counting
Counting is an act of determining the number of elements in a finite set of objects of the same type. Counting takes place in counting steps, often in steps of one, whereby the corresponding number sequence, as a sequence of number words , for example "one, two, three" or " two, four, six, seven ”is run through. The use of an ascending sequence is called "counting up", and that of a descending sequence is called "counting down". The determination of the number of distinguishable objects by addition , which is the basis of an ascending sequence of numbers, is called counting . The associated noun counting describes the counting process or its result (e.g. a census ). Counting the number of defined units ( normals ), objects or events is a form of measurement , the determination of a quantity .
Archaeological evidence shows that people have had counting methods for at least 50,000 years. Counting was already used in ancient cultures to record the number and completeness of social and economic counting objects such as group members, prey, possessions or debts. Counting led to the development of number notation , number systems and writing .
On the biosociology of counting
Counting is a linguistic skill that, in the strict sense, probably only humans acquired in the course of their biosocial phylogenesis (tribal development). According to this assumption, animals, such as birds, can probably notice that one thing is 'missing' with small numbers (e.g. their eggs), but they cannot yet count. Since - according to Dieter Claessens - for humans on this side of the animal-human transition field initially literally “no egg looked like the other”, counting requires a sharpened ability to abstract (see also biosociology ).
The fact that something occurs in pairs (eyes, ears, hands) did not necessarily have to lead people to counting with the help of numbers . Because first of all, the duplication had to impose itself physically and concretely on them as a special case, without necessarily requiring a numerical word for “two”. A linguistic alternative to counting is the paral or the dual , two forms of the "two number" that appear next to the singular (the "singular") and run through all noun and verb forms accordingly. It is assumed that the linguistic form of the paral or dual was initially closely tied to the axially symmetrical human body, like that of almost all animals. This and the general occurrence of the parals or duals in all Indo-European languages that have been developed in this respect suggests that at the time of its creation one could not " count to three " or only with difficulty beyond two . Adding a lot of things at once in order to then count them requires a more far-reaching abstraction. It is therefore assumed that the dual is historically older than the plural (the "plural").
Even if the “two number” proves to be insufficient in survival practice, the “invention” of the “plural” is still not necessarily obvious. In some languages, the “three number” (the trial ) and the “small plural” (the paukal ) were first developed as a number analogous to the dual . A “four number” (the quadral ), however, is not used in any language.
At the same time as being able to count, linguistic means were needed to denote concrete numbers. At first there was presumably a need for smaller numbers everywhere (one, two, three, four ...) and with increasing degree of civilization for increasingly higher numbers. Each ethnic group was faced with the challenge of either having to invent new numerals for higher numbers, or to develop a system with which higher numbers can be expressed on the basis of lower numerals. It emerged Quinärsysteme based on 5 Dezimalsysteme on the base 10 and Vigesimalsysteme 20 based It is believed that the counting with the fingers , with both hands , or with fingers and toes , the reason for the basis of these counting systems is . In other cultures, counting with the help of the phalanges arose, which led one-handed to duodecimal systems (with the base twelve) and two-handed to number systems with the base 60 (see One- and two-handed counting with phalanges and fingers ).
More detailed determination
In general, counting is used to determine the number of a finite set of objects by assigning the next natural number to each object , starting with 1 , until there are no more objects left (by means of a bijection ). The number assigned last provides the number you are looking for. Some people, especially children, use their hands to avoid miscounting. A hand counter can also be used as a mechanical counting aid .
The size of an infinite amount can no longer be determined by counting; the mathematical concept of thickness serves as a substitute . Mathematically, this aspect is dealt with in the article cardinal numbers .
Humans are able to capture several objects simultaneously without having to count them. This can be used to speed up counting. Here, groups of a fixed size (e.g. groups of two or five) are formed and from number to number, not 1, but the group size (e.g. 2 or 5) is added : "Five, ten, fifteen, twenty ..."
If the order or the rank of the objects is important in addition to the number , one speaks of ordinal numbers .
In numbering (as opposed to counting) numbers are used to distinguish, not to count, and sometimes it is useful to leave out numbers. However, the number of the object is then no longer identical to its rank. Example: In identification numbers for persons (insurance companies, identity cards, etc.), dates of birth are encoded in the number, such as 10000024121928. Numbers such as 10000032121928 are not assigned. Numbers assigned in this way form a nominal scale .
Count from 0
In some situations in mathematics and computer science it makes sense to start counting or numbering at 0 , for example with memory cells or with arrays in most programming languages . This is also known from houses: The first floor is above the ground floor (the 0th floor). Below is the basement (−1st floor). Room numbers within a floor start with 0 (00 = toilet). If you have 100 rooms per floor and you number the rooms on the bottom with 0 to 99 and those above with 100 to 199, the 100's place for the room numbers is the same as the floor without having to leave out numbers.
