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I've always been fascinated with division by zero, so I would really love to see a day when calculators gave you a way to have a division by zero that was defined. I am not trolling, I am just seeking the best way to treat zero, and want to know if I have: all the answers; none of the answers; or some of the answers. This is a yes/no question, but I prefer feedback and critism as I spent time preparing this large list of well-thought-out arguments and observations.

Here is the numbered list:

  1. If you put 0 things in a room 7 times: you have failed to change the contents of that room, 7 times.

  2. If you take 0 things out of a room 7 times, you still will not have altered the number of things in that room, despite taking things from a room 7 times.

  3. If you take 0 things out of 7 rooms: you will not have altered their contents.

  4. If you take 0 things out of 7 rooms: you are physically capable of doing so: forever, because they will not run out of zero things for you to take.

  5. If you take any number of things out of any number of rooms, 0 times: those rooms will be 100 percent unchanged by: any & all amounts, but 0 percent changed by: any & all amounts.

  6. If you take 0 things out of 0 rooms you have the entire previous contents of all the rooms, but with one of everything that you took out left over inside them.

  7. If you are told to take 1 amount of things from 1 amount of rooms until the 1 amount of contents inside that room are empty: the amount of steps required to do that [task] depend on the exact amount of each category, and the minimum amount of calculations required to deduce that amount of steps is undetermined; unless any given category's amount is equal to zero, in the case of any of the categories' amounts being equal to zero: the amount of calculations required to deduce the amount of steps is equal to 1.

In the case of 1 amount of rooms being equal to 1 amount of 0 rooms: it takes 1 calculatory step because you know that 0 rooms require 0 work to empty. in the case of it being 1 amount worth 0 things being taken out of an amount of rooms until that amount of rooms are all all empty: you know in one step that it will take a forever amount of work (because you will never empty a room by not changing its amount of things).

In the case of the 1 amount of contents inside that room containing 0 things: you know in one step that you have already completed the task despite not putting any work into physically undertaking the task to come to its solution/resolution (however, if you are required to put a certain amount of work into the solution: you can now/also determine how incorrect you would become for doing so (ie: negative numbers whose absolute value is exactly equal to the numerical representation of how far away from the correct answer you are in units of the 1 amount you were told to use)).

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  • $\begingroup$ See also this question. $\endgroup$ – Dietrich Burde Dec 24 '17 at 12:54
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    $\begingroup$ Holy wall of text, Batman! Do you think you might add some paragraph breaks? As it is, the question doesn't really invite spending the effort to decipher what you're saying. $\endgroup$ – hmakholm left over Monica Dec 24 '17 at 13:09
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    $\begingroup$ The usual characterisation of zero in mathematics is: (a) $a+0=a=0+a$ for any $a;$ (b) $0\times a =0= a\times 0$ for any $a;$ (c) If $a\times b=0$ then $a=0$ or $b=0$. Note that (c) is not true in every case where zero is used but is true for real/rational/complex numbers and integers. The usual definition of division is: If $b\ne 0$ then $\frac ab$ is the unique number $c$ such that $c\times b=a.$ The reason mathematicians don’t define division by zero is that that makes the other rules inconsistent which means proofs based on them can be wrong which is undesirable. $\endgroup$ – Dan Robertson Dec 24 '17 at 13:45
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    $\begingroup$ The consensus is one should prefer “every time you have division in a proof you check first prove you aren’t dividing by zero” to “every time you have division in a proof you need to either prove you aren’t dividing by zero or split your proof into the normal case and the weird inconsistent division by zero case.” Certainly there are cases when one defines a value for division by zero but great care must be taken. $\endgroup$ – Dan Robertson Dec 24 '17 at 13:54
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    $\begingroup$ For the record, @DanRobertson's proposed edit seems to have crossed mine. Since I had already done his improvements, and more, I rejected it. $\endgroup$ – hmakholm left over Monica Dec 24 '17 at 13:56
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Your observations seem to be true as far as they go -- I may have lost the thread a bit around (6) or (7), but there's nothing that looks wildly inaccurate at first sight.

