Anton Zeilinger: Difference between revisions

Latest revision as of 13:09, 6 January 2015

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Anton Zeilinger: Difference between revisions

Latest revision as of 13:09, 6 January 2015

Contents

Vibram Five Fingers Sydney finance

Polo Shirts Malwarebytes

Nike Air Force 1 Australia More often than not

Cheap Nike Roshe Australia Now reengined

Nike Free 5.0 which means this question

Nike Shox and unfortunately

Hollister Brisbane leading to updated plans in No4

Navigation menu

@@ Line 1: / Line 1: @@
-{| border="1" style="float: right; border-collapse: collapse;"
+== Vibram Five Fingers Sydney  finance ==
-| colspan="2" align="center" | {{Irrational numbers}}
-|-
-|[[Binary numeral system|Binary]]
-| 10.0110101000001001111...
-|-
-| [[Decimal]]
-| 2.4142135623730950488...
-|-
-| [[Hexadecimal]]
-| 2.6A09E667F3BCC908B2F...
-|-
-| [[Continued fraction]]
-| <math>2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{\ddots}}}}</math>
-|-
-| [[Algebraic number|Algebraic form]]
-| <math>{1+\sqrt{2}}</math>
-|}
-In [[mathematics]], two quantities are in the '''silver ratio''' if the [[ratio]] between the sum of the smaller plus twice the larger of those quantities and the larger one is the same as the ratio between the larger one and the smaller (see below). This defines the silver ratio as an [[irrational number|irrational]] [[mathematical constant]], whose value of one plus the [[square root of 2]] is approximately 2.4142135623. Its name is an allusion to the [[golden ratio]]; analogously to the way the golden ratio is the limiting ratio of consecutive [[Fibonacci number]]s, the silver ratio is the limiting ratio of consecutive [[Pell number]]s. The silver ratio is denoted by δ<sub>''S''</sub>.
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-[[Mathematician]]s have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the [[square root of 2]], its covergents, [[square triangular number]]s, [[Pell number]]s, [[octagon]]s and the like.
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-:<math> \frac{2a + b}{a} = \frac{a}{b} \equiv \delta_S\,.</math>
+== Nike Air Force 1 Australia More often than not ==
-The silver ratio can also be defined by the simple [[continued fraction]] [2; 2, 2, 2, ...]:
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-:<math>\delta_S = 2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}\, .</math>
+== Cheap Nike Roshe Australia Now reengined ==
-The [[Convergent (continued fraction)|convergents]] of this continued fraction (2/1, 5/2, 12/5, 29/12, 70/29, ...) are ratios of consecutive [[Pell number]]s. These fractions provide accurate [[Diophantine approximation|rational approximations]] of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive [[Fibonacci number]]s.
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-== Properties ==
+== Nike Free 5.0  which means this question ==
-=== Number-theoretic properties ===
-The silver ratio is a [[Pisot–Vijayaraghavan number]] (PV number), as its conjugate {{nowrap|1 − {{sqrt|2}} {{=}} <big>{{frac2|−1|δ<sub>''S''</sub>}}</big> ≈ −0.41}} has absolute value less than 1. In fact it is the second smallest quadratic PV number after the golden ratio. This means the distance from δ{{su|p=''n''|b=''S''}} to the nearest integer is {{nowrap|<big>{{frac2|1|δ{{su|p=''n''|b=''S''}}}}</big> ≈ 0.41<sup>''n''</sup>}}. Thus, the sequence of [[fractional part]]s of δ{{su|p=''n''|b=''S''}}, {{nowrap|''n'' {{=}} 1, 2, 3, ...}} (taken as elements of the torus) converges. In particular, this sequence is not [[equidistributed sequence|equidistributed mod 1]].
