Nominal number

Labelling a pair of referees as referees "1" and "2" is a use of nominal numbers. Any set of numbers (a subset of the natural numbers) will be consistent labels as long as a distinct number is uniquely used for each distinct term which needs to be labelled. Nonetheless, sequences of integers may naturally be used as the simplest way to begin labelling; for example, 1, 2, 3, and so on.

Definition

The term "nominal number" may be quite recent and of limited use. It appears to have originated in school textbooks derived from the statistical term "nominal data", defined as data indicating "merely statements of qualitative category of membership". This usage comes from the sense of nominal as "name".

Mathematically, nominal numbering could be thought of as a bijection —that is, an injective and surjective function— from a set of entities onto a set of numerals. Proving a mapping is a bijection involves these arguments:

• The mapping being described is well-defined as a function, since each entity has always assigned to it a single numeral; neither a null amount of numerals nor many numerals at once.

• It is injective, since the assumption is of different entities having different numerals, never sharing the same numeral.

• It is surjective, since for every numeral in the numerals set, the assumption is one can find an entity which is described by it.

In reality, the sets of entities and numerals change, and mappings morph with time or even with different speakers, making this a very strict interpretation. It also behooves oneself to think of how to deal with unrealized entities, only existing in language (such as a nonexistent bus route number). Alternative definitions can be considered without injection, without surjection, relaxing the definition of a function to allow for multi-valued functions, or with wholly different mathematical models.

"Nominal number" can be broadly defined as "any numeral used for identification, however it was assigned", or narrowly as "a numeral with no information other than identification".

For the purposes of naming, the term "number" is often used loosely to refer to any string (sequence of symbols), which may not consist entirely of digits—it is often alphanumeric. For instance, UK National Insurance numbers, some driver's licence numbers, and some serial numbers contain letters.

Use of nominal numbers

"Nominal" refers to the use of numbers: any nominal number can be used by its numerical value as an integer—added to another, multiplied, compared in magnitude, and so forth—but for nominal numbers these operations are not, in general, meaningful. For example, the ZIP code 11111 is less than the ZIP code 12345, but that does not necessarily mean that 11111 was issued before 12345 or that the region denoted by 11111 is further south than 12345, though it might be. Similarly, one can add or subtract ZIP codes, but this is meaningless: 12345 − 11111 does not have any meaning as a ZIP code.

In general, the only meaningful operation with nominal numbers is to compare two nominal numbers to see whether they are identical or not (whether they refer to the same object).