Counting is the process of determining the
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
of
elements of a
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
of objects, i.e., determining the
size
Size in general is the Magnitude (mathematics), magnitude or dimensions of a thing. More specifically, ''geometrical size'' (or ''spatial size'') can refer to linear dimensions (length, width, height, diameter, perimeter), area, or volume ...
of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
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
An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set. The precise requirements for an enumeration (fo ...
'' refers to uniquely identifying the elements of a
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
(combinatorial)
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
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, "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. 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
Accounting, also known as accountancy, is the measurement, processing, and communication of financial and non financial information about economic entities such as businesses and corporations. Accounting, which has been called the "language ...
). Notched bones were also found in the Border Caves in South Africa that may suggest that the concept of counting was known to humans as far back as 44,000 BCE. The development of counting led to the development of
mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...
,
numeral system
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner.
The same s ...
s, and
writing
Writing is a medium of human communication which involves the representation of a language through a system of physically Epigraphy, inscribed, Printing press, mechanically transferred, or Word processor, digitally represented Symbols (semiot ...
.
Forms of counting
![Hanakapiai Beach Warning Sign Only](https://upload.wikimedia.org/wikipedia/commons/f/fa/Hanakapiai_Beach_Warning_Sign_Only.jpg)
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, also called hash marks, are a unary numeral system ( arguably).
They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate ...
, 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. Tallying is base 1 counting; normal counting is done in base 10. Computers use
base 2 counting (0s and 1s), also known as
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
.
Counting can also be in the form of
finger counting
Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic n ...
, 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). Older finger counting used the four fingers and the three bones in each finger (
phalanges
The phalanges (singular: ''phalanx'' ) are digital bones in the hands and feet of most vertebrates. In primates, the thumbs and big toes have two phalanges while the other digits have three phalanges. The phalanges are classed as long bones.
...
) to count to the number twelve.
Other hand-gesture systems are also in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. By using
finger binary
Finger binary is a system for counting and displaying binary numbers on the fingers of either or both hands. Each finger represents one binary digit or bit. This allows counting from zero to 31 using the fingers of one hand, or 1023 using both: ...
(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
abacus
The abacus (''plural'' abaci or abacuses), also called a counting frame, is a calculating tool which has been used since ancient times. It was used in the ancient Near East, Europe, China, and Russia, centuries before the adoption of the Hin ...
es.
Inclusive counting
Inclusive counting is usually encountered when dealing with time in
Roman calendar
The Roman calendar was the calendar used by the Roman Kingdom and Roman Republic. The term often includes the Julian calendar established by the reforms of the Roman dictator, dictator Julius Caesar and Roman emperor, emperor Augustus in the ...
s and the
Romance language
The Romance languages, sometimes referred to as Latin languages or Neo-Latin languages, are the various modern languages that evolved from Vulgar Latin. They are the only extant subgroup of the Italic languages in the Indo-European languages, I ...
s.
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
A fortnight is a unit of time equal to 14 days (two weeks). The word derives from the Old English term , meaning "" (or "fourteen days," since the Anglo-Saxons counted by nights).
Astronomy and tides
In astronomy, a ''lunar fortnight'' is h ...
" is ''quinzaine'' (15
ays, and similar words are present in Greek (δεκαπενθήμερο, ''dekapenthímero''), Spanish (''quincena'') and Portuguese (''quinzena''). In contrast, the English word "fortnight" itself derives from "a fourteen-night", as the archaic "
sennight
A week is a unit of time equal to seven days. It is the standard time period used for short cycles of days in most parts of the world. The days are often used to indicate common work days and rest days, as well as days of worship. Weeks are ofte ...
" does from "a seven-night"; the English words are not examples of inclusive counting. In exclusive counting languages such as English, when counting eight days "from Sunday", Monday will be ''day 1'', Tuesday ''day 2'', and the following Monday will be the ''eighth day''. For many years it was
a standard practice in English law for the phrase "from a date" to mean "beginning on the day after that date": this practice is now deprecated because of the high risk of misunderstanding.
In the Roman calendar, the ''nones'' (meaning "nine") is 8 days before the ''ides''; more generally, dates are specified as inclusively counted days up to the next named day.