Counting distances
When counting distances within a sequence of elements, in contrast to counting the elements themselves, the usual procedure is to start counting with 1 for the second element. This will give you the correct distance. Example:
| Element: | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 | 10 |
| Distance to the first element: | (0) | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th | 9 |
The distance between an element and itself is 0. If the elements are numbered consecutively, the distance can also be calculated by calculating the difference between the two numbers ( subtraction ). A possible alternative to this is the historical inclusive census (see below).
The inclusive count
In the inclusive counting of distances and periods of time , which was used from antiquity to the post-medieval period , both the start and the end element of the sequence were counted (i.e. a count including both elements). This counting method sometimes causes confusion, especially when it comes to time periods: The Olympic Games , which take place every four years, were B. as penteterian (πεντητηρικός, 'to celebrate every five years'), competitions that take place every two years were described as trieter (τριετηρικός, 'to celebrate every three years'), etc. See also fence post problem .
This procedure, which is correct when counting things , produces values when counting distances that, according to today's understanding, are always 1 too large. You can work with the distances counted in this way as long as you remain aware of the inclusive counting and take into account their special properties - for example, that when adding two successive distances, 1 must be subtracted, otherwise the middle element will be counted twice.
Examples of the historical inclusive census that still determine our linguistic usage today are:
Counting days
Normally today, for information such as “in n days”, the current day is not included. For example, you don't say “in two days” when you mean “tomorrow”. In contrast, it is widespread in German-speaking countries to say "in eight days" when referring to a calendar week. There is an analogy in French with dans quinze jours , "in fifteen days", as a term for "in two weeks", as well as in Greek (δεκαπενθήμερο) and in Spanish (quincena) for the two-week period.
The current day of the week is included in the inclusive count:
| Weekday: | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | Monday |
|---|---|---|---|---|---|---|---|---|
| "Number" of the day: | 1. | 2. | 3. | 4th | 5. | 6th | 7th | 1. |
| Distance: | 1 day | 2 days | 3 days | 4 days | 5 days | 6 days | 7 days | |
| Distance with inclusive counting: | 2 days | 3 days | 4 days | 5 days | 6 days | 7 days | 8 days |
Further examples:
- In the Roman calendar , the term Nonen ("the ninth days", Nundinum ) referred to the ninth day before the Ides .
- In the liturgy , octave refers to a period of one week including the following eighth day, the octave day. For the dating of several festivals in the church year , counts of 40 or 50 days are given, including the day on which the festival falls, such as Ascension 40 days after Easter, representation of the Lord 40 days after Christmas. The ancient Greek name for Pentecost , pentēkostē , means "on the fiftieth day" (after Easter).
Counting of years and centuries
In the historical chronology there is the problem of inclusive counting. For example, the traditional years of rulers' reign cannot simply be added up, because the years on which a change of ruler took place were counted twice.
Intervals in music
In the case of musical intervals , both the beginning and the end tone are included in the naming. Therefore, the prime has a distance of 0 notes, the second has a distance of 1 note, the third has a distance of 2 notes, the fourth has a distance of 3 notes, the fifth has a distance of 4 notes, the sixth is a distance of 5 notes, the seventh is a distance of 6 notes and octave the distance of 7 tones.
In terms of language, it is possibly confusing that the Latin word intervallum means "space", which suggests an exclusive rather than an inclusive count.
| Note name: | C. | D. | E. | F. | G | A. | H | c |
| Tone name as a number: | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th |
| Distance to the keynote: | 0 | 1 | 2 | 3 | 4th | 5 | 6th | 7th |
| Distance with inclusive counting: | 1 | 2 | 3 | 4th | 5 | 6th | 7th | 8th |
The fact that the name of each interval, which is common in music, is one larger, can be seen, among other things, when adding intervals. A fourth and a fifth add up to an octave. But 4 + 5 is not 8 - rather 3 + 4 = 7. This coincides with the fact that the octave consists of 7 (and not about 8) root tones .
See also
literature
- August F. Pott : The linguistic difference in Europe in the numerals demonstrated as well as quinary and vigesimal counting methods. Halle an der Saale 1868; Reprint Amsterdamm 1971.
- H. Wiese: Numbers and numerals. An investigation into the correlation of conceptual and linguistic structures (studia grammatica 44). Berlin 1997.
- H. Wiese: Numbers, Language, and the Human Mind . Cambridge 2003.
- M. Wedell: Counting. Semantic and praxeological studies on numerical knowledge in the Middle Ages (historical semantics 14). Göttingen 2011.
Web links
Individual evidence
- ↑ “count” on www.duden.de
- ^ Howard Eves: An Introduction to the History of Mathematics. 6th edition, 1990, p. 9.
- ↑ Dieter Claessens: The concrete and the abstract. Sociological sketches for anthropology. Suhrkamp, Frankfurt am Main 1980, ISBN 3-518-07329-X .