Numerical tower

In Scheme and some other Lisp dialects, a numerical tower is the set of data types that represent numbers in a given programming language.

Each type in the tower conceptually "sits on" a more fundamental type, so an integer is a rational number and a number, but the converse is not necessarily true, i.e. not every number is an integer; this asymmetry implies that a language can allow implicit coercions of numerical types—without creating semantic problems—in only one direction: coercing an integer to a rational loses no information and does not affect the results of a function, but to coerce most reals to an integer could well result in a problem (for example, the real 1/3 does not equal any integer).

The Scheme programming language defines all its arithmetic within this model, as do other Lisp dialects. Some implementations may extend or adapt the tower. Kawa, for example, extends it with a Quantity type that is even more generic than Number. Smalltalk is another programming language that follows this model, but it has a Magnitude as superclass of Number. Another popular variant is having both exact and inexact versions of the tower or parts of it. Most languages and language implementations do not support a Scheme-like numerical tower. Some languages support it only in a limited way.