Counting in the context of Finiteness

A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in language with number words or by dedicated symbols called numerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, which can display any non-negative integer using a combination of ten symbols, called numerical digits. Numerals can be used for counting (as with cardinal number of a collection or set), labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$, and π, and complex numbers which extend the real numbers with a square root of −1, and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

Addition, usually denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or sum of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as "3 + 2 = 5", which is read as "three plus two equals five".

Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, and elements of additive groups.

Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and the development of both hardware and software. Computing has scientific, engineering, mathematical, technological, and social aspects. Major computing disciplines include computer engineering, computer science, cybersecurity, data science, information systems, information technology, and software engineering.

The term computing is also synonymous with counting and calculating. In earlier times, it was used in reference to the action performed by mechanical computing machines, and before that, to human computers.

In mathematics and empirical science, quantification (or quantitation) is the act of counting and measuring that maps human sense observations and experiences into quantities. Quantification in this sense is fundamental to the scientific method.

1 (one, unit, unity) is a number, numeral, and glyph. It is the first and smallest positive integer of the infinite sequence of natural numbers. This fundamental property has led to its unique uses in other fields, ranging from science to sports, where it commonly denotes the first, leading, or top thing in a group. 1 is the unit of counting or measurement, and represents a single thing. The representation of 1 evolved from ancient Sumerian and Babylonian symbols to the modern Arabic numeral. Linguistically, in English, "one" is a determiner for singular nouns and a gender-neutral pronoun.

In mathematics, 1 is the multiplicative identity, meaning that any number multiplied by 1 equals the same number. 1 is by convention not considered a prime number. In digital technology, 1 represents the "on" state in binary code, the foundation of computing. Philosophically, 1 symbolizes the ultimate reality or source of existence in various traditions.

Tally marks, also called hash marks, are a form of numeral used for counting. They can be thought of as a unary numeral system.

They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks, were also historically used for this purpose.

Obsessive–compulsive disorder (OCD) is a mental disorder in which an individual has intrusive thoughts (an obsession) and feels the need to perform certain behaviors (compulsions) repeatedly to relieve the distress caused by the obsession, to the extent where it impairs general function.

Obsessions are persistent unwanted thoughts, mental images, or urges that generate feelings of anxiety, disgust, or discomfort. Some common obsessions include fear of contamination, obsession with symmetry, the fear of acting blasphemously, sexual obsessions, and the fear of possibly harming others or themselves. Compulsions are repetitive actions performed in response to obsessions to reduce anxiety, such as washing, checking, counting, reassurance seeking, and situational avoidance.

Arithmomania (from Greek arithmós, "number", and maníā, "compulsion") is a mental disorder that may be seen as an expression of obsessive–compulsive disorder (OCD). Individuals experiencing this disorder have a strong need to count their actions or objects in their surroundings.

For example, individuals with arithmomania may feel compelled to count steps while climbing or descending stairs or to tally the number of letters in words. They often feel compelled to repeat an action a specific number of times to avert a perceived catastrophe. Other examples include counting tiles on floors or ceilings, lines on highways, or touching objects a specific number of times, such as doorknobs or tables.

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 (for example, whether the set must be finite, or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem.

Some sets can be enumerated by means of a natural ordering (such as 1, 2, 3, 4, ... for the set of positive integers), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics, the term enumeration is used more in the sense of counting – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements.

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science.

Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

In linguistics, and more precisely in traditional grammar, a cardinal numeral (or cardinal number word) is a part of speech used to count.

Examples in English are the words one, two, three, and the compounds three hundred [and] forty-two and nine hundred [and] sixty. Cardinal numerals are classified as definite, and are related to ordinal numbers, such as the English first, second, third, etc.