Binary Binary numeral system The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...	1.0101001100100000101…
Decimal Decimal The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....	1.32471795724474602596…
Hexadecimal Hexadecimal In mathematics and computer science, hexadecimal is a positional numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen...	1.5320B74ECA44ADAC1788…
Continued fraction Continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...	Note that this continued fraction is not periodic.
Algebraic form Algebraic number In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the plastic number ρ (also known as the plastic constant) is a mathematical constant

Mathematical constant

A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...

 which is the unique real solution of the cubic equation

Cubic function

In mathematics, a cubic function is a function of the formf=ax^3+bx^2+cx+d,\,where a is nonzero; or in other words, a polynomial of degree three. The derivative of a cubic function is a quadratic function...

It has the value
its decimal expansion begins with .

The plastic number is also sometimes called the silver number, but that name is more commonly used for the silver ratio

Silver ratio

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. This defines the silver ratio as an irrational mathematical constant,...

 .

Properties

The powers of the plastic number A(n) = ρⁿ 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

Padovan sequence

The Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...

 and the Perrin sequence

Perrin number

In mathematics, the Perrin numbers are defined by the recurrence relationandThe sequence of Perrin numbers starts withThe number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number.-History:...

, and bears the same relationship to these sequences as the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

 does to the Fibonacci sequence

Fibonacci number

In mathematics, the Fibonacci numbers are the numbers in the following integer sequence:0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots\; ....

 and the silver ratio

Silver ratio

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. This defines the silver ratio as an irrational mathematical constant,...

 does to the Pell number

Pell number

In mathematics, the Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers...

s.

Because the plastic number has minimal polynomial

Minimal polynomial (field theory)

In field theory, given a field extension E / F and an element α of E that is an algebraic element over F, the minimal polynomial of α is the monic polynomial p, with coefficients in F, of least degree such that p = 0...

  it is also a solution of the polynomial equation p(x) = 0 for every polynomial p that is a multiple of but not for any other polynomials with integer coefficients.

The plastic number satisfies the nested radical recurrence:


The plastic number is the smallest Pisot–Vijayaraghavan number. Its algebraic conjugates are
of absolute value

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

 ≈ 0.868837. This value is also because the product of the three roots of the minimal polynomial is 1.

History

The name plastic number (het plastische getal in Dutch

Dutch language

Dutch is a West Germanic language and the native language of the majority of the population of the Netherlands, Belgium, and Suriname, the three member states of the Dutch Language Union. Most speakers live in the European Union, where it is a first language for about 23 million and a second...

) was given to this number in 1928 by Dom Hans van der Laan. Unlike the names of the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

 and silver ratio

Silver ratio

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. This defines the silver ratio as an irrational mathematical constant,...

, 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. 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.

External links

• Tales of a Neglected Number by Ian Stewart
  Ian Stewart (mathematician)
  Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...