# Why do many floating point numbers appear compact in base16 and base32 representation? (IEEE 754)

Consider 3 randomly chosen floating point values and their base conversions:

Floating point        Base-16    Base-15
0.5898704528808594  = 0.9701c    0.8cac2d1d9bd5e9
0.21180343627929688 = 0.3638c    0.329c837b2b291b
0.8631706237792969  = 0.dcf8c    0.ce3302d46d553


The same issue occurs with Base-24, and 32: The representation of the float is very compact.

These were found by searching random IEE754 binary64 double numbers, isolating those with the criteria of less than 6 hex digits. To reduce the search, I forced the lower 33 bits of the mantissa to 0 prior to converting to double. I'm not entirely sure why, but this only seems to happen if the lower 33 bits of the mantissa are zero. I ended up with a list of only 1048225 that met the criteria after 2.2 billion numbers, which is strikingly close to 2^20.

In fraction form, these 3 numbers happen to have a denominator of 262144. Others I've found have denominators of 1048576, 65536, 262144, etc. - the denominator always seems to be a power of 2 when this happens.

Is there an explanation for this?