rational

rational System Class

Class Precedence List:

rational, real, number, t

Description:

The canonical representation of a rational is as an integer if its value is integral, and otherwise as a ratio.

The types integer and ratio are disjoint subtypes of type rational.

Compound Type Specifier Kind:

Abbreviating.

Compound Type Specifier Syntax:

(rational [lower-limit [upper-limit]])

Compound Type Specifier Arguments:

lower-limit, upper-limit—interval designators for type rational. The defaults for each of lower-limit and upper-limit is the symbol *.

Compound Type Specifier Description:

This denotes the rationals on the interval described by lower-limit and upper-limit.

Expanded Reference: rational

Type Checking

The rational type includes both integers and ratios. It represents exact numeric values (as opposed to the approximate representation of floats).

(typep 42 'rational)
=> T
(typep 2/3 'rational)
=> T
(typep 3.14 'rational)
=> NIL
(typep #C(1 2) 'rational)
=> NIL
(rationalp 42)
=> T
(rationalp 1/3)
=> T

Type Hierarchy

The rational type has two subtypes: integer and ratio. It is itself a subtype of real.

(subtypep 'integer 'rational)
=> T
=> T
(subtypep 'ratio 'rational)
=> T
=> T
(subtypep 'rational 'real)
=> T
=> T
(subtypep 'rational 'number)
=> T
=> T

Converting to Rational

The rational and rationalize functions convert real numbers to rationals, but they use different strategies for floats.

;; rational converts using exact float representation
(rational 0.5)
=> 1/2
(rational 0.1)
=> 13421773/134217728

;; rationalize finds a simpler rational
(rationalize 0.5)
=> 1/2
(rationalize 0.1)
=> 1/10

;; Integers pass through unchanged
(rational 42)
=> 42

Exact Arithmetic

Rationals provide exact arithmetic with no rounding errors, unlike floating-point numbers.

;; Float arithmetic has rounding errors
(- 1.0 0.9 0.1)
=> 2.2351742e-8

;; Rational arithmetic is exact
(- 1 9/10 1/10)
=> 0
(+ 1/3 1/3 1/3)
=> 1