Here Daniel mentions
... you pick any integer in [0, 2²⁴), and divide it by 2²⁴, then you can recover your original integer by multiplying the result again by 2²⁴. This works with 2²⁴ but not with 2²⁵ or any other larger number.
But when I tried
>>> b = np.divide(1, 2**63, dtype=np.float32)
>>> b*2**63
1.0
Although it isn't working for 2⁶⁴, but I'm left wondering why it's working for all the exponents from 24 to 63. And moreover if it's unique to numpy only.
In the context that passage is in, it is not saying that an integer value cannot be divided by 225 or 263 and then multiplied to restore the original value. It is saying that this will not work to create an unbiased distribution of numbers.
The text leaves some things not explicitly stated, but I suspect it is discussing taking a value of integer type, converting it to IEEE-754 single-precision, and then dividing it. This will not work for factors larger than 224 because the conversion from integer type to IEEE-754 single-precision will have to round the number.
For example, for 232, all numbers from 0 to 16,777,215 will convert to themselves with no error, and then dividing by 232 will produce a unique floating-point number for each. But both 16,777,216 and 16,777,217 will convert to 16,777,216, and then dividing by 232 will produce the same number for them (1/256). All numbers from 2,147,483,520 to 2,147,483,776 will map to 2,147,483,648, which then produces ½, so that is 257 numbers mapping to one floating-point number. But all the numbers from 2,147,483,777 to 2,147,484,031 map to 2,147,483,904. So this one has 255 numbers mapping to it. (The difference is due to the round-to-nearest-ties-to-even rule.) At the high end, the 129 numbers from 4,294,967,168 to 4,294,967,296 map to 4,294,967,296, for which dividing produces 1, which is out of the desired half-open interval, [0, 1).
On the other hand, if we use integers from 0 to 16,777,215 (224−1), there is no rounding, and each result maps from exactly one starting number and stays within the interval.
Note that “significand“ is the preferred term for the fraction portion of a floating-point representation. “Mantissa” is an old word for the fraction portion of a logarithm. Significands are linear. Mantissas are logarithmic. And the significand of the IEEE-754 single-precision format has 24 bits, not 23. The primary field used to encode the significand has 23 bits, but the exponent field provides another bit.