1/0 = 0 (2018)
hillelwayne.com
1/0 = 0 (2018)
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Re: 1/0 = 0 (2018)
#2I'd agree to some kind of 1//0=0 for ints; but for floats you'll take 1/0=inf from my cold, dead hands.
Re: 1/0 = 0 (2018)
#3As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking.
I might be wrong but I don't think it was addressed in the article either.
Re: 1/0 = 0 (2018)
#4https://xenaproject.wordpress.com/2020/07/05/division-by-zer...
explains Lean's behavior. Basically, you use a goofy alternate definition of division (and sqrt, and more), and to compensate you have to assume (or prove based on assumptions) that the things you will divide by are never zero.
Hillel's pedantry is ill-taken, though, because he starts off with a false accusation that the headline tweet was insulting anyone.
Also, 1/0=0" is sound only if you change the field axiom.of division, which is fine, but quite rather hiding the ball. If you add " 1/0=0" as an axiom to the usual field axioms, you do get an unsound system.
Re: 1/0 = 0 (2018)
#5Honestly this hurts my head but Hillel is inevitably correct. You can define an explicitly undefined operation to do whatever you like. But what’s the point? There’s no new mathematics you can do with it, no existing behaviours you can extend like this. Normally, when you divide by a small number, you get a large number. Now for some reason it goes through zero. Why not five? Why not seven?
Just because it’s formally consistent doesn’t mean it isn’t dumb.
Re: 1/0 = 0 (2018)
#6It’s unexpected and that makes it dangerous.
Re: 1/0 = 0 (2018)
#7Q on this post:
Is the field rule "Every element Except Zero has ... " (the 9th rule) defined with respect to the additive identity "zero" or the magical other undefined "Zero" that is the number we're all familiar with?
If so, how weirdly arbitrary that the additive zero is omitted for all multiplicative inverse definitions. (At least it seems to me). I always figured this was a consequence of our number systems, not of all fields.
Re: 1/0 = 0 (2018)
#8As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking. I might be wrong but I don't think it was addressed in the article either.
It's fine. Infinity isn't a real number, so 1/x isn't continuous at 0, so it doesn't matter what the value of 1/0 is. All your open sets still behave the way you expect. Whether you choose "this function is undefined here" vs "it's impossible to ever reach the value of this function at this value, under any assumptions I'll ever care about" is purely a matter of convenience.
Re: 1/0 = 0 (2018)
#9As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking. I might be wrong but I don't think it was addressed in the article either.
There's a great Radiolab episode[0] that talks about divide by zero in perhaps more conceptual terms.
KARIM ANI: If you take 10 and divide it by 10, you get one. 10 divided by five is two. 10 divided by half is 20. The smaller the number on the bottom, the number that you're dividing by, the larger the result. And so by that reasoning ...
LULU: If you divide by zero, the smallest nothingness number we can conceive of, then your answer ...
KARIM ANI: Would be infinity.
LULU: Why isn't it infinity? Infinity feels like a great answer.
KARIM ANI: Because infinity in mathematics isn't actually a number, it's a direction. It's a direction that we can move towards, but it isn't a destination that we can get to. And the reason is because if you allow for infinity then you get really weird results. For instance, infinity plus zero is ...
LATIF: Infinity.
KARIM ANI: Infinity plus two is infinity. Infinity plus three is infinity. And what that would suggest is zero is equal to one, is equal to two, is equal to three, is equal to four ...
STEVE STROGATZ: And that would break math as we know it. Because then, as your friend says, all numbers would become the same number.
Re: 1/0 = 0 (2018)
#10As long as lim(1/x)_x->0 = inf, 1/0 = 0 doesn't make a whole lot of sense, mathematically speaking. I might be wrong but I don't think it was addressed in the article either.
I was also looking for this. And would like to add: lim(-1/x)_x -> 0 = -inf
That is (in my opinion) the whole point why it is actually undefined. On one side of the y-axis it goes to infinity, on the other to minus infinity. I don't see a solution to this and therefore always have accepted that it is undefined.