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The term "nominal number" is quite recent and of limited use. It appears to have originated as a usage in school textbooks derived from the statistical term "nominal data", defined as data indicating "...[m]erely statements of qualitative category of membership." This usage comes from the sense of nominal as "name".
Mathematically, nominal numbering is a one-to-one and onto function from a set of objects being named to a set of numerals, which may change (typically growing) over time: it is a function because each object is assigned a single numeral, it is one-to-one because different objects are assigned different numerals, and it is onto because every numeral in the set at a given time has associated with it a single named object.
"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, National Insurance numbers, some driver's license 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 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: 11111 - 12345 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).
A great variety of numbers meet the broad definition, including:
- National identification numbers, such as:
- Routing numbers, such as:
- Bank codes and sort codes, like IBAN, and the ABA Routing transit number.
- Postal codes, such as ZIP codes (These are generally numeric in the United States, but other nations often use alphanumeric systems.)
- Telephone numbers, assigned by various telephone numbering plans, such as the ITU-T E.164 and the North American Numbering Plan (NANPA).
- Numbers of train or bus routes in public transport
- Car model names from some car manufactors, such as BMW or Peugeot, are plain numbers.
- Magic numbers
These are usually assigned either in some hierarchical way, such as how telephone numbers are assigned (in NANPA) as Country Code + Area Code + Prefix + Suffix, where the first 3 are geographically based, or sequentially, as in serial numbers; these latter are thus properly ordinal numbers.
Narrowly defined 
Numerical identifiers that are nominal numbers narrowly defined, viz, convey no information other than identity, are quite rare. These must be defined either arbitrarily or randomly, and most commonly arise in computer applications, such as dynamic IP addresses assigned by Dynamic Host Configuration Protocol. A more everyday example are sports squad numbers, which do not in general have any public meaning beyond identity, though they may be allocated based on some internal club or organization policy. In some settings these are based on position, but in others they are associated with an individual, being a proper nominal number. The naming function is demonstrated by retired numbers, where a club "retires" a number that has become associated with a particularly famous player, but reallocate others to new players when they become available.