Version 9th February 2011

A contribution to disambiguation of the concepts ‘number’, ‘time’ and ‘frequency’

Abhary K.*, Djukic D.**, Hsu H-Y.*, Kovacic Z.***, Mulcahy D.*, Spuzic S.*, Uzunovic F.****

*University of South Australia, **Massey University, New Zealand, ***Open Polytechnic, New Zealand,
****University of Zenica, Bosnia & Herzegovina


The antonym for the term 'knowledge' is not 'ignorance'. 'Ignorance' is a lack of education, unfamiliarity with otherwise existing knowledge. The opposite of ‘knowledge’ is an infinitely large, still unexplored, undiscovered, variety of relations, processes and something else within and beyond us. Are we hopelessly lost in unending space or are we infinitely rich because there are ceaseless resources around us? This depends on our ability to create, share and use the relevant knowledge.

The way we share theories affects efficiency of  knowledge verification and applications. Any successful application of knowledge provides  additional supporting evidence to its validity. Therefore, more attention should be devoted to the modes of knowledge presentation and sharing.

Development of specialised sources, [20-22], is useful as long as it does not promote definitions that are misaligned with other sources.

The differences in interpretations of concepts, which are criticized in this treatise, must be distinguished from the category of differences that result from the rivaling hypotheses, each attempting to explain the same phenomena, or even the same evidence. It is important to note that these competing hypotheses do not simply ignore the differences. On the contrary, all care is taken to explain these differences, and to present the preferred arguments in most clear fashion.  A transparent approach to differences has enabled advances such as the discovery of penicillin, x-rays, atoms, neoprene, and countless other achievements embedded in our history.

This treatise focuses on the cases where the differences are ignored. These differences can vary from the simple cases of homonymy to quite substantial conceptual misalignments.  None of them should be tolerated.  Defining knowledge in terms of its application and sharing has a significant impact on sustainability of our civilisation. 

Sources [2, 3] point at the damaging consequences of ambiguities associated with  the concepts of ‘number’, ‘time’ and ‘frequency’. We propose therefore the modified definitions that are intended to contribute to the disambiguation in the usage of these three concepts. These definitions are based on axioms outlined in [2] and [4].

Number, numeral, cipher and digit

The concepts of the ‘number’, 'numeral', 'cipher' and 'digit' can be derived from their hypernyms as follows:

A ‘figure’ is an arrangement of points made within two-dimensional space to present a visual static impression (a perception) of something (e.g., a figure printed on a book page, showing the front view of a home).

‘Metric’ is a relation used to enable a comparison of (the groups of) phenomena.

‘Comparison’ is a definition indicating whether (or to what degree) one phenomenon differs from other phenomena. When comparison indicates that some phenomena are sufficiently similar, such phenomena can be counted using numbers. When some group (set) of phenomena can be classified within the same category, their attributes can be compared using numbers (by quantifying the relevant levels at a corresponding numerical scale).

‘Number’ is a general metric, systematically ordered within mathematics. Every number has a unique position in the sequence which continues within one- or multi-dimensional series (typically, towards an infinity). Hypernyms for term 'number' include 'metric', 'norm', 'comparison' and 'measure'.

‘Numeral’ is (in English, and numerous other languages) any one of the elements that can be combined to represent numbers in a number system (e.g., decimal system, binary system, hexadecimal system, etc.). 

‘Digit’ is a figure representing a numeral; examples: "0", "1", "A", "i".

‘Cipher’  is a figure representing a number; examples: "10001001", "10,001.001", "p".

The usage of the concept of  ‘number’ must be specified with regard to purpose of that usage. For example, 'count' is the number denoting how many of otherwise separated individual events (or other phenomena) are observed to have some common attributes. 'Count' is hyponym of 'number'.

Further example of the use of 'numbers' is representing different values of a numerical variable (e.g. "temperature of 23 degree Celsius").

Other usages such as
- ordinal number; an identifier showing an order in a sequence of successors and predecessors  (e.g. "In the men's race Japan finished first while Canada was third"),
- outcomes of mathematical analyses (e.g. in multivariate calculus, differential geometry) and
- other mathematical comparators (e.g. topological modular forms),
should include modifiers attached to the term 'number'.

A statement "the frequency is the number of observed occurrences” [5] is ambiguous. 
A clearer definition of 'frequency' will be proposed in section "Frequency and frequence" below.

Time, duration and datum

The concept of  ‘time’ is discussed widely in many disciplines, including the historic views [6-9].

