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Professor Comes Up With a Way to Divide by Zero
Posted by
samzenpus
on Thu Dec 07, 2006 02:01 AM
from the it-seems-so-obvious-now dept.
from the it-seems-so-obvious-now dept.
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."
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Argh!!! (Score:5, Funny)
Re:Argh!!! (Score:5, Funny)
In fact, using proof-by-blatant-assertion,
if 0/0=14
then 0*14 must = 0
which it does
therefore 0/0=14
so there !
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Re:Argh!!! (Score:5, Funny)
Why is the algorithm producing that? Oh I introduced a nullity.
Furthermore, they shouldn't have called it a nullity. They should have called it a Bush.
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Re:Argh!!! (Score:5, Funny)
How many light bulbs does it take to change a light bulb?
...
One, if it knows its own Goedel number.
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Re:Argh!!! (Score:5, Funny)
None. It's a hardware problem
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Re:Argh!!! (Score:5, Funny)
O(1)
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Re:Argh!!! (Score:5, Funny)
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Re:Argh!!! (Score:5, Funny)
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Re:Argh!!! (Score:5, Funny)
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Re:Argh!!! (Score:5, Interesting)
Reading his stuff, he's proposing an abstract machine as an alternative to the universal turing machine (also an abstract machine) that solves the problem of exceptions in algebra. He's suggesting it has alot of philisophical implications somewhat aligned with the way conventional algebra does. I havent quite grokked the central thesis of it, as my maths is way rusty, but its actually quite interesting.
The 0/0 = nullity stuff is a tragic little misstatement of what he's getting at.
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Re:Argh!!! (Score:5, Funny)
http://archives.nesc.ac.uk/gcproposal-5/0080.html [nesc.ac.uk]
"It is simply a technical matter to extend this compiler to deal with the
whole of C. I could then cross-compile from Pop11, Lisp, or any other
language for which there is a C source version. At that point I would be
able to produce massive neural nets that implement operating systems, word
processors, compilers and the like. It would be relatively straight forward
to compile Linux into a neural net. This opens up the possibility of doing
research on massively large neural networks. We could then move away from
our toy implementations and start examining useful systems. "
Imagine a Beow...[Error in universe.pl line 15x10^9: Division by zero]
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Re:Argh!!! (Score:5, Funny)
No wonder the universe sucks, it's implemented in Perl!
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Re:Basic math (Score:5, Informative)
lim x->0+ (1/x) = inf
lim x->0- (1/x) = -inf
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Re:Argh!!! (Score:5, Funny)
wait, you paid $200 for a calculator?
b = $100
a = b
a^2 = ab
a^2-b^2 = ab-b^2
(a+b)(a-b) = b(a-b)
a+b = b
since a = b
b+b = b
2b = b
$200 = $100
They ripped you off. $200 is really only worth $100
Parent
Well, thats just nullty. (Score:5, Interesting)
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.
Re:Well, thats just nullty. (Score:5, Interesting)
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Re:Well, thats just nullty. (Score:5, Insightful)
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Re:Well, thats just nullty. (Score:5, Funny)
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Re:Well, thats just nullty. (Score:5, Insightful)
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.
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Re:Well, thats just nullty. (Score:5, Funny)
Your interpretation is correct but for proper mathematical representation it should be reduced to its simplest form.
While simpler reductions may be possible I believe the following best conveys the essence of the equation:
"Dr. Anderson is a pompous idiot."
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Re:Well, thats just nullty. (Score:5, Funny)
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Re:Well, thats just nullty. (Score:5, Funny)
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Re:Well, thats just nullty. (Score:5, Funny)
Although your number is higher than his.
So, perhaps i should say:
You must be new here, because i think you mean, "You must be new here."
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Re:Well, thats just nullty. (Score:5, Funny)
In any case, I'm not sure I see how nullity rectifies the problem.
"Good morning ladies and gentlemen, this is your captain speaking. We're nullity minutes into this flight, and we're cruising at nullity knots, at an altitude of nullity feet below sea level. We've got a nice tailwind blowing along an axis perpendicular to spacetime, so we hope to arrive at our destination (7i-4) minutes early."
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Re:Well, thats just nullty. (Score:5, Funny)
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Unspoken of, third sign (Score:5, Interesting)
Therefore, if there was 0 and -0, you could claim x/0 = (SIGN(x))*infinity and x/(-0) = -(SIGN(x))*infinity.
Perhaps nullity is used to address exactly this problem of zero's "third sign". There is also similar concept, "infinite complex number", where complex plane is mapped on Riemann's sphere, where south pole is mapped to zero, while north pole is considered "complex infinity". The nullity is "real numbers' only" version of that.
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Not everyone's happy (Score:5, Funny)
Umm... NaN? (Score:5, Funny)
Re:Umm... NaN? (Score:5, Insightful)
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
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Re:Umm... NaN? (Score:5, Interesting)
(nullity)^(-1) = nullity
So division by nullity is just nullity.