-=== Powers ===
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-The lower powers of the silver ratio are
-:<math> \!\ \delta_S^0 = [1] = 1</math>
+   <li>[http://enseignement-lsf.com/spip.php?article64#forum18083649 http://enseignement-lsf.com/spip.php?article64#forum18083649]</li>
-:<math> \delta_S^1 = \delta_S + 0 = [2;2,2,2,2,2,\dots] \approx 2.41421</math>
-:<math> \delta_S^2 = 2\delta_S + 1 = [5;1,4,1,4,1,\dots] \approx 5.82842</math>
+   <li>[http://115.28.106.0/forum.php?mod=viewthread&tid=3465 http://115.28.106.0/forum.php?mod=viewthread&tid=3465]</li>
-:<math> \delta_S^3 = 5\delta_S + 2 = [14;14,14,14,\dots] \approx 14.07107</math>
-:<math> \delta_S^4 = 12\delta_S + 5 = [33;1,32,1,32,\dots] \approx 33.97056 </math>
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-The powers continue in the pattern
+== Nike Shox  and unfortunately ==
-:<math> \!\ \delta_S^n = K_n\delta_S + K_{n-1} </math>
-where
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-:<math> \!\ K_n = 2 K_{n-1} + K_{n-2} </math>
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-:<math> \!\ \delta_S^5 = 29\delta_S + 12 = [82;82,82,82,\dots] \approx 82.01219 </math>
+   <li>[http://www.vitrinaweb.es/index.php?page=item&id=109836 http://www.vitrinaweb.es/index.php?page=item&id=109836]</li>
-Using {{nowrap|K<sub>0</sub> {{=}} 1}} and {{nowrap|K<sub>1</sub> {{=}} 2}} as initial conditions, a [[Binet's formula|Binet]]-like formula results from solving the recurrence relation
+   <li>[http://www.hanzhuangpifa.org/forum.php?mod=viewthread&tid=1486410&fromuid=73548 http://www.hanzhuangpifa.org/forum.php?mod=viewthread&tid=1486410&fromuid=73548]</li>
-:<math> \!\ K_n = 2 K_{n-1} + K_{n-2} </math>
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-which becomes
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-:<math> \!\ K_n = \frac{1}{2\sqrt{2}} {(\delta_S^{n+1} - {(2-\delta_S)}^{n+1})} </math>
+   <li>[http://www.histoirepassion.eu/spip.php?article1078/ http://www.histoirepassion.eu/spip.php?article1078/]</li>
-=== Trigonometric properties ===
+ </ul>
-{{see also|Exact trigonometric constants #22.5°: regular octagon}}
-The silver ratio is intimately connected to trigonometric ratios for {{nowrap|<big>{{frac2|π|8}}</big> {{=}} 22.5°}}.
-:<math>\textstyle \sin \frac {\pi}{8} = \cos \frac {3\pi}{8} = \frac{1}{2} \sqrt{2 - \sqrt 2}=\sqrt{ \frac{1}{2\delta_s}} </math>
-:<math>\textstyle \cos \frac {\pi}{8} = \sin \frac {3\pi}{8} = \frac{1}{2} \sqrt{2 + \sqrt{2}}=\sqrt{ \frac{\delta_s}{2}} </math>
-:<math>\textstyle \tan \frac {\pi}{8} = \cot \frac {3\pi}{8} = \sqrt{2}-1= \frac{1}{\delta_s} </math>
-:<math>\textstyle \cot \frac {\pi}{8} = \tan \frac {3\pi}{8} = \sqrt{2}+1=\delta_s </math>
-So the area of a regular [[octagon]] with side length ''a'' is given by
-:<math>A = \textstyle 2a^2 \cot \frac{\pi}{8} = 2(1+\sqrt{2})a^2 \simeq 4.828427 a^2.</math>
-== Silver means ==
-<table class="infobox" style="font-size: 90%; text-align: left;">
-<tr><td colspan="3" align="center">'''Silver means'''</td></tr>
-<tr><td>'''0:'''</td><td> {{sfrac|0 + {{sqrt|4}}|2}} </td><td>1</td></tr>
-<tr><td>'''1:'''</td><td> {{sfrac|1 + {{sqrt|5}}|2}} </td><td>[[golden ratio|1.618033989]]</td></tr>
-<tr><td>'''2:'''</td><td> {{sfrac|2 + {{sqrt|8}}|2}} </td><td>2.414213562</td></tr>
-<tr><td>'''3:'''</td><td> {{sfrac|3 + {{sqrt|13}}|2}} </td><td>3.302775638</td></tr>
-<tr><td>'''4:'''</td><td> {{sfrac|4 + {{sqrt|20}}|2}} </td><td>4.236067978</td></tr>
-<tr><td>'''5:'''</td><td> {{sfrac|5 + {{sqrt|29}}|2}} </td><td>5.192582404</td></tr>
