The BBC won't tell you, but the original paper is here.
I can't bear to read any more comments on Slashdot, Wikipedia or Digg from people who don't know as much maths as they think they do, so here is what's actually going on. Do not pay any attention to any other commentary on this subject that does not include the words "field" or "ring" in their mathematical senses.
The "transreal numbers" defined in this paper may very well be internally consistent, although it's not been peer-reviewed and it would be too much work to check it out thoroughly. However, it doesn't really matter, because by the author's own axioms A8 and A18, "nullity", "infinity" and "-infinity" do not have additive or multiplicative inverses, which means that the transreal numbers aren't a field (hell, they're not even a ring), which means that everything you ever thought you knew about arithmetic will have to be rigorously revalidated, and a lot of it will probably turn out not to work any more. This is a high price to pay for being able to work with the result of 0/0 arithmetically instead of just treating it as undefined, and so the transreal numbers are effectively useless.
[Edit: bohemiancoast points out that one of the other authors on the paper I reference above looks strangely familiar.]