Zero Division – Yes

On the subject of “Zero Division”, I would definitely agree that zero division is possible in a very general case. First, I would just like to give a short background to describe the motivation of my arguments. One can view the subject of Mathematics as the science of Sets whose members obey a group of laws called axioms. In particular, I would like to refer specifically to the set of real numbers as for the most part, it is the default set that we based all our common mathematical operations. Take any arbitrary real numbers say x, y and z then the following laws are assumed true:

(i) x+y = y+x is a real number (closure and commutative laws for addition)

(ii) their exist another real number -x (called the additive inverse) such that x + (-x) = 0 where 0 is called the additive identity which has the property 0 + x = x + 0 = x

(iii) x + (y + z ) = (x + y) + z (associative law)

(iv) x* y = y * x is a real number (closure and commutative law for multiplication)

(v) their exist another real number x^-1 (called the multiplicative inverse) such that for each x not equal to 0, x * x^-1 = x^-1 * x = 1 where 1 is called the multiplicative identity with the property that 1 * x = x * 1 = x

(vi) x* (y + z) = x*y + x*z and (x + y) * z = x*z + y*z (right and left distributive laws)

The above laws are called the Real Number Axioms and they govern the operations that real numbers can undergo. Two things to note:

1) All arithmetic and algebraic computations that we carry out on a daily basis, from simple to complex Mathematics, are accounted for by these axioms.

2) There is no stipulation for division by zero stated within these axioms since 0 is the only number excluded from having an inverse.

Now 1) and 2) above tells us that as long as we are working within the context of the real numbers or any of its subsets, division by zero is undefined and hence impossible within our scope! (since 0 has no inverse).

Now, as mathematician, we are always looking for structures, sub-structures and a way of developing generalizations of these structures. In real analysis, we are restricted in how to deal with cases where we are met with computations resulting in having to compute sqrt(-1). In real analysis, such value does not exist! However, by viewing the set of real numbers as a subset of a larger more general set called the set of Complex Numbers, we find that this restriction has been lifted and we are now able to perform computations involving sqrt(-1). In a very similar way, I do admit the possibility of dividing by 0. I think that to have this operation admitted as valid, we may well be required to again look for a larger more general set to work with, possessing the axiomatic framework that admits zero division and outlines the form in which such numbers will have.