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
Counting using tally marks at Hanakapiai Beach
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,
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 number of group members, prey animals, property, or debts
(i.e., accountancy). The development of counting led to the
development of mathematical notation, numeral systems, and writing.
1 Forms of counting
2 Inclusive counting
3 Education and development
Counting in mathematics
5 See also
7 External links
Forms of counting
Prehistoric numerals and Numerical digit
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. Tallying 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). Older finger
counting used the four fingers and the three bones in each finger
(phalanges) 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 (base 2 counting), 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.
Inclusive counting is usually encountered when dealing with time in
the Romance languages. In exclusive counting languages such as
English, 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 quinzaine (15 [days]), 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" does from "a seven-night";
the English words are not examples of inclusive counting.
Names based on inclusive counting appear 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.
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
Main article: Pre-math skills
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
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
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 induction, 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 integers
(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
numbers, 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 f: X → Y 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 f: X → Y 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
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.
Automated pill counter
Card reading (bridge)
Counting problem (complexity)
History of mathematics
Level of measurement
Subitizing and counting
Unary numeral system
List of numbers
List of numbers in various languages
Yan tan tethera (
Counting sheep in Britain)
^ An Introduction to the History of Mathematics (6th Edition) by
Howard Eves (1990) p.9
^ Macey, Samuel L. (1989). The Dynamics of Progress: Time, Method, and
Measure. Atlanta, Georgia: University of Georgia Press. p. 92.
^ James Evans, The History and Practice of Ancient Astronomy. Oxford
University Press, 1998. ISBN 019987445X. Chapter 4, page 164.
^ 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,
^ 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
^ 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.