Counting is the
mathematicalMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
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 functionIn mathematics, an injective function is a function that associates distinct arguments with distinct values; in other words, every unique argument produces a unique result...
from the remaining objects to the natural numbers starting from two), or for
well-orderIn mathematics, a well-order relation on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering.Equivalently, a well-ordering is a well-founded total order....
ed objects, to find the
ordinal numberIn set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
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 namesIn linguistics, number names are specific words in a natural language that represent numbers.In writing, numerals are symbols also representing numbers.-Numeral types:...
and the
numberA number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral, but in common usage the word number is used for both the abstract object and the symbol, as well as for the word for the number...
system. Sometimes the term
counting is used to mean the same as
enumerationIn mathematics and theoretical computer science, the broadest and most abstract definition of an enumeration of a set is an exact listing of all of its elements . The restrictions imposed on the type of list used depend on the branch of mathematics and the context in which one is working...
, i.e. finding the number of elements of a
finiteFinite is the opposite of infinite. It may refer to:* Having a finite number of elements: finite set* Being a finite number, so not equal to ; for example, all real numbers are 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.,
accountancyAccountancy or accounting is the art of communicating financial information about a business entity to users such as shareholders and managers...
).
The development of counting led to the development of
mathematical notationA mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics and the physical sciences, engineering and economics...
and
numeral systemA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
s.
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 marksTally marks are an implementation of the unary numeral system. They are a form of numeral used for counting. They allow updating written intermediate results without erasing or discarding anything written down...
, 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 countingFinger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by the Arabic numeral system, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught...
, 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 binaryFinger binary is a system for counting and displaying binary numbers on the fingers and thumbs of one or more hands. It is possible to count from 0 to 31 using the fingers of a single hand, or from 0 through 1023 if both hands are used.- How it works :In the binary number system, each numerical...
(base 2 place-value notation), it is possible to keep a finger count up to 1023 = 2
10 - 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
QuinquagesimaQuinquagesima is the name for the Sunday before Ash Wednesday. It was also called Quinquagesima Sunday, Shrove Sunday or Esto Mihi. The name originates from Latin quinquagesimus , referring to the fifty days before Easter Sunday using inclusive counting which counts both Sundays...
(meaning 50) is 49 days before Easter Sunday.
The Jewish people also counted inclusively. For instance,
JesusJesus of Nazareth —also known as Jesus Christ or occasionally Jesus the Christ—is the central figure of Christianity. Within most Christian denominations...
announced he would
die and resurrectThe death and resurrection of Jesus may refer to:*Crucifixion of Jesus*Empty tomb*Passion *Resurrection appearances of Jesus*Resurrection of Jesus...
"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
intervalIn music theory, the term interval describes the relationship between the pitches of two notes.Intervals may be described as:* vertical if the two notes sound simultaneously* linear , if the notes sound successively....
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 syndromeWilliams syndrome is a rare neurodevelopmental disorder caused by a deletion of about 26 genes from the long arm of chromosome 7...
often display serious delays in learning to count.
Mathematics
In mathematics, the study of counting is by definition simply the study of repeatedly adding one of something (in a context where it does not matter what that something is). In mathematical terms, the sets which have every mathematical property of repeatedly adding one (every mathematical property of counting) are called "
finiteFinite is the opposite of infinite. It may refer to:* Having a finite number of elements: finite set* Being a finite number, so not equal to ; for example, all real numbers are finite...
" and it is these sets which constitute the subject matter of the area of mathematics known as
combinatoricsCombinatorics is a branch of pure mathematics concerning the study of discrete objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics...
. 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.
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