One class of definitions indicates that time is a scalar quantity that can be correlated with a number line stretching from - ¥ to +¥. Advances in physics required introducing hypotheses where time is defined along the fourth axis of the space-time manifold. Relativity theory requires that the distance between the points on that axis depend on the relative speed of the phenomena observers. Special relativity postulates that time cannot be understood except as part of space-time, a combination of space and time. General relativity has further changed the notion of time by introducing the idea of curved space-time. [10-12] 

Philosophical considerations [6] add further questions to this concept of ‘time’. Although, nearly a century  after its publication, general relativity remains a highly successful model, there are indications that the theory is incomplete, and the question of the reality of space-time singularities remains open. [9, 13, 14]

It is proposed hereby to sustain this concept, termed ‘time’, in line with the advances in science. Its measure, ‘second’, defined by ISO standards [15], page 112, and [16], allows for both theoretical and practical applications and advances in relevant knowledge with a useful and continuously improving precision and accuracy. 'Second' is defined as an event in which 9 192 631 770 wavelengths occur due to the hyperfine electron transition in a stable caesium-133 atom.

However, it is recommended hereby to further distinguish the above concept of ‘time’ from its special case termed ‘duration’ (which is also measured in seconds). ‘Duration’ is a time measurement that allows for comparisons such as to which of the two observed phenomena takes less time. For example, if we denote time by t, then                                              

                                  duration = t2 – t1,    seconds

where ti = i-th point on the time number line (for example, corresponding to a start of some event). Point ti+1 (e.g. corresponding to an end of an event) occurs at any time after the point ti.

Moreover, the concept of ‘time’ is often used to refer to the hour of the day reckoned by the position of a celestial reference point relative to a corresponding celestial meridian. Time may be designated solar, lunar, or sidereal, as the reference is the sun, moon, or vernal equinox, respectively. Solar time may be further classified as mean or astronomical, etc.

Time may also be designated according to the reference meridian, either the local or Greenwich meridian or, additionally, in the case of mean solar time, a designated zone meridian. Standard and daylight saving time are variations of zone time. [9,  17]

Sources [18, 19] present further examples of usage of this term with the above meaning.

It is recommended herewith to associate this third meaning with the terms ‘datum’ and ‘instant’ defined in accordance to Coordinated Universal Time (UTC) unless otherwise stated. An example of recommended usage is: “The plenary session started at 2 pm on 24th August 2009." (Here "24th August 2009" is a datum). "At the same instant, in the adjacent room, a work session started devoted to coordination of multi-language translating services.”

Difference between the concepts of 'time', 'duration' and 'datum' can be compared to the difference between the concepts of 'length', 'displacement' and 'position' which are well defined in physics.

Hypernyms for the above terms 'time', 'duration' and 'datum' ('instant') are 'measure' 'norm',  'metrics', and 'dimension'.

Frequency and frequence

Application, dissemination and broadening of knowledge requires observations of phenomena (relations, events, systems etc). Some phenomena can be classified in a category that allows for counting their occurrences.

Frequency f is a ratio of the count and the count y
f = x/y,
where x = the count of actual specific occurrences and y = the total count of all possible events of relevance. 

‘Events of relevance’ are the observations of the phenomena that are classifiable within the analysed scope (within some boundaries defined by the purpose of observations). 

For example, if we observe a large container filled in with a mix of thousands of yellow, green and red balls, we can count how many red balls have been found in small boxes filled each time with ten balls chosen at random from the above large container. If all colours have been well mixed within the large container, then the frequency of red balls in each box can vary from 0 to 1.

Hypernyms for term 'frequency' are 'statistics' and 'number'.

At the same time, many authors us the term ‘frequency’ with another special meaning, particularly related to periodic phenomena (e.g. in so-called 'time series analysis in the frequency domain) and the concept of duration.  This homonymy itself can be eliminated, for example by introducing a term ‘frequence’ as a tentative solution until projects such as De-CITE point at a term favoured by the broader community.

It is proposed therefore to consider defining the concept, ‘frequence’, as follows: ‘frequence’ (f) is the ratio of the count of the observed cycles, e.g. wavelengths (nλ1), and  a multiple of the count of wavelengths of  some convenient, measurable, stable and standardized radiation (Nls) that occurs simultaneously.

For the convenience, rather than counting Nl each time we need to measure the frequence of some cycles, a metrological device termed a ‘clock’ is developed. Such a clock enables measuring Nls to be expressed in ‘seconds’ (s). [15, 16]

Frequence is measured in units termed 'hertz'  (Hz):

                      1 Hz = 1 s-1

Bearing in mind that Nlis subject to some level of  uncertainty, the value of the derived frequence f  is always just an approximation characterized by some estimate of an average value and variance. The current state of the art in sciences allows for reducing the relative uncertainty to the order of 10-15; [16].

In the above definition, the use of the concepts of ‘duration’ (and especially that of the ‘time’) was deliberately avoided, until it became necessary to introduce some observable metrics, namely  the second. It is instructive to emphasise that a ‘second’ is defined in terms of the count Nls.

Hypernym for 'frequence' is 'attribute'.


Differentiation of disciplines is useful as long as it does not prevent knowledge shareability.

