Why should we not divide by zero? | Hacker News Reader

Why should we not divide by zero?

blog.xevoryn.com

3 pointsxevoryn7 comments

This article argues that x ÷ 0 = x, mentions that x ÷ 1 also equals x only in passing (“This is the exact same result as if it would be divided by one”), and also doesn’t mention (likely doesn’t realize) how that breaks the relation between division and multiplication (e.g. a ÷ b = c implies b × c = a, so if one wants to keep that, multiplying by zero should produce any number or, maybe, a random one, or all numbers simultaneously, or be impossible.)

Either is a choice one can make, but not one that’s convenient.

It's often occurred to me it might be possible to resolve some of the more difficult problems in physics such as infinities and renormalization in QED if we had some other form of mathematics that treated the 1/0 matter differently.

Other than posing the question I've never pushed my brain any further on the matter as I've just assumed that those with much greater mathematical brainpower than I have have either ruled it out or considered it not worth pursuing.

x div 0 = infinity, which is inconvenient if not impossible to represent in most numerical systems and makes for pernicious frame lag in video games.

when we figure out how to represent the problem and state the solution in fine enough grain for accuracy we may find we have recreated the universe. i suspect no one has the patience to wait out the full run for to see the results.

It doesn't equal infinity, it is undefined. x/n approaches infinity as n goes to 0, but at n=0 there is no answer. One way to think of it is x/n = m therefore n * m = x. You can recover the original value by multiplying the result by the denominator. But when n=0, you can't multiply 0 by anything to get x. There is just no such number.

I think it's a matter of convention, and that all the maths I can think of survives the following definitions:

If infinity were to be defined as immeasurably large, and zero as immeasurably small, we would be saying infinity=(some finite number)/0 and 0=(some finite number/infinity

This still leaves us unable to find an definite answer to the multiplication of 0 and infinity, because we might have chosen any 2 finite numbers for the definitions above.

The problem is that the "infinitely small" is already defined, and not equal to zero: The infinitesimal. It is not so simple as defining what the result should be - one needs to make sure this is consistent with the rest of mathematics, or rather, with the mathematics generated by the rest of your axioms and definitions, without resulting in paradox.

If anything, division by zero generates the set of all numbers in your field. That is consistent with division being the operation of finding the multiplicative inverse, but inconsistent with the idea that arithmetic operations have only a single result, rather than a set - but this already exists in finding k-roots of numbers, so it is not completely unreasonable to treat some arithmetic operations as potentially resulting in sets.

Is it a useful definition? Unclear.

In IEEE 754, division by zero is well defined[1], and some algorithms rely on this[2].

Not sure how it relates to frame lag, some less-than-ideally-implemented iterative routine?

[1]: https://en.wikipedia.org/wiki/IEEE_754#Infinities

[2]: https://tavianator.com/2011/ray_box.html

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