Топ-100 ★ Non-integer representation - non-standard positional nume
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Non-integer representation
                                     

★ Non-integer representation

A non-integer representation uses non-integer numbers as the radix, or bases, of a positional numeral system. For a non-integer radix β >, 1, the value of

x = d n … d 2 d 1 d 0. d − 1 d − 2 … d − m {\displaystyle x=d_{n}\dots d_{2}d_{1}d_{0}.d_{-1}d_{-2}\dots d_{-m}}

it

x = β n d n + ⋯ + β 2 d 2 + β d 1 + d 0 + β − 1 d − 1 + β − 2 d − 2 + ⋯ + β − m d − m. {\displaystyle {\begin{aligned}x&,=\beta ^{n}d_{n}+\cdots +\beta ^{2}d_{2}+\beta d_{1}+d_{0}\\&,\qquad +\beta ^{-1}d_{-1}+\beta ^{-2}d_{-2}+\cdots +\beta ^{-m}d_{-m}.\end{aligned}}}

Figures d and non-negative integers less than β. It is also known as β-expansion, a notion introduced Renyi 1957 and was first studied in detail by parry 1960. Every real number has at least one, possibly infinite β-expansion.

There are applications of β-expansions in coding theory Kouts 1965 and models of quasicrystals Burdick et al. 1998, Thurston 1989.

                                     

1. Construction. (Строительство)

β-expansions are a generalization of the decimal expansions. While infinite decimal expansions are not unique, e.g., 1.000. = 0.999., all finite decimal expansions are unique. However, even the limited β-expansions are not necessarily unique, for example, φ 1 = φ 2 β = φ, the Golden ratio. The canonical choice for β-expansion of real numbers can be defined by the following greedy algorithm, mainly due to Renyi 1957 and is formulated as given here by Frougny 1992.

Let β > 1 to be the base and X is a nonnegative real number. Under we denote ⌊ X ⌋ the function of the integer part of number X, i.e. the largest integer less than or equal to X, and let { x } = x − under ⌊ x ⌋ be the fractional part of X. There is an integer k such that ≤ β K x < β to 1. Set

d k = ⌊ x / β k ⌋ {\displaystyle d_{k}=\lfloor x / \beta ^{k}\rfloor }

and

r k = { x / β k }. {\displaystyle r_{k}=\{x / \beta ^{k}\}.\,}

For K − 1 ≥ j and GT −∞, put

d j = ⌊ β r j + 1 ⌋, r j = { β r j + 1 }. {\displaystyle d_{j}=\lfloor \beta r_{j+1}\rfloor,\quad r_{j}=\{\beta r_{j+1}\}. }

In other words, the canonical β-expansion of X is determined by the choice of large D K D K such that β ≤ X, then choosing the largest D K -1, such that β K D β K D K -1 K -1 ≤ X, etc. Thus, he chooses the lexicographically largest string that represents H.

Integer base, it defines a common extension of the number system for the number of X. This construction extends the usual algorithm, possibly non-integer values of β.

                                     

2.1. Examples. Base √ 2. (Основание √ 2)

The base is √ 2 behaves in a very similar way, the base 2, as all you need to do to convert a number from binary to base √ 2 is a zero digit In each binary digit, for example, 1911 10 = 11101110111 101010001010100010101 2 becomes √ 2 and 5118 10 = 1001111111110 1000001010101010101010100 becomes 2 √ 2. This means that each integer can be expressed in base √ 2, without the decimal point. The base can also be used to show the relationship of the squares side to its diagonal of a square with side length 1 √ 2 will have a diagonal of 10 √ 2 and a square with a side length of 10 √ 2 will have a diagonal of 100 √ 2. Another option is the use of base-show the ratio of silver in its representation in base √ 2 is simply 11 √ 2. In addition, the area of a regular octagon with a side length of 1 √ 2 is 1100 √ 2, the Area of a regular octagon with side 10 √ 2 is 110000 √ 2, the Area of a regular octagon with a side of 100 √ 2 is 11000000 √ 2, etc.

                                     

2.2. Examples. Base e. (Основанием e)

With base E natural logarithm behaves like a logarithm as лн1 e = 0, ln10 E = 1, e ln100 = ln1000 2 and E = 3.

Base e is the most economical choice of the radix β > 1 Hayes 2001, when the economy is the number system is defined as the product of the base of the number system and the length of the string of characters required to Express a given range of values.

                                     

2.3. Examples. Base π. (База π)

Base π can be used to more easily show the relationship between the diameter of a circle to its circumference, which corresponds to the perimeter as the circumference = the diameter × π, a circle with diameter 1 is π will have a circumference 10 π, a circle with a diameter of 10 π will have a circumference of 100 π, etc. in addition, since area = π × radius 2 a circle with a radius of 1 π will have an area of 10 π, a circle with a radius of 10 π will have an area of 1000 π and the circle with radius of 100 π will have an area of 100000 π.

                                     

3. Properties. (Свойства)

In a positional numeral system number can be expressed uniquely. For example, in base ten, the number 1 has two representations: 1.000. and 0.999. Dialing with two different representations of the dense R Petkovsek 1990, but the issue of classification of real numbers with unique β-expansions is much more subtle than the number of bases Glendinning & Sidorov, 2001.

Another problem is the classification of real numbers whose β-expansion is periodic. Let β > 1, and β Q is the smallest extension of the field of rational numbers containing β. Then any real number in "0.1), the presence of periodic β-expansion must lie in Q β. On the other hand, the reverse is not necessarily true. To the contrary, however, if β is a Pisot number, Schmidt 1980, although necessary and sufficient conditions is not known.

                                     
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