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this post was submitted on 23 Oct 2023
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Asklemmy
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There are some good answers here already. I feel the need to add something, though.
If I gave you a number, 6, and multiplied it by 2 you’d get 12. If I asked you to “undo” the multiplication, you’d divide it by 2. So, you can think of division as the “inverse” of multiplication.
So: 12 * 1/2 = 6.
6, when doubled is 12, and 12, when halved, is 6. You can never double 6 and get 14. We say that multiplication between two (nonzero) numbers has a one-to-one relationship.
Then, let’s say I asked you what 0*6 is. And you’d say 0*6=0.
Then, let’s say I didn’t know what we started with. I give you this equation and ask you to find a value for x:
0*x=0
What is x? X can be anything here, 1, 17, pi, all numbers work. You can even choose 0.
Could you try 0*x*1/0=? How would you choose one number to be correct?
There is no “undo” button here. 1/0 is meaningless because we can’t assign it a unique value. A math person would say, “0 has no multiplicative inverse”.
Is it fair to suggest or infer that in order for division to be possible, the divisor must be partitive of the dividend? Like 6 can never be partitive of 14 in the whole number sense such that axiomatically 14/6 = undefined in some vague sense that I can be damned to elaborate or defend currently?
I can’t parse this paragraph.
14/6 has a solution, its 2.3333…..
Multiply 14/6 by 6 and you get 14 again. You can always go back to the initial state when you know what actions are taken, unless you’ve multiplied by 0.
If I am understanding what you're trying to say, then yes, 14/6 does not have a solution in the sets of natural numbers, whole numbers (natural numbers and zero), and integers (whole numbers and the negative numbers). This is because 14 is not a multiple of 6.
However it does have a solution in the set of rational numbers and by extension, the real numbers. Thus, if we're talking about real numbers, 14/6 does have a solution: 2.3̅3 (two point three three repeating).
I recognize that I'm veering very closely, if not already have gone to explaining this in axiomatic terms. I apologize.
At any rate, if you're wondering if there is any similar extensions to the real numbers that can accommodate 1/0, as far as I understand, such extensions, if they do exist in the way you want them to behave, would still run into the same problems other replies to your post have described. Because of those difficulties (among others), in the two extensions I've looked at (hyperreals and the surreals), division by zero (and thus, 1/0) is still impossible in general, but are allowable in certain circumstances corresponding to the situations where y/x at the limit as x→0 exists or is +infinity or -infinity (but not both). However, as far as I understand it, in the real projective line, where there is a "point at infinity" added, you can define 1/0 as being that point at infinity.
EDIT: added some important clarification.
You have nothing to apologize for, this is a very eclectic discussion from the get-go ♾️
Thanks! I hope my reply has helped you somehow. I think that "point at infinity" concept from the real projective line (which I added as an edit) is the closest to a "solution" to your question. It's likely not an end-all-be-all solution to 1/0 though, since, as far as I understand it, it "collapses" -infinity and +infinity to just one point at infinity, which might not be a desired property in certain circumstances.
I so wish I had taken like advanced maths + calculus and not had a narcissistic tormenter of a parent so I could actually think and digest math. None of the above "equation" would have been possible and I mourn that fact but I hope to incrementally refine my thinking and possibly approach that level over time if I can by diffusion like in this thread (a little bit)
It is possible to grasp (relatively) advanced math like in this discussion via self-study. I myself came from a different background (engineering) though I've been somehow interested in math since high school. I wish I can recommend some material for studying such fields of math from a non-math background (at least at the start) but maybe that should be saved for a separate thread. I do remember encountering such a thread in the other place, but it won't hurt much to ask for it here (if there isn't already such a thread here).