Guy Haworth wrote This is a deeper question than it appears ... and there are of course some well-informed books on the history of zero. A colleague of mine created an algebra J = {(a, b) | a, b in the Reals R} ... with (a, b) being analogous to a/b in R. Thus, addition, inverses, subtraction and multiplication can be defined naturally ... and division by (a, b) is defined as multiplication by (b, a) without any exceptions. There is also an equivalence relationship: (ca, cb) is equivalent to (a, b). Thus, he has elements in the algebra like (a, 0), (-a, 0) and (0, 0) with a > 0. He gives these the names 'infinity', '-infinity' and 'nullity'. Anything equivalent to nullity is also 'nullity'. Nullity is what I (and maybe others) would call an 'annihilator' ... if it is combined with anything, the result is 'nullity'. Let us add 0/0 as a notional 'number' to the set of numbers a/b in the Reals R. Let this algebra be R'. rwg> For a very informal treatment, www.tweedledum.com/rwg/rectarith12.pdf p7/14 <rwg Thus, there is a 1-1 mapping from R into J. Any expression not featuring 0/0 in R' maps in an obvious way to J ... and expressions featuring 0/0 in R' map to nullity in J. So J is equivalent to a number system with 0/0 being treated as a number rather than an exception. However, it does not follow that '2', i.e. 2/1, equals '1', i.e. 1/1, in this algebra. "All expressions have a value" might be a neat mantra ... but we haven't really created - IMHO - a number system which is different from what we have now. The history of new numbers - minus 1, zero, famously sqrt(2) and presumably the unfortunately named complex numbers ... is that they are not universally received with peace of mind. They had a troubled birth. I have not been able to persuade my colleague 'J' that he has not created anything new. One cannot simply ignore the fact that a number is zero if it is effectively being divided into something. Many mathematical methods have 'if zero' get-out clauses. The standard formula for the roots of a quadratic does not work if the coefficient of x^2 is zero. Maybe there is a neat proof that this concept of 'nullity' does not bring anything to the party: I sure wish there was ... and then I could save my colleague's time writing conference papers and being convinced that he is onto something. I believe his interest in 0/0 stemmed from an interest in Project Geometry, and then in Interval Arithmetic ... where he was trying to tackle the problem of Instability in computations. Guy