British mathematician divides by zero - solves 1200yr old problem

[heneganj]

Dr James Anderson, from the University of Reading's computer science department, says his new theorem solves an extremely important problem - the problem of nothing.

http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml
and
http://science.slashdot.org/article.pl?sid=06/12/07/0416223

To understand the derivation of 'nullity' you need to view the video (in the first link) which is in Real Video (RAM) format.

[heneganj]

Question is, if 'i' (sqrt(-1)) spawned a new field of mathematics - ie complex numbers, is there any way that nullity could be used to solve new problems?

More specifically, now with the 'new found' ability to raise 0 to the power 0 (or divide 0 by 0) is there any way classic attractors or fractal equations that escape to infinity (or zero) be tamed in some way to use nullity?

[lycium]

infinity is often taken to be part of the complex numbers, which completes the riemann sphere mapping of the complex plane.

as for diving by zero, he's basically discovered what a NaN is! he makes ridiculous analogies with planes crashing and hearts stopping, yet his scheme doesn't allow anything to be computed that couldn't be computed before...

[Sockratease]

Not being able to divide by zero never sat well with me.

Nor did the concept of infinity!

But I always felt they went together and you could not have one without the other, I just never had enough credentials to get anyone to listen!

Thanks for the link.

I'll check out the article when time permits...

[Nahee_Enterprises]

Jason Henegan wrote:
>
>    Dr James Anderson, from the University of Reading's computer
>    science department, says his new theorem solves an extremely
>    important problem - the problem of nothing.

   Maybe he can get with the people at Microsoft, since they have always had a problem with just defining "nothing":
      Null
      Null Character
      Nothing
      Zero Length
      Empty
      Void
      ""

           

[eNZedBlue]

More specifically, now with the 'new found' ability to raise 0 to the power 0 (or divide 0 by 0) is there any way classic attractors or fractal equations that escape to infinity (or zero) be tamed in some way to use nullity?

Perhaps the Circle Inversion Transformation could be useful in some way? Solutions to certain classic problems (e.g. the Apollonius 3-circle problem) can be found by inverting the geometry about a circle. This avoids some nasty divide by zeros and preserves various geometrical properties. For example, lines and circles that intersect before the transformation will intersect afterwards and vice versa. Basically you transform the problem into another spatial mapping, solve it there, and then transform it back.

This is relevant to the divide-by-zero problem because NaN/nullity etc is geometrically related to the inversion of the origin point using an inversion circle centred on the origin. To generalise, X/Y can be obtained by plotting Y and then inverting it using an inversion circle centred on the origin with radius X.

Division by zero of a real number r intuitively results in inifinty (becase you can take zero away from any number an infinite number of times). However it also results in negative infinity (since you can take zero away from -r as well). Inverting the origin about a unit circle to compute 1/0 yields a circle of infinite radius centred about the origin. Therefore the geometric meaning of NaN in the complex plane is not a single point but rather "infinity in all directions" (I think).

I wonder if circle inversion in the complex plane could be used to "tame" some fractal equations? I have generated some fractals using circle inversion a while ago but the results are basically simple 2-D distortions of the original fractal.

[lycium]

if you add something to nullity you still get nullity, so effectively you've got a weird symbol for a nan - it doesn't really assist any computation, just tells you when your answer is bogus

[David Makin]

I agree that he's not bringing in anything new - he's simply teaching the idea of "NAN" at a lower education level than normal.

My own theory is that:

   zero*infinity = 1
  -zero*infinity = -1
   zero*-infinity = -1
  -zero*-infinity = 1

(just kidding)

[lycium]

the interaction of infinite and infinitesimal variables has been rigorously worked out by conway (as documented by knuth) quite some time ago through the system of "surreal" numbers.

there's a good introduction at http://www.tondering.dk/claus/sur15.pdf but i can highly recommend knuth's book (info at http://www-cs-faculty.stanford.edu/~knuth/sn.html) since it's much more like a story than a normal mathematical exposition! you follow the development of the subject by two "desert island mathematician" protagonists, which is interesting because it sheds light on how the theorems are discovered - number systems are typically taught in the same way as religion, as canon!

it's not a terribly useful system for normal computation (although charles might find them a good substitute for fixed point numbers, who knows ), but it's an excellent read and taught me the basics of axiomatic set theory - a much nicer intro before doing it at university with its usual rigor.

[heneganj]

Charles is kicking the bare ass of the Mandel and Julia sets recently!  I have high hopes Charles is going to hit upon something huge!

And let's face it, we are getting lots more value for money out of Charles than Michael 'pump and dump' Barnsley...

Come on Charles!!   

[hoogamaphone]

I remember when I was in elementary school, I learned about i=sqrt(-1) and how it was this new number that was invented, which helped mathematicians solve some equations that they couldn't solve before.

It was at that point that I had the great idea of revolutionizing mathematics. I was going to invent a new number to solve the problem of how to divide by zero. I sat down and started to try to work out how I would carry out calculations with this number, and then realized that it was useless. This is not a new idea at all. I'm sure thousands of people have thought of it before.

[jehovajah]

The discussion of nullity  I am sure will be of suprise to many ordinary people but of course those who have to deal with the impracticality of some of our rules may benefit from following what  Anderson suggests. I too once laboured under the false assumption that mathematics was a natural element of my world instead of part of a constructed model of infinite possibility space as i refer to it. The constructed arithmetic model of the transreals for computational arithmetic has much to recommend it. However the concept of nothingness emptiness etc is not to be confused with the requirements of a pragmatic computational arithmetic for dealing with distinct valuations. The inclusion of nullity and infinity as fixed entities on or off, in or out of the set of reals is a welcome clarification of the constructed nature of these fractal metrics.

With reference to the imaginary valuation i : it was always an operator which we were taught was a valuation at a primary level. For anything to be a valuation it must be a reference to a perceived value or quantity. The reals are so far our culturally neutral quantifiers as are binary representations of the reals. What we do is to try to devise operators and so methods to map our value exeriences onto the reals. Thus millions of colours are now specifiable by a real quantifier which our machine methodology is able to map onto a screen as a colour value. Like i the nullity is an operator and because of machine methodology we can realise both of these to practical advantage.

How does it affect my notion of reality? It further identifies the constructed nature of that concept and the cultural forces that determine or maintain it.