Rational numerals
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The system of rational numerals is a positional binary numeral system for geometric representation of numbers. It was invented 1989 by Armands Strazds.^{[1]} The system is based upon three strokes, which are combined into six rational elements.^{[2]}Template:Rp Unlike in non-rational numeral systems, the symbols (glyphs) used to represent the system have quantitative equality with the system itself, resulting in a new concept of numerals as value with a sign, instead of sign with value.^{[2]}Template:Rp
Contents
Etymology
The system is called "rational numerals", because it is based on the ratio of geometric magnitudes.^{[3]}Template:Rp
Syllables
All numbers are expressed in rational numerals using three strokes: a vertical stroke | and two diagonal strokes / and \. The strokes represent a mutual shift of two quantitative binary digits: 2 units and 3 units long. The quantitative digits are called "syllables". The digit of magnitude 2 is called "short syllable", and the digit of magnitude 3 "long syllable".
Characters
A symbol set used to represent numbers in the rational numeral system is unchanging as long as the underlying ratio of the geometric magnitudes remains unchanged. The full system was first described by Armands Strazds in his book Suranadira.^{[3]}
Functionally, the rational numeral system is a further development of the early tally stick number recording principle, since it offers a method for representation and interpretation of the numerical values in a way, which is independent of any cultural convention. An improvement in comparison to the unary numeral system used with tally sicks is that the rational numeral system is non linear, but logarithmic, and thus suitable for the representation of large numbers.^{[3]}Template:Rp
Levels
Syllables are organized in levels. Levels correspond to the positions of the positional notation. The count of levels is unlimited. The sequence of syllables on the top level (L_{0}) is: short-long-short-long-short-long-.., or binary: 010101.. On the next deeper level (L_{1}) the sequence of syllables is: short-short-long-long-short-short-long-long-.., or binary: 00110011.. The count of short and long syllable runs on a specific level corresponds to 2 to the power of the level index (2^{n}), where n = 0..Template:Math.
Numerals 0 and 1
Two highest levels (L_{0} and L_{1}, bandwidth 1) together can represent the first two rational numerals: 0 and 1. The numeral 0 is produced by a short syllable on level 0 and a short syllable on level 1. Since both syllables are of the same type and are not mutually horizontally shifted, two vertical strokes ("I-strokes") are the geometric equivalent to the value of 0. The geometric form of two I-strokes with a horizontal distance of 2 units is called "H-form".
The left stroke of the numeral 1 is produced by a long syllable on level 0 and a short syllable on level 1. The difference in syllable length results in from top to left directed diagonal stroke, called "Z-stroke". The right stroke is produced by a short top syllable and long bottom syllable, resulting together in no mutual shift and thus the I-stroke. The geometric form of a Z-stroke and an I-stroke is called "J-form".
Numerals 0 to 3
Three highest levels (L_{0} to L_{2}, bandwidth 2) together can represent the first four rational numerals: 0 to 3. Numerals 0 and 1 are padded by a leading zero. The leading zero for the numeral 0 is an H-Form, and the leading zero for the numeral 1 is a new form called "A-form". With exception of numeral 0, leading zeroes always include exactly one A-form accompanied by the required number of H-forms. In split components^{[3]}Template:Rp leading zeroes also include a form called "Z-Form", which is discussed below. The behavior of the A-form is governed by the OEIS sequences Template:OEIS2C and Template:OEIS2C, and the behavior of the combined forms of the arbitrary bandwidth is governed by the OEIS sequence Template:OEIS2C.
Numeral 2 introduces a new form on level 1, which is called "V-form". The V-form has a short syllable at top and a long syllable at bottom. A connection of the V-form and one or more H-forms, like in the numeral 2, is called the "V-component".^{[3]}Template:Rp
Numeral 3 has two I-strokes on level 1, but unlike in an H-form the horizontal distance between them amounts to 3 units, since both the top and the bottom syllable is long, and the syllables have no mutual horizontal shift. This new form is called the "I-form".^{[3]}Template:Rp
Numerals 0 to 7
Four highest levels (L_{0} to L_{3}, bandwidth 3) together can represent the first eight rational numerals: 0 to 7. Numerals 0 to 3 are padded by a leading zero. The leading zero of numeral 3 has a form called "Z-form". Like with the I-form both the top and the bottom syllable of Z-form is long, unlike with the I-form the bottom syllable is shifted one unit to the left.