Knowledge can be thought to be composed of definitions, and appropriate and transparent definitions are infinitely shareable. Once proven useful, a definition can be rephrased, presented with more or less details (depending on the purpose), but it should not be modified in any way that would contradict the initial meaning. If new evidence, proving a definition is false, prevails, a new definition should be created and denoted by a new term.

Impedances to knowledge shareability range from relatively easily resolvable cases of homonymy and synonymy, to the substantial ambiguities that are usually a consequence of isolationism.

Homonymy (synonymy) can be resolved by attributing differing terms and definitions to differing concepts. 

More complex ambiguities can be ameliorated by promoting a cross-disciplinary scrutiny of relevant definitions, theories and hypotheses.

In this treatise, improvements to definitions of three exemplary concepts ('number', 'time' and 'frequency') are presented to suggest how the present ambiguities can be ameliorated. Projects such as De-CITE [4] aim to further disseminate awareness about similar problems related to other key concepts (e.g. definition, information, structure, ontology, technology and vector) and to contribute towards their disambiguation.  

Appropriate definitions are probability multipliers: they significantly increase probability of intended realisation (actualisation). In line with Occam's razor, a sufficient level of definition is reached when it enables a satisfactory increase in the probability of intended realisation.


[1] K. Abhary, H. K. Adriansen, F. Begovac, Z. Kovacic, C. N. Shpigelman, C. Stevens, S. Spuzic, F. Uzunovic and K. Xing "A Contribution to Transparency of Scientific and Engineering Concepts" The International Journal of Knowledge, Culture and Change Management, (2009) Volume 9, Issue 5, pp.93-106.

[2] S. Spuzic, K. Xing and K. Abhary "Some Examples of Ambiguities in Cross-disciplinary Terminology", The International Journal of Technology, Knowledge and Society (2008), Volume 4, Issue 2, pp.19-28

[3] S. Spuzic and F. Nouwens "A Contribution to Defining the Term ‘Definition’", Issues in Informing Science and Information Technology Education, Volume 1 (2004) pp 645 - 662,

[4] Axioms postulated within the project De-CITE     at   (mirror site   accessed 10th Sept 2010

[5] I. Diamond and J. Jefferies ”Beginning statistics: an introduction for social scientists” Sage Publications Ltd,  2000

[6] N. Markosian  “Time” Stanford Encyclopedia of Philosophy  accessed 10 Sept 2010.

[7] R. A. Serway and J. W. Jewett “Physics for Scientists and Engineers with Modern Physics”,  International Edition 8e,  BrooksCole, 2010 

[8] S. Hawking “A Brief History of Time”  Bantam 1998

[9]  “Time”    Wikipedia, Wikimedia Foundation,   (accessed 11th September 2010)

[10]  J. Barbour "The End of Time: The Next Revolution in Physics" Oxford University Press1999

[11] P. Davies "About Time: Einstein’s Unfinished Revolution"  Simon & Schuster, 1996

[12] R. Feynman "The Character of Physical Law" Cambridge (Mass): The MIT Press, 108-126, 1994

[13] J. Maddox “What Remains To Be Discovered”, Macmillan, 1998

[14]  R. Penrose  “The Road to Reality: A Complete Guide to the Laws of the Universe”  Vintage (2007)

[15] “The International System of  Units”, Organisation Intergouvernementale de la Convention du Mètre,  2006    (accessed 10th September 2010)

[16] “Second”    Wikipedia, Wikimedia Foundation,   &   (accessed 10th September 2010)

[17] R. Marks  "The New Dictionary & Handbook of Aerospace"  Praeger, 1969

[18] The UNESCO Thesaurus, (accessed on 13 October 2010)

[19] “UN glossaries and UN interpreters’ resource page…”,     & (accessed on 10th November 2010)

[20]  "Chemical Entities of Biological Interest", funded by the European Commission ) within Research Infrastructures of the FP7 Capacities Specific Programme  (accessed on 23rd December 2010)

[21] "The Specialist Lexicon",  The Lexical Systems Group, the Cognitive Science Branch of the Lister Hill Center for Biomedical Communications  (accessed on 23rd December 2010)

[22] "Unified Medical Language System" U.S. National Library of Medicine, National Institutes of Health, Health & Human Services (accessed on 23rd December 2010)

[23] Abhary K, Adriansen H K, Begovac F, Djukic D, Qin B, Spuzic S, Wood D and Xing K (2009) "Some Basic Aspects of Knowledge" Procedia - Social and Behavioral Sciences, Volume 1, Issue 1, 2009, Pages 1753-1758; presented at the World Conference on Educational Sciences, North Cyprus, 04-07 February, 2009;

[24] Matzen, L.E.  "Recommendations for Reducing Ambiguity in Written Procedures" Sandia National Laboratories (Sandia Report SAND2009-7522), 2009 (retrieved from  16th January 2011)


Make a Free Website with Yola.