In the Christian calendar,
Quinquagesima
Quinquagesima (), in the Western Christian Churches, is the last Sunday of Shrovetide, being the Sunday before Ash Wednesday. It is also called Quinquagesima Sunday, Quinquagesimae, Estomihi, Shrove Sunday, Pork Sunday, or the Sunday next before ...
(meaning 50) is 49 days before Easter Sunday.
Musical terminology also uses inclusive counting of
intervals
Interval may refer to:
Mathematics and physics
* Interval (mathematics), a range of numbers
** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets
* A statistical level of measurement
* Interval e ...
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
In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
''.
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, 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. 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.
[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.] In the meantime, children learn how to name cardinalities that they can
subitize
Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. The term was coined in 1949 by E. L. Kaufman et al., and is derived from the Latin adjective '' subitus'' (meaning "sudden") and captures ...
.
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
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
(or bijection) of the set with the subset of positive integers . A fundamental fact, which can be proved by
mathematical induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
, is that no bijection can exist between and unless ; 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
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
—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 for ''any''
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
''n''; these are called
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set th ...
s, while those sets for which such a bijection does exist (for some ''n'') are called
finite set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
:\
is a finite set with five elements. Th ...
s. 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
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
." 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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, that can be shown to be "too large" to admit a bijection with the natural numbers, and these sets are called "
uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
." Sets for which there exists a bijection between them are said to have the same
cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, 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 is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
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
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
, then it is also
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, and vice versa. A related fact is known as the
pigeonhole principle
In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...
, which states that if two sets ''X'' and ''Y'' have finite numbers of elements ''n'' and ''m'' with , 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
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
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.
The domain of
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. More generally, given an infin ...
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
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
s of for any natural number ''n''.
See also
*
Card reading (bridge)
In contract bridge, card reading (or counting the hand) is the process of inferring which remaining cards are held by each opponent. The reading is based on information gained in the bidding and the play to previous tricks. The technique is used by ...
*
Calculation
A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to th ...
*
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. Th ...
*
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
*
Count data
Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York: ...
*
Counting (music)
In music, counting is a system of regularly occurring sounds that serve to assist with the performance or audition of music by allowing the easy identification of the beat. Commonly, this involves verbally counting the beats in each measure as th ...
*
Counting problem (complexity)
In computational complexity theory and computability theory, a counting problem is a type of computational problem. If ''R'' is a search problem then
:c_R(x)=\vert\\vert \,
is the corresponding counting function and
:\#R=\
denotes the corre ...
*
Developmental psychology
Developmental psychology is the science, scientific study of how and why humans grow, change, and adapt across the course of their lives. Originally concerned with infants and children, the field has expanded to include adolescence, adult deve ...
*
Elementary arithmetic
The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type. ...
*
Finger counting
Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic n ...
*
History of mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...
*
Jeton
Jetons or jettons are tokens or coin-like medals produced across Europe from the 13th through the 18th centuries. They were produced as counters for use in calculation on a counting board, a lined board similar to an abacus. They also found use ...
*
Level of measurement
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scal ...
*
Mathematical quantity
*
Ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least n ...
*
Particle number
The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
*
Subitizing and counting
*
Tally mark
Tally marks, also called hash marks, are a unary numeral system ( arguably).
They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate r ...
*
Unary numeral system
The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number ''N'', a symbol representing 1 is repeated ''N'' times.
In the unary system, the number 0 (zero) is represented by the empty string, that ...
*
List of numbers
This is a list of notable numbers and articles about notable numbers. The list does not contain all numbers in existence as most of the number sets are infinite. Numbers may be included in the list based on their mathematical, historical or cu ...
*
Yan tan tethera
Yan Tan Tethera or yan-tan-tethera is a sheep-counting system traditionally used by shepherds in Northern England and some other parts of Britain. The words are numbers taken from Brythonic Celtic languages such as Cumbric which had died out in ...
(Counting sheep in Britain)
References
{{Reflist
Elementary mathematics
Numeral systems
Statistics
Mathematical logic
ca:Comptar