Earlier quoted context omitted.
What if a is negative?
Same. Unless you want to differentiate -0 and +0 (which make it more complicated), you can not distinguish infinity from negative infinity.
John Conway can
1/0 = 0 (2018)
131–140 of 245 posts
Re: 1/0 = 0 (2018)
#132Earlier quoted context omitted.
There's no "if" in the division operation. Division is not defined for b=0. a/0 is a nonsensical quantity because the zero directly contradicts the definition of division. maybe someday there will be a revelation where somebody proposes that it's a new class of numbers we've never considered before like how (1-1), (0-1) and sqrt(-1) used to be nonsensical values to past mathematicians. For now it's not defined.
Division by zero is perfectly well defined in floating point. x/0 = INF and INF*0 = NaN. That means b*(a/b) != a if b = 0. It's true that it's not defined for integer types, but that wouldn't make a = b*(a/b) true for them either. It's also common to define x/0 = infinity in the extended real numbers that floating point models.
The definitions in the floating point standard make much more sense when you look to 0/INF as "something so close to/far from 0 we cannot represent it", rather than the actual concepts of 0 and infinity.
Re: 1/0 = 0 (2018)
#133Consistency depends on your set of axioms. If you are willing to give up various nice properties of division, then you can obviously extend it however you like.
My gripe with arbitrary choices like this is that it pushes complexity from your proof's initial conditions ("we assume x != 0") into the body of your proof (every time you use division now you've split your proof into two cases). The former is a linear addition of complexity to a proof, whereas the latter can grow exponentially.
Of course, nothing is stopping you from using an initial condition anyway to avoid the case splitting, but if you were going to do that why mess with division in the first place?
Re: 1/0 = 0 (2018)
#134Earlier quoted context omitted.
NaN is a valid float, so are infinities
They're valid according to a spec that doesn't mean I want one showing up when I'm trying to calculate the area of a semicircle or whatever. In the context of getting one by surprise in simple arithmetic they are approximately as bad as zero. Either way you have to decide how to handle it and there are tradeoffs of different approaches, as the article discusses. It's not about someone just being ignorant of basic mat…
>In the context of getting one by surprise in simple arithmetic they are approximately as bad as zero.
I don't think so, because getting 0 in a larger expression might yield a result that looks plausible, leading to hidden bugs. Inf and NaN both are good because they necessarily propagate all the way up to the end result, making it obvious that something went wrong.
Re: 1/0 = 0 (2018)
#135Disagreed. 1/0 should be infinity, and computers should be able to handle these concepts. Just look into what is 1/0.00000000000[etc]1. And no is not an error, you find out with a very real and tangible example, when you are developing a 3D engine and you want to make the camera to look at vector [ 0, 0, 0 ]. Quick resume: You can't, you need to force add a slight displacement so you can skip this silly error.
Re: 1/0 = 0 (2018)
#136Earlier quoted context omitted.
you can't divide the apples among 0 people and then claim to still have them, because in that case you would have divided them among 1 people
Like everything in life, it depends... For example: Storage has 5 items that need to be processed. 5 items need to be split equaly between available processes. There are currently 0 available processes so 5 / 0 = 0 items to be processed is more correct than either 5 or Nan or infinity.
Your example is quite vague (e.g. are we dealing with an integer number of items and processes?) and in general if something looks kinda like a division it doesn't mean it is exactly division. Just like in math, we have the power to simply say: if COND -> divide normally, else -> do something else.
Re: 1/0 = 0 (2018)
#137My head-canon with dividing by zero is that 1/0 = undefined and 1/-0 = -undefined, and that's where I leave it because anything less funny than that seems like an impractical answer.
0 / 0 = Any number non-zero / 0 = No number
0/0 might be a specific number. You have to go and use l'hopitals rule to work out what it might be.
Re: 1/0 = 0 (2018)
#138Earlier quoted context omitted.
Reusing symbols like +, *, or / to define operations that aren't the + or the / you're used to is pretty common in math. It's just notation. At the end of the day, the / that we have in programming has the same problem as this article's /, almost all programming languages will return 5/2 = 2 when dividing integers, even though 2 * 2 is not 5! Division is not defined for all integers, but it's just convenient to exten…
> Reusing symbols like +, *, or / to define operations that aren't the + or the / you're used to is pretty common in math. It's just notation. Reusing symbols in a different context is pretty common; taking a symbol that is already broadly used in a specific way (in this case, that `a/b` is defined for elements in a field as multiplying `a` by the multiplicative inverse of `b`) is poor form and, frankly, a disingenuo…
I am a professor for algebra at a research university. I make a point out of teaching my students that `a/b` is NOT the same as multiplying `a` by the multiplicative inverse of `b`.
The standard example is that we have a well-defined and useful notion of division in the ring Z/nZ for n any positive integer even in cases were we "divide" by an element that has no multiplicative inverse. Easy example: take n=8 then you can "divide" 4+nZ by 2+nZ just fine (and in fact turn Z/nZ into a Euclidean ring), even though 2+nZ is not a unit, i.e. admits no multiplicative inverse.
Re: 1/0 = 0 (2018)
#139Sounds legit, infinity is singular and so is 0. I think one problem is also that division isn't the only mathematical operation which can produce dubious results. E.g. sqrt(x), arctan(x) which have multiple branches which is why there is often a separate arctan2(x, y) to select the correct branch. Oh well and then there's just addition which silently overflows in almost every programming language.
Without arbitrary precision numerics and functions which aren't explicit about corner cases it's always a simplification. However performance-/code-wise this is usually not feasible.
Re: 1/0 = 0 (2018)
#140Earlier quoted context omitted.
There's no "if" in the division operation. Division is not defined for b=0. a/0 is a nonsensical quantity because the zero directly contradicts the definition of division. maybe someday there will be a revelation where somebody proposes that it's a new class of numbers we've never considered before like how (1-1), (0-1) and sqrt(-1) used to be nonsensical values to past mathematicians. For now it's not defined.
Division by zero is perfectly well defined in floating point. x/0 = INF and INF*0 = NaN. That means b*(a/b) != a if b = 0. It's true that it's not defined for integer types, but that wouldn't make a = b*(a/b) true for them either. It's also common to define x/0 = infinity in the extended real numbers that floating point models.
The limit 1/x as x goes to zero diverges to plus or minus infinity depending on whether you approach from the right or the left. IEEE 754 uses a signed zero, so defining 1/+0 = +INF and 1/-0 = -INF makes sense. If you do not have a signed zero, arbitrarily picking either plus or minus infinity makes much less sense and picking their "average" zero seems more sensible. So x/0 is not actually +INF - even if you meant +0 and we forget about -0 - it is +INF or -INF depending on the sign of x and NaN if x is +0 or -0.