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MantissaExponent

Status: Stable

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

Description

MantissaExponent[x] gives a list {m, e} containing the mantissa and exponent of the real number x, such that x = m * 10^e and 1/10 <= |m| < 1 (or m = 0 when x = 0).
MantissaExponent[x, b] gives the base-b mantissa and exponent; the mantissa then lies in 1/b <= |m| < 1.
Works for exact (Integer, Rational) and approximate (Real, MPFR) numeric inputs.  For exact inputs the mantissa is an exact Rational; for inexact inputs the mantissa carries the same precision as x.  Currently only integer bases >= 2 are supported.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= MantissaExponent[3.4 10^30]
Out[1]= {0.34, 31}

In[2]:= MantissaExponent[456.1414]
Out[2]= {0.456141, 3}

In[3]:= MantissaExponent[123451]
Out[3]= {123451/1000000, 6}

In[4]:= MantissaExponent[1027, 2]
Out[4]= {1027/2048, 11}

In[5]:= MantissaExponent[2^100, 2]
Out[5]= {1/2, 101}

In[6]:= MantissaExponent[N[Pi, 30]]
Out[6]= {0.3141592653589793238462643383278, 1}

In[7]:= MantissaExponent[-3/2]
Out[7]= {-3/20, 1}

Implementation notes

builtin_mantissa_exponent returns {m, e} with x = m * b^e and 1/b <= |m| < 1 (default base 10). It classifies the input via rd_classify. For exact inputs (Integer/BigInt/Rational) it works in a signed mpq_t: it finds the natural exponent e with rd_rational_natural_exp, scales numerator or denominator by b^|e|, canonicalises, and emits an exact Rational mantissa. For machine reals it computes e = floor(log|x|/log b) + 1 then m = x / b^e with off-by-one corrections for log double-rounding; the MPFR path mirrors this at the input precision. MantissaExponent[0] is {0, 0}. Complex inputs emit MantissaExponent::realx; base < 2 emits ::ibase; non-integer bases are left unevaluated (only integer bases supported).

  • 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]:= MantissaExponent[123.45]
Out[1]= {0.12345, 3}

In[1]:= MantissaExponent[7/3]
Out[1]= {7/30, 1}

In[1]:= MantissaExponent[1024, 2]
Out[1]= {1/2, 11}

In[1]:= MantissaExponent[N[Pi, 30]]
Out[1]= {0.3141592653589793238462643383278, 1}

In[1]:= MantissaExponent[255, 16]
Out[1]= {255/256, 2}

Notes

MantissaExponent[x] returns {m, e} with x = m * 10^e and 1/10 <= |m| < 1 (or {0, 0} when x is 0). MantissaExponent[x, b] uses base b, so 1/b <= |m| < 1. Exact inputs keep an exact Rational mantissa (7/3 -> {7/30, 1}); inexact inputs keep their full working precision (the N[Pi, 30] mantissa carries 30 digits). Only integer bases >= 2 are currently supported.