Non-integer Base of Numeration

Get Non-integer Base of Numeration essential facts below. View Videos or join the Non-integer Base of Numeration discussion. Add Non-integer Base of Numeration to your PopFlock.com topic list for future reference or share this resource on social media.

## Construction

### Conversion

### Example implementation code

#### To base ?

#### From base ?

## Examples

### Base

### Golden base

### Base ?

### Base *e*

### Base ?

## Properties

## See also

## References

## Further reading

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Non-integer Base of Numeration

This article includes a list of general references, but it remains largely unverified because it lacks sufficient corresponding inline citations. (March 2019) |

Numeral systems |
---|

Hindu-Arabic numeral system |

East Asian |

American |

Alphabetic |

Former |

Positional systems by base |

Non-standard positional numeral systems |

List of numeral systems |

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

is

The numbers *d*_{i} are non-negative integers less than ?. This is also known as a **?-expansion**, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every real number has at least one (possibly infinite) ?-expansion. The set of all ?-expansions that have a finite representation is a subset of the ring **Z**[?,?^{-1}].

There are applications of ?-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998; Thurston 1989).

?-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even finite ?-expansions are not necessarily unique, for example *?* + 1 = *?*^{2} for *?* = *?*, the golden ratio. A canonical choice for the ?-expansion of a given real number can be determined by the following greedy algorithm, essentially due to Rényi (1957) and formulated as given here by Frougny (1992).

Let *?* > 1 be the base and *x* a non-negative real number. Denote by ?*x*? the floor function of *x*, that is, the greatest integer less than or equal to *x*, and let {*x*} = *x* - ?*x*? be the fractional part of *x*. There exists an integer *k* such that *?*^{k} x < *?*^{k+1}. Set

and

For *k* - 1 >= *j* > -?, put

In other words, the canonical ?-expansion of *x* is defined by choosing the largest *d*_{k} such that *?*^{k}*d*_{k} x, then choosing the largest *d*_{k-1} such that *?*^{k}*d*_{k} + ?^{k-1}*d*_{k-1} x, etc. Thus it chooses the lexicographically largest string representing *x*.

With an integer base, this defines the usual radix expansion for the number *x*. This construction extends the usual algorithm to possibly non-integer values of *?*.

Following the steps above, we can create a ?-expansion for a real number . The steps are identical for an , although n must first be multiplied by to make it positive, then the result must be multiplied by to make it negative again.

First, we must define our k value (the exponent of the nearest power of β greater than n, as well as the amount of digits in , where is n written in base β). The k value for n and β can be written as:

After a k value is found, can be written as d, where

for *k* - 1 >= *j* > -?. The first k values of d appear to the left of the decimal place.

This can also be written in the following pseudo-code:

```
function toBase(n, b) {
k = floor(log(b, n)) + 1
precision = 8
result = ""
for (i = k - 1, i > -precision-1, i--) {
if (result.length == k) result += "."
digit = floor((n / b^i) mod b)
n -= digit * b^i
result += digit
}
return result
}
```

^{[1]}

Note that the above code is only valid for and , as it does not convert each digits to their correct symbols or correct negative numbers. For example, if a digit's value is , it will be represented as instead of A.

- Javascript:
^{[1]}function toBasePI(num, precision = 8) { let k = Math.floor(Math.log(num)/Math.log(Math.PI)) + 1; if (k < 0) k = 0; let digits = []; for (let i = k-1; i > (-1*precision)-1; i--) { let digit = Math.floor((num / Math.pow(Math.PI, i)) % Math.PI); num -= digit * Math.pow(Math.PI, i); digits.push(digit); if (num <= 0) break; } if (digits.length > k) digits.splice(k, 0, "."); return digits.join(""); }

- Javascript:
^{[1]}function fromBasePI(num) { let numberSplit = num.split(/\./g); let numberLength = numberSplit[0].length; let output = 0; let digits = numberSplit.join(""); for (let i = 0; i < digits.length; i++) { output += digits[i] * Math.pow(Math.PI, numberLength-i-1); } return output; }

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

In the golden base, some numbers have more than one decimal base equivalent: they are **ambiguous**. For example:
11_{?} = 100_{?}.

There are also some numbers in base ? are also ambiguous. For example, 101_{?} = 1000_{?}.

With base *e* the natural logarithm behaves like the common logarithm as ln(1_{e}) = 0, ln(10_{e}) = 1, ln(100_{e}) = 2 and ln(1000_{e}) = 3.

The base *e* is the most economical choice of radix ? > 1 (Hayes 2001), where the radix economy is measured as the product of the radix and the length of the string of symbols needed to express a given range of values.

