivative of Concave Real Function is Non-Positive.
== Sources ==
* : Chapter $\text {B}.21$: The Cycloid: Problem $2$
Category:Cycloids
Arccosecant Logarithmic Formulation

== Theorem ==
Let $x$ be a real number. 
Let $x \in \hointl {-\infty} {-1} \cup \hointr 1 {\infty}$.
Then: 
:$\arccsc x = -i \map \Ln {\sqrt {1 - \dfrac 1 {x^2} } + \dfrac i x}$
where:
:$\arccsc$ is the arccosecant function
:$\Ln$ is the principal branch of the complex logarithm whose imaginary part lies in $\hointl {-\pi} \pi$. 
== Proof ==
== Also see ==
* Arcsine Logarithmic Formulation
* Arccosine Logarithmic Formulation
* Arctangent Logarithmic Formulation
* Arccotangent Logarithmic Formulation
* Arcsecant Logarithmic Formulation
Category:Arccosecant Function
Arccosecant of Reciprocal equals Arcsine

== Theorem ==
Everywhere that the function is defined:
:$\map \arccsc {\dfrac 1 x} = \arcsin x$
where $\arcsin$ and $\arccsc$ denote arcsine and arccosecant respectively.
== Proof ==
Category:Arcsine Function
Category:Arccosecant Function
Category:Reciprocals
Arccosine Function in terms of Gaussian Hypergeometric Function

== Theorem ==
:$\arccos x = 2 \sqrt {\dfrac {1 - x} 2} \map F {\dfrac 1 2, \dfrac 1 2; \dfrac 3 2; \dfrac {1 - x} 2}$
where:
:$x$ is a real number with $\size x \le 1$
:$\arccos$ denotes the arccosine function
:$\map F {a, b; c; x}$ is the Gaussian hypergeometric function of $x$: $\ds \sum_{k \mathop = 0}^\infty \dfrac {a^{\overline k} b^{\overline k} } {c^{\overline k} } \dfrac {x^k} {k!}$
== Proof ==
== Also see ==
* Arcsine Function in terms of Gaussian Hypergeometric Function
Category:Arccosine Function
Category:Gaussian Hypergeometric Function
Category:Hypergeometric Functions
Arccosine Logarithmic Formulation

== Theorem ==
For any real number $x: -1 \le x \le 1$:
:$\arccos x = \dfrac 1 i \map \ln {x + \sqrt {x^2 - 1} }$
where $\arccos x$ is the arccosine and $i^2 = -1$.
== Proof ==
Assume $y \in \R$ such that $0 \le y \le \pi$.
== Also see ==
* Arcsine Logarithmic Formulation
* Arctangent Logarithmic Formulation
* Arccotangent Logarithmic Formulation
* Arcsecant Logarithmic Formulation
* Arccosecant Logarithmic Formulation
Category:Arccosine Function
Arccosine in terms of Arctangent

== Theorem ==
:$\arccos x = 2 \map \arctan {\sqrt {\dfrac {1 - x} {1 + x} } }$
where $x$ is a real number with $-1 < x \le 1$.
== Proof ==
Let: 
:$\theta = \arccos x$
Then by the definition of arccosine:
:$x = \cos \theta$
and: 
:$0 \le \theta < \pi$
Then: 
== Also see ==
* Arcsine in terms of Arctangent
Category:Arccosine Function
Category:Arctangent Function
Arccosine of Reciprocal equals Arcsecant

== Theorem ==
Everywhere that the function is defined:
:$\map \arccos {\dfrac 1 x} = \arcsec x$
where $\arccos$ and $\arcsec$ denote arccosine and arcsecant respectively.
== Proof ==
== Sources ==
* : $\S 5$: Trigonometric Functions: $5.78$
Category:Arccosine Function
Category:Arcsecant Function
Category:Reciprocals
Arccotangent Logarithmic Formulation

== Theorem ==
For any real number $x$:
:$\arccot x = \dfrac 1 2 i \, \map \ln {\dfrac {1 + i x} {1 - i x} }$
where $\arccot x$ is the arccotangent and $i^2 = -1$.
== Proof ==
Assume $y \in \R$, $ -\dfrac \pi 2 \le y \le \dfrac \pi 2 $.
== Also see ==
* Arcsine Logarithmic Formulation
* Arccosine Logarithmic Formulation
* Arctangent Logarithmic Formulation
* Arcsecant Logarithmic Formulation
* Arccosecant Logarithmic Formulation
Category:Arccotangent Function
Arccotangent of Reciprocal equals Arctangent

== Theorem ==
Everywhere that the function is defined:
:$\map \arccot {\dfrac 1 x} = \arctan x$
where $\arctan$ and $\arccot$ denote arctangent and arccotangent respectively.
== Proof ==
Category:Arctangent Function
Category:Arccotangent Function
Category:Reciprocals
Archimedes' Cattle Problem

