Slashdot Mirror
Professor Comes Up With a Way to Divide by Zero
54mc writes "The BBC reports that Dr. James Anderson, of the University of Reading, has finally conquered the problem of dividing by zero. His new number, which he calls "nullity" solves the 1200 year old problem that niether Newton nor Pythagoras could solve, the problem of zero to the zero power. Story features video (Real Player only) of Dr. Anderson explaining the "simple" concept."
Not really. NaN is: Not a Number.
He proposes to define a new number that doesn't exist (or fit for that matter) in the current system.
But still it's useless, or at least I think it is.
100/0 != 10/0 != 1/0 != 0/0
but he uses the same identifier for all of them, so that would mean:
(100/0) / (1/0) = 1
That goes against the principle of:
infinity / (infinity - 1) != 1
0/0 should be a special case where dividing by zero actually yields a valid real number, and all other divisions by 0 are undefined.
Wrong.
0/x gives 0. Always. And x/x gives 1. Always. Now, try for x=0... That gives 0/0 = 0 and 1 at the same time. That's why it's undefined, usually called NaN (Not a Number).
Anything else divided by zero can be defined as giving infinity or -infinity, which can be used in further calculations just fine, even coming to the correct result.
Example: The angle of the vector (1,0): arctan(1/0)*180/pi = 90 degrees. Works just fine. Not so for NaN, any calculation involving NaN will continue giving NaN.
Wow. Looking over the guy's axioms, as soon as you introduce "nullity" the result of all of your computations is nullity:
- the sum of anything and nullity is nullity (his axiom A4)
- the product of nullity and anything is nullity (his axiom A15)
- the reprical of nullity is nullity (his axiom A22)
So, his arithmetic is normal arithmetic, but as soon as you hit nullity anywhere, it's a black hole you can never get out of. All he's essentially done is take the "error state" and add it into the system as an object. You still can't compute anything you couldn't compute before. So yes, he has truly discovered NaN.
And for him it is true; he's defined infinity to have these values. He very specifically wants a fixed value for infinity.
Nor does this work. Division, in his system, is not the multiplicative inverse, but the reciprocal. So, for him: 1/0 = infinity implies 0/1 = 1/infinity, which does in fact meet our expectations.
Basically, what he's done with his system is come up with a (completely consistent, as far as I can tell from scanning from his website) framework where singularities now have a defined value, which means that all functions are defined everywhere on the real line (or the transreal line, which is what he calls his infinity-and-nullity supplemented system). Which is great, as far as it goes. But there's a big trade-off for this: there is now no longer a guarantee that if both f(x) and the limit at x of f both exist, that they will have the same value. The example he himself gives is the hypebolic tangent at infinity; the limit is 1, but by direct evaluation, it ends up being nullity. To get around this, he proposes a hierarchy of value determinations; a function is defined at a point by its transreal arithmetic value only if a different value isn't suggested by analysis. So tanh(infinity) would be treated as 1, even though working through the definition of tanh requires the value to be nullity in his system.
So in summary, he's defined terms so that division by zero is consistent and workable, but the price is that even relatively simple calculus becomes a lot more complicated. Nor is it all clear that transreal arithmetic will hold up with higher mathematics at all (when infinity is valued rather than defined by limits, how does cardinality work?). So I think he's got to a better job selling it than "it's better than NaN or having values undefined," because I can't see how it is.
Yes, actually it is, and there are different sets of rules (aka axioms) that are used. For example, Euclid chose to include the Parallel Postulate among the axioms that define his geometry, but there are various well-developed -- and useful! -- non-Euclidean geometries that assume the parallel postulate is not true. There are many branches of mathematics that modify what most would consider the "normal" rules in various ways. Many of them prove to be useful in the real world, too.
Mathematicians realized a century ago that their work is a discipline of arbitrary rules, and that none of their theorems have any inherent real-world truth or falsehood. Math is simply an abstract model. By choosing the right set of axioms one can create a model that maps well onto various aspects of reality, making it useful for physics, engineering and much, much more. Sometimes the common rule set doesn't map well, and even physicists and engineers use the alternative rule sets mathematicians have devised.
This concept of "nullity" isn't something that mathematicians would call wrong. For it to be wrong, it would have to be inconsistent with the results of whatever other axioms Anderson has chosen to use. What mathematicians would call it, however, is an old, uninteresting idea. There have been many others that postulated a placeholder "value" for infinity and explored the results of that assumption. Some of the results are even occasionally useful in simplifying useful calculations. And sometimes the alternative system produces results that don't map well onto reality, and the distinction between the cases is well-explored and well-understood.
I may be stating that too strongly, though. It's possible that Anderson has adjusted his definition in a way that makes it useful for a broader set of problems. Honestly, though, I doubt it. This is thoroughly plowed-over terrain.
I think it's most likely that Anderson has discovered some specific, important problems in optics(which involves some very high-powered mathematics, BTW, much more so than most engineering disciplines) that can be simplified by postulating a nullity, and that he published the work in an appropriate journal to an appreciative audience.
Note to ACs: I usually delete AC replies without reading them. If you want to talk to me, log in.