Base ? can be used to more easily show the relationship between the diameter of a circle to its circumference, which corresponds to its perimeter; since circumference = diameter × ?, a circle with a diameter 1_{?} will have a circumference of 10_{?}, a circle with a diameter 10_{?} will have a circumference of 100_{?}, etc. Furthermore, since the area = ? × radius^{2}, a circle with a radius of 1_{?} will have an area of 10_{?}, a circle with a radius of 10_{?} will have an area of 1000_{?} and a circle with a radius of 100_{?} will have an area of 100000_{?}.^{[2]}

In no positional number system can every number be expressed uniquely. For example, in base ten, the number 1 has two representations: 1.000... and 0.999.... The set of numbers with two different representations is dense in the reals (Petkov?ek 1990), but the question of classifying real numbers with unique ?-expansions is considerably more subtle than that of integer bases (Glendinning & Sidorov 2001).

Another problem is to classify the real numbers whose ?-expansions are periodic. Let ? > 1, and **Q**(?) be the smallest field extension of the rationals containing ?. Then any real number in [0,1) having a periodic ?-expansion must lie in **Q**(?). On the other hand, the converse need not be true. The converse does hold if ? is a Pisot number (Schmidt 1980), although necessary and sufficient conditions are not known.

- Beta encoder
- Non-standard positional numeral systems
- Decimal expansion
- Power series
- Ostrowski numeration

- ^
^{a}^{b}^{c}https://decimalsystem.js.org **^**"Weird Number Bases".*DataGenetics*. Retrieved .

- Bugeaud, Yann (2012),
*Distribution modulo one and Diophantine approximation*, Cambridge Tracts in Mathematics,**193**, Cambridge: Cambridge University Press, ISBN 978-0-521-11169-0, Zbl 1260.11001 - Burdik, ?.; Frougny, Ch.; Gazeau, J. P.; Krejcar, R. (1998), "Beta-integers as natural counting systems for quasicrystals",
*Journal of Physics A: Mathematical and General*,**31**(30): 6449-6472, Bibcode:1998JPhA...31.6449B, CiteSeerX 10.1.1.30.5106, doi:10.1088/0305-4470/31/30/011, ISSN 0305-4470, MR 1644115. - Frougny, Christiane (1992), "How to write integers in non-integer base",
*LATIN '92*, Lecture Notes in Computer Science, 583/1992, Springer Berlin / Heidelberg, pp. 154-164, doi:10.1007/BFb0023826, ISBN 978-3-540-55284-0, ISSN 0302-9743. - Glendinning, Paul; Sidorov, Nikita (2001), "Unique representations of real numbers in non-integer bases",
*Mathematical Research Letters*,**8**(4): 535-543, doi:10.4310/mrl.2001.v8.n4.a12, ISSN 1073-2780, MR 1851269. - Hayes, Brian (2001), "Third base",
*American Scientist*,**89**(6): 490-494, doi:10.1511/2001.40.3268, archived from the original on 2016-03-24. - Kautz, William H. (1965), "Fibonacci codes for synchronization control",
*Institute of Electrical and Electronics Engineers. Transactions on Information Theory*, IT-11 (2): 284-292, doi:10.1109/TIT.1965.1053772, ISSN 0018-9448, MR 0191744. - Parry, W. (1960), "On the ?-expansions of real numbers",
*Acta Mathematica Academiae Scientiarum Hungaricae*,**11**(3-4): 401-416, doi:10.1007/bf02020954, hdl:10338.dmlcz/120535, ISSN 0001-5954, MR 0142719, S2CID 116417864. - Petkov?ek, Marko (1990), "Ambiguous numbers are dense",
*The American Mathematical Monthly*,**97**(5): 408-411, doi:10.2307/2324393, ISSN 0002-9890, JSTOR 2324393, MR 1048915. - Rényi, Alfréd (1957), "Representations for real numbers and their ergodic properties",
*Acta Mathematica Academiae Scientiarum Hungaricae*,**8**(3-4): 477-493, doi:10.1007/BF02020331, hdl:10338.dmlcz/102491, ISSN 0001-5954, MR 0097374, S2CID 122635654. - Schmidt, Klaus (1980), "On periodic expansions of Pisot numbers and Salem numbers",
*The Bulletin of the London Mathematical Society*,**12**(4): 269-278, doi:10.1112/blms/12.4.269, hdl:10338.dmlcz/141479, ISSN 0024-6093, MR 0576976. - Thurston, W.P. (1989), "Groups, tilings and finite state automata",
*AMS Colloquium Lectures*

- Sidorov, Nikita (2003), "Arithmetic dynamics", in Bezuglyi, Sergey; Kolyada, Sergiy (eds.),
*Topics in dynamics and ergodic theory. Survey papers and mini-courses presented at the international conference and US-Ukrainian workshop on dynamical systems and ergodic theory, Katsiveli, Ukraine, August 21-30, 2000*, Lond. Math. Soc. Lect. Note Ser.,**310**, Cambridge: Cambridge University Press, pp. 145-189, ISBN 978-0-521-53365-2, Zbl 1051.37007

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists