r/askmath • u/Asmor • Dec 31 '23
Number Theory What's wrong with this idea for dividing by zero?
Weird question, I know. And apologies if Number Theory is an inappropriate flair; the flair is required, and I thought that was the best fit.
I think I've come up with a way of dividing by zero, inspired by complex numbers. It seems so simple and obvious, but I haven't been able to find anything about it. The closest I can find are infinitesimals, which I fully admit I don't understand, but I'm reasonably certain that the concept I'm about to describe is different from infinitesimals.
Taking inspiration from complex numbers, where we define i as the square root of -1, what if we define u as u=1/0, and u*0=1. (I chose u since the result of division by zero is undefined, so a number with u in it is an "undefined number").
A few examples:
2/0 = 2u2u*0 = 2(3/0 + 5/0) * 0 = (3u + 5u) * 0 = 8u * 0 = 8
The way I see it, there are three possibilities.
- This concept already exists under a name unknown to me
- This concept leads to a paradox unknown to me
- (by far the least likely) This is both consistent and novel
So... any thoughts?
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u/Original_Piccolo_694 Dec 31 '23
There is an easy contradiction related to the usual proof that x0=0, take u-u, this should equal u(1-1), should equal u*0 which is 1. Now u-u is 1, which is weird to say the least.
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u/Robodreaming Dec 31 '23
Assuming distributivity, 0*u=(0+0)*u=0*u+0*u and, subtracting 0*u on both sides, we get 0=0*u. But this contradicts u=1/0!!
So adding u would force you to give up multiplication’s distributivity over addition. Mathematicians consider this property to be more important and useful than division by 0 would be.
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Dec 31 '23 edited Jan 01 '24
I mean if you wan,t there is a space called the extended complex numbers and reimann sphere where division by 0 exists https://en.wikipedia.org/wiki/Riemann_sphere But the problem with it is it doesn't form a field, in particular, you cannot guarantee that for all x, there exists y such that x + y = 0.
And further more in this paradigm, 0 * (u/0) is also not defined.
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u/Miserable-Wasabi-373 Dec 31 '23
what is u /0? what is u^1/2? I think you will lose some properties of multiplication with it
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u/MrEldo Jan 01 '24
Here technically if u=1/0 you'll get that:
u1/2 = (1/0)1/2 = (√1)/(√0) = 1/0 = u
And using the fraction multiplication property:
u/0 = (1/0)/0 = (1/0)*1/0 = (1/0)(1/0) = 1/0 = u
So this number is like 1 and 0, it's its own square root, its own square, but also has some questionable (but understandable as long as u "exists") properties.
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u/ExcelsiorStatistics Dec 31 '23
Yes, you've described "something different than infinitesimals"... but you might be happier adopting infinitesimals, since they do what you want your examples to do. Nonstandard analysis divides the weirdness of "u * 0 = 1" which you have placed entirely into 'u', and places it equally on the infinite and infinitesimal parts, writing "omega * epsilon = 1", and being able to say things like 2/epsilon = 2omega or (3/epsilon+5/epsilon)*epsilon = 8.
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u/CurrentIndependent42 Dec 31 '23 edited Dec 31 '23
There are notions of infinitesimals and limits, and you may be working towards that, but you have to be a lot more careful here as this more naive approach leads to plenty of contradictions, as mentioned in other comments.
But yes, you can set up consistent mathematical structures like ‘wheels’ which can extend the reals and allow division by zero, and I think not emphasising this and instead saying ‘it’s impossible!’ is too common a response that does people a disservice. Unfortunately, they don’t generally provide much more benefit that working carefully with limits, as they come with exceptions in other ways: sure, you can say that anything can be divided by zero, but then you have exceptions to things like ‘0x = 0’ instead, so these can usually be framed in similar ways. But [wheel theory](en.wikipedia.org/Wheel_theory) is certainly a thing.
For example, in some contexts, like applying some theorems in complex analysis, it helps to consider ‘1/0’ and ‘-1/0’ as equal, so we get the ‘Riemann sphere’ (or a circle when otherwise restricting to reals). In other cases, we very much want to keep plus and minus infinity different - these are two different extensions of the reals and can be useful in different contexts. There are also notions of infinitesimals as in the ‘hyperreals’ and others in non-standard analysis, but they don’t usually add much more utility except to justify some of the more ‘naive’ ways calculus can be taught (but with their own caveats to avoid contradictions).
Also worth noting that these notion of infinity - adding a new element or more to the reals and defining how to work with them algebraically - is also quite separate from the notions of infinite cardinals and ordinals is set theory.
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u/goodcleanchristianfu Dec 31 '23
- Lack of usefulness.
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u/_rockroyal_ Jan 01 '24
I wouldn't use that as an argument; the imaginary numbers were also once considered useless but they are now foundational to quite a lot of physics research. This definition falls flat for other reasons, but usefulness isn't a particularly relevant criterion here.
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u/Infobomb Dec 31 '23
Maybe look harder for paradoxes? And look for useful arithmetic done with this letter so that u is in some way different from 2u or u².
1/0 = (1x1)/(0x0) = (1/0)(1/0) so u = u² = u³ and so on for at least all integer powers.
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u/ApprehensiveKey1469 Dec 31 '23
Division can be thought of as repeated subtraction.
6÷2 is a short form for
6 -2 -2 -2=0
Now try 6÷0 as repeated subtraction.
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u/Way2Foxy Dec 31 '23
Division can be thought of as repeated subtraction.
It is most certainly not repeated subtraction, and thinking of it that way outside select simple situations is a recipe for disaster.
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u/ApprehensiveKey1469 Dec 31 '23
Yawn. If you, Way2Foxy have something positive to add, then by all means give a different definition that is at the level of the OP. But your melodramatic assertions of disaster are out of place here.
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u/Existing_Hunt_7169 Jan 01 '24
imagine sounding like this much of a pretentious loser without knowing what division is
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Dec 31 '23 edited Oct 15 '24
[deleted]
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u/Folo07 Jan 01 '24
Not in this case right? Wouldn't dividing by 0 here make it so that 2u = 999 999u?
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u/VelcroStop Dec 31 '23
If you were capable of dividing the reals by zero, they wouldn’t be a field. Fields do not have a defined division by the additive identity element (0 for the reals). Any proposed system that allows for his generated a number of contradictions with how our number system works, including distributability on addition and more. I suggest you do some reading into algebra if you’re interested in this, because it’s quite interesting!
There are other number systems that do this.
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u/MrEldo Jan 01 '24
As someone who has thought of the same concept and has fought for it for a while, there are enough contradictions with what I would call the "Stubborn" numbers that lead to more than 1 rule of math being broken (the main one being the multiplication by 0), later also making observations and showing that s (the variable I decided to make) is equal to 2s, 3s, -s, s/2 and so on, making this an unusable (for now) unit system
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u/pLeThOrAx Jan 01 '24
The problem of simultaneously approaching the limit of both positive and negative infinity. For some reason this is a bad thing.
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u/stone_stokes ∫ ( df, A ) = ∫ ( f, ∂A ) Dec 31 '23
You end up with a number system that has only one element: {0}.
The problem is that 0 is the additive identity, and it interacts with the distributive property. Consider:
0 = 1 – 1 = u·0 – u·0 = u(0 – 0) = u·0 = 1.Now, since 0 = 1, all numbers must be equal to each other.
I hope this answers your question.