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Nutters.org The Nutter Log
Division by Zero Entry id: div-zero
By The Famous Brett Watson
On Sun, 14 Sep 2003 00:15:00 +1000

If you know anything about mathematics at all, you're probably familiar with the fact that you can't divide by zero. It's one of those well-established rules, and I've never heard any mathematician claim that there's any way around it. In the space of this brief article, I intend to correct that oversight.

You may also be familiar with the anomaly of square roots and negative numbers. Multiply any negative number by another negative number, and you get a positive number, so it's not possible for any negative number to have a square root. (And, as an aside, all positive numbers have two square roots, although we generally only refer to the positive roots.) This doesn't stop mathematicians doing a whole bunch of mathematics based around the square roots of negative numbers, though: they just had to invent a new number to solve the problem. That number is called "i", and it's not a member of the set of Real Numbers. Any number which has "i" as a factor is an Imaginary Number; similarly, Complex Numbers are the sum of a Real component and an Imaginary component. Square "i", and you get "-1".

So why can't we pull the same trick on division by zero? There's no member of the Real Numbers or Imaginary Numbers sufficient to the task of solving division by zero, but that need not prevent us from inventing another number which has the right stuff. After all, that's all they did with the Imaginary Number, "i". To this end, I give you the number "k". (I'd use "j", but that's already commonly used by electrical engineers as a synonym for "i".) The number "k" is the reciprocal of zero: it has the property that when you multiply it by zero, you get "one". Given that it's not a Real or Imaginary Number, I'll have to invent a new category for it, which I shall name, "Absurd Numbers".

So what are the consequences of using the Absurd Number, "k"? We can say, "1/0 = k", or "N/0 = Nk", which solves the original problem of finding an answer to division by zero. On the other hand, we can't preserve some of the usual properties of multiplication when Absurds get involved. To demonstrate, consider the expression, "k multiplied by zero, multiplied by two". We've given that "k" multiplied by "zero" is "one", and we already know that "one" multiplied by "two" is "two", so the answer appears to be "two". Multiplication is normally an associative operation, meaning that it doesn't matter which multiplication we perform first; but if we multiply "zero" by "two" first, then we get "zero" (obviously), and if we multiply that by "k", we get "one" — a different result. So we could say that multiplication is associative for Real numbers, but not for heterogeneous operations with Absurds and Reals. It's messy and weird.

There is probably a lot more one could say about a calculus involving Absurd Numbers, but I won't embark on such a study here. In terms of technological application, I suspect that Absurd Numbers are not useful, whereas Imaginary Numbers are (in electrical engineering, for example). Where I see Absurd Numbers as having an application is in philosophy, and I don't think I need to develop the calculus itself any further for that purpose. Specifically, the Absurd "k" raises a question of ontology. Ontology is the question of "what exists?" It's a pretty tough question, and there are many aspects to it. One is the epistemological aspect of how we can know that something exists. Another is the metaphysical question of what it means for something to exist at all, and this is the aspect that I'd like to emphasise here.

It's often overlooked, but, in mathematics, the question of whether something "exists" is a transitive notion: something has to exist in something else. The containing domain is usually implicit: if I say, "there exists no solution to the relation 'x squared equals minus one'," then that's true if I'm implicitly talking about the Real Numbers, but false if I'm implicitly talking about Complex Numbers. Prior to my definition of the Absurd Numbers, it would have been fair to say, "there exists no solution to the relation, 'x multiplied by zero equals one'," because it was true of all the implicit domains to which we could have applied it. I've thrown a spanner in the works, making it so that we can no longer take that nonexistence for granted.

When we talk in general terms about something "existing", we usually mean the domain to be that of physical reality. Thus, I can say, "Santa Claus does not exist", despite the fact that there is so much in the way of conceptual backing for him (pictures, myths, advertising, impersonators, and so on). In the case of Santa, it seems you have everything but the real thing! There are also abstractions which we may want to give the status of "exists" or not. Does "love" exist, for example? How about "purpose"? Physical reality (or some substitute for it) would seem necessary but not sufficient for these things to have an existence: the implicit domain must surely include entities capable of acting with "intent" to even allow for their possibility. (That's not to say that "intent" is necessarily non-physical, but rather that the physical does not necessarily incorporate "intent".)

So, in questions of "what exists?", it seems vital to specify the domain in which the thing potentially exists. There exists no solution to the division by zero problem in the domains of Real or Imaginary Numbers, but that doesn't prevent me from inventing the Absurd Numbers in which the solution does exist. You can deny something existence in two ways: either by denying its actuality (as in the case of Santa Claus, a non-existent person), or by denying its possibility, which is to deny it a domain capable of containing it. My invention of Absurd Numbers overcame a "possibility" problem, since none of the available mathematical domains contained a solution. Indeed, in mathematics, there are no problems of actuality, only of possibility. In the physical world, on the other hand, once given the possibility of a thing, we still have to determine its actuality.

One final note: you could ask whether Absurd Numbers themselves actually exist, or whether they are merely a figment of my warped imagination, but if you heed my warning, then you should also specify the domain of your question. Without a domain, the question isn't well formulated. Or is it? Can we have an intransitive version of "exists", such that something exists (or not) in an absolute or ultimate sense? Well, yes, we can. The question of whether God exists can be a question of ultimate existence, as can the question of whether physical reality exists, but in the absence of a containing domain, what does it mean to exist? Judging by the history of metaphysics, this kind of question can not be answered in a universally-agreeable manner. As a matter of practical advice, therefore, use "exists" in the transitive sense, and specify a domain — if you want to make any progress towards an answer, that is.

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