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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Counting | 1/2 | https://en.wikipedia.org/wiki/Counting | reference | science, encyclopedia | 2026-05-05T13:54:33.881561+00:00 | kb-cron |
Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...). There is archaeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as the number of group members, prey animals, property, or debts (that is, accountancy). Notched bones were also found in the Border Caves in South Africa, which may suggest that the concept of counting was known to humans as far back as 44,000 BCE. The development of counting led to the development of mathematical notation, numeral systems, and writing.
== Forms of counting ==
Verbal counting involves speaking sequential numbers aloud or mentally to track progress. Generally such counting is done with base 10 numbers: "1, 2, 3, 4", etc. Verbal counting is often used for objects that are currently present rather than for counting things over time, since following an interruption counting must resume from where it was left off, a number that has to be recorded or remembered. Counting a small set of objects, especially over time, can be accomplished efficiently with tally marks: making a mark for each number and then counting all of the marks when done tallying. Tallying is base 1 counting. Finger counting is convenient and common for small numbers. Children count on fingers to facilitate tallying and for performing simple mathematical operations. Older finger counting methods used the four fingers and the three bones in each finger (phalanges) to count to twelve. Other hand-gesture systems are also in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. With finger binary it is possible to keep a finger count up to 1023 = 210 − 1. Various devices can also be used to facilitate counting, such as tally counters and abacuses.
== Inclusive counting == Inclusive/exclusive counting are two different methods of counting. For exclusive counting, unit intervals are counted at the end of each interval. For inclusive counting, unit intervals are counted beginning with the start of the first interval and ending with end of the last interval. This results in a count which is always greater by one when using inclusive counting, as compared to using exclusive counting, for the same set. Apparently, the introduction of the number zero to the number line resolved this difficulty; however, inclusive counting is still useful for some things. Refer also to the fencepost error, which is a type of off-by-one error. Inclusive counting is usually encountered when dealing with time in Roman calendars and the Romance languages. In the ancient Roman calendar, the nones (meaning "nine") is 8 days before the ides; more generally, dates are specified as inclusively counted days up to the next named day. In the Christian liturgical calendar, Quinquagesima (meaning 50) is 49 days before Easter Sunday. When counting "inclusively", the Sunday (the start day) will be day 1 and therefore the following Sunday will be the eighth day. For example, the French phrase for "fortnight" is quinzaine (15 [days]), and similar words are present in Greek (δεκαπενθήμερο, dekapenthímero), Spanish (quincena) and Portuguese (quinzena). In contrast, the English word "fortnight" itself derives from "a fourteen-night", as the archaic "sennight" does from "a seven-night"; the English words are not examples of inclusive counting. In exclusive counting languages such as English, when counting eight days "from Sunday", Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. For many years it was a standard practice in English law for the phrase "from a date" to mean "beginning on the day after that date": this practice is now deprecated because of the high risk of misunderstanding. Similar counting is involved in East Asian age reckoning, in which newborns are considered to be 1 at birth. Musical terminology also uses inclusive counting of intervals between notes of the standard scale: going up one note is a second interval, going up two notes is a third interval, etc., and going up seven notes is an octave.
== Education and development ==