19! = 121645100408831984

Factorial numbers are conspicuous for their characteristic large groups of trailing zeros. So I was puzzled when I recently stumbled upon the following using a Google Docs spreadsheet:

Of course 19! is not 121645100408831984, but 121645100408832000, off by 16. The puzzlement continued when I wrote:

Though the values at B2 and B3 are displayed differently, they compare the same. What's going on here? Actually, B3 is not holding a number but the string "121645100408832000", hence its left alignment --numbers are displayed right-aligned by default. Once we force "121645100408832000" to be internally translated to a number the glitch reappears; it suffices to enter the value as "=121645100408832000":

After thinking a bit about it I realized the value 19! has 57 binary digits, and this rang a bell: if values were stored in double precision IEEE 754 format, with fraction width of 52 bits, not all integers greater than 2⁵³ would have exact representation, and in fact 2⁵³ + 1 would be the first non-representable integer:

The figure above seems to confirm the hypothesis. For 2²⁴ +1 the representation is exact, which seems to rule out single precision IEEE 754 with its 23 bit fraction. In fact, I tested all the values 2ⁿ + 1 for n = 1,...,54, and the representation is exact except for n = 54. Further confirmation of the double precision IEEE 754 hypothesis comes from the following:

Does this explain it all? Well, not entirely. The binary representation of 19! is

110110000001010111001001100000110100010010000000000000000.

Thanks to the 16 trailing zeros, this is exactly representable in double precision IEEE 754 as:

19! = s·2^{e − 1023}·1.f,
s = 0,
e = 10000110111₂ = 1079 = 56 + 1023,
f = 1011000000101011100100110000011010001001000000000000₂.

So there must be additional problems apart from the loss of precision due to the use of floating point. We will investigate the issue further in a later entry.