On Tue, 24 May 2016 12:29 am, Ben Bacarisse wrote: > Ian Kelly <ian.g.ke...@gmail.com> writes: > >> On Mon, May 23, 2016 at 2:09 AM, Steven D'Aprano >> <steve+comp.lang.pyt...@pearwood.info> wrote: >>> Are you saying that the Egyptians, Babylonians and Greeks didn't know >>> how to work with fractions? >>> >>> http://mathworld.wolfram.com/EgyptianFraction.html >>> >>> http://nrich.maths.org/2515 >>> >>> Okay, it's not quite 4000 years ago. Sometimes my historical sense of >>> the distant past is a tad inaccurate. Shall we say 2000 years instead? >> >> Those links give dates of 1650 BC and 1800 BC respectively, so I'd say >> your initial guess was closer. > > Right, but this is to miss the point. Let's say that 4000 years have > defined 1/3 to be one third, but Python 3 (as do many programming > languages) defines 1/3 to be something very very very very close to one > third, and *that* idea is very very very very new! It's unfortunate > that the example in this thread does not illustrate the main problem of > shifting to binary floating point, because 1/2 happens to be exactly > representable.
That's not really the point. I acknowledge that floats do not represent all rational numbers (a/b) exactly. Neither do decimal floats -- most school children will learn that 0.333333333333 is not 1/3 exactly, and anyone who has used a calculator will experience calculations that give (say) 0.999999999 or 1.0000000001 instead of 1. And you know what? *People cope.* For all the weirdness of floating point, for all the rounding errors and violations of mathematical properties, floating point maths is *still* the best way to perform numerical calculations for many purposes. In fact, even IEEE-754 arithmetic, which is correctly rounded and therefore introduces the least possible rounding error, is sometimes "too good" -- people often turn to GPUs instead of CPUs for less accurate but faster bulk calculations. The point is, most people wouldn't really care that much whether 1/3 returned a binary 0.3333333333, or decimal 0.3333333333, or an exact rational fraction 1/3. Most people wouldn't care too much if they got 32-bits of precision or 64, or 10 decimal places or 18. When cutting (say) a sheet of paper into three equal pieces, they are unlikely to be able to measure and cut with an accuracy better than 1/2 of a millimeter, so 15 decimal places is overkill. But one thing is certain: very few people, Jon Ribbens being one of them, expects 1/3 to return 0. And that is why Python changed the meaning of the / operator: because using it for integer division was deeply unpopular and a bug magnet. -- Steven -- https://mail.python.org/mailman/listinfo/python-list
