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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."
So much for my $200 calculator.
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His new number, which he calls "nullity"
:-)
Well, thats just nullty.
Seriously though, as I understand it, this is simply another mathematical structure that allows a different scalar much like a real projective line, right? If that is the case, then there is nothing really new here and there can be no application or definition with real numbers or integers. Alternatively by interpreting this as a commutative ring, one might be able to extend this to where division by zero does not always get you in trouble, but the precise interpretation of "division" is fundamentally altered. This too is not a new concept.
However, all of that said, I am a bioscientist and my math skills are not as strong as a formally trained mathematician, so I will defer to those here who are stronger mathematicians than I if this interpretation is incorrect.
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The professors at 'Rithmetic State were non-plussed upon hearing the news.
Is it just me or does it sound like he thinks he's invented the NaN?
There's zero comments yet. Wonder how many comments that is per poster
I can make up numbers too...
What he did was assign the previously "undefined" integer with a defined symbol that means the same thing. Infinity in both directions.
While interesting, the concept has little use.
From the article "Imagine you're landing on an aeroplane and the automatic pilot's working," he suggests. "If it divides by zero and the computer stops working - you're in big trouble. If your heart pacemaker divides by zero, you're dead.".
Now, instead of getting an error message, the computer give a 0 with a line through it, and THEN an error message.
--sig fault--
mod original post up by 0/0 points :)
-- "Genius is 1% inspiration and 99% perspiration" - TAE --
It's sad that he teaches math and thinks this is a worthwhile concept.
For just one example of why it sucks, he BEGINS by defining: (infinity) = 1/0 and (-infinity) = -1/0.
My conclusion: (0)*(infinity)=1
So 2*0*infinity = 2*1
So 2 = 2*0*infinity = (2*0)*infinity = 0*infinity = 1
And once you know that 2 != 1 and 2 =1, it turns out you can prove quite a bit...
Total nonsense, and the BBC is encouraging it. *shakes head* Although, I've got to say, it's nice, for once in my life, to deservedly be a smug American.
Think! It ain't illegal yet!
George Clinton
Uh... are you joking?
Imaginary numbers (specifically, complex numbers, which consist of a sum of a real and an imaginary number, and which comprise the "complex plane") are INCREDIBLY important in the "real world."
I'm just a chemist, not a mathematician, but I am well aware that imaginary numbers are critical in the Fourier transforms used every time I take an IR or NMR spectrum.
Ever do electrical engineering? Circuit analysis is made a great deal easier when you can treat circuit elements in terms of complex numbers. All that "impedance" stuff you hear about capacitors and the like that makes it possible to apply Ohm's Law to LRC circuits.
These also are not merely made up properties, they are fundamental to mathematics and thus (if one believes that math is the language of the universe) physics. For example, certain integrals necessarily yield imaginary results. These integrals are not of some ethereal interest, but appear throughout quantum mechanics. This is why the amplitude of a wavefunction (used, for example, in molecular modeling that allows for practical achievements like better medicines) is not the square of the wave function (or, for that matter, its absolute value) but the product of the wavefunction and ITS COMPLEX CONJUGATE.
If you'd like more examples of the utility of complex numbers and other "random rules," check out Boas' "Mathematical Methods In The Physical Sciences."
Helpful little hint from the end of the video:
Yeah. It was that simple.
I'm just reminded of that proof from way-back-when that 2 = 1:
All this guy has done is provide another little fun "proof" that you can use to win bar bets. "Betcha I can divide by zero..."
The heavens do not fall for such a trifle.
Note: IAAM(athematician). You pose a good question. The game in mathematics, though, is not to "make up random rules so that something that occurs to them suddenly works". It's (broadly speaking) to make up new rules which are completely consistent with all the old rules which allow us to understand a previously mysterious example. This is where "imaginary" numbers succeed tremendously, and "nullity" fails miserably. See my post downthread for why nullity sucks.
"Imaginary" numbers are just the "thingys" which are solutions to polynomials. I.e., mathematicians find it useful to have an answer to the question "for what values of x does x^2 + 1 = 0?" The answers are useful, even though they aren't good at measuring length or breadth or depth or other one-dimensional concepts. They're useful because they allow mathematicians to develop a theory which has answered questions which couldn't be answered before. This is true even though both the question and the answer both lie in the realm of real numbers. Should there be an answer to every question of this type that doesn't use complex numbers? Perhaps, but it certainly doesn't have to be pretty, or easy to discover. Often the shortest path to a "real" truth lies on an "imaginary" line.
Think! It ain't illegal yet!
George Clinton
The first paper he describes as:
The second paper he says:
Seriously though this is the sort of thing that you don't want to sneeze at, it can sound both inane and brilliant. Anderson is not such a crackpot, I found a presentation of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge."). He seems to be a geometer stating his perspective in the first line of that presentation: "Aims: To unify projective geometry and the Turing machine".
He's a geek hero! Who knows if his nullity will end up just NaN with a British twang or the next best thing to sliced bread and i?
I was unable to hear the realaudio casts but from Book of Paragon, The Perspex Machine (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations (a mathematical treatise with applications to computer vision and robotics) just glancing I'd have to say the guy seems to be a real mathematician, geek and philosopher-king. I don't know if he's up there with Newton but he at least deserves an honorable mention for his wonderfully witty (and to me as yet inscrutable) naming of the Walnut Cake Theorem (see page 10 of Perspex.pdf). It seems that he was motivated to create nullity in order to make reliable advanced computers that would not barf when asked questions about the universe, and to him "Not-a-Number" is vomit. I'd say read some of his stuff before assigning him to the 9th Hell. Would like to hear what any mathematicians or other people with brain cells over the age of 12 have to think about it. It's okay if he reinvented something but it appears he is trying to make a machine that can handle infinities and other tough numerical concepts with ease, and that's worth something. Oh, that and his quantum computer looks neat.
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.
I will never forget when I was about 8 years old going up to the adding machine in my grandfather's home office. It was about twice the size of a toaster and made of that old typewriter metal. It looked like it weighed as much as a car and had probably cost as much new. Just to see what would happen I entered '0', '/' and '0'. Without hesitation it began producing line after line of '0', '0', '0' on the paper tape accompanied by a cacaphony of mechanical gears. It became apparent to me in a split second that it had no intention of stopping. Ever. It had come alive and was angry.
I yanked the plug from the wall socket and ran from the room in terror.
If he can make up numbers, then I cam make up words,
this whole thing is utterly stuipfluous.
Nah... It's more on the lines of "Not another NaN"... heh heh... Not another Nan!, recursive... gettit?
(returns to its corner)