- high-resolution terminology - matching measurements at high-resolution
Number
Description
A number N is a count NX [x] of elementary entity X divided by the unit-entity UX [x]. X must represent the same entity in both occurences. The counting-unit [x] cancels by division, such that numbers (for example, numbers 8 or 24) are abstracted from the counted entity (we write 8 and 24, although 8 xΒ·x-1 and 24 xΒ·x-1 would be equally correct; distinguished from a count of 8 x or 24 x if we count an entity-type X=apple). It is difficult to separate the concept of 'number' from the realization of number words or number symbols. The number symbols are called numerals; a numeral is the figure of a number, with different notation types used as a figure (VIII and 8 for Roman and Arabic numerals; ε « and ζ for practical and financial Chinese). Consider the symbol 9 written into MitoPedia as elementary entity X=9. Then counting "9 9 9 9 9 9 9 9" yields a count N9 = 8 x, and the count N9 [x] divided by the unit-entity U9 [x] yields the number N = 8, using the figure eight as the numberal in Arabic notation type. The human number concept has not only quantitative cardinal meaning related to the count (8 or 24 elementary entities), but is applied in expressing the ordinal rank of objects or events arranged in a sequence (in the Fibonacci-sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, .. the 8th number is 13, whereas in an older representation of the Fibonacci-sequence 1, 1, 2, 3, 5, 8, 13, 21, .. the 5th number is 5; the 24th day of a month), and in nominal labelling (drawing lot #24; serial number #8.007; DOI number doi10.26124bec2020-0001). Counting numbers (1, 2, 3, 4, 5, 6, 7, 8, ..) are unified multiplicities required for cardinal counting or ordinal nomination of the endpoint in a sequence. It is debatable, if one can have a zero count; a no-object, or an object that is not there to be counted. If this possibility is not denied, then counting numbers are equivalent to natural or whole numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, ..). Numbers are represented by numerals as words, iconic symbols, or entirely abstract symbols. The word 'snake', the numeral 'eight', the symbol '8' written in ink on a piece of white paper are as different from the real "object snake", as they differ from the "concept ////////", or "concept §§§§§§§§", or "concept 88888888", or "concept ββββββββ", or "concept 'number eight'". We are so deeply used to these symbols, that we easily take the iconic or abstract symbol 8 β that represents the number eight β as the number eight itself, without a need to give the symbol 8 an interpretation and ask for its meaning.The numeral has to be distinguished from it's interpretation as the number that the numeral represents.
Abbreviation: N
Communicated by Gnaiger Erich 2020-06-28
Formats and meanings of numbers
- It is useful to compare some terms related to count and number in different languages (Table 1).
Term English German count NX count (account) Anzahl (Konto) number N number Zahl numeral X numeral (notation, form, entity-type) Ziffer
- Figure 1: Formats and meanings of numbers.
- Counting and notation types: (1.1) dyce, (1.2) Roman numerals, (1.3) Mandarin-Chinese signs, (1.4) Arabic numerals, (1.5) English words. The dyce format requires hardly any interpretation, since the signal for counting is given in a series of linear expansion; this works well up to :::, but does not work for 66 or 666. Similarly, Roman and Mandarin symbols from I to III do not need interpretation due to the signal for counting, but IV to VI is more complex in Roman and Mandarin notations by compression required for extension towards higher numbers. Interpretation of Arabic numerals and English words needs learning from beginning with 1 and one, since there is no relation to counting; but these investments pay off β once Arabic numerals have been learned, these symbols can be recognized and distinguished most rapidly, be written most economically, and be extended to high numbers by combination in the decimal number system. English words are much less economical in writing, but they connect isomorphically the image of the written number-word with the acoustic form of the spoken number-word.
- Cardinal counting and ordinal ranking of dice: There are 15 dice in the figure. Dice 1 to 5 are in row 1; dice 6 to 10 are in row 2; dice 11 to 15 are in row 3.
- Nominal labelling: Dice of tye (1) are with single notation and positioned on the marging of the figure; (2) dice with single notation and positioned in the center of the figure; (3) dice with multiple notations and positioned on the margin of the figure; (4) dice with multiple notations and positioned in the center of the figure.
- Number magnitude and space: Dice with different notation types have an increasing magnitude from left to right. This spacial association is less pronounce for Mandarin notation type.
