Counting

Counting is the mathematical action of continually adding (or subtracting) one at a time, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with an injective function from the remaining objects to the natural numbers starting from two), or for well-ordered objects, to find the ordinal number of a particular object, or to find the object with a particular ordinal number. Counting is also used (primarily by children) to demonstrate knowledge of the number names and the number system. Sometimes the term counting is used to mean the same as enumeration, i.e. finding the number of elements of a finite set.)

Counting sometimes involves numbers other than one; for example, when counting money, counting out change, when "counting by twos" (2, 4, 6, 8, 10, 12…) or when "counting by fives" (5, 10, 15, 20, 25…).

There is archeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of economic data such as debts and capital (i.e., accountancy). The development of counting led to the development of mathematical notation and numeral systems.

Forms
Counting can occur in a variety of forms.

Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is often used to count objects that are present already, instead of counting a variety of things over time.

Counting can also be in the form of tally marks, making a mark for each number and then counting all of the marks when done tallying. This is useful when counting objects over time, such as the number of times something occurs during the course of a day.

Counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations. The most naive finger-counting uses unary notation (one finger = one unit), and is thus limited to counting 10. Other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary (base 2 place-value notation), it is possible to keep a finger count up to 1023 = 210 - 1.

Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.

Some different forms of counting can be explained by example. Sometimes counting appears in the form of "counting out"; for example, "counting out" a certain number of cards, or at least a certain number of cards, to give to somebody in a game of cards. Another different form of counting to this is "counting the"; for example, "counting the" the number of pieces of fruit in a fruit bowl. In this form, the "the" refers to some things that are already "given" in some sense. In counting the fruit, some things (the pieces of fruit) are already given. In counting out cards, there are no particular cards that have been given, as the counter can count out any cards he or she likes.

Inclusive counting
Inclusive counting is usually encountered when counting days in a calendar. Normally when counting 8 days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. 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 quinze jours (15 days), and similar words are present in Greek (δεκαπενθήμερο), Spanish (quincena) and Portuguese (quinzena). This practice appears in other calendars as well; in the Roman calendar the nones (meaning nine) is 8 days before the ides; and in the Christian calendar Quinquagesima (meaning 50) is 49 days before Easter Sunday.

The Jewish people also counted inclusively. For instance, Jesus announced he would die and resurrect "on the third day," i.e. two days later. Scholars most commonly place his crucifixion on a Friday afternoon and his resurrection on Sunday before sunrise, spanning three different days but a period of around 36–40 hours.

Musical terminology also uses inclusive counting of interval 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
Learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a child's very first step into mathematics, and constitutes the most fundamental idea of that discipline. However, some cultures in Amazonia and the Australian Outback whose languages have few words have no number words beyond "one" or "many", preferring to gesticulate, and although they can subitize, they are handicapped in dealing with larger quantities.

Many children at just 2 years of age have some skill in reciting the count list (i.e., saying "one, two, three..."). They can also answer questions of ordinality for small numbers, e.g., "What comes after THREE?". They can even be skilled at pointing to each object in a set and reciting the words one after another. This leads many parents and educators to a false belief that the child knows how to use counting to determine the size of a set. Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are done. In the mean time, children learn how to name cardinalities that they can subitize.

Children with Williams syndrome often display serious delays in learning to count.

Mathematics
In mathematics, the study of counting is by definition simply the study of adding one at a time, (in a context where it does not matter what "one" is). In mathematical terms, the sets which have every mathematical property of adding one at a time, starting from nothing (every mathematical property of counting) are called "finite" and it is these sets which constitute the subject matter of the area of mathematics known as combinatorics. Thus we have the study of counting on the one hand, and the mathematical study of the arbitrary finite set on the other. The latter is identical to the former, except it is dressed up in the language of mathematics.