> Are you saying that my students should memorize which numbers are actual floats and which are not?

Yes.

They shouldn’t “memorize” this per se, but it should take them only a few seconds to work out in their head.

Your students should be able to figure out if a computation is exact or not, because they should understand binary representation of numbers.

Tell them that they can only store integer powers of 2 and their sums exactly. 2^0 == 1. 2^-2 == .25. Then say it's the same with base 10. 10^-1 == 0.1. 1/9 isn't a power of 10, you you can't have an exact representation.

If they were taught what was representable and why they’d learn it quickly. And those that forget details later know to chase it down again if they need it. Making it voodoo hides that it’s learnable, deterministic, and useful to understand.

(Another small note.... 1.25 * 0.1 is not representable because 0.1 is not representable, so that doesn't divide by 10)

1.25 = 2^0 + 2^-2, so is representable.

0.125 = 2^-3, so is representable

1.25 / 10.0 = 0.125 so is representable. 10.0 = 2^3 + 2^1.

1.25 * 0.1 is not representable, because 0.1 is not representable, and those low order bits show up in the multiplication

0.1 is of course a real number, but let A \in R the set of actual numbers... (/s)

That's essentially what you already do for integer arithmetic.

Yes. A computer scientist should know how numbers are represented and not expect non-representable numbers in that format to be representable.

0.1 is just as non-representable in floating point as is pi as is 100^100 in a 32 bit integer.

Terminating dyadic rationals (up to limits based on float size) are the representable values.