However, they don't really seem to bring us any closer to "a day when calculators gave you a way to have a division by zero that was defined". If all you want is for your calculator to display something other than error when you divide by zero, you can just program it to display $42$. You're going to lose out on a lot of useful properties of division that way, but the point is that you're going to lose them anyway.

The problem with division by zero is not that we haven't figured out what it should be yet, but that we already know which kind of consequences defining it would have, and those consequences are too undesirable for any possible definition to win a significant following.

For example, a very important property of division -- arguably its most important property -- is its relation to multiplication:

For any numbers $x,y,z$ such that $x/y=z$ it holds that $x = y\cdot z$.

Conversely, for any numbers $x,y,z$ such that $x=y\cdot z$ and such that $x/y$ is defined at all, we have $x/y=z$.

At least one of those facts will necessarily be lost if you choose to define division by zero. Together they imply that $$ \frac{x\cdot y}{z\cdot y} = \frac xz $$ whenever both sides of this equality are defined, and therefore we must have $$ \frac{1}{0} = \frac{1\cdot 2}{0\cdot 2} = \frac{2}{0} $$ By then, by the first part of the rule, multiplying the common value of $1/0$ and $2/0$ by $0$ would need to produce both $1$ and $2$ at the same time!

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  • $\begingroup$ I appreciate the feedback. I am glad that you can verify I am [seemingly] not wrong. I wish there was a way I could keep from losing you at around 6 or 7, though, cos that's where/when it gets interesting in my oppinion. I've had these vague inklings of an idea that there might be other types of number, like: the containers, the contents, and the ammount of steps required to perform a given change to them; so atleast 3. But this is some ethereal whimsy... I entertain as an eccentric wannabe-engineer who has no qualifications yet... $\endgroup$ – user179283 Dec 25 '17 at 1:38
  • $\begingroup$ So I appreciate your honest. And I readily accept your points, even if they run counter-current to my hopes. But as a hobby I feel challenged to create a model that results in every equation having a list of answers dependant upon specific presepositional patterns, due to each 0-d variable being now: multifaceted... ...it's a crazy idea not ready yet to be taken seriously $\endgroup$ – user179283 Dec 25 '17 at 1:50
  • $\begingroup$ I only bring it up, because I'd like to see a world where in some given contexts: numbers broke the "X=Y*Z" in one way, but not multiple ways; and then within another/some-other context they didn't. So I want to see some weird version set theory that fixes that, by allowing each way it can break, to exist in a seperate 'continuity' of context, so that mixings of the logics only happen intentionally. And so that nothing is a fully broken/exploitable system, only partially. $\endgroup$ – user179283 Dec 25 '17 at 1:57
  • $\begingroup$ It's just an inkling, but I am not just trying to get zero included: I'm taking a long think about how to maybe include all of its "possible" outcomes as pieces within different "number continuity types" (for lack of a good name), where each one exists in a differently imperfect numberline, but selecting the right one for a given task preserves basic logic, and each is seperate, but proportionally connected somehow... IDK, it's just some fun math on the side, to make my life more interesting $\endgroup$ – user179283 Dec 25 '17 at 2:01
  • $\begingroup$ I didn't lead with that, cos it seemed irrelevant, and possibly inflammatory, but now it may provide much needed context. It's ambitious not-likely-to-succeed project, that I'm taking for a/some: Design; Philosophy; Mathematics; Software-engineering challenge/practice, and may most likely fizzle out, but if it doesn't the ammount of joy I would feel, would make me wanna do it all over again, so I am happy for trying, and apologize if my nativity comesoff as a DunningKrueger effect (it probably is), but this is the ecacr angle of my work/interests at present. Thank you. $\endgroup$ – user179283 Dec 25 '17 at 2:08

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