What is the theory for a number divided by zero?

Answer 1

The theory is simple: the answer for dividing by zero is NaN, "not a number". Zero cannot divide into any number, including itself. Conceptually, there is no correct finite answer because you can always take away more zeros from the numerator. To get this answer in mathematical terms, we instead look at what is called a limit as a function approaches a point that is NaN. For example, take the function f(x)=1/(x^2). This function has no value at x=0. However, if we look at the value of the function at x values very, very, very (actually, infinitely) close to x=0, we can see a trend and use that to understand what happens at x=0. In the case of 1/(x^2) the function approaches infinity at very small negative and very small positive values, so we conclude that the limit as x approaches 0 of 1/(x^2) is infinite, even though the function has no value at x=0.

The above applies to many, but not all, algebraic structures. Specifically, because it models the world we live in, it's true for fields. The axioms of a field require that any number multiplied by zero is equal to zero and that any number multiplied by its inverse is equal to one. Therefore, 0*0-1 equals both 1 and 0 according to these rules which is a contradiction, therefore making division by zero in general undefinable.

We reach a delema in our understanding of what the results are saying. If we look at the question as we approach zero from a positive side we see our limit is positive infinity and if we approach it for a negative direction we reach a limit of negative infinity. Here we throw up our hands and say this can not be and dismiss the answers as impossibles. What if our understanding of infinity is wrong as in thinking infinity must be a number and thus only one answer is allowed. What if infinity is an area of understanding that we do not yet possess. What if division by zero is just what the answers say. the only understanding we can derive from both being correct is that it represents a joining point of negative and positive numbers.

Let us look at Einstin's theory