Counting

[1] 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.

Verbal counting involves speaking sequential numbers aloud or mentally to track progress.

Children count on fingers to facilitate tallying and for performing simple mathematical operations.

Inclusive counting is usually encountered when dealing with time in Roman calendars and the Romance languages.

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).

[5] Similar counting is involved in East Asian age reckoning, in which newborns are considered to be 1 at birth.

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 do not count,[6][7] and their languages do not have number words.

This leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set.

[8] 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 performed.

The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well-understood set.

One important principle is that if two sets X and Y have the same finite number of elements, and a function f: X → Y is known to be injective, then it is also surjective, and vice versa.

Similar counting arguments can prove the existence of certain objects without explicitly providing an example.