Numerical tower

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

A representation of the numerical tower with five types of numbers

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 most other Lisp dialects.[1] Some implementations may extend or adapt the tower. Kawa, a Scheme implementation for JVM, 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; R⁷RS Scheme recommends but does not strictly require this of implementations. Most programming languages and language implementations do not support a Scheme-like numerical tower, though some languages provide limited or inconsistent support if implementation simplicity permits.