Math Forum: Ask Dr. Math FAQ: Dividing by Zero: "I want to use 'divide by zero' to indicate a physically impossible task. What does the phrase actually mean?
Well, division by zero is not so much 'physically impossible' as it is 'in violation of mathematical axioms.' You see, the phrase 'physically impossible' implies a task that cannot be done, no matter the amount of exertion of effort, whereas the phrase 'in violation of mathematical axioms' means that the operation contradicts certain basic assumptions regarding the system in question.
Numbers have certain properties and rules; for instance, we say that adding, subtracting, multiplying, and dividing two numbers will give another number. Subtraction is the opposite of addition, as division is the opposite of multiplication. Any number multiplied by zero gives zero.
There are several of these basic rules, called axioms, and in particular, the kinds of numbers we are familiar with, and do basic arithmetic with, form what mathematicians call a 'field.' In this field, these rules I have described are called 'field axioms.' (There are others as well.)
In essence, the field axioms lay down a set of rules, i.e., basic assumptions, about how to put numbers together to get other numbers. And so, division by zero can be shown to contradict these rules (this proof is usually taught in beginning algebra classes.)
Technically speaking, division by 0 is not impossible; rather, it is contradictory to assumption. As such, we disallow it as a valid operation on numbers. 'Physically impossible' is a more fitting description of a phenomenon, such as the creation of a perpetual motion machine, or the decrease in entropy of a closed system. Division by 0 is not so much a phenomenon as it is a supposed construction, which is provably contradictory t"