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Re:Umm... NaN? (Score:5, Funny)
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Re:YaNaN? (Score:5, Funny)
Nah... It's more on the lines of "Not another NaN"... heh heh... Not another Nan!, recursive... gettit?
(returns to its corner)
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Hmm (Score:5, Funny)
And this is important, why? (Score:5, Funny)
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.
mod post up by ... (Score:5, Funny)
testing, exception handling etc. (Score:5, Insightful)
This is computer programming ABC: you DONT allow undefined behavious to occur in your program! (especially if your doing MIL-STD Ada for avionics etc.) This guys 'method' is just a form of exception handling that any programmer with half-a-brain could implement.
Sad, really... (Score:5, Interesting)
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.
Nothing to see here, people... (Score:5, Funny)
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..."
Don't sneeze at it (Score:5, Interesting)
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 [bookofparagon.com] of his on optical computing and an introduction to its underlying theory called perspex algebra ( "Representing geometrical knowledge." [nih.gov]). 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 [bookofparagon.com] (Anderson mentions transreal arithmetic) and Exact Numerical Computation of the Rational General Linear Transformations [bookofparagon.com] (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.
He's just made "error" an object (Score:5, Insightful)
- 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.
If only we'd had this 30 years ago. (Score:5, Funny)
I yanked the plug from the wall socket and ran from the room in terror.
new things (Score:5, Funny)
this whole thing is utterly stuipfluous.
Re:Imaginary Numbers (Score:5, Informative)
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."
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Re:Imaginary Numbers (Score:5, Insightful)
People take mathematical tools and models and apply them to the real world because they are useful. However, that usefulness is a lucky accident.
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Re:Imaginary Numbers (Score:5, Interesting)
"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.
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Imaginary Numbers?! (Score:5, Insightful)
ALL Mathematics is COMPLETELY synthetic. That's the whole point -- that's the power of mathematics. You can define any set of rules, any set of axioms, any set of symbols, and start deducing. If the tools you need don't exist, you make them up. Nothing is more valuable in mathematics than a nice, clean, clear definition that increases the expressivity of math. Since math has no independent existence anyway, you can get away with pretty much anything so long as your new system has useful properties. Mathematicians with the guts to make things up as they go along end up with their names in textbooks and attached to great theorems, assuming what they made is conceptually useful (whether nullity is conceptually useful remains to be seen; a written description of the definitions would be nice).
Mathematicians that only do calculations that we already know about and are comfortable with? They're called accountants, and they have no friends. Seriously though -- since when did making up new ideas become a bad thing? I was under the (apparently mistaken) view that creativity was a praiseworthy trait.
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Dr. James Anderson's actual papers (Score:5, Informative)
The first paper [bookofparagon.com] he describes as:
The second paper [bookofparagon.com] he says:
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Re:Dr. James Anderson's actual papers (Score:5, Informative)
Axioms of Transreal Arithmetic:
- The majority of his proofs are done `mechanically' and not provided.
- He makes a big fuss about the validity of real arithmetic in the `Discussion'. Not a word about validity elsewhere.
- He seems to equate IEEE floating-point arithmetic with real arithmetic.
Transreal Analysis:
- This is an _Analysis_ paper with no mention of continuity or epsilon neighborhoods.
- Doesn't the isolated nullity value cause hell when doing analysis proofs with epsilon neighborhoods?
- How exactly does one define an epsilon neighborhood around nullity?
- A picture of the transreal `number line' does not constitute proof.
- Attempting to disprove other people's counter proofs is not proof in itself.
- Why not attempt all of the fun proofs and lemmas in an upper division real analysis course regarding continuity, differentiation and integration?
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Re:Even I knew this was wrong as a 10 year old (Score:5, Insightful)
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.
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It's Not Rubbish (Score:5, Insightful)
That's why he's defined a new arithmetic - he calls it transreal - where division by zero is defined. The PDFs on his website clearly explain what he's done.
It isn't rubbish. In second year high school mathematics they had us "invent" our own arithmetic. We could define whatever operations we like (eg, a funny symbol that would multiple the left hand value by 2 and add it to the inverse of the right hand value) and then we had to prove whether the operation was commutative, distributive, etc. This guy has done the same thing but with a new "number" he calls nullity. He has defined what happens when you add a real to nullity, when you multiply a real by nullity, when you divide nullity by nullity, etc. It's an internally consistent number system.
It's interesting for grade schoolers because it gets them thinking about number theory. Instead of thinking "you can't divide by zero" they instead think "oh, well that's just a law for the real numbers, but I'm not constrained by real numbers, I can invent a number system where division by zero is allowed". That is far more insightful and creative than "you can't divide by zero". A child who grasps that concept has the potential to become a great mathematician. A child who merely parrots "you can't divide by zero" will become a bus driver or a computer programmer :-P
It's hard to explain abstract concepts such as number theory. Congratulations to him for making it look like fun.
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