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Comment by andyjohnson0 | Hacker News Reader

From the article:

> There is a duality between zero and infinity, expressed in the elementary identity 1/0 = ∞.

I understand what they're saying here - but I was always under the impression that, rather than yielding infinity, division by zero was undefined. Wikipedia [1] says that this is true "In ordinary arithmetic", but goes on to say that it is "sometimes useful" to think of a "formal calculation" involving division by zero as evaluating to infinity. A formal calculation seems to be one that ignores whether the result is well defined.

Would anyone care to comment on whether the mathematical perspective above carries over into the physics domain?

[1] https://en.wikipedia.org/wiki/Division_by_zero

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Division by 0 is not possible in fields [1], which are the algebraic structure most commonly associated with “ordinary arithmetic.” For example, the real numbers comprise an ordered and complete field. You cannot divide by 0 in a field essentially by definition, so it’s undefined.

However, you can define meaningful algebras in which division by 0 is not only trivially allowed, but also equal to infinity. This is possible by extending the complexes, for example. More generally, see [2].

Note I’m speaking strictly of the math here. I can’t comment on this usage in physics specifically. But there’s no issue with defining 1/0 = ∞ if the rest of the theory remains fundamentally consistent.

______

1. https://en.m.wikipedia.org/wiki/Field_(mathematics)

2. https://en.m.wikipedia.org/wiki/Wheel_theory

I disagree, only because infinity is not an element of the real or the complex numbers. You can make extended number systems (the hyperreals, for instance) but it isn't commonly done. Infinity as a notation in the limit is not the same thing as infinity as an element of a set.

Can you clarify what you're disagreeing with? I don't think I stated anything controversial.

No, ∞ is not an element of the real or complex fields, that's correct. But it is an element of the extended real and complex number systems, according to the following rules:

    for all x ∈ ℝ:
        1) - ∞ < x < + ∞,
        2) x + ∞ = + ∞, x - ∞ = - ∞,
        3) (x / + ∞) = (x / - ∞) = 0,
        4) x > 0 ==> x(+ ∞) = + ∞, x(- ∞) = - ∞,
        5) x < 0 ==> x(+ ∞) = - ∞, x(- ∞) = + ∞

Infinity is not just an artifact of limit notation, because the algebraic rules defined here formalize the analytic theory underlying calculus. This set is distinct from the hyperreals for a number of reasons (most importantly, it's not a field). But just because it's not a field doesn't mean it's not a coherent algebra. Which really circles back to the original point: you can usefully define an algebra such that division by 0 is not undefined.

In particular, note that by augmenting ℝ with +∞, -∞, we can derive (x / 0) = ∞ from the third rule. This behavior is only very slightly pathological because 1) it's (mostly) contained to the infinities, and 2) it still mostly preserves the order, completeness, addition and multiplication properties of the real numbers themselves. It's just not a field because we cannot do some propositional things that follow from Peano arithmetic (e.g. cancellation) and we can't e.g. take square roots of infinity.

To make this point even more technical, we can choose to define numbers as cardinalities of sets under Zermelo-Fraenkel set theory instead of under Peano arithmetic. This allows us to neatly define arithmetic involving infinities via operations on infinite sets.

Upon rereading what you are saying, I realize I am not disagreeing with you. I am, however, easily triggered by this as an armchair student of math, because I think infinity forms a kind of a trap for amateurs like me. "Actual" infinity is not a member of the reals and it is not necessary to augment the reals to have a functioning real analysis. But you are right that it can be done, it can be added to the reals to form interesting algebraic structures and there are even books on nonstandard analysis that start from here (Elementary Calculus, which I have read part of but not finished).

Division by zero can certainly be given meaning in other mathematical contexts. The real projective line is one example [0]. There are other algebraic structures in ring theory that also have a division by zero, for example a ring with only a zero element. So I would say it depends on your context if something makes sense to do. You aren't ignoring or doing something ill defined, its the opposite (in mathematics). You can do it if its consistent mathematics.

Physicists have very good intuition but sometimes making their statements mathematically rigorous requires extra work. I think a good example is in the history of field theory. Over time the intuition was made into something more formal. The article in the OP discusses renormalisation.

[0] https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

You could think of approaching this result by dividing 1 by an ever smaller and smaller number and the result is getting bigger and bigger, with no end in sight, as you can always come up with a smaller number, never reaching 0.

There are several places in physics where you encounter infinities, the most prominent case probably is the aforementioned uncertainty principle, which leads to 'loops' in QED and need to be countered by renormalization[0].

[0] https://en.wikipedia.org/wiki/Renormalization

You could say that is impossible to start from something finite and completely destroy it to 0, and this analogous to the impossibility of creating infinite pieces. But another option is that there is simply nothing to start with.

It's been a long time since I've done any physics and there is absolutely no rigour here, but the (extremely useful) Dirac delta function comes to mind. You can think of it as being infinitely tall and infinitely thin, but with unit area. If you tried to divide the "area" (1) by the "width" (0), you'd get the height (infinity).

~$ node

> 1/0

Infinity

Q.E.D.

unless you want to claim that javascript is occasionaly illdefined.

the result is well defined in terms of a limit: lim 1/x as x -> 0 = ∞. I consider 1/0 = ∞ a practical shortcut notation.

Except it’s also negative infinity.

Actually, ∞. +∞.and -∞ are three different things.

∞ and +∞ are the same thing afaik.

Technically ∞ is an element of the projective real line: R U {∞}. +∞ and -∞ are distinct elements in the extended real number system: R U {-∞, +∞}.

It depends on the context. ∞ is sometimes defined as the point on the projective real line equal to both +∞ and -∞.

Oh, right!

lim (x->0⁺) 1/x = ∞

It is 0/0 that is undefined. For 1/0 the standard is either infinity (easily checkable in a language like JavaScript or C) or you get the usual talk that you can't mix infinity with numbers and the operation would be meaningless, etc.

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