-<tr><td>'''6:'''</td><td> {{sfrac|6 + {{sqrt|40}}|2}} </td><td>6.162277660</td></tr>
-<tr><td>'''7:'''</td><td> {{sfrac|7 + {{sqrt|53}}|2}} </td><td>7.140054945</td></tr>
-<tr><td>'''8:'''</td><td> {{sfrac|8 + {{sqrt|68}}|2}} </td><td>8.123105626</td></tr>
-<tr><td>'''9:'''</td><td> {{sfrac|9 + {{sqrt|85}}|2}} </td><td>9.109772229</td></tr>
-<tr><td>&nbsp; ⋮</td></tr>
-<tr><td>'''''n'':'''</td><td> {{sfrac|''n'' + {{sqrt|4 + ''n''<sup>2</sup>}}|2}} </td></tr>
-</table>
-The more general [[continued fraction|simple continued fraction]] expressions
-:<math>
-n + \cfrac{1}{n+\cfrac{1} {n+\cfrac{1} {n+\cfrac{1} {n+\ddots\,}}}}
-= [n;n,n,n,n,\dots] = \frac{1}{2}\left(n+\sqrt{n^2+4}\right)\,
-</math>
-are known as the '''silver means''' or '''metallic means'''<ref>‏[[Vera de Spinadel]] (1999)[http://www.mi.sanu.ac.rs/vismath/spinadel/index.html The Family of Metallic Means], ''Vismath'' 1(3) from Mathematical Institute of [[Serbian Academy of Sciences and Arts]]‏</ref> of the successive [[natural number]]s. The golden ratio is the silver mean between 1 and 2, while the silver ratio is the silver mean between 2 and 3.  The term "'''bronze ratio'''", or terms using other names of metals, are occasionally used to name subsequent silver means. The values of the first ten silver means are shown at right.<ref>[http://mathworld.wolfram.com/SilverRatio.html Table of silver means] from [[Wolfram MathWorld]]</ref> Notice that each silver mean is a root of the simple quadratic equation
-:<math> x^2 - nx = 1,\,</math> where ''n'' is any positive natural number.
-=== Properties ===
-These properties are valid only for integers m, for nonintegers the properties are similar but slightly different.
-The above property for the powers of the silver ratio is a consequence of a property of the powers of silver means.  For the silver mean ''S'' of ''m'', the property can be generalized as
-:<math> \!\ S_{m}^n = K_{n}S_{m} + K_{(n-1)} </math>
-where
-:<math> \!\ K_n = mK_{(n-1)} + K_{(n-2)}. </math>
-Using the initial conditions {{nowrap|K<sub>0</sub> {{=}} 1}} and {{nowrap|K<sub>1</sub> {{=}} ''m''}}, this recurrence relation becomes
-:<math> \!\ K_n = \frac{1}{\sqrt{m^2 + 4}} {(S_m^{n+1} - {(m-S_m)}^{n+1})}. </math>
-The powers of silver means have other interesting properties:
-:If ''n'' is a positive even integer:
-::<math> \!\ {S_m^n - \lfloor S_m^n \rfloor}  = 1 - S_m^{-n}. </math>
-Additionally,
-::<math> \!\ {1 \over {S_m^4 - \lfloor S_m^4 \rfloor}} + \lfloor S_m^4 - 1 \rfloor = S_{(m^4 + 4m^2 + 1)} </math>
-::<math> \!\ {1 \over {S_m^6 - \lfloor S_m^6 \rfloor }} + \lfloor S_m^6 - 1 \rfloor = S_{(m^6 + 6m^4 + 9m^2 +1)}. </math>
-Also,
-::<math> \!\ S_m^3 = S_{(m^3 + 3m)} </math>
-::<math> \!\ S_m^5 = S_{(m^5 + 5m^3 + 5m)} </math>
-::<math> \!\ S_m^7 = S_{(m^7 + 7m^5 + 14m^3 + 7m)} </math>
-::<math> \!\ S_m^9 = S_{(m^9 + 9m^7 + 27m^5 + 30m^3 + 9m)} </math>
-::<math> \!\ S_m^{11} = S_{(m^{11} + 11m^9 + 44m^7 + 77m^5 + 55m^3 + 11m)}. </math>
-In general:
-::<math> \!\ S_m^{2n+1} = S_{\sum_{k=0}^n {{2n+1} \over {2k+1}} {{n+k} \choose {2k}} m^{2k+1}}.</math>
-The silver mean ''S'' of ''m'' also has the property that
-:<math> \!\ 1/S_m = S_m - m </math>
-meaning that the inverse of a silver mean has the same decimal part as the corresponding silver mean.
-:<math> \!\ S_m = a + b </math>
-where ''a'' is the integer part of ''S'' and ''b'' is the decimal part of ''S'', then the following property is true:
-:<math> \!\ S_m^2 = a^2 + mb + 1. </math>
-Because (for all ''m'' greater than 0), the integer part of {{nowrap|''S<sub>m</sub>'' {{=}} ''m''}}, {{nowrap|''a'' {{=}} ''m''}}. For {{nowrap|m > 1}}, we then have