== Classic Problem ==
The sun god had a herd of cattle consisting of bulls and cows, divided into $4$ herds according to their colour:
:white
:black
:dappled
:yellow.
(Accounts vary as to their actual colours, which are arbitrary.)
Among the bulls:
:the number of white ones was one half plus one third the number of the black greater than the yellow
:the number of the black ones was one quarter plus one fifth the number of the dappled greater than the yellow
:the number of the dappled ones was one sixth and one seventh the number of the white greater than the yellow.
Among the cows:
:the number of white ones was one third plus one quarter of the total black cattle
:the number of the black ones was one quarter plus one fifth the total of the dappled cattle
:the number of dappled cattle was one fifth plus one sixth the total of the yellow cattle
:the number of the yellow cattle was one sixth plus one seventh the total of the white cattle.
What was the composition of the herd?
=== Difficult Version ===
== Solution ==
Let:
:$W$ be the number of white bulls
:$B$ be the number of black bulls
:$D$ be the number of dappled bulls
:$Y$ be the number of yellow bulls
:$w$ be the number of white cows
:$b$ be the number of black cows
:$d$ be the number of dappled cows
:$y$ be the number of yellow cows.
Then the conditions can be expressed as:
This is a set of $7$ simultaneous equations in $8$ variables, and thus indeterminate.
From $(1)$ to $(3)$ we can find:
Because $\gcd \set {891, 1580} = 1$, we have that $Y$ must be a multiple of $891$: $G$, say.
Hence:
Substituting into equations $(4)$ to $(7)$, we obtain:
Solving these $4$ equations gives:
Because $4657$ is a divisor of none of the numbers on the , it must be the case that
:$4657 \divides G$
where $\divides$ denotes divisibility.
Hence the smallest solution is:
== Also known as ==
This problem is also known as:
:the bovinum problema (that is, cattle problem in Latin)
:Archimedes' Reverse
== Historical Note ==
== Sources ==
* 
* 
* (Translated by )
* : Archimedes' Cattle Problem: $16$
* 
Category:Archimedes' Cattle Problem
Category:Diophantine Problems
Category:Classic Problems
Archimedes' Cattle Problem/Difficult Version

== Classic Problem ==
In addition:
:All the white and black bulls were arranged in a square
:All the dappled and brown bulls were arranged in a triangle.
What was the composition of the herd?
== Solution ==
Reduction of the system of simultaneous equations results in the Pell's equation:
:$t^2 - 4 \, 729 \, 494 u^2 = 1$
The smallest solution set to this equation is found to be:
In this case, the total number of cattle is approximately:
:$W + B + D + Y + w + b + d + y = 7 \cdotp 760271 \ldots \times 10^{206 \, 544}$
== Historical Note ==
== Sources ==
* 
* 
* : $4,729,494$
* : Archimedes' Cattle Problem: $16$
* : $4,729,494$
* 
Category:Archimedes' Cattle Problem
Archimedes' Cattle Problem/Historical Note

== Historical Note on Archimedes' Cattle Problem ==
There are various interpretations of this problem, as the initial wording can be construed as ambiguous.
In particular, it is not completely clear in the difficult version whether the white and black bulls need to be arranged in a square or a rectangle.
While the problem is attributed to , it is far from certain that he was the one who originated it.
It appears, from the arbitrary nature of the numbers selected, and the difficulty of finding a solution, that the problem was contrived with no consideration of what the solution might be, or indeed whether there is one, in the same way that a lecturer in mathematics invents a problem, on the fly, as an illustration of the use of a particular solution technique, without first having checked that the solution will be immediately tractable.
The smallest solution to the Pellian equation $t^2 - 4 \, 729 \, 494 u^2 = 1$ was discovered by and in $1880$, but the corresponding total number of cattle was not actually calculated until $1965$.
This was done by computer. It took $7$ hours and $49$ minutes of computer time, and its printout filled $46$ pages.
This was checked in $1981$ on a more powerful machine, and this took $10$ minutes.
The full solution was published in the .
== Sources ==
* 
* 
* 
* : $4,729,494$
* : Archimedes' Cattle Problem: $16$
* : $4,729,494$
* 
Category:Historical Notes
Archimedes' Cattle Problem/Mistake

== Source Work ==
:
:The Dictionary
::$4,729,494$
:
:The Dictionary
::$4,729,494$
== Mistake ==
:... in this case the total number of cattle is a number of $206,545$ digits, starting $7766 \ldots$
According to other sources, the number in fact starts $7760271 \ldots$
== Sources ==
* : $4,729,494$
* : $4,729,494$
Category:Mistakes/Curious and Interesting Numbers
Archimedes' Limits to Value of Pi

== Theorem ==
The value of $\pi$ lies between $3 \frac {10} {71}$ and $3 \frac 1 7$:
:$3 \dfrac {10} {71} < \pi < 3 \dfrac 1 7$
== Archimedes' Iterative Proof ==
== Trigonometric Proof ==
== Historical Note ==
== Sources ==
* 
* : Chapter $\text{II}$: Modern Minds in Ancient Bodies
Category:Archimedes' Limits to Value of Pi
Category:Pi
Category:Circles
Archimedes' Limits to Value of Pi/Archimedes' Iterative Proof/Historical Note