- Sex of numbers: Even numbers such as 6 are associated with female sex, whereas odd numbers such as 1 connotate masculinity. Isn't it odd?
Does a number make sense?
- Something heavy makes sense. We feel the weight on the basis of interpreting sensory signals. Mass does not make sense. Even if we are familiar with the concept of mass, we tend to say 'weight' for some kg of potatoes, and tend to forget what we learned about weight as a force in contrast to the base quantity mass. Even if we tend to forget similarly what we learned about irrational or imaginary numbers, we do not forget what we learned about numbers in general. This is perhaps due to the high probability, that we never learned to think deeply about the sense of numbers, not in the sense of numerology, but in the sense of a sensory organ for numerosity.
A taxonomy of numbers
Class Example Comment counting number, natural number 1, 2, 3, 4, .. Positive integers. All natural numbers are whole numbers. whole number 0, 1, 2, 3, 4, .. Natural numbers including zero. All whole numbers are integers. negative integer -1, -2, -3, -4, .. Negative integers are integers excluding whole numbers. integer .., -4, -3, -2, -1, 0, 1, 2, 3, 4, .. Negative integers and whole numbers. Orders: (1) even numbers, (2) odd numbers, (3) prime numbers. All integers are rational numbers. rational number -2/1, -0.5, 0.0, 0.3, 1/2, 7.8 Any number that can be written as a ratio of two integers β where the denominator must not be zero β, or as the resulting fraction written with decimal digits after the decimal dot. All rational numbers are real numbers, and are not imaginary numbers. irrational number square root of 2; β2 Irrational numbers cannot be written as a ratio of two integers; the have an infinite number of decimal places. real number Real numbers include all types of numbers listed above and are defined as points on a line from -β to +β. immaginary number square root of -1 An imaginary number squared yields a negative real number. complex number Complex numbers are combinations of real and imaginary numbers.
References
| Bioblast link | Reference | Year |
|---|---|---|
| Agrillo 2020 PLOS ONE | Agrillo Christian, Piffer Laura, Bisazza Angelo, Butterworth Brian (2020) Evidence for two numerical systems that are similar in humans and guppies. PLOS ONE 7:e31923. | 2020 |
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| Bell 1999 Springer | Bell John L (1999) The art of the intelligible. An elementary survey of mathematics in its conceptual development. Springer Science+Business Media Dordrecht:249 pp. | 1999 |
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| Brown 2021 Metrologia | Brown RJC (2021) A metrological approach to quantities that are counted and the unit one. Metrologia 58:035014. https://doi.org/10.1088/1681-7575/abf7a4 | 2021 |
| Bureau International des Poids et Mesures 2019 The International System of Units (SI) | Bureau International des Poids et Mesures (2019) The International System of Units (SI). 9th edition:117-216. ISBN 978-92-822-2272-0 | 2019 |
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| Gnaiger 2020 MitoFit x | Gnaiger E (2021) The elementary unit β canonical reviewer's comments on: Bureau International des Poids et Mesures (2019) The International System of Units (SI) 9th ed. https://doi.org/10.26124/mitofit:200004.v2 | 2021 |
| Gong 2019 J Numerical Cognition | Gong Tianwei, Li Baichen, Teng Limei, Zhou Zijun, Gao Xuefei, Jiang Ting (2019) The association between number magnitude and space is dependent on notation: evidence from an adaptive perceptual orientation task. J Numerical Cognition 5:38β54. | 2019 |
| Grosholz 2007 Oxford Univ Press | Grosholz Emily R (2007) Representation and productive ambiguity in mathematics and the sciences. Oxford Univ Press 312 pp. | 2007 |
| Kadosh 2015 Oxford Univ Press | Kadosh Roi Cohen, Dowker Ann, ed (2015) The Oxford handbook of numerical cognition. Oxford Univ Press:1185 pp. | 2015 |
| Singh 1997 Fourth Estate | Singh Simon (1997) Fermat's last theorem. Fourth Estate, London 340 pp. | 1997 |
| Spiegelhalter 2015 Profile Books | Spiegelhalter David (2015) Sex by numbers: What statistics can tell us about sexual behaviour. Profile Books, London 368 pp. | 2015 |
| Wilkie 2015 Front Psychol | Wilkie James EB, Bodenhausen Galen V (2015) The numerology of gender: gendered perceptions of even and odd numbers. Front Psychol 6:810. | 2015 |
MitoPedia concepts:
Ergodynamics
