Counting
Counting is the process of determining the
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.
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.
Various devices can also be used to facilitate counting, such as tally counters and abacuses.
Inclusive counting
Inclusive/exclusive counting are terms used for counting intervals. For inclusive counting the starting point is one; for exclusive counting the starting point is zero. Inclusive counting is usually encountered when dealing with time in
Similar counting is involved in
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
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 do not count,[6][7] and their languages do not have number words.
Many children at just 2 years of age have some skill in reciting the count list (that is, saying "one, two, three, ..."). They can also answer questions of ordinality for small numbers, for example, "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 the conclusion that the child knows how to use counting to determine the size of a set.
Counting in mathematics
In mathematics, the essence of counting a set and finding a result n, is that it establishes a
Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for any natural number n; these are called infinite sets, while those sets for which such a bijection does exist (for some n) are called finite sets. Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets.
The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some well-understood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called "
Counting, mostly of finite sets, has various applications in mathematics. 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
The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of permutations of {1, 2, ..., n} for any natural number n.
See also
- Calculation
- Card reading (bridge)
- Cardinal number
- Combinatorics
- Count data
- Counting (music)
- Counting problem (complexity)
- Developmental psychology
- Elementary arithmetic
- Finger counting
- History of mathematics
- Jeton
- Level of measurement
- List of numbers
- Mathematical quantity
- Ordinal number
- Particle number
- Subitizing and counting
- Tally mark
- Unary numeral system
- Yan tan tethera (Counting sheep in Britain)
References
- ^ An Introduction to the History of Mathematics (6th Edition) by Howard Eves (1990) p.9
- ^ "Early Human Counting Tools". Math Timeline. Retrieved 2018-04-26.
- ISBN 978-0-8203-3796-8.
- ^ ISBN 019987445X.
- ^ "Drafting bills for Parliament". gov.uk. Office of the Parliamentary Counsel. 18 June 2020. See heading 8.
- ^ Butterworth, B., Reeve, R., Reynolds, F., & Lloyd, D. (2008). Numerical thought with and without words: Evidence from indigenous Australian children. Proceedings of the National Academy of Sciences, 105(35), 13179–13184.
- ^ Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499.
- ^ Fuson, K.C. (1988). Children's counting and concepts of number. New York: Springer–Verlag.
- ^ Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.
- ^ Le Corre, M., Van de Walle, G., Brannon, E. M., Carey, S. (2006). Re-visiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52(2), 130–169.