Some programming languages provide a built-in (primitive) rational data type to represent rational numbers like 1/3 and -11/17 without rounding, and to do arithmetic on them. Examples are the ratio
type of Common Lisp, and analogous types provided by most languages for algebraic computation, such as Mathematica and Maple. Many languages that do not have a built-in rational type still provide it as a library-defined type.
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Transcription
Representation
A variable or value of that type is usually represented as a fraction m/n where m and n are two integer numbers, either with a fixed or arbitrary precision. Depending on the language, the denominator n may be constrained to be non-zero, and the two numbers may be kept in reduced form (without any common divisors except 1).
Languages that support a rational data type usually provide special syntax for building such values, and also extend the basic arithmetic operations ('+', '−', '×', '/', integer powers) and comparisons ('=', '<', '>', '≤') to act on them — either natively or through operator overloading facilities provided by the language. These operations may be translated by the compiler into a sequence of integer machine instructions, or into library calls. Support may also extend to other operations, such as formatting, rounding to an integer or floating point value, etc.. As in mathematics, those languages often interpret an integer value as equivalent to a rational value with a unit denominator.
Language support
Built-in or core library:
- C++ has included support for compile-time rational arithmetic in the form of the contents of its standard library's
<ratio>
header since its 2011 revision. - Clojure can perform arithmetic on rational numbers and offers a literal form to represent them.
- Go provides rational numbers in the standard library, in the
math/big
package. - J provides rational numbers in the base language. For example,
1r3
is one-third. Rationals in J use arbitrary precision integers for both the numerator and denominator, allowing arbitrary precision non-integers. For instance,12683021339465478347804472r7322545784478161858100577
represents the square root of three to 50 decimal digits.[1] - Julia provides rational numbers with the rational operator,
//
. For example,6//9 == 2//3 && typeof(-4//9) == Rational{Int64}
.[2] - Haskell provides a
Rational
type, which is really an alias forRatio Integer
(Ratio
being a polymorphic type implementing rational numbers for anyIntegral
type of numerators and denominators). The fraction is constructed using the % operator.[3] - OCaml's Num library implements arbitrary-precision rational numbers.
- Perl:
Math::BigRat
core module implements arbitrary-precision rational numbers. Thebigrat
pragma can be used to turn on transparent BigRat support. - Raku: use by default
Rat
[4] type (rational numbers with limited-precision).FatRat
[5] data type implements arbitrary-precision rational numbers. - Python: The standard library includes a
Fraction
class in the modulefractions
.[6] - Ruby: native support using special syntax.
- Smalltalk represents rational numbers using a
Fraction
class in the formp/q
wherep
andq
are arbitrary size integers. Applying the arithmetic operations*
,+
,-
,/
, to fractions returns a reduced fraction.
With external libraries:
- For C & C++ there are projects like the GNU Multiple Precision Arithmetic Library.
- The Apache Commons Math library provides rational numbers for Java with its
Fraction
class.
Common Lisp
Common Lisp provides a numeric data type for arbitrarily sized rational numbers: RATIO.[7]
1/3
⇒ 1/3
The type of a rational number is RATIO
:
(type-of 1/3)
⇒ RATIO
Dividing two integers may return a rational number and the multiplication of a rational number may return an integer number:
(/ 6 8)
⇒ 3/4
(* 3/4 16)
⇒ 12
The numerator and denominator may be obtained using the homonymous functions, that reduce a rational to canonical form and compute the numerator or denominator of that form respectively:[8]
(numerator 12/16)
⇒ 3
(denominator 12/16)
⇒ 4
Computing with large integers returning a large rational number:
(/ (1- (expt 2 200)) (1- (expt 2 43)))
⇒ 1606938044258990275541962092341162602522202993782792835301375/8796093022207
Clojure
(print (+ 1/10 2/10)) ⇒ 3/10
Julia
julia> 1//10 + 2//10
3//10
Haskell
In module Data.Ratio
(1 % 10) + (2 % 10) ⇒ 3 % 10
Racket (PLT Scheme)
> (+ 1/10 2/10)
3/10
Raku
Raku provides Rat
type by default.
my $v = 0.2;
say "{$v} is {$v.^name} and has numerator {$v.numerator} and denominator {$v.denominator}";
# ⇒ 0.2 is Rat and has numerator 1 and denominator 5
say 0.1 + 0.2
# ⇒ 0.3
say (0.1 + 0.2 - 0.3).fmt("%.17f")
# ⇒ 0.00000000000000000
say 1 / (0.1 + 0.2 - 0.3)
# ⇒ Attempt to divide by zero when coercing Rational to Str
Ruby
Using special syntax in 2.1 or newer:
irb(main):001:0> puts 1/10r + 2/10r
3/10
=> nil
References
- ^ "Vocabulary/NumericPrecisions - J Wiki".
- ^ "Complex and Rational Numbers — Julia Language development documentation". docs.julialang.org. Archived from the original on 2012-07-15.
- ^ "The Haskell 98 Library Report: Rational Numbers".
- ^ "Class Rat".
- ^ "Class FatRat".
- ^ "Fractions — Rational numbers — Python 3.10.0 documentation".
- ^ Common Lisp HyperSpec: RATIO
- ^ Function NUMERATOR, DENOMINATOR at the Common Lisp HyperSpec
- Donald Knuth, The Art of Computer Programming, vol. 2. Addison-Wesley.
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