They meant using the equal operator, because floating point is inexact and can produce different representations for the same number depending on how it was obtained. Greater and less than are fine. Equal is usually implemented by seeing if a number fits inside a tight range.
Every finite floating point value precisely represents an exact quantity. Arithmetic isn't lossless though. This isn't a special property of floats. Decimals behave the same way.
Second point: Combing FPs isn't lossless, across most/all arithmetic. Clearly true, the root of our discussion
Third point: decimal arithmetic is lossy too. Strongly disagree. The system of representation is what is lossy - floating point. Arithmetic (between two decimal numbers) is clearly not lossy in and of itself, only as implemented with computers.
Decimal is only lossless if you track an indefinitely-growing suffix to your number with a big bar over it to indicate "repeating". It's basically working with rational numbers but more painful.
Decimal numbers with any limit on digits, even a hundred, are lossy.
Decimal numbers as used by humans outside of computers have limits on digits, so they are lossy.
Maybe my point is uninteresting, but I feel you haven't really understood it.
Combining two FP numbers with arithmetic leads to possible errors that you can't represent with FP on every operation.
Combining two decimal numbers with non-division arithmetic never leads to that case. With division, sure, things are bad, but that's more of an exception than a rule to me.
This is because decimal numbers don't really come with any inherent limit on digits, and it's a bit strange to add a clause to the other's claims before making a counterargument.
Well in part I didn't understand your point because you didn't mention division as an exception.
But even then, multiplication causes an explosion in number length if you do it repeatedly.
When you specifically talk about numbers not being in computers, I think it's fair to talk about digit limits. Most real-world use of decimals is done with less precision than the 16 digits we default to in computers. Let alone growing to 50, 100, 200, etc as you keep multiplying numbers together to perform some kind of analysis. Nobody uses decimal like that. Real-world decimal is lossy for a large swath of multiplication.
I agree that if you're doing something like just adding numbers repeatedly in decimal, and those numbers have no repeating digits, then you have a nice purity of never losing precision. That's worth something. But on the other hand if you started with the same kind of numbers in floating point, let's say about 9 digits long, you could still add a million of them without losing precision.
And nobody has said anything about irrational numbers as you dismissed in your other comment.
So in summary: decimal division, usually lossy; decimal multiplication, usually lossy; decimal addition and subtraction, lossless but with the same kind of source numbers FP is usually lossless too
> Combining two decimal numbers with non-division arithmetic never leads to that case.
Most types are subject to overflow that has similar effects. Most of the FP error people encounter is actually division error. For example, the constant 0.1 contains an implicit division. The "tenths" place is defined by division by 10. I think that almost all perceptions of floating point lossiness come from this fact.
Likewise, bit-limited integers are lossy as they are incomplete representations of ℤ, and are not reciprocal with division. The point being that, just like floating-point representations of ℝ, division will always be lossy in the general case unless you choose a symbolic algebra instead of numeric.
that's slightly incoherent and reflective of the same confusion i identified; there is strictly speaking, no such thing as a decimal number, only a decimal numeral. normally the difference is subtle, but in this case it's the essence of the issue
'decimal' means 'base 10'
numbers, the abstract entities we do arithmetic on, are not decimal or in any other base; it is correct that arithmetic on them is not in any sense lossy
decimal is a system of representation that represents these abstract numbers as finite strings of digits; these are called 'numerals'. it can represent any integer, but only some fractions, those whose denominator has no prime factors other than 2 or 5. such fractions arise non-lossily as the result of the arithmetic operator of division when, for example, the dividend is 3. representing these in decimal requires rounding them, so decimal is lossy
binary floating point is lossy in the same way, with the additional limitation to only being able to represent fractions whose denominator is a power of 2, whose numerator does not have too many significant bits (53 most often), and which are not too positive, too negative, or too small
there are other systems of represention for numbers that are not lossy in this way. for example, improper fractions, mixed numbers, finite continued fractions, and decimal augmented with overbar to indicate repeated digits