Numeral 4 introduces a new component called "J-component". J-component has a J-form as the bottom element, and none or more H-forms above it.^{[3]}Template:Rp
Numeral 5 has a J-form as the top element, and a new component called "U-component" below it. U-component has a V-form as the bottom element, an A-form as the top element, and none or more H-forms in the middle.^{[3]}Template:Rp
Numerals 0 to 15
Five highest levels (L_{0} to L_{4}, bandwidth 4) together can represent the first sixteen rational numerals: 0 to 15^{[a]}
Numeral 9 has a J-form as the top element, and a new component called "O-component" below it. O-component has a V-form as the bottom element, an A-form as the top element, and one or more H-forms in the middle.^{[3]}Template:Rp
Numeral 10 has a V as the top component, and a new component called "N-component" below it. Like the U-component e.g. in numerals 5 or 13, N-component has a V-form as the bottom element, an A-form as the top element, and one or more H-forms in the middle. Unlike in the U-component, where the bottom V-element is connected using the left shift, in the N-component it is connected using the right shift.^{[3]}Template:Rp
The vertical extent of the individual components of the rational numerals corresponds to the OEIS sequence Template:OEIS2C^{[2]}Template:Rp
Rational numerals for larger numbers can be generated by adding more levels.
Encoding
To express a number as rational numeral, the following two steps are required:
- Step 1: Converting a binary number into delta numeral.
- Step 2: Converting the delta numeral into rational numeral.
Example
To express the binary number 1100100010 (decimal 802) as a rational numeral:
- Step 1: 1100100010 → [1, 100, 1000, 10] = [1, 3, 4, 2]
- Step 2: 2 → V_{2}, 4 → N_{4}, 3 → O_{3}, 1 → I_{1} = V_{2}N_{4}O_{3}I_{1} (View)
Decoding
To decode a rational numeral into number, the following two steps are required:
- Step 1: Converting a rational numeral into delta numeral.
- Step 2: Converting the delta numeral into binary number.
Example
To decode the rational numeral V_{2}N_{4}O_{3}I_{1} into (binary) number:
- Step 1: I_{1} → 1, O_{3} → 3, N_{4} → 4, V_{2} → 2 = [1, 3, 4, 2]
- Step 2: [1, 3, 4, 2] → [1, 100, 1000, 10] = 1100100010
Forms
The following table lists the syntax rules, which determine the type of individual forms in a rational numeral, based on the following three parameters: delta value, level (number or parity), and the predecessor form. All the rules are exclusive, and their conditions are evaluated sequentially starting at rule 1.
Rule # | Delta value | Level | Previous | Form |
---|---|---|---|---|
1 | even | 0 | none | V |
2 | odd | 0 | none | J |
3 | >1 | even | IJ_{1}, I_{2}, IU_{2} | Z |
4 | >1 | odd | I_{2}, IN_{2} | S |
5 | even | even | any | N |
6 | even | odd | any | U |
7 | 1 | any | any | I |
8 | odd | any | any | O |
Usage
The first public presentation of the rational numeral system was made 2016 by Modris Tenisons as a part of his master classes in Moscow, Russia.^{[4]}
Simona Orinska and Christian Zanési used rational numerals in their performance at Festival Butô Acousmatique 2016 in Paris, France.^{[5]}
Rational numeral system has been described in doctoral dissertations on philosophy, mathematics, informatics, and musicology.^{[6]}
Notes
- a.^ Figure shows shorthand characters for numerals 3 and 11. See also Strazds, Armands (2017) (in German). Suranadira: Der Fluss des Himmels und der Töne (1st ed.). urn:nbn:de:101:1-201702133788. http://d-nb.info/1125385642/34., pp. 126-127.
See also
Template:Col-begin Template:Col-2
References
- ↑ Dzenītis, Andris (2006). "Zelta grāmatas rakstītājs" (in Latvian). Mūzikas Saule (2): 8–11. http://www.lmic.lv/core.php?id=9227&pageId=726&subPageId=759&pageAction=showSubpage&fromComp=true.
- ↑ ^{2.0} ^{2.1} ^{2.2} Strazds, Armands (2016). "Quantitative Digit in a Decoupled Positional System: A New Method of Understanding and Teaching Numbers" (PDF). International Journal of Engineering Science Invention 5 (9): 29–33. ISSN 2319–6734. http://www.ijesi.org/papers/Vol(5)9/G05902933.pdf.
- ↑ ^{3.0} ^{3.1} ^{3.2} ^{3.3} ^{3.4} ^{3.5} ^{3.6} ^{3.7} ^{3.8} ^{3.9} Strazds, Armands (2017) (in German). Suranadira: Der Fluss des Himmels und der Töne (1st ed.). urn:nbn:de:101:1-201702133788. http://d-nb.info/1125385642/34.
- ↑ Template:Cite AV media
- ↑ Template:Cite AV media
- ↑ Template:Cite thesis, pp. 74-76