-:<math> \!\ S_m^2 = ma + mb + 1 </math>
-:<math> \!\ S_m^2 = m(a+b) + 1 </math>
-:<math> \!\ S_m^2 = m(S_m) + 1. </math>
-Therefore the silver mean of m is a solution of the equation
-:<math> \!\ x^2 - mx - 1 = 0.</math>
-It may also be useful to note that the silver mean ''S'' of −''m'' is the inverse of the silver mean ''S'' of ''m''
-:<math> \!\ 1/S_m = S_{(-m)} = S_m - m. </math>
-Another interesting result can be obtained by slightly changing the formula of the silver mean. If we consider a number
-:<math> \!\ \frac{1}{2}\left(n+\sqrt{n^2+4c}\right) = R </math>
-then the following properties are true:
-:<math> \!\ R - \lfloor R \rfloor = c/R </math> if ''c'' is real,
-:<math> \!\ \left({1 \over R}\right)c = R - \lfloor \operatorname{Re}(R) \rfloor </math> if ''c'' is a multiple of ''i''.
-The silver mean of ''m'' is also given by the integral
-:<math> S_m = \int_0^m {\left( {x \over {2\sqrt{x^2+4}}} + {{m+2} \over {2m}} \right)} \, dx. </math>
-== Paper sizes and silver rectangles ==
-<!-- linked from redirect [[Silver rectangles]] -->
-[[File:A size illustration2.svg|150px|right]]
-The [[paper size]]s under [[ISO 216]] are [[rectangle]]s in the proportion 1:{{sqrt|2}} (approximately 1:1.4142135 decimal), sometimes called "A4 rectangles". Removing a largest possible square from a sheet of such paper leaves a [[rectangle]] with proportions 1:{{sqrt|2}}−1 which is the same as 1+{{sqrt|2}}:1, the silver ratio. Removing a largest square from one of ''these'' sheets leaves one again with aspect ratio 1:{{sqrt|2}}. A rectangle whose aspect ratio is the silver ratio is sometimes called a '''silver rectangle''' by analogy with [[golden rectangle]]s. Confusingly, "silver rectangle" can also refer to the paper sizes specified by [[ISO 216]].
-[[File:Octagon1 svg.png|150px|right]]
-Removing the largest possible square from either kind yields a silver rectangle of the other kind, and then repeating the process once more gives a rectangle of the original shape but smaller by a linear factor of 1+{{sqrt|2}}.<ref name="kapusta">{{citation|last=Kapusta|first=Janos|title=The square, the circle, and the golden proportion: a new class of geometrical constructions|journal=Forma|volume=19|year=2004|pages=293–313|url=http://www.scipress.org/journals/forma/pdf/1904/19040293.pdf}}.</ref>
-However, only the [[Lichtenberg ratio]] rectangles (rectangles with the shape of ISO 216 paper) have the property that by cutting the rectangle in half across its long side produces two smaller rectangles of the same aspect ratio.
-The silver rectangle is connected to the regular [[octagon]]. If a regular octagon is partitioned into two isosceles trapezoids and a rectangle, then the rectangle is a silver rectangle with an aspect ratio of 1:δ<sub>''S''</sub>, and the 4 sides of the trapezoids are in a ratio of 1:1:1:δ<sub>''S''</sub>.
-If the edge length of a regular octagon is ''t'', then the [[inscribed figure|inradius]] of the octagon (the distance between opposite sides) is δ<sub>''S''</sub>''t'', and the area of the octagon is 2δ<sub>''S''</sub>t<sup>2</sup>.<ref name="kapusta"/>
-== References ==
-{{Reflist}}
-== External links ==
-*{{mathworld|urlname=SilverRatio|title=Silver Ratio}}
-*[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfINTRO.html#silver Explanation of Silver Means]
-{{DEFAULTSORT:Silver Ratio}}
-[[Category:Algebraic numbers]]
-[[Category:Continued fractions]]
-[[Category:Irrational numbers]]
-[[Category:Mathematical constants]]
-[[Category:Ratios]]

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