(07-18-2018 08:31 PM)BartDB Wrote:  

(07-18-2018 04:05 PM)Vtile Wrote:  (a/b)/(a/b) = 1
...
(2a/b)*(b/a) = 2

Not for all values of a and b.

More generally \(\frac{x}{x}= 1\: for \: all \:x\neq 0\)

The discontinuities must always be excluded. For example in equations with fractions, the points where the denominator(s) = 0 must be excluded (have exception handling).

(07-18-2018 08:49 PM)Vtile Wrote:  Yes you are correct many such cases done with proper accepted rules (edit. btw. again exception in definition), but iirc isn't NaN (as infinity many results/logic brakepoint/undefined infinity)* and +-inf (numerical infinity)* and +-0 (infinitesimals/zero as inverse of infinity) used for such to continue the analysis in logical way.

*as I see/understand them.

(07-19-2018 06:28 PM)Claudio L. Wrote:  

(07-18-2018 08:49 PM)Vtile Wrote:  Yes you are correct many such cases done with proper accepted rules (edit. btw. again exception in definition), but iirc isn't NaN (as infinity many results/logic brakepoint/undefined infinity)* and +-inf (numerical infinity)* and +-0 (infinitesimals/zero as inverse of infinity) used for such to continue the analysis in logical way.

*as I see/understand them.

In a way I tend to agree with you (I guess engineer's brains are wired like that), if we accept the "undefined" and stop the calculation there, then we don't need to define Infinity or NaN. Just throw a "Divide by zero" exception, end of discussion.
Thing is, some calculators (and most computers) have the option NOT to throw an error on infinity, and then you need a set of consistent definitions to deal with this, where the only goal is to keep the calculation going in a meaningful way as far as possible. Having said that, I don't necessarily agree with 0/0=1 but it's practical for some summations, etc. so it's a valid choice if all you want is to be able to keep going with those summations.
For other uses, that choice might be bad, so you cannot use it universally, hence 0/0=Undefined in most modern "universal" math packages and calculators.
Other definitions, as arbitrary as they might seem (like +/-0), are more universally usable, hence they made it into our calculators/computers.
In reality, once you get Infinity in any calculation, that thing propagates so fast to other results that you end up with matrices completely filled with NaN in just a couple of steps, and that's just as useless as stopping the calculation at the first exception.

(I must admit this is as stupid one could think to get while written in the middle of the night. There should be a timer when one can write here)