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Rationalize

Status: Stable

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

Description

Rationalize[x]
    converts an approximate number x to a nearby rational with small denominator.
Rationalize[x, dx]
    yields the rational number with smallest denominator that lies within dx of x.

Rationalize[x] yields x unchanged if there is no rational number close enough to x to satisfy |p/q - x| < c/q^2, with c = 10^-4.
Rationalize[x, dx] works with exact numbers x: the value is first numericalised, then rationalised.
Rationalize[x, 0] forces conversion of any inexact number x to rational form, using a tolerance derived from the precision of x.
Rationalize threads over compound expressions and Complex[re, im], so e.g. Rationalize[1.2 + 6.7 x] gives 6/5 + (67 x)/10.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_rationalize dispatches on arity into internal_rationalize_expr with one of three modes. Rationalize[x] (RATIONALIZE_DEFAULT) walks the convergents of the continued-fraction expansion of x and returns the first p/q whose error satisfies |p/q − x| < 10^-4 / q^2 (Mathematica's threshold); if none qualify it returns the value unchanged. Rationalize[x, dx] with dx > 0 (RATIONALIZE_TOLERANCE) solves the "simplest rational in an interval" problem by the Stern–Brocot / continued-fraction descent: it extracts integer parts of the lower and upper bounds until they diverge, at which point the smallest integer in the lower interval gives the minimal-denominator p/q with |p/q − x| ≤ dx. dx == 0 (RATIONALIZE_ZERO) is treated as a thin ulp-derived tolerance so e.g. Rationalize[N[Pi], 0] yields 245850922/78256779 rather than the bit-exact dyadic. GMP is used throughout so denominators exceeding int64_t are handled. A negative or non-numeric dx returns NULL (left unevaluated).

Threading. Rationalize descends through expression structure: a compound subexpression that is itself NumericQ (e.g. Sqrt[2], Exp[Sqrt[2]]) is numericalised via the N pipeline and then rationalised end-to-end; otherwise the head and arguments are recursively rationalised and re-evaluated so Rationalize[1.2 + 6.7 x] becomes 6/5 + (67 x)/10.

Attributes: Protected.

Implementation status

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

References

Notes & additional examples

Worked examples

In[1]:= Rationalize[0.5]
Out[1]= 1/2

In[2]:= Rationalize[N[Pi], 10^-4]
Out[2]= 333/106

In[3]:= Rationalize[1.2 + 6.7 x]
Out[3]= 6/5 + 67/10 x

Tightening the tolerance recovers the continued-fraction convergents. With the golden ratio it returns a ratio of consecutive Fibonacci numbers, and with π the celebrated convergent 355/113 and beyond:

In[1]:= Rationalize[N[GoldenRatio], 10^-6]
Out[1]= 1597/987

In[2]:= Rationalize[N[Pi, 40], 10^-10]
Out[2]= 312689/99532

Rationalize[x, 0] forces conversion of an inexact number using a tolerance derived from its precision:

In[1]:= Rationalize[N[E], 0]
Out[1]= 325368125/119696244

Notes

Rationalize[x] finds a nearby rational with small denominator (within c/q^2, c = 10^-4); the two-argument form Rationalize[x, dx] returns the smallest-denominator rational within dx of x. It threads over compound expressions.