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Rational

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Rational[n, d]

represents the rational number n/d.

Notes When n and d are integers, Rational auto-reduces by gcd, normalises the sign onto the numerator, and collapses to an Integer when d == 1. Rationals propagate through Plus / Times exactly via GMP.

Examples (6)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (1)

In[1]:= Rational[15, 5]
Out[1]= 3

Applications (5)

In[2]:= Rational[6, 4]
Out[2]= 3/2

In[3]:= Rational[10, 2]
Out[3]= 5

In[4]:= Rational[-3, -9]
Out[4]= 1/3

In[5]:= 1/2 + 1/3 + 1/6
Out[5]= 1

In[6]:= Sum[1/k^2, {k, 1, 10}]
Out[6]= 1968329/1270080

Implementation notes

Rational[n, d] is the internal head for exact rationals. builtin_rational only fires for two integer arguments: it calls make_rational(n, d) to reduce to lowest terms with a positive denominator. If the input is already in canonical form (no reduction happened) it returns NULL so the structural Rational[n, d] is left as-is; otherwise it returns the reduced form (an EXPR_INTEGER when the denominator becomes 1). Division by zero emits Power::infy and returns ComplexInfinity (or Indeterminate for 0/0). Non-integer arguments return NULL.

  • Automatically simplifies to lowest terms (e.g. Rational[15, 5] evaluates to 3, Rational[2, 4] evaluates to Rational[1, 2]).
  • Returns Indeterminate when n and d are both 0 (e.g. Rational[0, 0]).
  • Returns ComplexInfinity when n is non-zero and d is 0 (e.g. Rational[1, 0]).

Attributes: Protected.

References

Notes & additional examples

Notes

Rational[n, d] represents the rational number n/d. With integer arguments it reduces to lowest terms, moves the sign to the numerator, and becomes an Integer when d divides n. The head of any non-integer fraction such as 1/2 is Rational.