|
|
| Line 1: |
Line 1: |
| {{for|numbers stamped on [[plastic]]s for recycling identification|Resin identification code}}
| | The name of the author is Jayson. To climb is something I really appreciate performing. He is an information officer. For a while I've been in Mississippi but now I'm considering other options.<br><br>Also visit my web page :: psychic readings online ([http://www.socialairforce.com/groups/hobby-advice-and-tips-completely-from-the-pros/ socialairforce.com]) |
| | |
| {| border="1" style="float: right; border-collapse: collapse; margin-left:1em;"
| |
| | colspan="2" align="center" | {{Irrational numbers}}
| |
| |-
| |
| |[[Binary numeral system|Binary]]
| |
| | 1.0101001100100000101…
| |
| |-
| |
| | [[Decimal]]
| |
| | 1.32471795724474602596…
| |
| |-
| |
| | [[Hexadecimal]]
| |
| | 1.5320B74ECA44ADAC1788…
| |
| |-
| |
| | [[Continued fraction]]<ref>Sequence {{OEIS2C|A072117}} in the [[OEIS]]</ref>
| |
| | {{nowrap|[1; 3, 12, 1, 1, 3, 2, 3, 2, 4, 2, 141, 80 ...]}}<br /><small>Note that this continued fraction is not [[Periodic continued fraction|periodic]]. (Shown in [[continued fraction#Notations for continued fractions|linear notation]])</small>
| |
| |-
| |
| | [[Algebraic number|Algebraic form]]
| |
| | <math>\frac{\sqrt[3]{108 + 12\sqrt{69}} + \sqrt[3]{108 - 12\sqrt{69}}}{6}</math>
| |
| |}
| |
| In [[mathematics]], the '''plastic number''' ''ρ'' (also known as the '''plastic constant''') is a [[mathematical constant]] which is the unique real solution of the [[cubic function|cubic equation]]
| |
| :<math>x^3=x+1\, .</math>
| |
| | |
| It has the value<ref>{{MathWorld |title=Plastic Constant |id=PlasticConstant}}</ref>
| |
| :<div style="display: inline-block; width: 300px;"><math>\rho = \frac{\sqrt[3]{108 + 12\sqrt{69}} + \sqrt[3]{108 - 12\sqrt{69}}}{6}\, .</math></div>
| |
| Its decimal expansion begins with {{gaps|1.32471|79572|44746|02596|09088|54...}}.<ref>Sequence {{OEIS2C|id=A060006}} in the [[OEIS]].</ref> and at least 10,000,000,000 decimal digits have been computed.<ref>http://www.komsta.net/computations</ref>
| |
| | |
| The plastic number is also sometimes called the '''silver number,''' but that name is more commonly used for the [[silver ratio]] <math>1 + \sqrt{2}</math>. | |
| | |
| == Properties ==
| |
| | |
| The powers of the plastic number ''A''(''n'') = ''ρ''<sup>''n''</sup> satisfy the recurrence relation ''A''(''n'') = ''A''(''n'' − 2) + ''A''(''n'' − 3) for ''n'' > 2. Hence it is the limiting ratio of successive terms of any (non-zero) integer sequence satisfying this recurrence such as the [[Padovan sequence]] and the [[Perrin number|Perrin sequence]], and bears the same relationship to these sequences as the [[golden ratio]] does to the [[Fibonacci number|Fibonacci sequence]] and the [[silver ratio]] does to the [[Pell number]]s.
| |
| | |
| Because the plastic number has [[Minimal polynomial (field theory)|minimal polynomial]] {{nowrap|''x''<sup>3</sup> − ''x'' − 1 {{=}} 0,}} it is also a solution of the polynomial equation ''p''(''x'') = 0 for every polynomial ''p'' that is a multiple of {{nowrap|''x''<sup>3</sup> − ''x'' − 1,}} but not for any other polynomials with integer coefficients.
