> > Surely it would be more sensible to define x/0 to be infinity for all real cases of x.
> >
>
> x/0=0 would lead to 0*0=x, which would mean that if you multiply 0 by 0, you'd get any real value that you want. That's nice, but I think it's not the way anything works.
>
> x/0=Infinity would mean that a*0 is not always 0, which also doesn't work.
>
> > winter"If it matters"mute
>
> Fuzzpilz

In both cases, extending division leads to the problem that multiplication and division aren't inverses (they can't be since 0*0 wouldn't be uniquely determined).

However, the rationale behind defining x/0 = ComplexInfinity is that in the complex plane, this makes division a continuous function of the denominator (with fixed non-zero numerator) at *all* numbers, including zero. This is pretty important if you do calculations in the complex plane. If we defined x/0 = 0, you wouldn't have continuity at zero.

In more explicit terms, if {a0, a1, a2, ...} is a sequence which converges to zero, then for any fixed numerator x, we have:

lim n->inf (x/an) = x/(lim n->inf (an)),

since the left-hand side is ComplexInfinity, and the right hand-side is x/0. [Read a0, a1, a2, an as a-subscript-0, etc.]

Note that you can't have continuity of division as a function of either the numerator or the denominator at *all* numbers, since you'd be forced to have 0/0 = 0 for denominator continuity, and 0/0 = ComplexInfinity for numerator continuity.

In either case, defining x/0 = 0 or x/0 = ComplexInfinity, you can't fix the essential problem at 0/0. It's best to just leave 0/0 undefined.

[Aside note on other infinity calculations: in real analysis, it's convenient for abstract integration purposes to define 0 * Infinity = 0, and to let Infinity - Infinity stay undefined. It makes the formulas come out simpler.]