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ratio

ratio System Class

Class Precedence List:

ratio, rational, real, number, t

Description:

A ratio is a number representing the mathematical ratio of two non-zero integers, the numerator and denominator, whose greatest common divisor is one, and of which the denominator is positive and greater than one.

See Also:

Figure 2–9, Section 2.3.2 (Constructing Numbers from Tokens), Section 22.1.3.1.2 (Printing Ratios)

Expanded Reference: ratio​

Type Checking​

A ratio is the type of rational numbers that are not integers. A ratio is always stored in reduced form with a positive denominator.

(typep 1/2 'ratio)
=> T
(typep 2/3 'ratio)
=> T
(typep 42 'ratio)
=> NIL
(typep 0.5 'ratio)
=> NIL

Automatic Canonicalization​

Common Lisp automatically reduces ratios to lowest terms and ensures the denominator is positive. If the result is a whole number, it becomes an integer instead of a ratio.

4/6
=> 2/3
;; -3/-4
;; => 3/4
6/3
=> 2
0/5
=> 0

(typep 6/3 'ratio)
=> NIL
(typep 6/3 'integer)
=> T

Accessing Numerator and Denominator​

Use numerator and denominator to extract the parts of a ratio. They always reflect the canonical form.

(numerator 2/3)
=> 2
(denominator 2/3)
=> 3
(numerator -2/3)
=> -2
(denominator -2/3)
=> 3
(numerator 4/6)
=> 2
(denominator 4/6)
=> 3

Type Hierarchy​

The ratio type is a subtype of rational, disjoint from integer. Together they partition the rational type.

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

Ratio Arithmetic​

Arithmetic on ratios produces exact results. A ratio is only produced when the result is not a whole number.

(+ 1/3 1/6)
=> 1/2
(* 2/3 3/4)
=> 1/2
(/ 1 3)
=> 1/3
(- 5/3 2/3)
=> 1
(typep (- 5/3 2/3) 'integer)
=> T