| |
| | |
| The plastic number satisfies the [[nested radical]] recurrence:<ref>{{MathWorld|urlname=PlasticConstant|title=Plastic Constant|author=Piezas, Tito III; van Lamoen, Floor; and Weisstein, Eric W.}}</ref>
| |
| | |
| :<math>\rho = \sqrt[3]{1 + \sqrt[3]{1 + \sqrt[3]{1 + \cdots}}} \, .</math> | |
| | |
| The plastic number is the smallest [[Pisot–Vijayaraghavan number]]. Its [[algebraic conjugate]]s are
| |
| :<math>\left(-\frac12\pm\frac{\sqrt3}2i\right)\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\left(-\frac12\mp\frac{\sqrt3}2i\right)\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}\approx -0.662359 \pm 0.56228i | |
| ,</math>
| |
| of [[absolute value]] ≈ 0.868837 {{OEIS|A191909}}. This value is also <math>\frac{1}{\sqrt{\rho}}</math> because the product of the three roots of the minimal polynomial is 1.
| |
| | |
| == History ==
| |
| | |
| The name ''plastic number'' (''het plastische getal'' in [[Dutch language|Dutch]]) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the [[golden ratio]] and [[silver ratio]], the word plastic was not intended to refer to a specific substance, but rather in its adjectival sense, meaning something that can be given a three-dimensional shape.<ref>{{harvtxt|Padovan|2002}};{{harvtxt|Shannon|Anderson|Horadam|2006}}.</ref> This is because, according to Padovan, the characteristic ratios of the number, 3/4 and 1/7, relate to the limits of human perception in relating one physical size to another.
| |
| | |
| ==Notes==
| |
| {{reflist|2}}
| |
| | |
| ==References==
| |
| {{refbegin}}
| |
| *{{citation|first=Midhat J.|last=Gazalé|title=Gnomon|year=1999|publisher=Princeton University Press}}.
| |
| *{{citation
| |
| | url = http://www.nexusjournal.com/conferences/N2002-Padovan.html
| |
| | contribution = Dom Hans Van Der Laan And The Plastic Number
| |
| | last = Padovan | first = Richard | authorlink = Richard Padovan
| |
| | title = Nexus IV: Architecture and Mathematics
| |
| | pages = 181–193 | publisher = Kim Williams Books | year = 2002}}.
| |
| *{{citation
| |
| | last1 = Shannon | first1 = A. G.
| |
| | last2 = Anderson | first2 = P. G.
| |
| | last3 = Horadam | first3 = A. F.
| |
| | title = Properties of Cordonnier, Perrin and Van der Laan numbers
| |
| | journal = International Journal of Mathematical Education in Science and Technology
| |
| | volume = 37 | issue = 7 | year = 2006 | pages = 825–831
| |
| | doi = 10.1080/00207390600712554}}.
| |
| *{{citation
| |
| | last1 = Aarts | first1 = J.
| |
| | last2 = Fokkink | first2 = R.
| |
| | last3 = Kruijtzer | first3 = G.
| |
| | title = Morphic numbers
| |
| | journal = Nieuw Arch. Wiskd.
| |
| | series = 5
| |
| | volume = 2 | issue = 1 | year = 2001 | pages = 56–58
| |
| | url = http://www.nieuwarchief.nl/serie5/pdf/naw5-2001-02-1-056.pdf}}.
| |
| {{refend}}
| |
| | |
| ==External links==
| |
| *[http://wayback.archive.org/web/20120320051231/http://members.fortunecity.com/templarser/padovan.html Tales of a Neglected Number] by [[Ian Stewart (mathematician)|Ian Stewart]]
| |
| | |
| [[Category:Irrational numbers]]
| |
| [[Category:Mathematical constants]]
| |
The name of the author is Jayson. To climb is something I really appreciate performing. He is an information officer. For a while I've been in Mississippi but now I'm considering other options.
Also visit my web page :: psychic readings online (socialairforce.com)