[PaulHoule]

That is overthinking it horribly.

The problem is that the user wants to write 1/10 and 2/20 and 3/10 but those numbers aren't really in the binary system.

The user gets some numbers (let's call them A, B and C) that aren't the same but they fool people at first because they not only deserialize as 0.1 but the they also serialize from 0.1. Trouble is that A + B != C but some other number.

Excel tries to hide it but the real answer is to keep the exponent in base 10 if you plan to read and write numbers like 137.036 or 9.1E-31. How the mantissa is doesn't matter, it could be base 7 for all I care -- it is just an integer.

Interval math is for much tougher problems like recursion of

  k * x * (1-x)

is easily proven to have periodic orbits of infinitely long period, but if you are using 32-bit floats you can't have a period longer than 4 billion. That kind of qualitatively difference means that there's no scientific value in iterating that function with floats, although you can do accurate grid samples with interval arithmetic.