Counting
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Counting is the mathematical action of repeatedly adding (or subtracting) one, 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. In mathematics the term counting or enumeration also means 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[1]. 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.
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[edit] 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.
[edit] 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 word for fortnight is quinze jours (15 days), and similar words are present in Greek (δεκαπενθήμερο) and Spanish (quincena). 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.
[edit] Psychology
By age 3, most children acquire the ability to count, and the majority of 3 year olds can count up to 10 objects correctly[2]. Children eventually come to understand the following 5 counting principles[2]:
- One-to-one correspondence: Each object must be labeled by a single number word.
- Stable order: The number should always be recited in the same order.
- Cardinality: The number of objects in the set is equal to the last number stated.
- Order irrelevance: Objects can be counted left to right, right to left, or in any order.
- Abstraction: Any set of discrete objects can be counted.
[edit] See also
[edit] References
- ^ An Introduction to the History of Mathematics (6th Edition) by Howard Eves (1990)p.9
- ^ a b Siegler, Robert (2006). How Childred Develop, Exploring Child Develop Student Media Tool Kit & Scientific American Reader to Accompany How Children Develop. New York: Worth Publishers. ISBN 0716761130.
