Skip to content

RealDigits

Status: Stable

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

Description

RealDigits[x] gives a list {digits, exp} of the digits in the approximate real number x together with the exponent such that the first digit is the coefficient of 10^(exp - 1).
RealDigits[x, b] gives base-b digits.
RealDigits[x, b, len] gives len digits.
RealDigits[x, b, len, n] gives len digits starting from the coefficient of b^n.
For rationals with non-terminating expansions the digit list ends in a nested list of the recurring block.  For inexact (machine or MPFR) reals, digits beyond the available precision are returned as Indeterminate.  The sign of x is discarded.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= RealDigits[123.55555]
Out[1]= {{1, 2, 3, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0}, 3}

In[2]:= RealDigits[Pi, 10, 25]
Out[2]= {{3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3}, 1}

In[3]:= RealDigits[19/7]
Out[3]= {{2, {7, 1, 4, 2, 8, 5}}, 1}

In[4]:= RealDigits[5.635, 10, 20]
Out[4]= {{5, 6, 3, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Indeterminate, Indeterminate, Indeterminate, Indeterminate}, 1}

In[5]:= RealDigits[Pi, 10, 20, -5]
Out[5]= {{9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3}, -4}

In[6]:= RealDigits[1.234, 2, 15]
Out[6]= {{1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1}, 1}

Implementation notes

Algorithm. builtin_realdigits returns the digit list of a real number, in the Mathematica {digits, exponent} form. It accepts RealDigits[x], RealDigits[x, b], RealDigits[x, b, len], RealDigits[x, b, len, p] (1–4 args; wrong count emits RealDigits::argb). x is classified by rd_classify; concrete non-real Complex input emits RealDigits::realx, and symbolic constants (Pi, E, …) are numericalised only once enough precision context (base and length) is known. The base defaults to 10, must be an integer ≥ 2 (RealDigits::ibase otherwise) and fit in unsigned long. Digits are extracted by repeated scaled-floor / MPFR shifting in the requested base, honouring the optional length and starting-position arguments.

Data structures. GMP mpz_t for the base and integer parts; MPFR for the fractional digit extraction when built. Output is a List of digits paired with an integer exponent.

  • Protected, Listable. Threads over lists in any argument position.
  • Works for Integer, BigInt, Rational, machine Real, and (under

Attributes: Listable, 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]:= RealDigits[123.456]
Out[1]= {{1, 2, 3, 4, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 3}

For an exact rational with a non-terminating decimal expansion, the digit list ends in a nested list giving the recurring block — here the period-6 cycle of 1/7, and the mixed 22/7 = 3.142857142857…:

In[1]:= RealDigits[1/7]
Out[1]= {{{1, 4, 2, 8, 5, 7}}, 0}

In[2]:= RealDigits[22/7]
Out[2]= {{3, {1, 4, 2, 8, 5, 7}}, 1}

High-precision constants expose their digits directly. Thirty digits of π from a 30-digit MPFR value, and the first ten significant base-10 digits of a 40-digit π:

In[1]:= RealDigits[N[Pi, 30]]
Out[1]= {{3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 8}, 1}

In[2]:= RealDigits[N[Pi, 40], 10, 10]
Out[2]= {{3, 1, 4, 1, 5, 9, 2, 6, 5, 3}, 1}

RealDigits also works in other bases — the binary expansion of 255 is eight ones:

In[1]:= RealDigits[255, 2]
Out[1]= {{1, 1, 1, 1, 1, 1, 1, 1}, 8}

Notes

RealDigits[x] gives {digits, exp} where the first digit is the coefficient of 10^(exp - 1). RealDigits[x, b] uses base b; RealDigits[x, b, len] returns len digits; RealDigits[x, b, len, n] starts from the coefficient of b^n. For rationals with non-terminating expansions the digit list ends in a nested list of the recurring block. For inexact reals, digits beyond the available precision are returned as Indeterminate. The sign of x is discarded.