Dividing by Zero

Dear sci/math,

I have a problem i would like to understand. Everybody knows you can't divide by zero, like everybody knows that binary codes just consist of 0 and 1.
First question: Why do you take a variable that is not defined at
for such an important system?

Second question: Would there be any difference if zero was defined for the casual use of Computers?

Maybe someone can help me just imagining how zero could be defined a little, if my questions make too little sense.

Thanks anon.

>>5
>>6
I remember reading the BBC article on James Anderson and specifically the "nullity" concept to come out of his transreal arithmetic.  It's been a long while, but I remember the impression I got after reading a little further than the article was that Anderson's theory got a lot of undeserved hostility simply because the BBC did a terrible job presenting it.  (This is unfortunately bound to happen every so often when reporters attempt to cover subjects they have no background in.)  It's not really suggesting division by zero is possible, it is just alternative way of having a computer handle the division by zero problem.

"Zero" is usually used to mean the additive identity of a ring (an algebraic structure with addition, subtraction, and multiplication, but not necessarily division, ahd the operations must satisfy some familiar requirements), and in this context division by it is only possible if the ring is trivial.  (If 0 is the only element, then of course you can say 0/0 = 0 without any problems.)

For any a in a given ring, a + 0 = a.  Multiplying both sides by a gives a(a + 0) = a*a.  Distributivity of multiplication over addition gives a*a + a*0 = a*a (= a*a + 0).  Subtracting a*a from both sides gives a*0 = 0.  In particular 0*0 = 0
If 0 has a multiplicative inverse, call it 0' (0*0' = 1), then 0*0*0' = 0*0', which yields (0 =) 0*1 = 1.  Then for a an arbitrary element of the ring, multiplying both sides of the previous equation by a gives (0 =) a*0 = a*1 (= a), so in fact the only element of this ring is 0.

The upshot is that division by zero isn't really possible unless you're not using a conventional notion of zero (or unless 0 is your only element).  Some people say wheel theory allows division by zero, and to be honest I haven't looked that far into it, but it's always looked to me like wheel theory has defined a new operation and simply labeled it as "division by zero" without it actually being directly related.