Counting
Encyclopedia
Counting is the action of finding the number of elements of a finite set of objects. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration
refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.
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
, numeral system
s and writing
.
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 base 1 counting; normal counting is done in base 10. Computers use base 2 counting (0's and 1's).
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. Finger-counting uses unary notation (one finger = one unit), and is thus limited to counting 10 (unless you start in with your toes). 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 counting), it is possible to keep a finger count up to .
Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.
(meaning 50) is 49 days before Easter Sunday.
The Jewish people also counted days 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 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.
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 the conclusion 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 performed. 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.
, is that no bijection can exist between {1, 2, ..., n} and {1, 2, ..., m} unless n = m; this fact (together with the fact that two bijections can be composed
to give another bijection) ensures that counting the same set in different ways can never result in different numbers (unless an error is made). This is the fundamental mathematical theorem that gives counting its purpose; however you count a (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite) combinatorics
—hence (finite) combinatorics is sometimes referred to as "the mathematics of counting."
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 "countably infinite." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all integer
s (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless there are sets, such as the set of real number
s, that can be shown to be "too large" to admit a bijection with the natural numbers, and these sets are called "uncountable." Sets for which there exists a bijection between them are said to have the same cardinality, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory
that explicitly studies possible cardinalities).
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 is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with n > m, then any map is not injective (so there exist two distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S, f(x) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example; for instance there must exists real numbers that are not computable number
s, because the latter set is only countably infinite, but by definition a non-computable number cannot be precisely specified.
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 permutation
s of {1, 2, ..., n} for any natural number n.
See also: counting games
Enumeration
In 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...
refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.
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
Accountancy
Accountancy is the process of communicating financial information about a business entity to users such as shareholders and managers. The communication is generally in the form of financial statements that show in money terms the economic resources under the control of management; the art lies in...
). The development of counting led to the development of mathematical notation
Mathematical notation
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics...
, numeral system
Numeral system
A 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 and writing
Writing
Writing is the representation of language in a textual medium through the use of a set of signs or symbols . It is distinguished from illustration, such as cave drawing and painting, and non-symbolic preservation of language via non-textual media, such as magnetic tape audio.Writing most likely...
.
Forms of counting
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
Tally marks
Tally marks, or hash marks, are a 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 base 1 counting; normal counting is done in base 10. Computers use base 2 counting (0's and 1's).
Counting can also be in the form of finger counting
Finger counting
Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in...
, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations. Finger-counting uses unary notation (one finger = one unit), and is thus limited to counting 10 (unless you start in with your toes). 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
Finger binary
Finger 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.- Mechanics :In the binary number system, each numerical...
(base 2 counting), it is possible to keep a finger count up to .
Various devices can also be used to facilitate counting, such as hand tally counters and abacuses.
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, dekapenthímero), Spanish (quincena) and Portuguese (quinzena) - whereas "a fortnight" derives from "a fourteen-night", as the archaic "a senight" does from "a seven-night". 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
Quinquagesima is the name used in the Western Church for the Sunday before Ash Wednesday. It was also called Quinquagesima Sunday, Quinquagesimae, Estomihi, or Shrove Sunday...
(meaning 50) is 49 days before Easter Sunday.
The Jewish people also counted days inclusively. For instance, Jesus
Jesus
Jesus of Nazareth , commonly referred to as Jesus Christ or simply as Jesus or Christ, is the central figure of Christianity...
announced he would die and resurrect
Death and Resurrection of Jesus
The Christian belief in the resurrection of Jesus states that Jesus returned to bodily life on the third day following his death by crucifixion. It is a key element of Christian faith and theology and part of the Nicene Creed: "On the third day he rose again in fulfillment of the Scriptures"...
"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 intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
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 the conclusion 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 performed. In the mean time, children learn how to name cardinalities that they can subitize.
Children with Williams syndrome
Williams syndrome
Williams syndrome is a rare neurodevelopmental disorder characterized by a distinctive, "elfin" facial appearance, along with a low nasal bridge; an unusually cheerful demeanor and ease with strangers; developmental delay coupled with strong language skills; and cardiovascular problems, such as...
often display serious delays in learning to count.
Counting in mathematics
In mathematics, the essence of counting a set and finding a result n, is that it establishes a one to one correspondence (or bijection) of the set with the set of numbers {1, 2, ..., n}. A fundamental fact, which can be proved by mathematical inductionMathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
, is that no bijection can exist between {1, 2, ..., n} and {1, 2, ..., m} unless n = m; this fact (together with the fact that two bijections can be composed
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
to give another bijection) ensures that counting the same set in different ways can never result in different numbers (unless an error is made). This is the fundamental mathematical theorem that gives counting its purpose; however you count a (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite) combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
—hence (finite) combinatorics is sometimes referred to as "the mathematics of counting."
Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for any natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
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 "countably infinite." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless there are sets, such as the set of real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, that can be shown to be "too large" to admit a bijection with the natural numbers, and these sets are called "uncountable." Sets for which there exists a bijection between them are said to have the same cardinality, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
that explicitly studies possible cardinalities).
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 is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with n > m, then any map is not injective (so there exist two distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S, f(x) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example; for instance there must exists real numbers that are not computable number
Computable number
In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
s, because the latter set is only countably infinite, but by definition a non-computable number cannot be precisely specified.
The domain of enumerative combinatorics
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations...
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 permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
s of {1, 2, ..., n} for any natural number n.
See also: counting games