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IntegerLength

Status: Stable

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

Description

IntegerLength[n]

gives the number of decimal digits in the integer n.

IntegerLength[n, b]

gives the number of base b digits in n.

Notes IntegerLength ignores the sign of n; IntegerLength\[0\] is 0.

Examples (9)

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

Basic examples (3)

In[1]:= IntegerLength[123456789]
Out[1]= 9

In[2]:= IntegerLength[100!, 2]
Out[2]= 525

In[3]:= Table[IntegerLength[100!, n], {n, 2, 20}]
Out[3]= {525, 332, 263, 227, 204, 187, 175, 166, 158, 152, 147, 142, 138, 135, 132, 129, 126, 124, 122}

Scope (2)

In[4]:= IntegerLength[] IntegerLength::argt: IntegerLength called with 0 arguments; 1 or 2 arguments are expected.

In[5]:= IntegerLength[1, 2, 3, 4] IntegerLength::argt: IntegerLength called with 4 arguments; 1 or 2 arguments are expected.

Applications (4)

In[6]:= IntegerLength[12345]
Out[6]= 5

In[7]:= IntegerLength[2^1000]
Out[7]= 302

In[8]:= IntegerLength[2^1000, 2]
Out[8]= 1001

In[9]:= IntegerLength[100!]
Out[9]= 158

Options & behaviour

Arity diagnostics (IntegerLength::argt). Wrong-arity calls emit the diagnostic and echo the call back unevaluated, matching Mathematica:

Non-integer-argument diagnostic (IntegerLength::int). A concrete non-integer numeric (Real, Rational, Complex) at position 1 or 2 emits the diagnostic and echoes the call back unevaluated; pure symbolic arguments flow through silently:

Packed arrays. The one-argument form runs on an int64 buffer and answers with a packed result.

Implementation notes

builtin_integerlength returns the number of base-b digits of |n| (default base 10). For bases <= 62 it uses GMP's mpz_sizeinbase, which is exact for power-of-two bases and at most one too large otherwise ā€” corrected by comparing |n| against base^(s-1) (intlen_count_digits). For arbitrary-precision bases it counts via repeated mpz_tdiv_q. IntegerLength[0] is 0. Validates arity (IntegerLength::argt), numeric non-integer n (::int), and base >= 2 (::ibase); symbolic n returns NULL.

  • Protected, Listable. Threads element-wise over a list of integers in either argument position, e.g. IntegerLength[{1, 10, 100}] or IntegerLength[8, {2, 3, 4}].
  • IntegerLength[0] is 0 in any base (zero has no significant digits).
  • IntegerLength[n, b] is effectively an efficient version of Floor[Log[b, |n|]] + 1 -- it never converts through floating-point and is exact for arbitrarily large n.
  • Works for both machine integers and bignums. The fast path uses GMP's mpz_sizeinbase for bases 2..62 (with an exact verification step for non-power-of-2 bases); arbitrary-precision bases fall back to repeated division.

Attributes: Listable, Protected.

References

Notes & additional examples

Notes

IntegerLength[n] returns the number of decimal digits of n, and IntegerLength[n, b] the number of base-b digits. Working at arbitrary precision, it answers questions that overflow fixed-width arithmetic: 2^1000 has 302 decimal digits but exactly 1001 binary digits, and 100! is a 158-digit number. The sign of n is ignored and IntegerLength[0] is 0.