== Historical Note on Archimedes' Limits to Value of Pi: Archimedes' Iterative Proof ==
This proof is presented in the same way that presented it.
He started the upper bound analysis by finding the perimeter of a circumscribed regular hexagon.
Then, using symmetry, one side of this hexagon is equated to twice the tangent of a right triangle with central angle $30 \degrees$ in a unit circle. 
The lower bound analysis, on the other hand, starts by finding the perimeter of an inscribed regular hexagon as the sine of a right triangle with inscribed angle of $60 \degrees$ in a circle with unit diameter.
Using this technique it is seen that:
:$(1): \quad$ the analyses both start with a regular hexagon, enabling the same sequence of polygons to be used
:$(2): \quad$ the estimates of $\sqrt 3$ are initial inputs to both parts of the algorithm.
This serves to simplify the development.
Category:Historical Notes
Archimedes' Limits to Value of Pi/Historical Note

== Historical Note on Archimedes' Limits to Value of Pi ==
 demonstrated the limits to the value of $\pi$ in his . 
He does not say where the two estimates $\dfrac {265} {153}$ and $\dfrac {1351} {780}$ for $\sqrt 3$ come from.
They can be easily derived from the continued fraction representation, namely, $\sqbrk {1, \sequence {1, 2} }$, but whether he knew this, or had another method, is unclear.
== Sources ==
* : Chapter $\text{II}$: Modern Minds in Ancient Bodies
* : Chapter $\text {A}.5$: Archimedes (ca. B.C.)
Category:Historical Notes
Archimedes' Limits to Value of Pi/Lemma 2

== Lemma for Archimedes' Limits to Value of Pi ==
Let:
:$\cot \phi = \dfrac p q$
Then:
:$\csc \phi = \dfrac 1 q \cdot \sqrt {p^2 + q^2}$
== Proof ==
We have:
Let:
:$\cot \phi = \dfrac p q$
Then:
Category:Archimedes' Limits to Value of Pi
Archimedes' Limits to Value of Pi/Lemma 3

== Lemma for Archimedes' Limits to Value of Pi ==
:$\pi < \dfrac {22} 7$
== Proof ==
;Upper bound
:
Let $AC$ be tangent to circle $O$ at $A$.
We define the following angles:
Thus we have that $\triangle AOC$ has angles $30 \degrees$, $60 \degrees$ and $90 \degrees$.
Therefore:
Therefore, $\triangle AOC$ has sides in the ratio:
:$AC : OC : OA = 1 : 2 : \sqrt 3$
Construct $C'$ on the extended tangent $AC$ such that $C'C = 2 AC$. 
$C'C$ subtends $\angle C'OC$ which is equal to $60 \degrees$.
Thus $C'C = 2 AC$ is one side of a regular polygon of $6$ sides.
The total perimeter $p$ of the circumscribed hexagon is:
:$p = 6 \cdot CC' = 12 \cdot AC$
The ratio of the perimeter of the circumscribed hexagon to the diameter of the circle is:
:$\dfrac p {AB} = \dfrac {12 AC} {2 AO} = 6 \cdot \dfrac {AC} {AO}$
Since $AB = 1$, the perimeter of the circumscribed hexagon is:
:$p = 6 \cdot \dfrac 1 {\sqrt 3} = 2 \sqrt 3 < 3.4642$
This is an initial upper bound on $\pi$.
We now continue by doubling the number of sides of the circumscribed regular polygon to refine our estimate.
We first note that:
We will now use the following rational approximation for $\sqrt 3$, whose decimal value is a little greater than $1.73205$:
:$\dfrac {OA} {AC} = \cot \theta_1 > \dfrac {265} {153}$
This approximation is about $1.73203$, which is slightly less than $1.73205$, and so less than $\sqrt 3$.
Thus, the initial upper bound estimate for $\pi$ is:
;Second Iteration Upper Bound Calculation
We see that:
:$591 \tfrac 1 8 = 591.125$ is lower than the square root of $349 \, 450$ which is approximately $591.143$.
Thus, our second upper bound estimate for $\pi$ is:
;Third Iteration Upper Bound Calculation
We see that:
:$1172 \tfrac 1 8 = 1172.125$ is lower than the square root of $1 \, 373 \, 943.516$ which is approximately $1172.153$.
Thus, our third upper bound estimate for $\pi$ is:
;Fourth Iteration Upper Bound Calculation
We see that:
:$2339 \tfrac 1 4 = 2339.25$ is lower than the square root of $5 \, 472 \, 132.0625$ which is approximately $2339.259$.
Thus, our fourth upper bound estimate for $\pi$ is:
;Fifth Iteration Upper Bound Calculation
Thus, our fifth upper bound estimate for $\pi$ is:
Category:Archimedes' Limits to Value of Pi