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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."
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.
Visit Jonesblog and say hello.
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
Mr. L'Hopital would have something to say against this.
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
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.
0/0 is special, explained:
Think of a division as the reverse of multiplication:
6 / 2 = 3, which means 3 * 2 = 6
With a division by 0, this does not hold:
6 / 0 = x, there is no possible x for which x * 0 = 6
X can be no real number
However, 0/0 is different:
0 / 0 = x, but no matter what you fill in for x, x * 0 = 0
X can be any real or imaginary number, 0 * x is always 0
This is why A / 0 has no solution, unless A = 0, then A / 0 does have a solution, an infinite number of solutions in fact: all numbers are a correct solution.
This professor didn't invent it by the way. He just seems to be the first to bother explaining it to school children.
Sure he did. He said the reciprocal of nullity was nullity:
(nullity)^(-1) = nullity
So division by nullity is just nullity.
This was a question posed in a book I read a while ago, by some reknown mathematician...for all his accomplishments, he couldn't help but wonder...was any of it really helping to describe the universe better and broadening our knowledge of it (thus, a discovery), or was more of it simply a figment of his stretched imagination?
So Nullity may now 'officially' mean n/0 but what does it mean really? Is it just another term for, say, infinity or undefined?
Actually Im going to retract unreservedly the crank comment right now...
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.
Excuse the Unicode crap in my posts. That's an apostrophe, and slashdot is busted.
I think the big difference is that IEEE numbers were designed for practical use (if you got x=NaN you do not want if(x=y) to work) while his definition is designed for ease of teaching - it is probably easier to explain the rule for 0/0 rather than tell the students that in this case you have think what to do.
His example with f(x)=sin(x)/x is the best illustration - his arithmetic happily produces f(0)=NULL while in practice you should never assume that a floating point number is exact and thus the best definition is where f(x) is continuous in 0 and f(0)=1 and if the code is missing this special case it should return an error.
On the other hand, I have never seen an equivalent of NaN or NULL in analytic computation, so it might be a convenient shorthand after all in the similar way how +infinity is so convenient in measure theory. Of course, one big reason for doing analytic computation is that one can use continuity arguments and since NULL or NaN has to be an isolated point this would likely just introduce a bunch of combinatorics into derivations and make everything more complex.
The problem with trying to abstract is that 0 holds no sign. It poses no problem when you multiply with 0, because you don't need to ask about the sign of resulting 0. However, when dividing finite with 0, you know that you have two possible and distant infinite outcomes.
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.
Here's one from the "young whippersnapper" department.
When I was a boy, we programmed air/space craft simultations for NASA.
Not the just abstract videogame types, but actual mechanically-linked 3D motion simulators
that jerked (jerk is a derivative of acceleration, in turn a derivative of velocity, thence a
derivative of position) human test pilots in a shaker cockpit.
Aside: the computer coding involved aviation control math models -> Ratfor -> FORTRAN-> real-time
assembly language -> custom digital I/O in the simulation cockpit, debugged via toggle switch
breakpoints set on a Xerox Sigma 9 console, later supplanted by Foonly machine efforts.
To make a long story short, the aerospace models often attempted divide-by-zero, either from
outright programming bugs or ill-conditioned equations.
So, did we then smash the test pilot into the cabin walls at a high rate-of-change?
No, the intrepid project mechanical engineers, who grokked servo mechanisms and could care less
about snotnose Unix-head punks simply used "mechanical rate limiters" to
overcome and smooth over these "divide-by-zero" disasters.
I'm telling you, even Professor Kahan's IEEE floating-point NAN nomenclature
for calculations didn't save the day for renormalizing these infinities -- how could it,
no self-respecting kernel (Unix or otherwise) has ever executed FP operations, which still
doesn't absolve integer div-zero horrors and concomitant analog duct tape patchwork
to save the day.
You can't do quantum mechanics without complex numbers. The Schrödinger equation has a fat i right in the middle of it. Complex numbers were discovered, not invented.
i think it is wrong, given his axioms (as defined here: http://www.bookofparagon.com/Mathematics/PerspexMa chineVIII.pdf).
(inf) = 1/0 [A20]
= 1/(-1 * 0) [T77]
= -1 * (1/0) [A13]
= -1 * (inf) [A20]
= -(inf) [A24]
which contradicts his axiomatic supposition of (inf) and -(inf) as unique entities [T41]
lysergically yours